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# Spectrum invariance dilemma for nonuniformly kinematically similar systems

Claudio A. Gallegos∗ Universidad de Chile, UCH, Facultad de Ciencias,
Departamento de Matemáticas, Casilla 653, Santiago, Chile.
<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Néstor Jara∗∗

###### Abstract.

We unveil instances where nonautonomous linear systems manifest distinct
nonuniform $\mu$-dichotomy spectra despite admitting nonuniform
$(\mu,\varepsilon)$-kinematic similarity. Exploring the theoretical
foundations of this lack of invariance, we discern the pivotal influence of
the parameters involved in the property of nonuniform $\mu$-dichotomy such as
in the notion of nonuniform $(\mu,\varepsilon)$-kinematic similarity. To
effectively comprehend these dynamics, we introduce the stable and unstable
optimal ratio maps, along with the $\varepsilon$-neighborhood of the
nonuniform $\mu$-dichotomy spectrum. These concepts provide a framework for
understanding scenarios governed by the noninvariance of the nonuniform
$\mu$-dichotomy spectrum.

###### Key words and phrases:

Nonautonomous differential equations, Nonautonomus hyperbolicity, Nonuniform
$\mu$-dichotomy, Nonuniform dichotomy spectrum, Kinematic similarity

###### 2020 Mathematics Subject Classification:

Primary: 37D25.; Secondary: 34C41, 37C60.

∗ This author was partially supported by ANID/FONDECYT Postdoctorado No
3220147.

∗∗ This author was partially supported by ANID, Beca de Doctorado Nacional
21220105.

##  1\. Introduction

Consider the nonautonomous linear differential equation

$\dot{y}=A(t)y(t),$ (1.1)

where $t\mapsto A(t)\in\mathbb{R}^{d\times d}$ is a locally integrable matrix
valued function. As usual, we denote by
$\Phi\colon\mathbb{R}\times\mathbb{R}\to\mathbb{R}^{d\times d}$,
$(t,s)\mapsto\Phi(t,s)$, the evolution operator –or the transition matrix–
associated to (1.1). That means, for every $s\in\mathbb{R}$ and
$x_{0}\in\mathbb{R}^{d}$, the function $\Phi(\cdot,s)x_{0}$ solves the initial
value problem (1.1) with initial condition $y(s)=x_{0}$.

###  1.1. kinematic similarity and dichotomy spectrum

Inspired by the notion of dichotomy spectrum for skew product flows introduced
in [16], S. Siegmund defined the concept of dichotomy spectrum for system
(1.1) as the set

$\Sigma(A)\mathrel{\mathop{\mathchar
58\relax}}=\\{\gamma\in\mathbb{R}\mathrel{\mathop{\mathchar
58\relax}}\dot{y}=[A(t)-\gamma\mathrm{Id}]y(t)\text{ admits no exponential
dichotomy}\\},$

demonstrating that is a finite union of closed intervals (spectral intervals),
which could be unbounded but with compactness ensured if system (1.1) exhibits
the property of bounded growth, see [18, Theorem 3.1]. In the subsequent work
of S. Siegmund [19], leveraging the spectral dichotomy theory developed in
[18], a pivotal outcome was achieved: a compelling reducibility result for
system (1.1), see [19, Theorem 3.2]. This result elucidates that system (1.1)
is kinematically similar to a system which is block-diagonalized, where each
block –of dimension less than $d$– precisely corresponds to a spectral
interval of $\Sigma(A)$. This interesting result relies on a fundamental
property: if two systems are kinematically similar, then they have the same
dichotomy spectrum, see [19, Corollary 2.1].

The formulation of the dichotomy spectrum incorporates the notion of
exponential dichotomy for nonautonomous systems, which in simple terms
emulates the idea of hyperbolic equilibrium from the autonomous framework, see
e.g. [7, 14]. In 2005, L. Barreira and C. Valls [4], motivated to consider
broader categories of hyperbolic dynamics, introduced the notion of nonuniform
exponential dichotomy for system (1.1). This concept extends the classical
exponential dichotomy, and henceforth, the academic community have adopted to
add the term “uniform” when referring to the classic one. We suggest to the
reader the book [2], where several contributions to the development of the
nonuniform theory are presented.

Following the ideas of S. Siegmund, the authors X. Zhang [23] and J. Chu et
al. [8] independently developed a spectral theory by considering the notion of
nonuniform exponential dichotomy (slightly modificated definitions are
utilized on each article). As in the classical case but considering the
assumption of nonuniform bounded growth, it is proved that the nonuniform
dichotomy spectrum can be decomposed as finite union of compact intervals, see
[8, Corollary 2.11] and [23, Theorem 2.1]. In addition, a reducibility theorem
is obtained by considering a fundamental property: the invariance of
nonuniform dichotomy spectra via nonuniform kinematic similarity.

In a recent work, C. Silva [17] introduced the notion of nonuniform
$\mu$-dichotomy, providing a unifying framework that extends and integrates
the discussed exponential dichotomy concepts (see Definition 2.2 in Section
2). Considering this nonuniform $\mu$-dichotomy, a spectral theory is
developed and a reducibility theorem is derived by once again leveraging an
essential property: the invariance of the nonuniform $\mu$-dichotomy spectra
through nonuniform $(\mu,\varepsilon)$-kinematic similarity.

###  1.2. Main results and contributions

The aim of this article is to defy the conventional idea of invariance of the
nonuniform $\mu$-dichotomy spectra for nonautonomous linear systems which are
nonuniformly $(\mu,\varepsilon)$-kinematically similar. This assertion is
explained on Remark 3.7 and supported by Examples 3.8 and 3.9, in which we
exhibit nonuniformly $(\mu,\varepsilon)$-kinematically similar systems
presenting distinct nonuniform $\mu$-dichotomy spectra. We emphasize that the
lack of invariance of the nonuniform dichotomy spectrum does not depend on the
specific notion of nonuniform dichotomy under consideration. In all cases,
noninvariance is detected; see Remark 3.11.

In this article, we will provide a theoretical foundation for this
noninvariance phenomenon. In this context, we observe that the parameters
involved in the notions of nonuniform $\mu$-dichotomy,
$({\mathrm{N}\mu},\epsilon)$-growth, and $(\mu,\varepsilon)$-kinematic
similarity, significantly contribute in these dynamics. For that reason, we
introduce some novel concepts with the aim of appropriately managing the
above-mentioned parameters:

  * •

The region of stable and unstable constants, see Definition 2.5.

  * •

On every spectral gap, we define the maps of optimal stable and unstable ratio
along with its underlying properties, see Definition 2.15 together with the
subsequent results from Section 2.

  * •

We introduce the $\varepsilon$-neighborhood of the nonuniform $\mu$-dichotomy
spectrum and the $\varepsilon$-interior of the nonuniform $\mu$-resolvent for
system (1.1), see Definition 3.14. In addition, we elucidate the relationship
between this concept of spectrum and the classic one for systems that admit
nonuniform $(\mu,\varepsilon)$-kinematic similarity, see Corollary 3.16.

By introducing this new concepts, our objective is to enhance the
understanding in scenarios characterized by the noninvariance of the
nonuniform $\mu$-dichotomy spectrum.

###  1.3. Consequences about the noninvariance of the nonuniform spectra

Reducibility results are relevant since they allow for an easier
interpretation of nonautonomous equations, but also as a crucial initial step
toward several results. For instance, a remarkable application of these type
of construction, which typically needs dichotomy spectrum preservation, is a
normal form result, which states that under a polynomial transformation, it is
possible to eliminate suitable Taylor terms of a nonlinear perturbation. The
first of these results for the nonautonomous framework was given by S.
Siegmund in [20].

Furthermore, even a normal form result is just a tool in order to find a local
smooth linearization, as given originally by H. Poincaré [15] for autonomous
analytic complex equations, and by S. Sternberg [21, 22] for real equations.
In the nonautonomous context, L. V. Cuong, S. T. Doan and S. Siegmund [9] were
able to find a local smooth linearization for systems admitting the classic
exponential dichotomy. Once again, for this construction, the invariance of
the dichotomy spectrum is required.

##  2\. Optimal dichotomy constants

In this section, we begin by recalling key concepts related to the nonuniform
$\mu$-dichotomy presented in [17]. Following this, we introduce the optimal
stable and unstable ratio functions, providing a detailed exposition of their
fundamental properties.

######  Definition 2.1.

A strictly increasing function $\mu\colon\mathbb{R}\to(0,+\infty)$ is said to
be a growth rate, if $\mu(0)=1$, $\mu(t)\to+\infty$ as $t\to+\infty$, and
$\mu(t)\to 0$ as $t\to-\infty$. Moreover, if $\mu$ is differentiable, it is
said a differentiable growth rate.

######  Definition 2.2.

Let $\mu\mathrel{\mathop{\mathchar 58\relax}}\mathbb{R}\to(0,+\infty)$ be a
growth rate. The system (1.1) admits nonuniform $\mu$-dichotomy if there exist
an invariant projector $t\mapsto\mathrm{P}(t)\in\mathbb{R}^{d\times d}$, that
means $\mathrm{P}$ satisfies

$\mathrm{P}(t)\Phi(t,s)=\Phi(t,s)\mathrm{P}(s),\qquad\text{for all
$t,s\in\mathbb{R}$,}$

constants $K\geq 1$, $\alpha<0$, $\beta>0$ and $\theta,\nu\geq 0$ satisfying
$\alpha+\theta<0$ and $\beta-\nu>0$, such that

$\left\\{\begin{array}[]{lc}\|\Phi(t,s)\mathrm{P}(s)\|\leq
K\left(\dfrac{\mu(t)}{\mu(s)}\right)^{\alpha}\mu(s)^{\mathrm{sgn}(s)\theta},&\forall\,\,t\geq
s,\\\ \\\ \|\Phi(t,s)[\mathrm{Id}-\mathrm{P}(s)]\|\leq
K\left(\dfrac{\mu(t)}{\mu(s)}\right)^{\beta}\mu(s)^{\mathrm{sgn}(s)\nu},&\forall\,\,t\leq
s.\end{array}\right.$ (2.1)

Moreover, if $\theta=\nu=0$, then we say the system (1.1) admits uniform
$\mu$-dichotomy.

Throughout the text, we abbreviate the notions of nonuniform $\mu$-dichotomy
by ${\mathrm{N}\mu\mathrm{D}}$ and uniform $\mu$-dichotomy by
${\mu\mathrm{D}}$. Usually, when we refer to a system having
${\mathrm{N}\mu\mathrm{D}}$, we will emphasize that admits
${\mathrm{N}\mu\mathrm{D}}$ with parameters
$(\mathrm{P};\alpha,\beta,\theta,\nu)$. We omit $K$ in the parameters because
is not fundamental for our later purposes. For the case
$\mathrm{P}=\mathrm{Id}$, there are no constants $\beta,\nu$, hence we denote
$(\mathrm{Id};\alpha,*,\theta,*)$ and for $\mathrm{P}=0$ we write
$(0;*,\beta,*,\nu)$.

The preceding concept broadens various dichotomies based on the growth rate
$\mu$ and the parameters involved in the estimations:

  * •

Define $\mu(t)=e^{t}$, for all $t\in\mathbb{R}$. Consider $\beta=-\alpha$ and
$\theta=\nu=0$. In this case we recover the notion of exponential dichotomy
($\mathrm{ED}$) defined by O. Perron in [14]. Moreover, in the absence of
constraints on the parameters, we arrive at the concept of nonuniform
exponential dichotomy ($\mathrm{NED}$), a topic extensively studied in [3, 8,
10, 23].

  * •

Consider an arbitrary strictly increasing surjective function
$\nu\colon[0,+\infty)\to[1,+\infty)$. This function $\nu$ induces a growth
rate $\mu$ defined as

$\mu(t)\mathrel{\mathop{\mathchar
58\relax}}=\left\\{\begin{array}[]{lcc}\nu(t)&\text{ if }&t\geq 0,\\\ \\\
\frac{1}{\nu(|t|)}&\text{ if }&t\leq 0.\end{array}\right.$

For instance, by considering a function $\nu$ defined by $\nu(t)=t+1$, for all
$t\geq 0$, we obtain a growth rate $\mu$ associated to the nonuniform
polynomial dichotomy ($\mathrm{NPD}$). If moreover $\theta=\nu=0$, we obtain
the polynomial dichotomy ($\mathrm{PD}$), see e.g. [1, 11, 12].

For $\gamma\in\mathbb{R}$ and a differentiable growth rate $\mu$, consider the
system

$\dot{y}=\left[A(t)-\gamma\frac{\mu^{\prime}(t)}{\mu(t)}\mathrm{Id}\right]y(t),$
(2.2)

where $t\mapsto A(t)$ is the matrix valued function from system (1.1). We said
that system (2.2) is the $\gamma$-shifted system associated to (1.1). Clearly,
the transition matrix $\Phi_{\gamma}$ of system (2.2) is given by

$\Phi_{\gamma}(t,s)=\Phi(t,s)\left(\frac{\mu(t)}{\mu(s)}\right)^{-\gamma},\quad
t,s\in\mathbb{R},$

where $\Phi(t,s)$ is the transition matrix of system (1.1). We call
$\Phi_{\gamma}(t,s)$ as the $\gamma$-shifted operator.

The $\gamma$-shifted system (2.2) serves a crucial role to the following
concept, see [17, p. 623].

######  Definition 2.3.

Let $\mu\colon\mathbb{R}\to(0,+\infty)$ be a differentiable growth rate. The
nonuniform $\mu-$dichotomy spectrum of system (1.1) is the set defined by

$\Sigma_{\mathrm{N}\mu\mathrm{D}}(A)\mathrel{\mathop{\mathchar
58\relax}}=\left\\{\gamma\in\mathbb{R}\mathrel{\mathop{\mathchar
58\relax}}\text{ \eqref{615} does not admit
}{\mathrm{N}\mu\mathrm{D}}\right\\}\,.$

Moreover, the nonuniform $\mu$-resolvent of system (1.1) is the set defined by

$\rho_{\mathrm{N}\mu\mathrm{D}}(A)\mathrel{\mathop{\mathchar
58\relax}}=\mathbb{R}\setminus\Sigma_{\mathrm{N}\mu\mathrm{D}}(A),$

i.e. the complement set of $\Sigma_{\mathrm{N}\mu\mathrm{D}}(A)$.

######  Remark 2.4.

Analogously, the uniform $\mu-$dichotomy spectrum and the uniform
$\mu$-resolvent of system (1.1) is defined as above by simply replacing the
${\mathrm{N}\mu\mathrm{D}}$ by ${\mu\mathrm{D}}$.

For $\gamma\in\rho_{{\mathrm{N}\mu\mathrm{D}}}(A)$, we know the
$\gamma$-shifted system admits nonuniform $\mu$-dichotomy. In general, we do
not know the constants involved in said dichotomy and they are by no means
unique. Indeed, if the $\gamma$-shifted system (2.2) admits
${\mathrm{N}\mu\mathrm{D}}$ with parameters
$(\mathrm{P};\alpha,\beta,\theta,\nu)$ satisfying

$\left\\{\begin{array}[]{lc}\|\Phi_{\gamma}(t,s)\mathrm{P}(s)\|\leq
K\left({\frac{\mu(t)}{\mu(s)}}\right)^{\alpha}\mu(s)^{\mathrm{sgn}(s)\theta},&\forall\,\,t\geq
s\\\ \\\ \|\Phi_{\gamma}(t,s)[\mathrm{Id}-\mathrm{P}(s)]\|\leq
K\left({\frac{\mu(t)}{\mu(s)}}\right)^{\beta}\mu(s)^{\mathrm{sgn}(s)\nu},&\forall\,\,t\leq
s,\end{array}\right.$

then for every $\widetilde{\alpha}\in(\alpha,-\theta)$,
$\widetilde{\beta}\in(\nu,\beta)$,
$\widetilde{\theta}\in(\theta,-\widetilde{\alpha})$, and
$\widetilde{\nu}\in(\nu,\widetilde{\beta})$, the following estimations are
verified as well

$\left\\{\begin{array}[]{lc}\|\Phi_{\gamma}(t,s)\mathrm{P}(s)\|\leq
K\left({\frac{\mu(t)}{\mu(s)}}\right)^{\widetilde{\alpha}}\mu(s)^{\mathrm{sgn}(s)\widetilde{\theta}},&\forall\,\,t\geq
s\\\ \\\ \|\Phi_{\gamma}(t,s)[\mathrm{Id}-\mathrm{P}(s)]\|\leq
K\left({\frac{\mu(t)}{\mu(s)}}\right)^{\widetilde{\beta}}\mu(s)^{\mathrm{sgn}(s)\widetilde{\nu}},&\forall\,\,t\leq
s,\end{array}\right.$

and thus, this defines once more a ${\mathrm{N}\mu\mathrm{D}}$ with parameters
$(\mathrm{P};\widetilde{\alpha},\widetilde{\beta},\widetilde{\theta},\widetilde{\nu})$
for the $\gamma$-shifted system (2.2). This situation motivates us to define
an optimal set of constants each time that a system admits a nonuniform
$\mu$-dichotomy.

In order to provide some general concepts, let us consider an arbitrary
nonautonomous linear system

$\dot{z}=B(t)z,$ (2.3)

where $t\mapsto B(t)$ is a locally integrable matrix valued function. In
addition, let $\Psi(t,s)$ be the transition matrix associated to (2.3).

######  Definition 2.5.

Assume the system (2.3) admits ${\mathrm{N}\mu\mathrm{D}}$ with an invariant
projector $\mathrm{P}$. The region of stable constants for (2.3) is the set
defined by

$\mathrm{St}_{\mathrm{P}}\mathrel{\mathop{\mathchar
58\relax}}=\\{(\alpha,\theta)\in\mathbb{R}^{2}\mathrel{\mathop{\mathchar
58\relax}}\text{ \eqref{601} admits }{\mathrm{N}\mu\mathrm{D}}\text{ with
parameters }(\mathrm{P};\alpha,\beta,\theta,\nu),\text{ for some
}\beta,\nu\\},$

and the region of unstable constants for (2.3) is the set defined by

$\mathrm{Un}_{\mathrm{P}}\mathrel{\mathop{\mathchar
58\relax}}=\\{(\beta,\nu)\in\mathbb{R}^{2}\mathrel{\mathop{\mathchar
58\relax}}\text{ \eqref{601} admits }{\mathrm{N}\mu\mathrm{D}}\text{ with
parameters }(\mathrm{P};\alpha,\beta,\theta,\nu),\text{ for some
}\alpha,\theta\\}.$

######  Remark 2.6.

Note that these regions allow for a characterization of those systems with
uniform dichotomy as those for which
$\mathrm{St}_{\mathrm{P}}\cap(\mathbb{R}\times\\{0\\})\neq\emptyset$ and
$\mathrm{Un}_{\mathrm{P}}\cap(\mathbb{R}\times\\{0\\})\neq\emptyset$.

Now we dedicate some remarks to make an estimation of these regions.

######  Remark 2.7.

Clearly, when system (2.3) admits ${\mathrm{N}\mu\mathrm{D}}$ with parameters
$(\mathrm{P};\alpha,\beta,\theta,\nu)$, the regions $\mathrm{St}_{\mathrm{P}}$
and $\mathrm{Un}_{\mathrm{P}}$ are nonempty sets, and the following inclusions
follow

$\mathrm{St}_{\mathrm{P}}\subset\\{(\alpha,\theta)\in\mathbb{R}^{2}\mathrel{\mathop{\mathchar
58\relax}}\alpha<0\leq\theta;\,\alpha+\theta<0\\},$

and

$\mathrm{Un}_{\mathrm{P}}\subset\\{(\beta,\nu)\in\mathbb{R}^{2}\mathrel{\mathop{\mathchar
58\relax}}0\leq\nu<\beta;\,\beta-\nu>0\\}.$

On the other hand, from the estimations of the ${\mathrm{N}\mu\mathrm{D}}$,
for $t=s$, we obtain

$\|\mathrm{P}(s)\|\leq
K\mu(s)^{\mathrm{sgn}(s)\theta},\quad\text{and}\quad\|\mathrm{Id}-\mathrm{P}(s)\|\leq
K\mu(s)^{\mathrm{sgn}(s)\nu},\quad\forall\,s\in\mathbb{R},$

which means that the regions of stable and unstable constants can be estimated
by the projector involved in the dichotomy. Expressly, for a projector
$\mathrm{P}$ we can define

$\widehat{\theta}_{\mathrm{P}}=\inf\left\\{\theta\geq
0\mathrel{\mathop{\mathchar 58\relax}}\|\mathrm{P}(s)\|\leq
K\mu(s)^{\mathrm{sgn}(s)\theta},\,\forall\,s\in\mathbb{R},\text{ for some
}K>0\right\\},$

and

$\widehat{\nu}_{\mathrm{P}}=\inf\left\\{\nu\geq 0\mathrel{\mathop{\mathchar
58\relax}}\|\mathrm{Id}-\mathrm{P}(s)\|\leq
K\mu(s)^{\mathrm{sgn}(s)\nu},\,\forall\,s\in\mathbb{R},\text{ for some
}K>0\right\\}.$

Therefore, we can refine the above inclusions as

$\mathrm{St}_{\mathrm{P}}\subset\\{(\alpha,\theta)\in\mathbb{R}^{2}\mathrel{\mathop{\mathchar
58\relax}}\alpha<0\leq\widehat{\theta}_{\mathrm{P}}\leq\theta;\,\alpha+\theta<0\\},$

and

$\mathrm{Un}_{\mathrm{P}}\subset\\{(\beta,\nu)\in\mathbb{R}^{2}\mathrel{\mathop{\mathchar
58\relax}}0\leq\widehat{\nu}_{\mathrm{P}}\leq\nu<\beta;\,\beta-\nu>0\\}.$

However, note that this does not mean that there are some $\alpha$ and $\beta$
such that $(\alpha,\widehat{\theta}_{\mathrm{P}})\in\mathrm{St}_{\mathrm{P}}$
and $(\beta,\widehat{\nu}_{\mathrm{P}})\in\mathrm{Un}_{\mathrm{P}}$. Consider
for example a nonuniform contraction, i.e. a system which admits
${\mathrm{N}\mu\mathrm{D}}$ but does not admit ${\mu\mathrm{D}}$ and the
projector is the identity. In this case, clearly
$\widehat{\theta}_{\mathrm{Id}}=\widehat{\nu}_{\mathrm{Id}}=0$, see e.g. [2,
Proposition 2.3]. In other words, the regions of possible constants can be
estimated by the projector, but do not depend exclusively of it, and also
depend on the system itself.

In the next, we recall te notion of $\mu$-bounded growth for the nonuniform
context [17].

######  Definition 2.8.

The system (2.3) has nonuniform $\mu$-bounded growth with parameter
$\epsilon>0$, or just $({\mathrm{N}\mu},\epsilon)$-growth, if there are
constants $\widehat{K}\geq 1$, $a\geq 0$ such that

$\|\Psi(t,s)\|\leq\widehat{K}\left(\frac{\mu(t)}{\mu(s)}\right)^{\mathrm{sgn}(t-s)a}\mu(s)^{\mathrm{sgn}(s)\epsilon},\quad\forall\,t,s\in\mathbb{R}.$

Moreover, if $\epsilon=0$, it is said that the system (2.3) has uniform
$\mu$-bounded growth o just $\mu$-growth.

######  Remark 2.9.

It is immediate to note that if the system (1.1) has
$({\mathrm{N}\mu},\epsilon)$-growth with constants $\widehat{K}\geq 1$ and
$a\geq 0$, then the $\gamma$-shifted system (2.2) also has
$({\mathrm{N}\mu},\epsilon)$-growth, with constants $\widehat{K}$ and
$a+|\gamma|$.

The following result, inspired by [5, Lemma 1], relates the parameters of the
${\mathrm{N}\mu\mathrm{D}}$ with the constants of the
$({\mathrm{N}\mu},\epsilon)$-growth of a system.

######  Lemma 2.10.

Assume the system (2.3) has $({\mathrm{N}\mu},\epsilon)$-growth with constants
$a\geq 0$, $\widehat{K}\geq 1$, and also admits ${\mathrm{N}\mu\mathrm{D}}$
with parameters $(\mathrm{P};\alpha,\beta,\theta,\nu)$. For $\mathrm{P}\neq
0$, we have $-(a+\epsilon)\leq\alpha$. In particular

$\mathrm{St}_{\mathrm{P}}\subset\\{(\alpha,\theta)\in\mathbb{R}^{2}\mathrel{\mathop{\mathchar
58\relax}}-(a+\epsilon)\leq\alpha\\}.$

Analogously, for $\mathrm{P}\neq\mathrm{Id}$, we have $\beta\leq a+\epsilon$.
In particular

$\mathrm{Un}_{\mathrm{P}}\subset\\{(\beta,\nu)\in\mathbb{R}^{2}\mathrel{\mathop{\mathchar
58\relax}}\beta\leq a+\epsilon\\}.$

###### Proof.

We prove only the first inequality, since the other case follows similarly.
Let $0<s<t$. Since $\mathrm{P}\neq 0$, we can chose
$\xi\in\mathrm{P}(s)\mathbb{R}^{d}$ with $\xi\neq 0$. Then we have the
following estimation

$\displaystyle\|\xi\|=\|\Psi(s,t)\Psi(t,s)\xi\|$
$\displaystyle\leq\|\Psi(s,t)\|\cdot\|\Psi(t,s)\mathrm{P}(s)\xi\|$
$\displaystyle\leq\widehat{K}\left(\frac{\mu(s)}{\mu(t)}\right)^{\mathrm{sgn}(s-t)a}\mu(t)^{\mathrm{sgn}(t)\epsilon}K\left(\frac{\mu(t)}{\mu(s)}\right)^{\alpha}\mu(s)^{\mathrm{sgn}(s)\theta}\|\xi\|,$

from which we deduce

$1\leq\widehat{K}K\mu(t)^{a+\alpha+\epsilon}\mu(s)^{\theta-a-\alpha},\quad\forall\,t>s.$

Note that considering $a+\alpha+\epsilon<0$ and taking the limit as
$t\to+\infty$, the right-hand side tends to zero, which is clearly a
contradiction. Thus, we must have $-(a+\epsilon)\leq\alpha$. ∎

######  Remark 2.11.

Taking into account Remark 2.7 and Lemma 2.10, if we assume the system (2.3)
has $({\mathrm{N}\mu},\epsilon)$-growth with constants $a\geq 0$ and
$\widehat{K}\geq 1$, and also admits ${\mathrm{N}\mu\mathrm{D}}$ with
invariant projector $\mathrm{P}$, then we deduce

$\mathrm{St}_{\mathrm{P}}\subset\\{(\alpha,\theta)\in\mathbb{R}^{2}\mathrel{\mathop{\mathchar
58\relax}}-(a+\epsilon)\leq\alpha<0\leq\widehat{\theta}_{\mathrm{P}}\leq\theta;\,\alpha+\theta<0\\},$

and

$\mathrm{Un}_{\mathrm{P}}\subset\\{(\beta,\nu)\in\mathbb{R}^{2}\mathrel{\mathop{\mathchar
58\relax}}0\leq\widehat{\nu}_{\mathrm{P}}\leq\nu<\beta\leq
a+\epsilon;\,\beta-\nu>0\\}.$

Therefore, both $\mathrm{St}_{\mathrm{P}}$ and $\mathrm{Un}_{\mathrm{P}}$ are
contained in bounded sets, as illustrated on Figure 1.

(a) Region $\mathrm{St}_{\mathrm{P}}$ (b) Region $\mathrm{Un}_{\mathrm{P}}$

Figure 1. Contentions from Remark 2.11

The preceding remark motivates us to introduce the concept of optimal
parameters associated with the ${\mathrm{N}\mu\mathrm{D}}$.

######  Definition 2.12.

Assume the system (2.3) admits $({\mathrm{N}\mu},\epsilon)$-growth and
${\mathrm{N}\mu\mathrm{D}}$ with an invariant projector $\mathrm{P}$.

  * •

The optimal stable ratio is defined by

$\mathfrak{st}_{\mathrm{P}}\mathrel{\mathop{\mathchar
58\relax}}=\inf\\{\alpha+\theta\mathrel{\mathop{\mathchar
58\relax}}(\alpha,\theta)\in\mathrm{St}_{\mathrm{P}}\\}.$

  * •

The optimal unstable ratio is defined by

$\mathfrak{un}_{\mathrm{P}}\mathrel{\mathop{\mathchar
58\relax}}=\sup\\{\beta-\nu\mathrel{\mathop{\mathchar
58\relax}}(\beta,\nu)\in\mathrm{Un}_{\mathrm{P}}\\}.$

In the next, we recall a characterization of the nonuniform $\mu$-dichotomy
spectrum

######  Theorem 2.13.

([17, Theorem 8]) Assume the system (1.1) has
$({\mathrm{N}\mu},\epsilon)$-growth. Then there exists some
$n\in\\{1,\dots,d\\}$ such that the nonuniform $\mu$-dichotomy spectrum of
(1.1) is nonempty, compact and has the form

$\Sigma_{\mathrm{N}\mu\mathrm{D}}(A)=\bigcup_{i=1}^{n}[a_{i},b_{i}],$

where $a_{i}\leq b_{i}<a_{i+1}$.

The compact intervals $[a_{i},b_{i}]$, for $i=1,\dots,n$, described on Theorem
2.13, are called spectral intervals. Each open interval that compose the
resolvent set is termed a spectral gap. Let us denote each spectral gap by
$(b_{i},a_{i+1})$, for $i=0,\dots,n+1$, where $b_{0}=-\infty$ and
$a_{n+1}=+\infty$.

######  Remark 2.14.

For any element in the resolvent set, the correspondent shifted system admits
${\mathrm{N}\mu\mathrm{D}}$. In [17, Lemma 5, Lemma 6] it was established that
for any two numbers $\gamma,\zeta$ contained in the same spectral gap
$(b_{i},a_{i+1})$, the projector associated to the dichotomy of both the
$\gamma$-shifted and the $\zeta$-shifted systems are the same. In other words,
every spectral gap has a unique invariant projector $\mathrm{P}$ associated to
it.

It is immediately deduced from the construction of the spectrum [17], that on
every bounded spectral gap, the associated projector $\mathrm{P}$ is neither
$0$ nor $\mathrm{Id}$. Nevertheless, on the unbounded spectral gap
$(-\infty,a_{1})$, if it is contained on the resolvent set, the projector is
always $0$. Similarly, if the spectral gap $(b_{n},+\infty)$ is contained on
the resolvent set, its associated projector is always $\mathrm{Id}$. Note that
by Theorem 2.13, if a system has $({\mathrm{N}\mu},\epsilon)$-growth, then it
is guaranteed that both unbounded spectral gaps are contained in its resolvent
set.

Let $\gamma$ be an element in the spectral gap $(b_{i},a_{i+1})$. Since the
$\gamma$-shifted system admits ${\mathrm{N}\mu\mathrm{D}}$, it follows that at
least one of the regions of constants is nonempty. Specifically, for
$i=1,\dots,n$, the region of stable constants is nonempty and we denote it by
$\mathrm{St}_{\mathrm{P}}^{\gamma}$, while for $i=0,\dots,n-1$, the region on
unstable constants is nonempty and we denote it by
$\mathrm{Un}_{\mathrm{P}}^{\gamma}$. Hence, for the correspondent index $i$,
we can define two functions, $\mathrm{St}_{\mathrm{P}}\colon(b_{i},a_{i+1})\to
2^{\mathbb{R}}$, $\gamma\mapsto\mathrm{St}_{\mathrm{P}}^{\gamma}$ and
$\mathrm{Un}_{\mathrm{P}}\colon(b_{i},a_{i+1})\to 2^{\mathbb{R}}$,
$\gamma\mapsto\mathrm{Un}_{\mathrm{P}}^{\gamma}$.

From Remarks 2.9 and 2.11, we deduce

$\emptyset\neq\mathrm{St}_{\mathrm{P}}^{\gamma}\subset\left\\{(\alpha,\theta)\in\mathbb{R}^{2}\mathrel{\mathop{\mathchar
58\relax}}-(a+\epsilon+|\gamma|)\leq\alpha<0\leq\widehat{\theta}_{\mathrm{P}}\leq\theta;\,\alpha+\theta<0\right\\},$

for $i=1,\dots,n$, and

$\emptyset\neq\mathrm{Un}_{\mathrm{P}}^{\gamma}\subset\left\\{(\beta,\nu)\in\mathbb{R}^{2}\mathrel{\mathop{\mathchar
58\relax}}0\leq\widehat{\nu}_{\mathrm{P}}\leq\nu<\beta\leq
a+\epsilon+|\gamma|;\,\beta-\nu>0\right\\},$

for $i=0,\dots,n-1$. Note that in both cases, the sets on the right-hand side
of the previous inclusions are bounded. Moreover, in the case that the
spectral gap $(b_{i},a_{i+1})$ is bounded, i.e. for $i=1,\dots,n-1$, the sets
on the right-hand side are uniformly bounded, that is

$\emptyset\neq\mathrm{St}_{\mathrm{P}}^{\gamma}\subset\left\\{(\alpha,\theta)\in\mathbb{R}^{2}\mathrel{\mathop{\mathchar
58\relax}}-(a+\epsilon+\max\\{|b_{i}|,|a_{i+1}|\\})\leq\alpha<0\leq\widehat{\theta}_{\mathrm{P}}\leq\theta;\,\alpha+\theta<0\right\\},$

and

$\emptyset\neq\mathrm{Un}_{\mathrm{P}}^{\gamma}\subset\left\\{(\beta,\nu)\in\mathbb{R}^{2}\mathrel{\mathop{\mathchar
58\relax}}0\leq\widehat{\nu}_{\mathrm{P}}\leq\nu<\beta\leq
a+\epsilon+\max\\{|b_{i}|,|a_{i+1}|\\};\,\beta-\nu>0\right\\},$

therefore, the functions $\gamma\mapsto\mathrm{St}_{\mathrm{P}}^{\gamma}$ and
$\gamma\mapsto\mathrm{Un}_{\mathrm{P}}^{\gamma}$ will be bounded on every
bounded spectral gap.

Similarly, in view of Definition 2.12, it makes sense to define the following
maps

######  Definition 2.15.

Assume the system (2.3) admits $({\mathrm{N}\mu},\epsilon)$-growth. Let
$(b_{i},a_{i+1})$ be an spectral gap.

  * •

For $i=1,\dots,n$, the function
$\mathfrak{st}_{\mathrm{P}}\colon(b_{i},a_{i+1})\to\mathbb{R}$ defined as

$\gamma\mapsto\mathfrak{st}_{\mathrm{P}}^{\gamma}\mathrel{\mathop{\mathchar
58\relax}}=\inf\\{\alpha+\theta\mathrel{\mathop{\mathchar
58\relax}}(\alpha,\theta)\in\mathrm{St}_{\mathrm{P}}^{\gamma}\\},$

is called function of optimal stable ratio.

  * •

For $i=0,\dots,n-1$, the function
$\mathfrak{un}_{\mathrm{P}}\colon(b_{i},a_{i+1})\to\mathbb{R}$ defined as

$\gamma\mapsto\mathfrak{un}_{\mathrm{P}}^{\gamma}\mathrel{\mathop{\mathchar
58\relax}}=\sup\\{\beta-\nu\mathrel{\mathop{\mathchar
58\relax}}(\beta,\nu)\in\mathrm{Un}_{\mathrm{P}}^{\gamma}\\},$

is called function of optimal unstable ratio.

In the remainder of this section, we will delve into the properties of the
functions $\mathfrak{st}_{\mathrm{P}}$ and $\mathfrak{un}_{\mathrm{P}}$.

######  Lemma 2.16.

On every spectral gap, the maps $\mathfrak{st}_{\mathrm{P}}$ and
$\mathfrak{un}_{\mathrm{P}}$ are decreasing.

###### Proof.

We prove that $\mathfrak{st}_{\mathrm{P}}$ is decreasing and
$\mathfrak{un}_{\mathrm{P}}$ follows similarly. Let
$\gamma,\zeta\in(b_{i},a_{i+1})$ with $\gamma<\zeta$. We will establish that
$\mathrm{St}_{\mathrm{P}}^{\gamma}\subset\mathrm{St}_{\mathrm{P}}^{\zeta}$,
from which it follows that
$\mathfrak{st}_{\mathrm{P}}^{\zeta}\leq\mathfrak{st}_{\mathrm{P}}^{\gamma}$.
For $(\alpha,\theta)\in\mathrm{St}_{\mathrm{P}}^{\gamma}$, there exists a
constant $K>0$ such that

$\|\Phi_{\gamma}(t,s)\mathrm{P}(s)\|\leq
K\left(\frac{\mu(t)}{\mu(s)}\right)^{\alpha}\mu(s)^{\mathrm{sgn}(s)\theta},\quad\forall\,t\geq
s.$

Now, as $\zeta>\gamma$, then for $t\geq s$ we have
$\left(\frac{\mu(t)}{\mu(s)}\right)^{\gamma-\zeta}<1$. Moreover, we have the
identity
$\Phi_{\zeta}(t,s)=\Phi_{\gamma}(t,s)\left(\frac{\mu(t)}{\mu(s)}\right)^{\gamma-\zeta}$,
for all $t,s\in\mathbb{R}$. Thus, we get the estimation

$\displaystyle\|\Phi_{\zeta}(t,s)\mathrm{P}(s)\|$ $\displaystyle\leq
K\left(\frac{\mu(t)}{\mu(s)}\right)^{\alpha+\gamma-\zeta}\mu(s)^{\mathrm{sgn}(s)\theta}$
$\displaystyle\leq
K\left(\frac{\mu(t)}{\mu(s)}\right)^{\alpha}\mu(s)^{\mathrm{sgn}(s)\theta},\quad\forall\,t\geq
s,$

which implies that $(\alpha,\theta)\in\mathrm{St}_{\mathrm{P}}^{\zeta}$.
Therefore, we infer that
$\mathrm{St}_{\mathrm{P}}^{\gamma}\subset\mathrm{St}_{\mathrm{P}}^{\zeta}$,
obtaining the desired result. ∎

######  Proposition 2.17.

On every spectral gap, the maps $\mathfrak{st}_{\mathrm{P}}$ and
$\mathfrak{un}_{\mathrm{P}}$ are continuous.

###### Proof.

We will prove $\mathfrak{st}_{\mathrm{P}}$ is continuous and
$\mathfrak{un}_{\mathrm{P}}$ will follow analogously.

As $\mathfrak{st}_{\mathrm{P}}$ is a decreasing function, any discontinuity is
a jump discontinuity. Let $\gamma\in(b_{i},a_{i+1})$ and suppose
$\mathfrak{st}_{\mathrm{P}}$ is not left-continuous at $\gamma$. Then, there
is some $\epsilon_{1}>0$ such that
$\mathfrak{st}_{\mathrm{P}}^{\zeta}>\mathfrak{st}_{\mathrm{P}}^{\gamma}+\epsilon_{1}$
for every $\zeta<\gamma$. Chose $\epsilon_{2}>0$ with
$\epsilon_{2}<\epsilon_{1}/3$. Chose now
$(\alpha^{\gamma},\theta^{\gamma})\in\mathrm{St}_{\mathrm{P}}^{\gamma}$ such
that
$\mathfrak{st}_{\mathrm{P}}^{\gamma}<\alpha^{\gamma}+\theta^{\gamma}<\mathfrak{st}_{\mathrm{P}}^{\gamma}+\epsilon_{2}$.
Then, there exists $K>0$ such that

$\|\Phi_{\gamma}(t,s)\mathrm{P}(s)\|\leq
K\left(\frac{\mu(t)}{\mu(s)}\right)^{\alpha^{\gamma}}\mu(s)^{\mathrm{sgn}(s)\theta^{\gamma}},\quad\forall\,t\geq
s.$

Chose now $\epsilon_{3}>0$ with
$\epsilon_{3}<\min\\{-(\alpha^{\gamma}+\theta^{\gamma}),\epsilon_{1}/3\\}$.
For $\zeta\in(\gamma-\epsilon_{3},\gamma)$ we have

$\displaystyle\|\Phi_{\zeta}(t,s)\mathrm{P}(s)\|$ $\displaystyle\leq
K\left(\frac{\mu(t)}{\mu(s)}\right)^{\alpha^{\gamma}+\gamma-\zeta}\mu(s)^{\mathrm{sgn}(s)\theta^{\gamma}}$
$\displaystyle\leq
K\left(\frac{\mu(t)}{\mu(s)}\right)^{\alpha^{\gamma}+\epsilon_{3}}\mu(s)^{\mathrm{sgn}(s)\theta^{\gamma}},\quad\forall\,t\geq
s.$

In addition, since $\alpha^{\gamma}+\epsilon_{3}+\theta^{\gamma}<0$, we infer
that
$(\alpha^{\gamma}+\epsilon_{3},\theta^{\gamma})\in\mathrm{St}_{\mathrm{P}}^{\zeta}$.
However, we also have the estimation

$\mathfrak{st}_{\mathrm{P}}^{\zeta}\leq\alpha^{\gamma}+\epsilon_{3}+\theta^{\gamma}<\mathfrak{st}_{\mathrm{P}}^{\gamma}+\epsilon_{3}+\epsilon_{2}<\mathfrak{st}_{\mathrm{P}}^{\gamma}+\epsilon_{1}<\mathfrak{st}_{\mathrm{P}}^{\zeta},$

which is evidently contradictory. Thus, $\mathfrak{st}_{\mathrm{P}}$ must be
left-continuous.

Suppose now that $\mathfrak{st}_{\mathrm{P}}$ is not right-continuous at
$\gamma$. Then, there is some $\epsilon_{4}>0$ such that
$\mathfrak{st}_{\mathrm{P}}^{\zeta}<\mathfrak{st}_{\mathrm{P}}^{\gamma}-\epsilon_{4}$
for every $\zeta>\gamma$. Chose $\epsilon_{5}>0$ with
$\epsilon_{5}<\epsilon_{4}/3$, and $\zeta>\gamma$ with
$\zeta-\gamma<\epsilon_{5}$. Chose now $\epsilon_{6}>0$ with
$\epsilon_{6}<\epsilon_{4}/3$, and a pair
$(\alpha^{\zeta},\theta^{\zeta})\in\mathrm{St}_{\mathrm{P}}^{\zeta}$ such that
$\mathfrak{st}_{\mathrm{P}}^{\zeta}<\alpha^{\zeta}+\theta^{\zeta}<\mathfrak{st}_{\mathrm{P}}^{\zeta}+\epsilon_{6}$.
This implies that there exists $K>0$ such that

$\|\Phi_{\zeta}(t,s)\mathrm{P}(s)\|\leq
K\left(\frac{\mu(t)}{\mu(s)}\right)^{\alpha^{\zeta}}\mu(s)^{\mathrm{sgn}(s)\theta^{\zeta}},\quad\forall\,t\geq
s.$

This leads us to deduce

$\displaystyle\|\Phi_{\gamma}(t,s)\mathrm{P}(s)\|$ $\displaystyle\leq
K\left(\frac{\mu(t)}{\mu(s)}\right)^{\alpha^{\zeta}+\zeta-\gamma}\mu(s)^{\mathrm{sgn}(s)\theta^{\zeta}}$
$\displaystyle\leq
K\left(\frac{\mu(t)}{\mu(s)}\right)^{\alpha^{\zeta}+\epsilon_{5}}\mu(s)^{\mathrm{sgn}(s)\theta^{\zeta}},\quad\forall\,t\geq
s.$ (2.4)

Note that

$\alpha^{\zeta}+\theta^{\zeta}+\epsilon_{5}<\mathfrak{st}_{\mathrm{P}}^{\zeta}+\epsilon_{6}+\epsilon_{5}<\mathfrak{st}_{\mathrm{P}}^{\gamma}-\epsilon_{4}+\epsilon_{6}+\epsilon_{5}<\mathfrak{st}_{\mathrm{P}}^{\gamma}\leq
0.$

First, this implies $\alpha^{\zeta}+\theta^{\zeta}+\epsilon_{5}<0$, hence (2)
implies
$(\alpha^{\zeta}+\epsilon_{5},\theta^{\zeta})\in\mathrm{St}_{\mathrm{P}}^{\gamma}$.
Nevertheless, we also have
$\alpha^{\zeta}+\theta^{\zeta}+\epsilon_{5}<\mathfrak{st}_{\mathrm{P}}^{\gamma}$,
which contradicts the minimality of $\mathfrak{st}_{\mathrm{P}}^{\gamma}$.
Thus, $\mathfrak{st}_{\mathrm{P}}$ must be right continuous. ∎

The previous results imply that both $\mathfrak{st}_{\mathrm{P}}$ and
$\mathfrak{un}_{\mathrm{P}}$ have lateral limits on every bounded spectral
gap. Similarly, in the case of the spectral gaps $(-\infty,a_{1})$ and
$(b_{n},+\infty)$, although the maps might be unbounded, the lateral limits
towards $a_{1}$ and $b_{n}$ still exist.

Next, we present the principal property of functions
$\mathfrak{st}_{\mathrm{P}}$ and $\mathfrak{un}_{\mathrm{P}}$, which will be
crucial to establish the novelty and the main result of the subsequent
section.

######  Theorem 2.18.

Assume the system (2.3) admits $({\mathrm{N}\mu},\epsilon)$-growth. For every
bounded spectral gap $(b_{i},a_{i+1})$, we have

$\lim_{\gamma\to b_{i}^{+}}\mathfrak{st}_{\mathrm{P}}^{\gamma}=\lim_{\gamma\to
a_{i+1}^{-}}\mathfrak{un}_{\mathrm{P}}^{\gamma}=0.$

Moreover, on the unbounded spectral gaps we have $\displaystyle\lim_{\gamma\to
b_{n}^{+}}\mathfrak{st}_{\mathrm{Id}}^{\gamma}=\lim_{\gamma\to
a_{1}^{-}}\mathfrak{un}_{0}^{\gamma}=0$.

###### Proof.

We will prove $\lim_{\gamma\to
b_{i}^{+}}\mathfrak{st}_{\mathrm{P}}^{\gamma}=0$ for a bounded spectral gap
(the other limits follow similarly). Since the map
$\mathfrak{st}_{\mathrm{P}}$ is decreasing, negative and continuous, it
follows that the lateral limits exist and are non-positive .

Suppose $\lim_{\gamma\to b_{i}^{+}}\mathfrak{st}_{\mathrm{P}}^{\gamma}=M<0$.
We chose $\epsilon_{1}>0$ such that $M+\epsilon_{1}<0$. Moreover, for each
$\gamma\in(b_{i},a_{i+1})$ we chose
$(\alpha^{\gamma},\theta^{\gamma})\in\mathrm{St}_{\mathrm{P}}^{\gamma}$ such
that

$\mathfrak{st}_{\mathrm{P}}^{\gamma}<\alpha^{\gamma}+\theta^{\gamma}<\mathfrak{st}_{\mathrm{P}}^{\gamma}+\epsilon_{1}.$

Note that this defines a new function
$\widetilde{\mathfrak{st}}_{\mathrm{P}}\mathrel{\mathop{\mathchar
58\relax}}(b_{i},a_{i+1})\to\mathbb{R}$,
$\gamma\mapsto\widetilde{\mathfrak{st}}_{\mathrm{P}}^{\gamma}\mathrel{\mathop{\mathchar
58\relax}}=\alpha^{\gamma}+\theta^{\gamma}$ which verifies
$\mathfrak{st}_{\mathrm{P}}^{\gamma}<\widetilde{\mathfrak{st}}_{\mathrm{P}}^{\gamma}<\mathfrak{st}_{\mathrm{P}}^{\gamma}+\epsilon_{1}$.
Then

$\liminf_{\gamma\to
b_{i}^{+}}\widetilde{\mathfrak{st}}_{\mathrm{P}}^{\gamma}\mathrel{\mathop{\mathchar
58\relax}}=\widetilde{M}\leq M+\epsilon_{1}<0.$

Let us consider $\epsilon_{2}\in(0,-\widetilde{M}/4)$. The previous lower
limit implies that there is $\delta_{1}=\delta_{1}(\epsilon_{2})>0$ such that
for every $\delta\in(0,\delta_{1})$ there is some
$\gamma\in(b_{i},b_{i}+\delta)$ such that
$\widetilde{\mathfrak{st}}_{\mathrm{P}}^{\gamma}\in(\widetilde{M}-\epsilon_{2},\widetilde{M}+\epsilon_{2})$.

Chose now $\delta_{2}=\min\\{\delta_{1}(\epsilon_{2}),-\widetilde{M}/4\\}$.
Then, there is some $\zeta\in(b_{i},b_{i}+\delta_{2})$ such that
$\widetilde{\mathfrak{st}}_{\mathrm{P}}^{\zeta}\in(\widetilde{M}-\epsilon_{2},\widetilde{M}+\epsilon_{2})$.

As by definition
$(\alpha^{\zeta},\theta^{\zeta})\in\mathrm{St}_{\mathrm{P}}^{\zeta}$, then the
$\zeta$-shifted system admits ${\mathrm{N}\mu\mathrm{D}}$ with parameters
$(\mathrm{P};\alpha^{\zeta},\beta,\theta^{\zeta},\nu)$, for some $\beta$ and
$\nu$. In other words

$\left\\{\begin{array}[]{lc}\|\Phi_{\zeta}(t,s)\mathrm{P}(s)\|\leq
K\left({\frac{\mu(t)}{\mu(s)}}\right)^{\alpha^{\zeta}}\mu(s)^{\mathrm{sgn}(s)\theta^{\zeta}},&\forall\,\,t\geq
s\\\ \\\ \|\Phi_{\zeta}(t,s)[\mathrm{Id}-\mathrm{P}(s)]\|\leq
K\left({\frac{\mu(t)}{\mu(s)}}\right)^{\beta}\mu(s)^{\mathrm{sgn}(s)\nu},&\forall\,\,t\leq
s.\end{array}\right.$

Moreover, we have the identity

$\Phi_{b_{i}}(t,s)=\Phi_{\zeta}(t,s)\left(\frac{\mu(t)}{\mu(s)}\right)^{\zeta-
b_{i}},\qquad\text{ for all $t,s\in\mathbb{R}$},$

from which we obtain

$\left\\{\begin{array}[]{lc}\|\Phi_{b_{i}}(t,s)\mathrm{P}(s)\|\leq
K\left({\frac{\mu(t)}{\mu(s)}}\right)^{\alpha^{\zeta}+(\zeta-
b_{i})}\mu(s)^{\mathrm{sgn}(s)\theta^{\zeta}},&\forall\,\,t\geq s\\\ \\\
\|\Phi_{b_{i}}(t,s)[\mathrm{Id}-\mathrm{P}(s)]\|\leq
K\left({\frac{\mu(t)}{\mu(s)}}\right)^{\beta+(\zeta-
b_{i})}\mu(s)^{\mathrm{sgn}(s)\nu},&\forall\,\,t\leq s,\end{array}\right.$

where clearly $\beta+(\zeta-b_{i})-\nu>\beta-\nu>0$, and

$\alpha^{\zeta}+(\zeta-
b_{i})+\theta^{\zeta}<\widetilde{\mathfrak{st}}_{\mathrm{P}}^{\zeta}+\delta_{2}<\widetilde{M}/2<0.$

Therefore, the $b_{i}$-shifted system admits ${\mathrm{N}\mu\mathrm{D}}$. This
is a contradiction due to $b_{i}$ belongs to
$\Sigma_{\mathrm{N}\mu\mathrm{D}}(A)$. In consequence, we infer that $M=0$. ∎

######  Remark 2.19.

Since the function $\gamma\mapsto\mathfrak{st}_{\mathrm{P}}^{\gamma}$ is
decreasing, negative and continuous on every spectral gap, if there is some
number $b$ such that $\lim_{\gamma\to
b^{+}}\mathfrak{st}_{\mathrm{P}}^{\gamma}=0$, then $b$ must be a boundary
point of said spectral gap, i.e. $b$ is exactly one of the $b_{i}$, with
$i=1,\dots,n$, of the spectrum, as tailored on Theorem 2.13. Similarly, if
$\lim_{\gamma\to a^{-}}\mathfrak{un}_{\mathrm{P}}^{\gamma}=0$, for some $a$,
then $a$ must be a boundary point in the spectrum. Observe that this assertion
can be regarded as the reciprocal to the previous theorem.

######  Corollary 2.20.

For every $\varepsilon>0$, there exists $\delta>0$ such that

$\gamma\in(b_{i},b_{i}+\delta)\Longrightarrow|\mathfrak{st}_{\mathrm{P}}^{\gamma}|<\varepsilon,$

and

$\gamma\in(a_{i}-\delta,a_{i})\Longrightarrow\mathfrak{un}_{\mathrm{P}}^{\gamma}<\varepsilon,$

for all $i=1,\dots,n$.

######  Remark 2.21.

For systems that present uniform $\mu$-dichotomy, the previous result implies
that the constants $\alpha^{\gamma}$ or $\beta^{\gamma}$ tend to zero when
$\gamma$ approaches the spectrum.

To conclude this section, we address a property of the optimal ratio maps on
the unbounded spectral gaps.

######  Lemma 2.22.

Assume the system (1.1) has $({\mathrm{N}\mu},\epsilon)$-growth with constants
$\widehat{K}\geq 1$ and $a\geq 0$. Then

$\lim_{\gamma\to-\infty}\mathfrak{un}_{0}^{\gamma}=-\lim_{\gamma\to+\infty}\mathfrak{st}_{\mathrm{Id}}^{\gamma}=+\infty.$

###### Proof.

We will prove
$\lim_{\gamma\to+\infty}\mathfrak{st}_{\mathrm{Id}}^{\gamma}=-\infty$ and the
other limit will follow similarly. As $\mathfrak{st}_{\mathrm{Id}}$ is
decreasing and continuous, it is enough to prove that the map cannot be
bounded. Suppose that $\mathfrak{st}_{\mathrm{Id}}$ is bounded, i.e. there is
some $N<0$ such that $N<\mathfrak{st}_{\mathrm{Id}}^{\gamma}<0$ for every
$\gamma>b_{n}$. Consider $t\geq s$. By the assumption of nonuniform
$\mu$-bounded growth, we have

$\|\Phi(t,s)\|\leq\widehat{K}\left(\frac{\mu(t)}{\mu(s)}\right)^{a}\mu(s)^{\mathrm{sgn}(s)\epsilon}.$

Thus, for $\gamma$ large enough, the estimation

$\|\Phi_{\gamma}(t,s)\|\leq\widehat{K}\left(\frac{\mu(t)}{\mu(s)}\right)^{a-\gamma}\mu(s)^{\mathrm{sgn}(s)\epsilon},$

defines a dichotomy with parameters $(\mathrm{Id};a-\gamma,*,\epsilon,*)$.
Moreover, for $\gamma>a+\epsilon-N$, we have
$\mathfrak{st}_{\mathrm{Id}}^{\gamma}\leq a-\gamma+\epsilon<N$, which is a
contradiction. ∎

##  3\. Kinematic similarity and spectra noninvariance

In this section, we aim to clarify certain inconsistencies that have emerged
from earlier studies regarding the preservation of the nonuniform
$\mu$-dichotomy spectra for systems which are nonuniformly kinematically
similar.

Let us commence by recalling the concept of kinematic similarity frequently
used in the nonuniform context, see e.g. [8, 13, 17, 23].

######  Definition 3.1.

Let $\mu\colon\mathbb{R}\to(0,+\infty)$ be a growth rate and let
$\varepsilon\geq 0$ be given. The systems (1.1) and (2.3) are nonuniformly
$(\mu,\varepsilon)$-kinematically similar, if there is a constant
$M_{\varepsilon}>0$, and a differentiable matrix valued function
$S\colon\mathbb{R}\to GL_{d}(\mathbb{R})$ satisfying the following properties:

  * (i)

$\|S(t)\|\leq M_{\varepsilon}\mu(t)^{\mathrm{sgn}(t)\varepsilon}$, for all
$t\in\mathbb{R}$.

  * (ii)

$\|S(t)^{-1}\|\leq M_{\varepsilon}\mu(t)^{\mathrm{sgn}(t)\varepsilon}$, for
all $t\in\mathbb{R}$.

  * (iii)

If $t\mapsto y(t)$ is a solution for (1.1), then $t\mapsto
z(t)\mathrel{\mathop{\mathchar 58\relax}}=S(t)^{-1}y(t)$ is a solution for
(2.3).

  * (iv)

If $t\mapsto z(t)$ is a solution of (2.3), then $t\mapsto
y(t)\mathrel{\mathop{\mathchar 58\relax}}=S(t)z(t)$ is a solution of (1.1).

In the particular case that $\varepsilon=0$, it is said that systems (1.1) and
(2.3) are uniformly kinematically similar.

######  Remark 3.2.

Every matrix valued function $S\colon\mathbb{R}\to GL_{d}(\mathbb{R})$
satisfying (i) and (ii) for some $\varepsilon>0$ is called a _nonuniform
Lyapunov matrix function_ with respect to $\mu$, and the change of variables
$y(t)=S(t)z(t)$ is called a _nonuniform Lyapunov transformation_ with respect
to $\mu$. As usual, in the case $\varepsilon=0$ the term “nonuniform” is
replaced with “uniform”.

The next result, which is a nonuniform version borrowed from [19, Lemma 2.1],
characterizes the nonuniform $(\mu,\varepsilon)$-kinematic similarity through
special cases of nonuniform Lyapunov matrix functions.

######  Lemma 3.3.

([17, Lemma 13]) Let $S(t)$ be a nonuniform Lyapunov matrix function with
respect to $\mu$. Then, the following statements are equivalent:

  * (a)

$S(t)$ verifies (iii) and (iv) from Definition 3.1.

  * (b)

The identity $\Phi(t,s)S(s)=S(t)\Psi(t,s)$ holds, for all $t,s\in\mathbb{R}$,
where $\Phi$ and $\Psi$ are the evolution operators of (1.1) and (2.3),
respectively.

  * (c)

$S(t)$ is a solution of $S^{\prime}=A(t)S-SB(t)$.

The following result is a simple but useful consequence of the previous lemma.

######  Corollary 3.4.

Assume the systems (1.1) and (2.3) are nonuniformly
$(\mu,\varepsilon)$-kinematically similar. Then for every
$\gamma\in\mathbb{R}$, the $\gamma$-shifted systems from both (1.1) and (2.3)
are nonuniformly $(\mu,\varepsilon)$-kinematically similar with the same
Lyapunov transformation and the same $\varepsilon$.

###### Proof.

This follows from the second equivalence on Lemma 3.3, as both $\Phi_{\gamma}$
and $\Psi_{\gamma}$ are obtained by multiplying $\Phi$ and $\Psi$ by the same
scalar evolution. ∎

In the next, we slightly reformulate a result established in [17, Lemma 14].
We include some extra details related with the parameters of the
${\mathrm{N}\mu\mathrm{D}}$.

######  Lemma 3.5.

Assume the system (1.1) admits ${\mathrm{N}\mu\mathrm{D}}$ with parameters
$(\mathrm{P};\alpha,\beta,\theta,\nu)$, where $\mathrm{P}$ is neither $0$ nor
$\mathrm{Id}$, and is nonuniformly $(\mu,\varepsilon)$-kinematically similar
to the system (2.3) with a Lyapunov function $S$, and

$\varepsilon<\frac{1}{3}\min\\{-(\alpha+\theta),\beta-\nu\\}.$ (3.1)

Then, the system (2.3) admits ${\mathrm{N}\mu\mathrm{D}}$ with parameters
$(\mathrm{Q};\alpha+\varepsilon,\beta-\varepsilon,\theta+2\varepsilon,\nu+2\varepsilon)$,
where $Q$ is the invariant projector for (2.3) defined by
$\mathrm{Q}(s)=S(s)^{-1}\mathrm{P}(s)S(s)$, for all $s\in\mathbb{R}$.

###### Proof.

We have

$\left\\{\begin{array}[]{lc}\|\Phi(t,s)\mathrm{P}(s)\|\leq
K\left({\frac{\mu(t)}{\mu(s)}}\right)^{\alpha}\mu(s)^{\mathrm{sgn}(s)\theta},&\forall\,\,t\geq
s,\\\ \\\ \|\Phi(t,s)[\mathrm{Id}-\mathrm{P}(s)]\|\leq
K\left({\frac{\mu(t)}{\mu(s)}}\right)^{\beta}\mu(s)^{\mathrm{sgn}(s)\nu},&\forall\,\,t\leq
s.\end{array}\right.$

From the identity $\Psi(t,s)=S(t)^{-1}\Phi(t,s)S(s)$, we obtain

$\displaystyle\|\Psi(t,s)\mathrm{Q}(s)\|$
$\displaystyle=\|S(t)^{-1}\Phi(t,s)\mathrm{P}(s)S(s)\|$ $\displaystyle\leq
KM_{\varepsilon}^{2}\left(\frac{\mu(t)}{\mu(s)}\right)^{\alpha+\varepsilon}\mu(s)^{\mathrm{sgn}(s)(\theta+2\varepsilon)},\qquad\forall\,t\geq
s,$

and

$\displaystyle\|\Psi(t,s)[\mathrm{Id}-\mathrm{Q}(s)]\|$
$\displaystyle=\|S(t)^{-1}\Phi(t,s)[\mathrm{Id}-\mathrm{P}(s)]S(s)\|$
$\displaystyle\leq
KM_{\varepsilon}^{2}\left(\frac{\mu(t)}{\mu(s)}\right)^{\beta-\varepsilon}\mu(s)^{\mathrm{sgn}(s)(\nu+2\varepsilon)},\qquad\forall\,t\leq
s.$

Therefore, as (3.1) is verified, the system (2.3) admits
${\mathrm{N}\mu\mathrm{D}}$ with parameters
$(\mathrm{Q};\alpha+\varepsilon,\beta-\varepsilon,\theta+2\varepsilon,\nu+2\varepsilon)$.
∎

######  Remark 3.6.

The preceding lemma remains valid when the projector is either $0$ or
$\mathrm{Id}$. In such cases, we only have to consider the constants $\beta$
and $\nu$ or $\alpha$ and $\theta$, respectively, due to the absence of stable
or unstable manifolds.

In the statement of [17, Lemma 14] is asserted that under assumptions of
nonuniform $(\mu,\varepsilon)$-kinematic similarity for systems (1.1) and
(2.3), with $\varepsilon$ satisfying (3.1), the nonuniform $\mu$-dichotomy
spectrum is invariant, i.e.
$\Sigma_{{\mathrm{N}\mu\mathrm{D}}}(A)=\Sigma_{{\mathrm{N}\mu\mathrm{D}}}(B)$,
or equivalently,
$\rho_{{\mathrm{N}\mu\mathrm{D}}}(A)=\rho_{{\mathrm{N}\mu\mathrm{D}}}(B)$.
However, the origin of the parameters $\alpha$, $\beta$, $\theta$ and $\nu$
involved in (3.1) is not explicitly clarified on [17, Lemma 14].

######  Remark 3.7.

Let us explain the dilemma with the preservation of the nonuniform
$\mu$-dichotomy spectrum for systems (1.1) and (2.3) which are nonuniformly
$(\mu,\varepsilon)$-kinematically similar: note that, from Corollary 3.4, the
corresponding $\gamma$-shifted systems associated to (1.1) and (2.3) are
nonuniformly $(\mu,\varepsilon)$-kinematically similar. Hence, for
$\gamma\in\rho_{{\mathrm{N}\mu\mathrm{D}}}(A)$, and in light of Lemma 3.5, it
becomes evident that in the nonuniform case, the fulfillment of condition
(3.1) is imperative for establishing the ${\mathrm{N}\mu\mathrm{D}}$ of the
$\gamma$-shifted system (2.3), and this dichotomy holds precisely with
parameters
$(\mathrm{Q};\alpha+\varepsilon,\beta-\varepsilon,\theta+2\varepsilon,\nu+2\varepsilon)$.
Nevertheless, the parameters in (3.1) strictly depend of the election of
$\gamma\in\mathbb{R}$, i.e. for a different $\gamma$ in
$\rho_{{\mathrm{N}\mu\mathrm{D}}}(A)$ we will have distinct constants
$\alpha,\beta,\theta$ and $\nu$. Moreover, in sight of Theorem 2.18, for every
$\varepsilon>0$, if $\gamma$ is close enough to the spectrum, we obtain either
$\mathfrak{st}_{\mathrm{P}}^{\gamma}+3\varepsilon>0$ or
$\mathfrak{un}_{\mathrm{P}}^{\gamma}-3\varepsilon<0$. Consequently, for every
$(\alpha,\theta)\in\mathrm{St}_{\mathrm{P}}^{\gamma}$ or
$(\beta,\nu)\in\mathrm{Un}_{\mathrm{P}}^{\gamma}$ we would obtain
$\alpha+\varepsilon+\theta+2\varepsilon>0$ or $\beta-\nu-3\varepsilon<0$,
which implies
$(\alpha+\varepsilon,\theta+2\varepsilon)\not\in\mathrm{St}_{\mathrm{Q}}^{\gamma}$
or
$(\beta-\varepsilon,\nu+2\varepsilon)\not\in\mathrm{Un}_{\mathrm{Q}}^{\gamma}$,
independently of the projector $\mathrm{Q}$. In other words, for elements
$\gamma$ which are close enough to the spectrum of the first system, the
previous lemma does not allow for a proof that such element lies on the
resolvent set of the second system.

Considering the aforementioned explanation, [17, Lemma 14] is apparently
contradictory in light of the following examples.

######  Example 3.8.

Consider the systems $\dot{x}=-10x(t)$ and $\dot{y}=-12y(t)$. Take the growth
rate $\mu(t)=e^{t}$, for all $t\in\mathbb{R}$. Define the nonuniform Lyapunov
function $S(t)=e^{2t}$, for all $t\in\mathbb{R}$. Clearly, we obtain the
estimations $|S(t)|,|S(t)^{-1}|\leq\mu(t)^{\mathrm{sgn}(t)2}$, for all
$t\in\mathbb{R}$. In addition, we get the identity

$e^{-10(t-s)}e^{2s}=e^{2t}e^{-12(t-s)},\qquad\text{ for all
$t,s\in\mathbb{R}$}.$

Based on the preceding information, it follows that the earlier systems are
nonuniformly $(\mu,2)$-kinematically similar. Nevertheless, note that

$\Sigma_{\mu\mathrm{D}}(-10)=\Sigma_{\mathrm{N}\mu\mathrm{D}}(-10)=\\{-10\\}\neq\\{-12\\}=\Sigma_{\mathrm{N}\mu\mathrm{D}}(-12)=\Sigma_{\mu\mathrm{D}}(-12).$

Therefore, the $\mu$-dichotomy spectrum is not preserved.

In the next, inspired by [2, Prop. 2.3], we provide a second illustrative
example of nonuniformly $(\mu,\varepsilon)$-kinematically similar systems
which do not have the same nonuniform $\mu$-dichomoty spectra. Throughout the
example, we will consider the growth rate as $\mu(t)=e^{t}$, for all
$t\in\mathbb{R}$.

######  Example 3.9.

Let $\varepsilon>0$ and choose positive constants $\omega_{1}$, $\omega_{2}$
and $\kappa$ such that

$\omega_{1}>\omega_{2}>3\kappa\quad\text{ and
}\quad\omega_{1}-\omega_{2}<\varepsilon.$ (3.2)

Consider the nonautonomous scalar systems

$\dot{u}=(-\omega_{1}-\kappa t\sin t)u,$ (3.3) $\dot{v}=(-\omega_{2}-\kappa
t\sin t)v,$ (3.4)

with evolution operators given by

$\Phi(t,s)=e^{-\omega_{1}(t-s)+\kappa t\cos t-\kappa s\cos s-\kappa\sin
t+\kappa\sin s},$

and

$\Psi(t,s)=e^{-\omega_{2}(t-s)+\kappa t\cos t-\kappa s\cos s-\kappa\sin
t+\kappa\sin s},$

respectively.

Define the function $S(t)=e^{(\omega_{2}-\omega_{1})t}$, for all
$t\in\mathbb{R}$. Clearly, the function $S$ verifies:

  * •

$|S(t)|\leq e^{(\omega_{1}-\omega_{2})|t|}\leq e^{\varepsilon|t|}$, for all
$t\in\mathbb{R}$.

  * •

$|S(t)^{-1}|\leq e^{(\omega_{1}-\omega_{2})|t|}\leq e^{\varepsilon|t|}$, for
all $t\in\mathbb{R}$.

  * •

$\Phi(t,s)S(s)=S(t)\Psi(t,s)$, for all $t,s\in\mathbb{R}$.

From Lemma 3.3, we deduce that systems (3.3) and (3.4) are nonuniformly
$(\mu,\varepsilon)$-kinematically similar.

On the other hand, we have

$|\Phi(t,s)|\leq
e^{2\kappa}e^{(-\omega_{1}+\kappa)(t-s)}e^{2\kappa|s|},\quad\,t\geq s.$ (3.5)

From [2, Prop. 2.3], we infer that the parameters involved on the above
estimation are optimal in the sense that
$\mathfrak{st}_{\mathrm{Id}}=-\omega_{1}+3\kappa$. Similarly, we obtain

$|\Phi(t,s)|\leq
e^{2\kappa}e^{(-\omega_{1}-\kappa)(t-s)}e^{2\kappa|t|},\quad\,s\geq t,$ (3.6)

which in conjunction with (3.5) implies

$|\Phi(t,s)|\leq
e^{2\kappa}e^{(\omega_{1}+3\kappa)|t-s|}e^{2\kappa|t|},\quad\forall\,s,t\in\mathbb{R}.$

In other words, the system (3.3) has nonuniform bounded growth, and thus its
spectrum is compact and nonempty. Moreover, as (3.3) is one-dimensional, its
spectrum is just one interval, hence it has the form $[a,b]$. Now we want to
compute explicitly this spectrum.

From Eq. (3.5), the absolute value of the $\gamma$-shifted operator can be
estimated as

$|\Phi_{\gamma}(t,s)|\leq
e^{2\kappa}e^{(-\omega_{1}-\gamma+\kappa)(t-s)}e^{2\kappa|s|},\quad\,t\geq s.$

Therefore, if $\gamma>-\omega_{1}+3\kappa$, then $\gamma$ belongs to the
nonuniform $\mu$-resolvent of the system (3.3) and
$\mathfrak{st}_{\mathrm{Id}}^{\gamma}=-\omega_{1}-\gamma+3\kappa$. Moreover,
from Remark 2.19, we deduce that $b=-\omega_{1}+3\kappa$.

Similarly, from Eq. (3.6), we derive the estimation

$|\Phi_{\gamma}(t,s)|\leq
e^{2\kappa}e^{(-\omega_{1}-\gamma-\kappa)(t-s)}e^{2\kappa|t|},\quad\,s\geq t.$

Therefore, if $\gamma<-\omega_{1}-3\kappa$, then $\gamma$ belongs to the
nonuniform $\mu$-resolvent of the system (3.3) and
$\mathfrak{un}_{0}^{\gamma}=-\omega_{1}-\gamma+-3\kappa$. Now, from Remark
2.19, we infer that $a=-\omega_{1}-3\kappa$.

Following this argument, it is deduced that the nonuniform $\mu$-dichotomy
spectrum of system (3.3) is given by

$\Sigma_{\mathrm{N}\mu\mathrm{D}}(u)=[-\omega_{1}-3\kappa,-\omega_{1}+3\kappa].$

Analogously to the previous analysis, we deduce that the nonuniform
$\mu$-dichotomy spectrum of the system (3.4) is given by
$\Sigma_{\mathrm{N}\mu\mathrm{D}}(v)=[-\omega_{2}-3\kappa,-\omega_{2}+3\kappa]$.
In summary,
$\Sigma_{\mathrm{N}\mu\mathrm{D}}(u)\neq\Sigma_{\mathrm{N}\mu\mathrm{D}}(v)$,
notwithstanding the nonuniform $(\mu,\varepsilon)$-kinematic similarity
between systems (3.3) and (3.4).

Finally, we emphasize that condition $\omega_{1}>\omega_{2}>3\kappa$ is added
just to ensure that the spectra are contained on the nonpositive semi axis,
but the computations follow the same if we drop this assumption.

######  Remark 3.10.

A similar conclusion to [17, Lemma 14], regarding the invariance of the
nonuniform $\mu$-dichotomy spectrum, was presented in the nonuniform
exponential case in [8, Lemma 3.6], [8, Corollary 3.7], and also in [23, Lemma
3.2]. As observed in Example 3.8 and 3.9, there is an inconsistency in
affirming the invariance of the nonuniform dichotomy spectrum through
nonuniform kinematic similarity. This inconsistency directly challenges the
assertions made in the aforementioned references, as well as any conclusions
that relied on their assumed validity.

######  Remark 3.11.

In literature, see e.g. [2, 13], some authors consider a slightly different
definition of nonuniform dichotomy, where the estimations (2.1) are once again
considered, but conditions $\alpha+\theta<0$ and $\beta-\nu>0$ are not
required, while still taking $\alpha<0<\beta$ and $\theta,\nu\geq 0$. We call
this a slow nonuniform $\mu$-dichotomy (${\mathrm{sN}\mu\mathrm{D}}$). This
dichotomy induces a different definition of the spectrum, but as every
dichotomy is in particular a slow dichotomy, we obtain
$\Sigma_{\mathrm{sN}\mu\mathrm{D}}\subset\Sigma_{\mathrm{N}\mu\mathrm{D}}\subset\Sigma_{\mu\mathrm{D}}$,
and indeed the spectra can be different sets.

For instance, consider the system (3.3) and the growth rate $\mu(t)=e^{t}$,
for all $t\in\mathbb{R}$. Therefore, we have

$\Sigma_{\mathrm{sN}\mu\mathrm{D}}(u)=[-\omega_{1}-\kappa,-\omega_{1}+\kappa]\subset\Sigma_{\mathrm{N}\mu\mathrm{D}}(u)=[-\omega_{1}-3\kappa,-\omega_{1}+3\kappa]\subset\Sigma_{\mu\mathrm{D}}(u)=\mathbb{R}.$

Nevertheless, under a similar construction that we give here, it is proved
that the slow spectrum is once again noninvariant under nonuniform kinematic
similarities. Indeed, in our Example 3.9, we still have the nonuniform
kinematic similarity between (3.3) and (3.4) while
$\Sigma_{\mathrm{sN}\mu\mathrm{D}}(v)=[-\omega_{2}-\kappa,-\omega_{2}+\kappa]$,
which shows that the problems with the spectra invariance do not come from the
specific definition of dichotomy.

In the remainder of this section, we will attempt to propose a way to address
the issue of noninvariance of the nonuniform $\mu$-dichotomy spectrum. For
this purpose, we will employ the optimal stable and unstable ratio functions
established in the previous section.

######  Definition 3.12.

The function of global optimal ratio is
$\mathfrak{Gor}\mathrel{\mathop{\mathchar
58\relax}}\rho_{\mathrm{N}\mu\mathrm{D}}\to\mathbb{R}$, defined by

$\mathfrak{Gor}^{\gamma}=\left\\{\begin{array}[]{lcc}\mathfrak{un}_{0}^{\gamma},&\text{
for }&\gamma\in(-\infty,a_{1}),\\\
\min\\{-\mathfrak{st}_{\mathrm{P}}^{\gamma},\mathfrak{un}_{\mathrm{P}}^{\gamma}\\},&\text{
for }&\gamma\in(b_{i},a_{i+1}),\,i\in\\{1,\dots,n-1\\},\\\
-\mathfrak{st}_{\mathrm{Id}}^{\gamma},&\text{ for
}&\gamma\in(b_{n},+\infty).\end{array}\right.$

The global optimal ratio function is defined on the $\mu$-resolvent set of a
system having $({\mathrm{N}\mu},\epsilon)$-growth (which is open). This
function is continuous (Proposition 2.17), strictly positive, tends to zero
when $\gamma$ approaches to the spectrum (Theorem 2.18) and to infinity when
$|\gamma|$ does (Lemma 2.22).

######  Proposition 3.13.

Assume the system (1.1) has $({\mathrm{N}\mu},\epsilon)$-growth. Assume
further that systems (1.1) and (2.3) are nonuniformly
$(\mu,\varepsilon)$-kinematically similar. If
$\gamma\in\rho_{\mathrm{N}\mu\mathrm{D}}(A)$ and

$3\varepsilon<\mathfrak{Gor}^{\gamma},$

then $\gamma\in\rho_{\mathrm{N}\mu\mathrm{D}}(B)$.

###### Proof.

Assume without loss of generality that $\gamma$ lies on a bounded spectral
gap, so the shifted system admits ${\mathrm{N}\mu\mathrm{D}}$ with some non-
trivial projector $\mathrm{P}$. As the system (1.1) has nonuniform
$\mu$-bounded growth, the functions of optimal stable and unstable ratio
$\mathfrak{st}_{\mathrm{P}}$ and $\mathfrak{un}_{\mathrm{P}}$ are well
defined. Since
$3\varepsilon<\mathfrak{Gor}^{\gamma}\leq-\mathfrak{st}_{\mathrm{P}}^{\gamma}$,
we can chose $\delta>0$ with
$\delta<-\mathfrak{st}_{\mathrm{P}}^{\gamma}-3\varepsilon$. By definition of
$\mathfrak{st}_{\mathrm{P}}^{\gamma}$, there is some
$(\alpha_{0},\theta_{0})\in\mathrm{St}_{\mathrm{P}}^{\gamma}$ such that

$\mathfrak{st}_{\mathrm{P}}^{\gamma}<\alpha_{0}+\theta_{0}<\mathfrak{st}_{\mathrm{P}}^{\gamma}+\delta<-3\varepsilon.$

Hence, we get $-(\alpha_{0}+\theta_{0})>3\varepsilon$. Analogously we can find
$(\beta_{0},\nu_{0})\in\mathrm{Un}_{\mathrm{P}}^{\gamma}$ such that
$\beta_{0}-\nu_{0}>3\varepsilon$.

From Corollary 3.4 and Lemma 3.5, we infer that the $\gamma$-shifted system
associated to (2.3) also admits ${\mathrm{N}\mu\mathrm{D}}$, implying
$\gamma\in\rho_{\mathrm{N}\mu\mathrm{D}}(B)$. ∎

In the next, we shall proceed to define a novel notion of nonuniform
$\mu$-dichotomy spectrum. This concept aims to offer a more effective approach
in addressing the specific challenge posed by the noninvariance through
nonuniform $(\mu,\varepsilon)$-kinematic similarity illustrated on Examples
3.8 and 3.9.

######  Definition 3.14.

Assume the system (1.1) has $({\mathrm{N}\mu},\epsilon)$-growth. For every
$\varepsilon\geq 0$, the $\varepsilon$-interior of the nonuniform
$\mu$-resolvent for system (1.1) is the set defined by

$\rho_{\mathrm{N}\mu\mathrm{D}}^{\varepsilon}(A)\mathrel{\mathop{\mathchar
58\relax}}=\left\\{\gamma\in\rho_{\mathrm{N}\mu\mathrm{D}}\mathrel{\mathop{\mathchar
58\relax}}\mathfrak{Gor}^{\gamma}>\varepsilon\right\\},$

and the $\varepsilon$-neighborhood of the nonuniform $\mu$-dichotomy spectrum
for system (1.1) is the set defined by

$\Sigma_{\mathrm{N}\mu\mathrm{D}}^{\varepsilon}(A)\mathrel{\mathop{\mathchar
58\relax}}=\mathbb{R}\setminus\rho_{\mathrm{N}\mu\mathrm{D}}^{\varepsilon}(A).$

Note that for $\varepsilon=0$, we have
$\rho_{\mathrm{N}\mu\mathrm{D}}^{\varepsilon}(A)=\rho_{\mathrm{N}\mu\mathrm{D}}(A)$
and
$\Sigma_{\mathrm{N}\mu\mathrm{D}}^{\varepsilon}(A)=\Sigma_{\mathrm{N}\mu\mathrm{D}}(A)$.

######  Proposition 3.15.

Assume the system (1.1) has $({\mathrm{N}\mu},\epsilon)$-growth. For every
$\varepsilon>0$, the $\varepsilon$-neighborhood of the nonuniform
$\mu$-dichotomy spectrum is a finite union of compact intervals. Moreover, it
is indeed a neighborhood of the closed set
$\Sigma_{\mathrm{N}\mu\mathrm{D}}(A)$, i.e. there is an open set $U$ such that
$\Sigma_{\mathrm{N}\mu\mathrm{D}}(A)\subset
U\subset\Sigma^{\varepsilon}_{\mathrm{N}\mu\mathrm{D}}(A)$.

###### Proof.

As mentioned earlier, the function $\mathfrak{Gor}$ is continuous and defined
on the open set $\rho_{\mathrm{N}\mu\mathrm{D}}(A)$, thus the
$\varepsilon$-interior of the nonuniform $\mu$-resolvent is by definition an
open set contained on $\rho_{\mathrm{N}\mu\mathrm{D}}(A)$, which implies that
$\Sigma_{\mathrm{N}\mu\mathrm{D}}^{\varepsilon}(A)$ is closed. Moreover, by
Corollary 2.20, on every bounded spectral gap $(b_{i},a_{i+1})$ there are two
positive numbers $\delta_{1},\delta_{2}>0$ such that

$\rho_{\mathrm{N}\mu\mathrm{D}}^{\varepsilon}(A)\cap(b_{i},a_{i+1})=(b_{i}+\delta_{1},a_{i+1}-\delta_{2}),$
(3.7)

where in the case that $a_{i+1}-\delta_{2}<b_{i}+\delta_{1}$, we consider the
right-hand side of (3.7) as the empty set.

Now, for the spectral gaps $(-\infty,a_{1})$ and $(b_{n},+\infty)$, from Lemma
2.22, we know both $-\mathfrak{st}_{\mathrm{P}}$ and
$\mathfrak{un}_{\mathrm{P}}$ tend to $+\infty$ when $|\gamma|\to+\infty$, thus
there are two positive numbers $\delta_{3},\delta_{4}>0$ such that

$\rho_{\mathrm{N}\mu\mathrm{D}}^{\varepsilon}(A)\cap(-\infty,a_{1})=(-\infty,a_{i+1}-\delta_{3}),$

and

$\rho_{\mathrm{N}\mu\mathrm{D}}^{\varepsilon}(A)\cap(b_{n},+\infty)=(b_{n}+\delta_{4},+\infty),$

which implies that $\Sigma_{\mathrm{N}\mu\mathrm{D}}(A)^{\varepsilon}$ is
bounded, and thus compact. Moreover, the previous intersections, in
conjunction to (3.7), imply that
$\Sigma_{\mathrm{N}\mu\mathrm{D}}^{\varepsilon}(A)$ is indeed a neighborhood
of $\Sigma_{\mathrm{N}\mu\mathrm{D}}(A)$. ∎

We finalize this section by presenting an important consequence from
Proposition 3.13. In this statement, we just assume
$({\mathrm{N}\mu},\epsilon)$-growth for system (1.1), since this property is
preserved through nonuniform $(\mu,\varepsilon)$-kinematic similarity.

######  Corollary 3.16.

Assume the system (1.1) has $({\mathrm{N}\mu},\epsilon)$-growth. If the
systems (1.1) and (2.3) are nonuniformly $(\mu,\varepsilon)$-kinematically
similar, then

$\rho_{\mathrm{N}\mu\mathrm{D}}^{3\varepsilon}(A)\subset\rho_{\mathrm{N}\mu\mathrm{D}}(B)\quad\text{
and
}\quad\rho_{\mathrm{N}\mu\mathrm{D}}^{3\varepsilon}(B)\subset\rho_{\mathrm{N}\mu\mathrm{D}}(A),$

or equivalently,

$\Sigma_{\mathrm{N}\mu\mathrm{D}}(A)\subset\Sigma_{\mathrm{N}\mu\mathrm{D}}^{3\varepsilon}(B)\quad\text{
and
}\quad\Sigma_{\mathrm{N}\mu\mathrm{D}}(B)\subset\Sigma_{\mathrm{N}\mu\mathrm{D}}^{3\varepsilon}(A).$

######  Remark 3.17.

Note that the previous corollary is consistent with Examples 3.8 and 3.9. For
instance, let us consider the systems (3.3) and (3.4). Since
$\omega_{1}-\omega_{2}<\varepsilon$, we infer the following inclusion

$\Sigma_{\mathrm{N}\mu\mathrm{D}}(u)=[-\omega_{1}-3\kappa,-\omega_{1}+3\kappa]\subset[-\omega_{2}-3\varepsilon-3\kappa,-\omega_{2}+3\varepsilon+3\kappa]=\Sigma_{\mathrm{N}\mu\mathrm{D}}^{3\varepsilon}(v)\,.$

##  4\. Endgame remarks

In this section, we present a synthesis of final remarks derived from our main
results:

  * •

In light of the preceding Corollary 3.16, we are unable to establish, using
the available tools, that if systems (1.1) and (2.3) are nonuniformly
$(\mu,\varepsilon)$-kinematically similar, then
$\rho_{\mathrm{N}\mu\mathrm{D}}(A)=\rho_{\mathrm{N}\mu\mathrm{D}}(B)$, nor
even
$\rho_{\mathrm{N}\mu\mathrm{D}}^{3\varepsilon}(B)=\rho_{\mathrm{N}\mu\mathrm{D}}^{3\varepsilon}(A)$.

  * •

In [17, Lemma 14], it was stated that the demonstration we give here for Lemma
3.5 is enough to conclude equality of the $\mu$-resolvent sets, or
equivalently, the equality of the nonuniform $\mu$-dichotomy spectra. However,
our Theorem 2.18 reveals that for every $\varepsilon>0$, the set
$\rho_{\mathrm{N}\mu\mathrm{D}}^{\varepsilon}(A)$ is strictly contained in
$\rho_{\mathrm{N}\mu\mathrm{D}}(A)$, hence the best possible conclusion from
Lemma 3.5 is our Proposition 3.13 or our Corollary 3.16, i.e. contention of
the $\varepsilon$-interior, but not equality of the resolvent sets or spectra.
Consequently, our results highlight certain inconsistencies in the proofs of
[17, Lemma 14], [8, Corollary 3.7] and [23, Lemma 3.2], where the assertion of
the invariance of the nonuniform dichotomy spectrum through nonuniform
kinematic similarity is affirmed. Thereupon, our findings call into question
as well the results that depend on the assumption of the spectrum invariance.
It could be interesting to revisit them.

  * •

The result about the invariance of uniform exponential dichotomy espectra
under classic uniform kinematic similarity [19, Corollary 2.1] still holds,
since in the case of a classic kinematic similarity, we have a Lyapunov matrix
function verifying (i) and (ii) from Definition 3.1 with $\varepsilon=0$.
Thus, the same proof we give here for Lemma 3.5 shows the spectra invariance.
In addition, we can provide an invariance result of the nonuniform
$\mu$-dichotomy spectrum considering uniform kinematically similar systems:

######  Corollary 4.1.

Assume the systems (1.1) and (2.3) are uniformly kinematically similar. Then
$\rho_{\mathrm{N}\mu\mathrm{D}}(A)=\rho_{\mathrm{N}\mu\mathrm{D}}(B)$, or
equivalently,
$\Sigma_{\mathrm{N}\mu\mathrm{D}}(B)=\Sigma_{\mathrm{N}\mu\mathrm{D}}(A)$.

###### Proof.

It follows from Lemma 3.5. ∎

######  Example 4.2.

Define the functions $\mathfrak{a}(t)=-\omega-\kappa t\sin t$, and
$\mathfrak{b}(t)=-\kappa t\sin t$, for all $t\in\mathbb{R}$, where
$\omega>6\kappa>0$. Consider the planar equations $\dot{x}=A(t)x$ and
$\dot{y}=B(t)y$, where

$A(t)=\begin{pmatrix}\mathfrak{a}(t)&0\\\
0&\mathfrak{b}(t)\end{pmatrix},\qquad t\in\mathbb{R},$

and

$B(t)=\begin{pmatrix}\mathfrak{a}(t)\cos^{2}t+\mathfrak{b}(t)\sin^{2}t&\omega\cos
t\sin t+1\\\ \omega\cos t\sin
t-1&\mathfrak{a}(t)\sin^{2}t+\mathfrak{b}(t)\cos^{2}t\end{pmatrix},\qquad
t\in\mathbb{R}.$

The evolution operators $\Phi_{A}$ and $\Psi_{B}$ are given by

$\Phi_{A}(t,s)=\begin{pmatrix}e^{-\omega(t-s)+k(t,s)}&0\\\
0&e^{k(t,s)}\end{pmatrix},$

where $k(t,s)=\kappa t\cos t-\kappa s\cos s-\kappa\sin t+\kappa\sin s$, for
all $t,s\in\mathbb{R}$, and

$\Psi_{B}(t,s)=\begin{pmatrix}a_{11}(t,s)&a_{12}(t,s)\\\
a_{21}(t,s)&a_{22}(t,s)\end{pmatrix},\qquad t,s\in\mathbb{R},$

where

    * $a_{11}(t,s)=\cos t\cos s\,e^{-\omega(t-s)+k(t,s)}+\sin t\sin s\,e^{k(t,s)}$,

    * $a_{12}(t,s)=-\cos t\sin s\,e^{-\omega(t-s)+k(t,s)}+\sin t\cos s\,e^{k(t,s)}$,

    * $a_{21}(t,s)=-\sin t\cos s\,e^{-\omega(t-s)+k(t,s)}+\cos t\sin s\,e^{k(t,s)}$,

    * $a_{22}(t,s)=\sin t\sin s\,e^{-\omega(t-s)+k(t,s)}+\cos t\cos s\,e^{k(t,s)}$.

Let us define

$S(t)=\begin{pmatrix}\cos t&-\sin t\\\ \sin t&\cos t\end{pmatrix},\qquad
t\in\mathbb{R}.$

Clearly, $\|S(t)\|=\|S(t)^{-1}\|=1$, for all $t\in\mathbb{R}$. In addition,
the identity

$S(t)\Psi_{B}(t,s)=\Phi_{A}(t,s)S(s)$

holds, for all $t,s\in\mathbb{R}$. In consequence, the planar systems are
uniformly kinematically similar.

Proceeding analogously as we computed the spectrum on Example 3.9, it is
deduced that
$\Sigma_{\mathrm{N}\mu\mathrm{D}}(A)=[-\omega-3\kappa,-\omega+3\kappa]\cup[-3\kappa,3\kappa]$.
Thus, by virtue of Corollary 4.1, we obtain
$\Sigma_{\mathrm{N}\mu\mathrm{D}}(B)=[-\omega-3\kappa,-\omega+3\kappa]\cup[-3\kappa,3\kappa]$.

  * •

For a similar reason, Silva’s reducibility result [17, Theorem 12] still
holds, but only for the uniform $\mu$-dichotomy case, that is, when the system
verifies uniform $\mu$-bounded growth.

  * •

Even if the spectrum can change, by the continuity of the global optimal ratio
map $\mathfrak{Gor}$, some characteristics can still be held for nonuniformly
kinematically similar systems. Consider for instance the following conclusions
immediately obtained from Corollary 3.16.

######  Corollary 4.3.

Assume the system (1.1) has $({\mathrm{N}\mu},\epsilon)$-growth and verifies
that $\Sigma_{\mathrm{N}\mu\mathrm{D}}(A)\subset(-\infty,0)$. Then, there is
some $\varepsilon>0$ such that every system which is nonuniformly
$(\mu,\varepsilon)$-kinematically similar to (1.1) also has its spectrum
contained on $(-\infty,0)$.

The previous conclusion is relevant since the condition of having the spectrum
contained in one of the semiaxis is a usual characterization of stable or
unstable systems. A similar strategy can be employed to prove the following
result regarding the notion of nonresonant spectra [6, 9, 20], which, as
mentioned on the introduction, has important applications in the realm of
linearization of nonautonomous systems.

######  Corollary 4.4.

Assume the system (1.1) has $({\mathrm{N}\mu},\epsilon)$-growth and its
spectrum
$\Sigma_{\mathrm{N}\mu\mathrm{D}}(A)=[a_{1},b_{1}]\cup\cdots\cup[a_{n},b_{n}]$
is nonresonant up to a certain order $p\in\mathbb{N}$, i.e. for every
$j=1,\dots,n$ and every $(k_{1},\dots,k_{n})\in(\mathbb{N}\cup\\{0\\})^{n}$
with $2\leq\sum_{i=1}^{n}k_{i}\leq p$, we have

$[a_{j},b_{j}]\cap\left[\sum_{i=1}^{n}k_{i}a_{i},\sum_{i=1}^{n}k_{i}b_{i}\right]=\emptyset,$

then, there is some $\varepsilon>0$ such that every system which is
nonuniformly $(\mu,\varepsilon)$-kinematically similar to (1.1) also has a
nonresonant spectrum up to the order $p\in\mathbb{N}$.

  * •

A question we leave open is the rate at which the convergence in Theorem 2.18
is attained. Note that in all our examples we have the estimations
$|\mathfrak{st}_{\mathrm{P}}^{\gamma}|\leq|\gamma-b_{i}|$ and
$\mathfrak{un}_{\mathrm{P}}^{\gamma}\leq|a_{i+1}-\gamma|$, for $\gamma$ in the
correspondent spectral gap. We conjecture that this estimation is always
valid, but we have not been able to prove it or find a counterexample.

## References

  * [1] Backes, L.; Dragičević, D.: Smooth linearization of nonautonomous dynamics under polynomial behaviour, preprint available at https://doi.org/10.48550/arXiv.2210.04804.
  * [2] Barreira L.; Valls, C.: Stability of Nonautomous Differential Equations, Lecture Notes in Mathematics, Springer, 2008.
  * [3] Barreira, L.; Valls, C.: A Grobman-Hartman theorem for nonuniformly hyperbolic dynamics, J. Differential Equations 228 (2006), no. 1, 285–310.
  * [4] Barreira, L; Valls, C.: Smoothness of invariant manifolds for nonautonomous equations, Comm. Math. Phys. 259 (2005), no.3, 639–677.
  * [5] Castañeda, Á.; Huerta, I.; Robledo, G.: The dichotomy spectrum approach for a global nonuniform asymptotic stability problem: triangular case via uniformization, preprint available at https://doi.org/10.48550/arXiv.2210.01943.
  * [6] Castañeda, Á.; Jara, N.: A generalization of Siegmund’s normal forms theorem to systems with $\mu$-dichotomies, preprint available at https://doi.org/10.48550/arXiv.2312.04544.
  * [7] Coppel, W. A.: Dichotomies in Stability Theory, Lecture Notes in Mathematics, vol. 629, Springer, 1978.
  * [8] Chu, J.; Liao, F-F.; Siegmund, S.; Xia, Y.; Zhang, W.: Nonuniform dichotomy spectrum and reducibility for nonautonomous equations, Bull. Sci. Math. 139 (2015), no.5, 538–557.
  * [9] Cuong, L. V.; Doan, T. S.; Siegmund, S.: A Sternberg theorem for nonautonomous differential equations, J. Dynam. Differential Equations 31 (2019), no. 3, 1279–1299.
  * [10] Dragičević, D.; Zhang, W., Zhang, W.: Smooth linearization of nonautonomous differential equations with a nonuniform dichotomy, Proc. Lond. Math. Soc. 121 (2020), 32–50.
  * [11] Dragičević, D.: Admissibility and nonuniform polynomial dichotomies, Math. Nachr. 293 (2020), 226–243.
  * [12] Dragičević, D.; Sasu, A. L.; Sasu, B.: On polynomial dichotomies of discrete nonautonomous systems on the half-line, Carpathian J. Math. 38 (2022), 663–680.
  * [13] Huerta, I.: Linearization of a nonautonomous unbounded system with nonuniform contraction: a spectral approach, Discrete Contin. Dyn. Syst. 40(2020), no.9, 5571–5590.
  * [14] Perron, O.: Die Stabilitätsfrage bei Differentialgleichungen, Math. Z. 32 (1930), no.1, 703–728.
  * [15] Poincaré, H.: Mémoire sur les courbes définies par une équation différentielle, Thesis 5 (1879); Also Oeuvres I, Gauthier—Villars, Paris, 1928, pp. 59–129.
  * [16] Sacker, R. J.; Sell, G. R.: A spectral theory for linear differential systems, J. Differential Equations 27 (1978), no. 3, 320–358.
  * [17] Silva, C. M.: Nonuniform $\mu$-dichotomy spectrum and kinematic similarity, J. Differential Equations 375 (2023), 618-652.
  * [18] Siegmund, S.: Dichotomy spectrum for nonautonomous differential equations, J. Dynam. Differential Equations 14 (2002), no.1, 243–258
  * [19] Siegmund, S.: Reducibility of nonautonomous linear differential equations, J. London Math. Soc. (2) 65 (2002), no. 2, 397–410.
  * [20] Siegmund, S.: Normal forms for nonautonomous differential equations, J. Differential Equations 178 (2002), no.2, 541–573.
  * [21] Sternberg, S.: Local contractions and a theorem of Poincaré, Am. J. Math. 79 (1957), 809-824.
  * [22] Sternberg, S.: On the structure of local homeomorphisms of Euclidian $n-$space, I. Am. J. Math. 80 (1958), 623-631.
  * [23] Zhang, X.: Nonuniform dichotomy spectrum and normal forms for nonautonomous differential systems, J. Funct. Anal. 267 (2014), no.7, 1889–1916.


# Identifying Distributional Differences in Convective Evolution Prior to
Rapid Intensification in Tropical Cyclones

Trey McNeely 1 Galen Vincent 1 Rafael Izbicki 2 Kimberly M. Wood 3 Ann B. Lee
1

1Department of Statistics & Data Science, Carnegie Mellon University
2Department of Statistics, Federal University of São Carlos, São Carlos,
Brazil
3Department of Geosciences, Mississippi State University
<EMAIL_ADDRESS>

###### Abstract

Tropical cyclone (TC) intensity forecasts are issued by human forecasters who
evaluate spatio-temporal observations (e.g., satellite imagery) and model
output (e.g., numerical weather prediction, statistical models) to produce
forecasts every 6 hours. Within these time constraints, it can be challenging
to draw insight from such data. While high-capacity machine learning methods
are well suited for prediction problems with complex sequence data, extracting
interpretable scientific information with such methods is difficult. Here we
leverage powerful AI prediction algorithms and classical statistical inference
to identify patterns in the evolution of TC convective structure leading up to
the rapid intensification of a storm, hence providing forecasters and
scientists with key insight into TC behavior.

##  1 Introduction

Background. Tropical cyclones (TCs) are powerful, organized atmospheric
systems that draw energy from the upper levels of the ocean. They rank among
the deadliest and costliest natural disasters in the United States, and rising
ocean heat content promises to amplify their danger over time [12]. TC
intensity and motion forecasting alike have improved since the 1990s, but
intensity forecasting has lagged behind [3]. Cases of rapid intensification
(RI), defined for this work as an intensification of at least 25 knots within
24 hours, are especially difficult to predict [7, 8, 9, 20]. Some of the most
impactful TCs in recent years underwent RI prior to landfall (e.g., Hurricanes
Irma [2017], Dorian [2019], and Ida [2021]). Understanding TC evolution in the
lead-up to RI events is therefore a critical component of forecasting and
damage mitigation efforts. In this work, we leverage powerful machine learning
(ML) methods to provide _scientific insight_ into the evolution of TCs leading
up to and during such events.

Multiple internal and external factors drive TC behavior [9]. This work
focuses on a key internal factor: the spatio-temporal structure of deep
convection—that is, vertical circulation of air—within the storm. Convective
patterns within TCs serve as reliable indicators of intensity and drivers of
intensity change; for example, identifying convective patterns is critical to
the Dvorak Technique for TC intensity estimation [4, 15, 16, 5]. Since
temperature generally decreases with height in the troposphere, the
temperature of a cloud top is a proxy for its altitude. Thus taller clouds
have colder tops, implying deeper (and thus stronger) convection. Stronger
convection, particularly near the TC center, tends to be associated with
stronger TCs. Infrared (IR) imagery from geostationary satellites such as the
Geostationary Operational Environmental Satellites (GOES) provide high spatio-
temporal resolution estimates of cloud-top temperature ($10.4\mu m$
wavelength) [19].

[capbesideposition=left,top,capbesidewidth=.31]figure[]

Figure 1: From Observation to Machine Learning to Science. Meteorologists seek
to connect observations and summaries of TC evolution to processes such as RI
(blue path). We pose this scientific question as a hypothesis test (bottom
right). By casting the hypothesis test as a prediction problem, we are able to
leverage high-capacity AI techniques, here ANNs (top right). Such black-box
models often struggle to extract scientifically meaningful insight (red path).
Our framework facilitates insight on labeled sequences of functional data via
local results in the original space of scientific functional features (black
path).

Scientific Goals. TC intensity forecasts are ultimately issued by human
experts who synthesize domain knowledge with the outputs of many forecasting
tools to issue a single forecast. It is thus unproductive to provide another
statistical model which only predicts TC intensity directly from IR imagery;
such black-box models are difficult for forecasters to utilize. Our goal is
instead to provide meteorologists with an enhanced understanding of how
convective evolution relates to RI events (Figure 1, blue arrow). Currently
operational statistical forecast tools are generally limited to a handful of
scalar summary statistics from each GOES image to aid forecasters in quickly
digesting convective behavior [2, 1, 9]. Conversely, we work with the rich
suite of ORB functions, a dictionary of scientifically meaningful one-
dimensional functional features [14]. In this paper, we illustrate our methods
on one such ORB function—the radial profile—but our framework can generalize
to jointly analyze multiple functional inputs. The radial profile, defined as
$\overline{T}(r)=\frac{1}{2\pi}\int_{0}^{2\pi}T_{b}(r,\theta)d\theta$,
captures the structure of cloud-top temperatures $T_{b}$ as a function of
radius $r$ from the TC center and serves as an easily interpretable
description of the depth and location of convection near the TC core [14, 18].
A single profile describes the instantaneous convective structure
($\mathbf{X}_{t}$; Figure 2, center) of a TC, whereas a 24-hour (48
observation) sequence of profiles provides a concise, human-readable summary
of evolution over time; we call the latter sequence a _structural trajectory_
($\mathbf{S}_{t}=\\{\mathbf{X}_{t},\mathbf{X}_{t-1},\dots,\mathbf{X}_{t-48}\\}\in\mathcal{S}$;
Figure 2, right).

Contribution. We seek to answer the meteorological questions: “Are the
sequences $\mathbf{S}_{t}$ leading up to RI events different from those
leading up to non-RI events?” and, if so, “How do RI events differ from non-RI
events over $\mathcal{S}$?” We pose this question as a hypothesis test
comparing the distributions of $\mathbf{S}_{t}$ on $\mathcal{S}$ between
classes (Equation 1). By recasting the hypothesis as a prediction problem, we
are able to leverage artificial neural networks (ANNs) to assess the
relationship between labeled sequence data
$\\{(\mathbf{S}_{t},Y_{t})\\}_{t\geq 0}$ and TC behavior. Within this
framework, shown in Figure 1, we not only detect the differing behavior of
convection leading up to RI, but we are able to identify locations in
$\mathcal{S}$ corresponding to RI, thus bridging the gap from ML back to
meteorological insight (Figures 3,4). To our knowledge, this is the only
principled framework that can detect distributional differences in sequences
of images, and in addition provide local diagnostics which can identify
spatio-temporal patterns commonly found in the lead-up to RI events.

##  2 Data

[capbesideposition=left,top,capbesidewidth=.3]figure[]

Figure 2: Structural Trajectories as Image Data. The raw data for each
trajectory $\mathbf{S}_{t}$ is a sequence of cloud-top temperature images from
GOES ($T_{b}$). We convert each of these into a radial profile
($\mathbf{X}_{t}$). The 24-hour sequence of radial profiles are combined into
a _structural trajectory_ or image ($\mathbf{S}_{t}$). These images are the
high-dimensional inputs to $\widehat{m}_{\text{post}}(\mathbf{s}_{t})$.

Inputs. Raw GOES image data are drawn from NOAA’s MERGIR database [6], which
contains 30-minute$\times$4-km resolution IR imagery dating back to 2000
(Figure 2, left). The NOAA HURDAT2 database provides TC best track data: the
best estimate of TC intensity and location produced after each season using
all available data, including data not available in real time [13]. HURDAT2
has a 6-hour resolution.

RI Labels. We attach labels $Y_{t}\in\\{0,1\\}$ to structural trajectories
$\mathbf{S}_{t}$ at each synoptic time (0000 UTC, 0600 UTC, 1200 UTC, and 1800
UTC) based on whether the TC was undergoing RI at time $t$ (where $Y=1$
indicates RI) according to Algorithm 2. The non-synoptic times are labeled
$Y_{t}=1$ if they fall between two consecutive synoptic $Y_{t}=1$
observations, and $Y_{t}=0$ otherwise.

Sample Size. We train our ANN on 33,873 labeled structural trajectories from
2000-2012 (further divided into 60% train/40% validation). We hold out 14,447
labeled trajectories from 2013-2019 as test data. When only labels $Y_{t}$ —
and not trajectories $\mathbf{S}_{t}$ — are required
($\widehat{m}_{\text{seq}}$; Alg. 1(3)), we train on 31,203 synoptic best
track entries from 1979-2012; we did not use data prior to 1979 due to reduced
availability of satellite observations to support post-season analysis and
thus the HURDAT2 best track database.

##  3 Methods

Testing Hypotheses. Our goal is to detect distributional differences in
labeled sequence data $\\{(\mathbf{S}_{t},Y_{t})\\}_{t\geq 0}$, where the
“labels” $Y_{t}\in\\{0,1\\}$ are binary, and each covariate
$\mathbf{S}_{t}\in\mathcal{S}$ is a sequence of functional quantities (our
structural trajectory, Figure 2). Formally, we test the hypothesis

$\displaystyle H_{0}:p(\mathbf{s}_{t}|Y_{t}=1)$
$\displaystyle=p(\mathbf{s}_{t}|Y_{t}=0)\ \text{for all}\
\mathbf{s}_{t}\in\mathcal{S}\ \text{versus}$ (1) $\displaystyle\
H_{1}:p(\mathbf{s}_{t}|Y_{t}=1)$ $\displaystyle\neq p(\mathbf{s}_{t}|Y_{t}=0)\
\text{for some}\ \mathbf{s}_{t}\in\mathcal{S}.$

In our setting, there are two methodological challenges: (i) Equation 1 is a
high-dimensional two-sample test for dependent sequence data, and (ii) to
glean scientific insight, an answer of whether we reject/fail-to-reject
$H_{0}$ is not enough: if we reject $H_{0}$, then we need to identify the
_regions_ in $\mathcal{S}$ where the two distributions differ.

From Tests to Prediction. For independent and identically distributed (IID)
data $\\{\mathbf{S}_{i}\\}_{i=1}^{n}$, Equation 1 can be re-cast as a
classification problem via Bayes Theorem, resulting in an equivalent test of
${H_{0}:\mathbb{P}(Y_{i}=1|\mathbf{s}_{i})=\mathbb{P}(Y_{i}=1),\>\forall\,\mathbf{s}_{i}\in\mathcal{S}}$.
This idea is at the core of the recent classification accuracy [11] and
regression-based [10] two-sample tests. The latter regression test goes beyond
classification to provide _local_ results, indicating where in $\mathcal{S}$
the two distributions differ. The key quantity to estimate in this procedure
is the _class posterior_
$m_{\text{post}}(\mathbf{s}_{i}):={\mathbb{P}}(Y_{i}=1|\mathbf{s}_{i})$. For
this, one can leverage any probabilistic classification technique; hence
converting a challenging inference and testing problem to a prediction
problem. We reject $H_{0}$ if the test statistic
$\lambda:=\sum_{t\in\mathcal{V}}(\widehat{m}_{\text{post}}(\mathbf{s}_{i})-\widehat{m}_{\text{prior}})^{2}$
is large on a held-out evaluation set $\mathcal{V}$ (not used in training).
Here $\widehat{m}_{\text{prior}}$ is the proportion of $Y_{i}=1$ cases in the
train sample, and $\widehat{m}_{\text{post}}$ is the estimated class
posterior.

Accounting for Dependence. However, existing high-dimensional tests do not
extend to dependent sequence data. In particular, the regression test [10]
relies on a permutation test and the assumption of exchangeability — an
assumption that is generally violated for spatio-temporal processes. Here we
propose a generalization of the regression test from IID to dependent data by
replacing the permutation test with a Monte Carlo (MC) test. Our MC test
estimates the label distribution
$m_{\text{seq}}(y_{t-1},\dots,y_{1}):=\mathbb{P}(Y_{t}=1|Y_{t-1}=y_{t-1},\dots,Y_{1}=y_{1})$
under $H_{0}$ in place of permutation. The MC test is asymptotically valid
(that is, it controls Type I errors), as long as the estimator
$\widehat{m}_{\text{seq}}$ converges to $m_{\text{seq}}$. See Algorithm 1 for
a full description of the testing procedure.

Local Importance. If we reject $H_{0}$, then the magnitude of the terms
$\lambda_{t}:=\widehat{m}_{\text{post}}(\mathbf{s}_{t})-\widehat{m}_{\text{prior}}$
in the test statistic $\lambda$ provide information as to which types of
structural trajectories contributed the most to the rejection. Hence, we
assign a local importance score
$I(\mathbf{s}_{t}):=\text{sign}(\lambda_{t})/\widehat{p}_{t}$ to each sequence
$\mathbf{s}_{t}$, where $\widehat{p}_{t}$ is an approximate local $p$-value
associated with $\lambda_{t}$, computed according to Algorithm 1. Positive
scores $I(\mathbf{s}_{t})\gg 0$ indicate
$p(\mathbf{s}_{t}|Y_{t}=1)>p(\mathbf{s}_{t}|Y_{t}=0)$, and negative scores
$I(\mathbf{s}_{t})\ll 0$ the reverse. In other words, our test and diagnostics
can identify 24-hour trajectories which are more common to the lead-up to RI
events.

##  4 Results

Testing Hypotheses. We execute Algorithm 1, estimating $m_{\text{seq}}$ with a
Markov chain model with 48-hour memory and $m_{\text{post}}$ with a
convolutional neural network (CNN). The CNN classifier achieved $\sim$75%
balanced accuracy on test data; note that classifier performance affects the
power of the test, but not validity. We find that the distributions of RI vs
non-RI trajectories are different at significance level $\alpha=0.01$.

Local Importance. Figure 3 (left) projects
$\mathbf{S}_{t}\in\mathcal{S}\subset\mathbb{R}^{48\times 120}$ onto their
first two principal components. We use principal component analysis (PCA) for
visualization only; $\widehat{m}_{\text{post}}$ is trained on the full
structural trajectories. In Figure 3 (left), a single point represents an
_entire_ 24-hour trajectory. Temporally adjacent trajectories are highly
correlated in PCA space, as evidenced by the “strands" in Figure 3. We divide
the importance scores $I(\mathbf{s}_{t})$ for each trajectory into 3
categories: negative ($I(\mathbf{s})<-40$; blue, RI less likely), null
($-40<I(\mathbf{s})<40$; gray), and positive ($I(\mathbf{s})>40$; red, RI more
likely). Positive importance scores tend to have low values of the first
principal component score, corresponding to cold trajectories whose
temperatures are decreasing.

Scientific Insight. Six sample trajectories from two example TCs (Hurricanes
Nicole [2016] and Jose [2017]) are labeled in Figure 3 (left) and displayed in
Figure 3 (right). The PCA map clearly show two patterns: (i) core convection
is stronger for low values of PCA1 and (ii) eye-eye wall structures are
present for low values of PCA2.

We show the raw images $\mathbf{S}_{i}$ for each point in Figure 3 (right)
along with Saffir-Simpson intensity categories. Out method is able to identify
the structural patterns which lead up to RI: In all three examples with
positive importance scores (B, C, F), cloud-top temperatures are cold and
getting colder. Trajectory F also shows the formation of an eye, during which
time the TC intensified by 40 kt. The examples with negative importance scores
(A, D, E) have warmer cloud-top temperatures with little indication of cooling
in the core. This analysis corroborates other studies that indicate strong,
deepening convection is crucial for RI, particularly near the TC core where
energy is most likely to result in strengthening surface winds.[17]

Figure 3: Differences in Distribution. (_Left_) PCA map, where each point,
colored by its importance score, represents a 24h structural trajectory. In
general, importance decreases from left to right, while current intensity
increases from top to bottom. Note that PCA is only used for visualization
purposes; the test is performed on the entire trajectory $\mathbf{S}_{t}$.
(_Right_) Panels of selected trajectories or points for Hurricane Jose [2017]
and Hurricane Nicole [2016]. The inset letter indicates the location and local
importance in the PCA map. Positive cases tend to have cold cloud tops near
the core, growing in coverage and depth of convection with time.

Case Study: Hurricane Nicole [2016]. We examine the full series
$\\{(\mathbf{S}_{t},Y_{t})\\}_{t\geq 0}$ for Hurricane Nicole [2016] in Figure
4. When $I(\mathbf{s}_{t})\gg 0$, we indeed find that TC intensity is
increasing (center; points A, C, D). More interesting still, the trajectories
_prior_ to points A and C — the _onset_ of RI — already begin to indicate that
RI is likely; there are obviously markers of RI in the radial profiles which
may be of use to forecasters. Future work will identify archetypal
trajectories of this sort: $I(\mathbf{s})\gg 0$ prior to intensification.

Figure 4: Case Study: Hurricane Nicole [2016]. (_Left_) Full set of
trajectories $\\{\mathbf{S}_{t}\\}_{t\geq 0}$ for Hurricane Nicole [2016],
with select trajectories labeled; again, each point represents a 24-hour
trajectory. The physical track of the storm across the North Atlantic is shown
inset, with the same trajectories labeled (here represented as segments of the
track). (_Center_) Storm intensity over time, where solid points mark RI
events. (_Right_) Full trajectory of Hurricane Nicole over 2 weeks. Note that
the importance scores capture rapidly intensifying periods (A, C, D),
collapsing convection (B), and the decay of the TC eye (E).

##  5 Conclusion

Our method successfully detects distributional differences in 24-hour
“structural trajectories” (derived from GOES-IR images) leading up to periods
of rapid intensification. A preliminary partitioning of trajectories by our
proposed “local importance score” shows several known archetypal structural
trajectories, laying the groundwork for forecasting tools. Further extending
this framework to trajectories of other high-dimensional data promises to
provide the insight necessary to understand and predict these increasingly
deadly events.

##  6 Acknowledgments

This work is supported in part by NSF DMS-2053804, and the NSF AI Institute:
Physics of the Future, NSF PHY-2020295. K. M. Wood was supported by the
Mississippi State University Office of Research and Economic Development. R.
Izbicki was supported by FAPESP (grant 2019/11321-9) and CNPq (grant
306943/2017-4).

## References

  * [1] Mark DeMaria.  A simplified dynamical system for tropical cyclone intensity prediction.  Monthly Weather Review, 137(1):68–82, 2009.
  * [2] Mark DeMaria and John Kaplan.  An updated statistical hurricane intensity prediction scheme (SHIPS) for the Atlantic and Eastern North Pacific basins.  Weather and Forecasting, 14(3):326–337, 1999.
  * [3] Mark DeMaria, Charles R. Sampson, John A. Knaff, and Kate D. Musgrave.  Is tropical cyclone intensity guidance improving?  Bulletin of the American Meteorological Society, 95(3):387–398, 2014.
  * [4] Vernon F. Dvorak.  Tropical cyclone intensity analysis and forecasting from satellite imagery.  Monthly Weather Review, 103(5):420–430, 1975.
  * [5] Liang Hu, Elizabeth A Ritchie, and J Scott Tyo.  Short-term tropical cyclone intensity forecasting from satellite imagery based on the deviation angle variance technique.  Weather and Forecasting, 35(1):285–298, 2020.
  * [6] John Janowiak, Bob Joyce, and Pingping Xie.  NCEP/CPC L3 half hourly 4km global (60S - 60N) merged IR V1, 2020.
  * [7] John Kaplan and Mark DeMaria.  Large-scale characteristics of rapidly intensifying tropical cyclones in the North Atlantic basin.  Weather and forecasting, 18(6):1093–1108, 2003.
  * [8] John Kaplan, Mark DeMaria, and John A. Knaff.  A revised tropical cyclone rapid intensification index for the Atlantic and Eastern North Pacific basins.  Weather and Forecasting, 25(1):220–241, 2010.
  * [9] John Kaplan, Christopher M Rozoff, Mark DeMaria, Charles R Sampson, James P Kossin, Christopher S Velden, Joseph J Cione, Jason P Dunion, John A Knaff, Jun A Zhang, et al.  Evaluating environmental impacts on tropical cyclone rapid intensification predictability utilizing statistical models.  Weather and Forecasting, 30(5):1374–1396, 2015.
  * [10] Ilmun Kim, Ann B Lee, and Jing Lei.  Global and local two-sample tests via regression.  Electronic Journal of Statistics, 13(2):5253–5305, 2019.
  * [11] Ilmun Kim, Aaditya Ramdas, Aarti Singh, and Larry Wasserman.  Classification accuracy as a proxy for two-sample testing.  The Annals of Statistics, 49(1):411–434, 2021.
  * [12] Philip J. Klotzbach, Steven G. Bowen, Roger Pielke, and Michael Bell.  Continental U.S. hurricane landfall frequency and associated damage: Observations and future risks.  Bulletin of the American Meteorological Society, 99(7):1359–1376, 2018.
  * [13] Christopher W. Landsea and James L. Franklin.  Atlantic hurricane database uncertainty and presentation of a new database format.  Monthly Weather Review, 141(10):3576–3592, 2013.
  * [14] Trey McNeely, Ann B Lee, Kimberly M Wood, and Dorit Hammerling.  Unlocking GOES: A statistical framework for quantifying the evolution of convective structure in tropical cyclones.  Journal of Applied Meteorology and Climatology, 59(10):1671–1689, 2020.
  * [15] Timothy L Olander and Christopher S Velden.  The advanced dvorak technique (adt) for estimating tropical cyclone intensity: Update and new capabilities.  Weather and Forecasting, 34(4):905–922, 2019.
  * [16] Elizabeth A Ritchie, Kimberly M Wood, Oscar G Rodríguez-Herrera, Miguel F Piñeros, and J Scott Tyo.  Satellite-derived tropical cyclone intensity in the North Pacific ocean using the deviation-angle variance technique.  Weather and forecasting, 29(3):505–516, 2014.
  * [17] Robert Rogers, Paul Reasor, and Sylvie Lorsolo.  Airborne doppler observations of the inner-core structural differences between intensifying and steady-state tropical cyclones.  Monthly Weather Review, 141(9):2970–2991, 2013.
  * [18] Elizabeth R. Sanabia, Bradford S. Barrett, and Caitlin M. Fine.  Relationships between tropical cyclone intensity and eyewall structure as determined by radial profiles of inner-core infrared brightness temperature.  Monthly Weather Review, 142(12):4581–4599, 2014.
  * [19] Timothy J Schmit, Paul Griffith, Mathew M Gunshor, Jaime M Daniels, Steven J Goodman, and William J Lebair.  A closer look at the ABI on the GOES-R series.  Bulletin of the American Meteorological Society, 98(4):681–698, 2017.
  * [20] Kimberly M Wood and Elizabeth A Ritchie.  A definition for rapid weakening of North Atlantic and Eastern North Pacific tropical cyclones.  Geophysical Research Letters, 42(22):10–091, 2015.

##  Appendix A Algorithms

0: type of test (MC=TRUE for MC test; MC=FALSE for permutation test); train
data $\\{Y_{t},\mathbf{S}_{t}\\}_{t\in\mathcal{T}_{1}}$ and regression method
for estimating
$m_{\text{post}}(\mathbf{s}):=\mathbb{P}(Y_{t}=1|\mathbf{S}_{t}=\mathbf{s})$;
for MC test, additional train data $\\{Y_{t}\\}_{t\in\mathcal{T}_{2}}$ and
regression method for estimating
$m_{\text{seq}}(Y_{t-1},\dots,Y_{t-k}):=\mathbb{P}(Y_{t}=1|Y_{t-1},\dots,Y_{t-k})$;
number of repetitions $B$; test points
$\\{\mathbf{s}_{t}\\}_{t\in\mathcal{V}}$.

0: p-value for testing
$H_{0}:p(\mathbf{s}_{t}|Y_{t}=1)=p(\mathbf{s}_{t}|Y_{t}=0),\ \text{for all}\
t$ and $\mathbf{s}_{t}$, local importance scores
$\\{I(\mathbf{s}_{t})\\}_{t\in\mathcal{V}}$

// Estimate underlying probability distributions

(1) Estimate $m_{\text{prior}}:=\mathbb{P}(Y_{t}=1)$ with class proportion
$\widehat{m}_{\text{prior}}=\frac{1}{\lvert\mathcal{T}_{1}\rvert}\sum_{t\in\mathcal{T}_{1}}Y_{t}$.

(2) Regress $Y_{t}$ on $\mathbf{S}_{t}$ using $\mathcal{T}_{1}$ to compute
$\widehat{m}_{\text{post}}$.

(3) if MC then Regress $Y_{t}$ on $Y_{t-1},\dots,Y_{t-k}$ using
$\mathcal{T}_{2}$ to compute $\widehat{m}_{\text{seq}}$.

// Compute test statistic and estimate its null distribution

(4) Compute test statistic
$\mathbf{\lambda}=\sum_{t\in\mathcal{V}}\lambda_{t}^{2},\text{ where
}\lambda_{t}=\widehat{m}_{\text{post}}(\mathbf{s}_{t})-\widehat{m}_{\text{prior}}$.

(5) for  _$b\in\\{1,2,\dots,B\\}$_ do

\- Draw new train labels $\\{\widetilde{Y}_{t}\\}_{t\in\mathcal{T}_{1}}$ under
$H_{0}$:

if MC then

\- Initialize $\widetilde{Y}_{1},\dots,\widetilde{Y}_{k}$ to random sequence.

\- Draw sequence of length 100$\times k$ from
$\widetilde{Y}_{t}\sim\text{Binom}(\widehat{m}_{\text{seq}}(\widetilde{Y}_{t-k},\dots,\widetilde{Y}_{t-1}))$
for burn-in.

\- Draw new labels
$\widetilde{Y}_{t}\sim\text{Binom}(\widehat{m}_{\text{seq}}(\widetilde{Y}_{t-k},\dots,\widetilde{Y}_{t-1}))$,
for $t\in\mathcal{T}_{1}\setminus\mathcal{T}_{0}$.

else

\- Permute original labels $\\{Y_{t}\\}_{t\in\mathcal{T}_{1}}$.

end if

\- Regress $\widetilde{Y}_{t}$ on $\mathbf{S}_{t}$ using $\mathcal{T}_{1}$ to
compute $\widehat{m}_{\text{post}}^{(b)}$.

\- Recompute test statistic
$\widetilde{\lambda}^{(b)}=\sum_{t\in\mathcal{V}}\widetilde{\lambda}_{t}^{(b)2}\text{
where
}\widetilde{\lambda}_{t}^{(b)}=\widehat{m}^{(b)}_{\text{post}}(\mathbf{s}_{t})-\widehat{m}_{\text{prior}}$.

// Compute approximate p-values

(6) Compute $p$-values according to
$\widehat{p}_{\text{global}}=\frac{1}{B+1}\left(1+\sum_{b=1}^{B}I\left(\widetilde{\lambda}^{(b)}>\lambda\right)\right)\text{
and
}\widehat{p}_{t}=\frac{1}{B+1}\left(1+\sum_{b=1}^{B}I\left(|\widetilde{\lambda}_{t}^{(b)}|>|\lambda_{t}|\right)\right).$

(7) Define local importance scores by
$I(\mathbf{s}_{t})=\frac{\text{sign}{(\lambda_{t})}}{\widehat{p}_{t}}$

return $\widehat{p}_{\text{global}},\
\\{I(\mathbf{s}_{t})\\}_{t\in\mathcal{V}}$

Algorithm 1 Test for distributional differences in labeled sequence data.

Require: Sequence of contiguous intensity (i.e. maximum wind speed)
observations $\\{W_{i}\\}_{t=1}^{T}$ for a storm and an intensity change
threshold $c$.

Initialize: $\mathbf{Z}=\mathbf{0}_{T,T}$, $Y=\mathbf{0}_{T}$

for  _$t$ in $1:T-1$_ do

$\Delta_{t,1}=W_{t+1}-W_{t}$; // lead-1 intensity change

if  _$t\leq T-4$_ then

$\Delta_{t,4}=\max(W_{t},\dots,W_{t+4})-W_{t}$; // lead-4 intensity change

end if

end for

$\mathcal{A}=\\{t:\Delta_{t,4}\geq c\\}$; // 24-hour windows containing RI

$\mathcal{B}=\\{t:\Delta_{t,1}>0\\}$; // 6-hour windows containing
intensification

for  _$t$ in $1:T-4$_ do

if  _$t\in\mathcal{A}$_ then // label 24-hour RI window

$Z_{t,t},\dots,Z_{t,t+4}=1$;

for  _$h$ in $4:1$_ do

if  _$t+h-1\notin\mathcal{B}$_ then // trim non-intensification from end of
event

$Z_{t,t+h}=0$;

else

break;

end if

end for

for  _$h$ in $0:3$_ do

if  _$t+h\notin\mathcal{B}$_ then // trim non-intensification from start of
event

$Z_{t,t+h}=0$;

else

break;

end if

end for

end if

end for

for  _$t$ in $1:T$_ do

$Y_{t}=\max(Z_{t,1},\dots,Z_{t,T})$; // points only need to be valid for one
start

end for

Output: $\\{Y_{i}\\}_{t=1}^{T}$

Note: The above algorithm identifies rapid intensification. To identify rapid
weakening instead, reverse the input sequence $\\{W_{i}\\}_{t=1}^{T}$ at
initialization, then reverse the output sequence $\\{Y_{i}\\}_{t=1}^{T}$ at
output.

Algorithm 2 Identifying observations that fall within rapid intensification
events.