Stäckel Equivalence of Non-Degenerate Superintegrable Systems, and Invariant
Quadrics

Stäckel Equivalence of Non-Degenerate
Superintegrable Systems, and Invariant Quadrics

Andreas VOLLMER †‡

A. Vollmer

† Institute of Geometry and Topology, University of Stuttgart, 70550
Stuttgart, Germany ‡ Dipartimento di Scienze Matematiche (DISMA), Politecnico
di Torino,
‡ Corso Duca degli Abruzzi, 24, 10129 Torino, Italy<EMAIL_ADDRESS><EMAIL_ADDRESS>
Received October 09, 2020, in final form February 02, 2021; Published online
February 17, 2021

A non-degenerate second-order maximally conformally superintegrable system in
dimension 2 naturally gives rise to a quadric with position dependent
coefficients. It is shown how the system’s Stäckel class can be obtained from
this associated quadric. The Stäckel class of a second-order maximally
conformally superintegrable system is its equivalence class under Stäckel
transformations, i.e., under coupling-constant metamorphosis.

Stäckel equivalence; quadrics; superintegrable systems

14H70; 70H06; 30F45

##  1 Introduction

Superintegrable systems in dimension 2 are a classical subject of mathematical
research. They have interrelations, for instance, with special functions [11,
21], quadratic algebras [8, 23, 32], degenerate Poisson pencils [27], and
constant divisors in the context of the Riemann–Roch theorem [37], see also
[35, 36]. In particular, second-order (maximally) superintegrable systems in
dimension 2 (2D) are classified [13, 14, 20, 22], for Euclidean 2-space even
algebraic geometrically [16, 24]. In this context, superintegrable systems are
considered up to isometries, i.e., locally up to coordinate transformations.

Another viewpoint is to consider second-order superintegrable systems up to
Stäckel (i.e., conformal) transformations [5], also known as coupling-constant
metamorphism [12]. This viewpoint appears, for instance, in [6, 9, 23, 26].
The equivalence classes resulting from this identification are called _Stäckel
classes_. The equivalence classes of non-degenerate second-order
superintegrable systems under Stäckel transforms are classified [9, 23].
Specifically, in [23] a method is developed that allows one to identify the
Stäckel class of a non-degenerate superintegrable system from the properties
of its associated quadratic algebra, see also [32]. The present paper presents
an alternative method to determine the Stäckel class of a given non-degenerate
second-order 2D superintegrable system. It also applies to conformally
superintegrable systems. This new method, presented below in Theorem 4.3, is
applicable to 2D non-degenerate second-order superintegrable systems of any
curvature. Specifically we do not require constant sectional curvature.

On the other hand, we shall see that the proof for this new method is
extremely simple, since we do not have to start our proof “from scratch”.
Instead, we shall exploit the _known_ flat second-order systems that realise
the _known_ Stäckel classes. With this in mind, the purpose of this paper is
not to present a tedious, complicated proof, but rather to report an efficient
tool that determines which Stäckel class a given non-degenerate second-order
2D (conformally) superintegrable system belongs to.

##  2 Preliminaries

Let $g$ be a (pseudo-)Riemannian metric on a $2$-dimensional manifold $M$ and
consider the Hamiltonian $H(x,p)=g^{ij}(x)p_{i}p_{j}+V(x)$. Here $x$ and $p$
stand, respectively, for position coordinates $x^{i}$ and canonical momenta
(fibre coordinates) $p_{i}$ on the cotangent space $T^{*}M$. Note that in what
follows we shall consider two Hamiltonians $H_{1}$, $H_{2}$ to be equal if
they are constant multiples as functions on $T^{*}M$.

A _second-order integral_ (of motion) for $H$ is a function
$F(x,p)=K^{ij}(x)p_{i}p_{j}+W(x)$ such that $H$ and $F$ commute w.r.t. the
canonical Poisson bracket on $M$ (Einstein’s summation convention applies),

$\\{F,H\\}=\frac{\partial F}{\partial x^{i}}\frac{\partial H}{\partial
p_{i}}-\frac{\partial H}{\partial x^{i}}\frac{\partial F}{\partial p_{i}}=0.$
(2.1)

We remark that condition (2.1) is equivalent to requiring that the
coefficients $K_{ij}$ (indices are lowered using $g$) form the components of a
Killing tensor $K$ and that $K$ is compatible with $V$ according to the
_Bertrand–Darboux condition_ [1, 7]

$\nabla_{[i}K\indices{{}_{j]}^{a}}\nabla_{a}V=K\indices{{}^{a}_{[i}}\nabla^{2}_{j]a}V,$
(2.2)

where square brackets denote antisymmetrisation; $\nabla$ is the Levi-Civita
connection for $g$ and $\nabla^{2}$ denotes the Hessian. Equation (2.2) is the
integrability condition for $W$; it is obtained as follows: First solve the
homogeneous linear component (in momenta) of the polynomial equation (2.1) for
the differential ${\rm d}W$, obtaining $\nabla_{a}W=K_{ab}\nabla^{b}V$. Then
take a further derivative; the integrability condition for $W$, i.e. ${\rm
dd}W=0$, is (2.2).

######  Remark 2.1.

Note that we work over the field $\mathbb{C}$ of complex numbers unless
otherwise indicated.

######  Definition 2.2.

A (2D maximally) _second-order superintegrable system_ is a Hamiltonian $H$
together with its space of second-order integrals (2.1), which are required to
contain two elements $F_{1}$ and $F_{2}$ such that $(H,F_{1},F_{2})$ are
functionally independent.

######  Remark 2.3.

Since from the knowledge of the superintegrable Hamiltonian we can reconstruct
the system, we typically specify only the Hamiltonian.

Within the theory of superintegrable systems, _non-degeneracy_ is a well-known
natural property [13, 18, 24]. In Definition 2.4, we complement the standard
definition (i) [20, 22, 25] by a helpful distinction in (ii).

######  Definition 2.4.

  1. (i)

A second-order superintegrable system in dimension $n$ is _non-degenerate_ if
equation (2.2) admits a $(n+2)$-dimensional space $\mathcal{U}$ of solutions
(for $V$).

  2. (ii)

We can consider the Hamiltonian with a fixed potential $V\in\mathcal{U}$ or
with the full $(n+2)$-parameter family. In the latter case we write
$V^{\mathcal{U}}$ for clarity. Note that $V^{\mathcal{U}}$ denotes a specific
parametrisation of $\mathcal{U}$.

######  Remark 2.5.

It is in place to remark on the system [E15] of [20], which appears alongside
the non-degenerate systems in [23]. The reader will notice that the system
[E15] does not appear in Table 1, while it appears in the corresponding table
of [23]. This is correct, and a closer look confirms that the system [E15]
does not meet the prerequisites of Definition 2.4. Indeed, the potential of
[E15] depends on a functional parameter, inconsistent with non-degeneracy.
Note that the system [E15] is also not included in the algebraic-geometric
classification of non-degenerate second-order superintegrable systems on flat
2-space, see [24].

The following equivalence relation of second-order maximally superintegrable
systems is well known.

######  Definition 2.6.

Let $\langle H,F_{i}\rangle$ be a second-order non-degenerate superintegrable
system with the Hamiltonian $H$ and integrals $F_{i}$. Let $U\in\mathcal{U}$
be one of its compatible potentials. Then
$\big{\langle}\tilde{H},\tilde{F}_{i}\big{\rangle}$ with

$\tilde{H}=U^{-1}H,\qquad\tilde{F}_{i}=F_{i}+(1-W_{i})U^{-1}H$ (2.3)

is called the _Stäckel transform_ of $\langle H,F_{i}\rangle$.

Often, since $U\in\mathcal{U}$, lives in a linear space of admissible
potentials, the Stäckel transform is considered as a transformation involving
a coupling parameter $\alpha$, i.e., the Hamiltonian
$H_{\alpha}=H_{\alpha}(x,y)=H+\alpha U$ is taken as dependent on a linear
parameter $\alpha$. Then, Stäckel transform can be interpreted as a change of
the Hamiltonian such that the roles of the energy and of the coupling
parameter are interchanged [14, 31]. A multi-parameter version of this
approach to Stäckel transforms has been developed by Sergyeyev and Błaszak
[33], see also [2, 3]. In our definition, the coupling parameters are
naturally geometrically replaced by the freedom to choose another element $U$
from the linear space $\mathcal{U}$.

######  Remark 2.7.

  1. (i)

The integrals $\tilde{F}_{i}$ are indeed integrals for $\tilde{H}$ and,
provided they are functionally independent, the Stäckel transform is again a
second-order superintegrable system.

  2. (ii)

In the context of second-order superintegrable systems, the _Stäckel
transform_ is also known under the name _coupling constant metamorphosis_ [5,
12], but these two concepts are not the same in other contexts [31].

######  Definition 2.8.

  1. (i)

We say that the non-degenerate second-order superintegrable systems $\langle
H,F_{i}\rangle$ and $\big{\langle}\tilde{H},\tilde{F}_{i}\big{\rangle}$ are
_Stäckel equivalent_ if they are Stäckel transforms.

  2. (ii)

The equivalence class of a non-degenerate second-order superintegrable system
under Stäckel equivalence is called its _Stäckel class_ $\mathcal{S}$.

Note that Stäckel transform is “conformal” in the following sense: For two
systems $h,\tilde{h}\in\mathcal{S}$, belonging to the same Stäckel class, the
underlying metrics $g_{h}$ and $g_{\tilde{h}}$ are conformally equivalent.

##  3 Method

The aim of the current section is to construct a certain variety $\mathcal{Q}$
that is invariant under Stäckel transform. It is encoded in a quadric, for a
given (non-degenerate) 2D second-order superintegrable system. In the
following section we shall relate this variety to the set of all flat
realisations of a given Stäckel class, i.e., those non-degenerate 2D second-
order superintegrable systems inside the given Stäckel class that are realised
on a flat geometry. In order to prevent misunderstandings, it is worthwhile to
state a clarification: The present paper seeks a method that determines the
Stäckel class of _any_ given 2-dimensional non-degenerate second-order
superintegrable system, i.e., for manifolds of arbitrary (including non-
constant) curvature. We base this method on an invariant variety $\mathcal{Q}$
inside an invariant projective space $\mathcal{W}$. The variety $\mathcal{Q}$
is defined by requiring flatness for objects in $\mathcal{W}$. We would like
to highlight that this does not imply any restriction to flat manifolds as far
as Theorem 4.3, and Theorem 5.2, are concerned. It is merely a technical
trick, i.e., a good choice, which enables us to reduce the problem under
consideration to a simple computation.

######  Definition 3.1.

A _quadric_ in projective space $\mathbb{P}^{m}$ is the subset defined by the
zero set of a homogeneous quadratic polynomial equation in $m+1$ variables.

Note that we do not require the polynomial equation to be irreducible. The
following observation is the basis for the technique developed further below.

######  Observation 3.2.

Consider two Stäckel equivalent, non-degenerate second-order superintegrable
Hamiltonians $H=g^{ij}p_{i}p_{j}+V$ and
$\tilde{H}=\tilde{g}^{ij}p_{i}p_{j}+\tilde{V}$. By Definition 2.4, they give
rise to $(n+2)$-dimensional spaces $\mathcal{U}$ and $\tilde{\mathcal{U}}$
solving (2.2), respectively. We have, according to Definition 2.6, that

$\mathcal{U}g=\frac{\mathcal{U}}{U}Ug=\tilde{\mathcal{U}}\tilde{g},$

where $U$ is a solution of (2.2) for the superintegrable system arising from
$H$. This is true in any dimension $n\geq 2$. As a result, for a non-
degenerate second-order superintegrable Hamiltonian $H$, the
$(n+2)$-dimensional linear space $\mathcal{V}$,

$\mathcal{V}=\mathcal{U}g,$

is invariant under Stäckel transformations.

We comment that an analogous observation holds for conformally superintegrable
Hamiltonians, but then $U$ can be any function (see details in Section 5.2).

Elements $g\in\mathcal{V}$ are typically metrics, since $\det(Ug)\neq 0$ for
$U\neq 0$. Clearly the origin never corresponds to a metric. We observe that
constant multiples of $U$ give rise to proportional Hamiltonians
$H=\frac{1}{U}\big{(}g^{ij}p_{i}p_{j}+V^{\mathcal{U}}\big{)}$. It is therefore
useful to reconsider the $(n+2)$-dimensional linear space $\mathcal{V}$ as an
$(n+1)$-dimensional projective space, which we denote by

$\mathcal{W}=(\mathcal{V}\setminus\\{0\\})/{\sim},$

where $h\sim k$ for $h,k\in\mathcal{V}$ if $h=ak$ for constant $a\neq 0$.

######  Remark 3.3.

Note that while we shall perform the computations using a specific
parametrisation $V^{\mathcal{U}}$ of the potential and particular coordinates
on $M$, the space $\mathcal{W}$ is indeed a geometric object. It is
independent of the specific parametrisation and the choice of coordinates.

######  Example 3.4.

We emphasize that elements $q\in\mathcal{W}$ are (classes of) symmetric
2-tensors (and in fact metrics). For instance, take the 2-sphere with the
round metric

$g=\frac{{\rm d}x{\rm d}y}{(xy+4)^{2}}.$ (3.1)

For the Hamiltonian we take that of the system [S4] in [20], i.e.,

$H=(xy+4)^{2}p_{x}p_{y}+a_{1}\frac{(xy+4)^{2}}{y^{2}}-a_{2}\frac{xy-4}{\sqrt{xy}}-a_{3}\frac{(xy+4)^{2}}{y\sqrt{xy}}+a_{4}.$
(3.2)

It is then straightforward to compute

$\mathcal{W}=\left\\{\left(a_{1}\frac{1}{y^{2}}-a_{2}\frac{xy-4}{(xy+4)^{2}\sqrt{xy}}-a_{3}\frac{1}{y\sqrt{xy}}+\frac{a_{4}}{(xy+4)^{2}}\right){\rm
d}x{\rm d}y\right\\}/{\sim},$ (3.3)

where $\sim$ denotes the identification111In order not to overload notation,
we tacitly adopt the convention that $(A/{\sim})=(A\setminus\\{0\\})/{\sim}$.
$g_{1}\sim g_{2}\Leftrightarrow g_{1}=kg_{2}$, $k\neq 0$. The elements of
$\mathcal{W}$ are (almost everywhere) non-singular 2-tensors up to
multiplication by a non-zero constant.

We shall now define a key object of the present paper – the subvariety
$\mathcal{Q}$ within $\mathcal{W}$ whose elements have vanishing Riemann
curvature,

$\mathcal{Q}=\\{q\in\mathcal{W}\colon\mathrm{Riem}(q)=0\\}.$

By construction, this is invariant under Stäckel transformations.

######  Remark 3.5.

  1. (i)

For elements $q\in\mathcal{W}$, the vanishing of their curvature tensor
$\mathrm{Riem}(q)$ is indeed independent of the choice of representative for
$q$.

  2. (ii)

Requiring the elements of $\mathcal{Q}$ to have vanishing curvature is a
choice. Another choice would be, for instance, to consider the subset of
elements $q\in\mathcal{W}$ such that the sectional curvature is
constant,222However, note that requiring a specific constant would not be an
invariant criterion. i.e., elements $q$ that live on the complex sphere.
However, the (non-degenerate) Stäckel classes (3,0), (0,11), (21,0) are shown
in [23] to _not_ admit realisations on the complex 2-sphere. Therefore such a
choice would not lead to an unambiguous characterisation of the Stäckel
classes considered here.

The definition of $\mathcal{Q}$ is possible in any dimension $n$. We now
restrict to $n=2$.

######  Lemma 3.6.

In dimension $2$, the variety $\mathcal{Q}\subset\mathcal{W}$ is defined by
one homogeneous quadratic polynomial equation whose unknowns are the
parameters of $V^{\mathcal{U}}$. The coefficients of this equation depend on
the position $x\in M$ on the underlying manifold $M$.

###### Proof.

In 2D the Riemannian curvature tensor is determined by its (unique) sectional
curvature or, alternatively, by its scalar curvature. Moreover, in suitable
local coordinates, any 2D metric can be written as $g=\phi^{2}{\rm d}x{\rm
d}y$, such that the requirement of vanishing Riemannian curvature becomes

$V^{\mathcal{U}}V^{\mathcal{U}}_{xy}\phi^{2}+2\big{(}V^{\mathcal{U}}\big{)}^{2}\phi_{xy}\phi-V^{\mathcal{U}}_{x}V^{\mathcal{U}}_{y}\phi^{2}-2\big{(}V^{\mathcal{U}}\big{)}^{2}\phi_{x}\phi_{y}=0,$
(3.4)

where we recall that $V^{\mathcal{U}}$ is a parametrisation of $\mathcal{U}$;
the subscripts $x$, $y$ denote usual derivatives. Therefore (3.4) is
homogeneously quadratic in the $n+2=4$ parameters of the potential
$V^{\mathcal{U}}$, with coefficients depending on the position. ∎

######  Example 3.7.

For the Hamiltonian of the Harmonic Oscillator,

$H=p_{x}p_{y}+a_{3}xy+a_{2}(x+y)+a_{1}(x-y)+a_{0},$

we find

$\mathcal{Q}=\big{\\{}\big{(}a_{3}xy+a_{2}(x+y)+a_{1}(x-y)+a_{0}\big{)}{\rm
d}x{\rm d}y\colon a_{1}^{2}-a_{2}^{2}=a_{0}a_{3}\big{\\}}.$

This is special since the condition on the parameters $a_{i}$ does not depend
on the position $(x,y)$. In general the restriction will depend on the
position (see below).

While here the main concern is about _non-degenerate_ second-order 2D
superintegrable systems, it may be pointed out that our definitions are not
restricted to the non-degenerate setting.

######  Example 3.8.

The Kepler–Coulomb system has the Hamiltonian
$H=p_{x}p_{y}+\frac{a_{1}}{xy}+a_{0}$. We find, where $a_{0}xy+a_{1}\neq 0$,
that

$\mathcal{F}=\left\\{\left(\frac{a_{1}}{xy}+a_{0}\right){\rm d}x{\rm d}y\colon
a_{0}a_{1}=0\right\\}/{\sim}$

consisting of two distinct projective points. Note however that Theorem 4.3
below does not apply to such degenerate systems (and neither does [23], which
however contains one exceptional case).

##  4 Results

We implement the method set out in the previous section for all cases of the
classification [20], and conclude that $\mathcal{Q}$ carries enough
information to identify the Stäckel class of a non-degenerate system. For this
purpose we introduce the following geometric object:

######  Definition 4.1.

Let $g$ be a 2-dimensional metric (of arbitrary curvature) that admits a non-
degenerate second-order superintegrable potential $V$. Denote by
$\mathcal{S}=\mathcal{S}(g,V)$ the Stäckel class of the system defined by the
Hamiltonian $H=g^{ij}p_{i}p_{j}+V$, i.e., the set of all (non-degenerate
second-order) superintegrable systems $h\in\mathcal{S}$ equivalent to the
initial one under Stäckel transform. We call

$\mathcal{F}(\mathcal{S})=\\{h\in\mathcal{S}\colon\mathrm{Riem}(g_{h})=0\\}/{\sim}$

the space of _flat realisations_ of $\mathcal{S}$. Here $g_{h}$ is the metric
underlying $h\in\mathcal{S}$, and $\sim$ stands for identification under
multiplication with a non-zero constant factor (i.e., we work projectively).
Note that $g_{h}$ is, in particular, a metric conformally equivalent to the
initial metric $g$.

As mentioned earlier, the flat realisations of any (non-degenerate) Stäckel
class $\mathcal{S}$ are known from [23].

######  Lemma 4.2.

The space $\mathcal{F}(\mathcal{S})$ is isomorphic to the intersection
$\bigcap_{N}\mathcal{Q}$ over a neighborhood $N\subset M$.

Note: We do not claim that the finding of Lemma 4.2 is new. Implicitly the
statement is found in the literature, e.g., [14, 31]. The author is not aware
of any reference discussing the invariance of $\mathcal{Q}$, or its associated
quadrics, however.

###### Proof.

By a suitable choice of coordinates, we can put any flat realisation of a
Stäckel class $\mathcal{S}$ into its normal form [Em]; we follow the
terminology of [20].333We denote the 2D Harmonic Oscillator system by [E3],
following [20], where however it is not written with all $n+2$ parameters. The
full potential is given in [23], for example, where the full system is
distinguished by a prime, [E3′]. The Hamiltonian $H_{m}$ of [Em] then has the
following form (note that only non-degenerate systems are listed):

$\displaystyle\begin{split}&H_{1}=p_{x}p_{y}+a_{3}xy+\tfrac{a_{2}}{(x+y)^{2}}+\tfrac{a_{1}}{(x-y)^{2}}+a_{0},\\\
&H_{2}=p_{x}p_{y}+a_{3}\big{(}4(x+y)^{2}-(x-y)^{2}\big{)}+a_{2}(x+y)+\tfrac{a_{1}}{(x-y)^{2}}+a_{0},\\\
&H_{3}=p_{x}p_{y}+a_{3}xy+a_{2}x+a_{1}y+a_{0},\\\
&H_{7}=p_{x}p_{y}+a_{3}xy+a_{2}\tfrac{y}{\sqrt{y^{2}-c^{2}}}+a_{1}\tfrac{x}{\sqrt{y^{2}-c^{2}}(y+\sqrt{y^{2}-c^{2}})^{2}}+a_{0},\\\
&H_{8}=p_{x}p_{y}+\tfrac{a_{3}x}{y^{3}}+\tfrac{a_{2}}{y^{2}}+a_{1}xy+a_{0},\\\
&H_{9}=p_{x}p_{y}+\tfrac{a_{3}}{\sqrt{y}}+a_{2}(x+y)+\tfrac{a_{1}(x+3y)}{\sqrt{y}}+a_{0},\\\
&H_{10}=p_{x}p_{y}a_{3}y+a_{2}\big{(}x-\tfrac{3}{2}y^{2}\big{)}+a_{1}\big{(}xy-\tfrac{1}{2}y^{3}\big{)}+a_{0},\\\
&H_{11}=p_{x}p_{y}+a_{3}x+\tfrac{a_{2}x}{\sqrt{y}}+\tfrac{a_{1}}{\sqrt{y}}+a_{0},\\\
&H_{16}=p_{x}p_{y}+\tfrac{1}{\sqrt{xy}}\left(a_{3}+\tfrac{a_{2}}{\tfrac{1}{2}(x+y)+\sqrt{xy}}+\tfrac{a_{1}}{\tfrac{1}{2}(x+y)-\sqrt{xy}}\right)+a_{0},\\\
&H_{17}=p_{x}p_{y}+\tfrac{a_{3}}{\sqrt{xy}}+\tfrac{a_{2}}{x^{2}}+\tfrac{a_{1}}{x\sqrt{xy}}+a_{0},\\\
&H_{19}=p_{x}p_{y}+\tfrac{a_{3}y}{\sqrt{y^{2}-4}}+\tfrac{a_{2}}{\sqrt{x(y+2)}}+\tfrac{a_{1}}{\sqrt{x(y-2)}}+a_{0},\\\
&H_{20}=p_{x}p_{y}+\tfrac{1}{\sqrt{xy}}\left(a_{3}+a_{2}\sqrt{\tfrac{1}{2}(x+y)+\sqrt{xy}}+a_{1}\sqrt{\tfrac{1}{2}(x+y)-\sqrt{xy}}\right)+a_{0}.\end{split}$
(4.1)

For each of these, the position-dependent quadric
$Q=Q_{(x,y)}(a_{3},a_{2},a_{1},a_{0})$ defining $\mathcal{Q}$ is polynomial in
the coordinates or at least a sum of almost everywhere linearly independent
functions $f_{j}(x,y)$. Thus, $Q=\sum_{j}\beta_{j}f_{j}(x,y)$ is a polynomial
with constant coefficients $\beta_{j}$ depending quadratically on the
parameters of $V^{\mathcal{U}}$. This proves the claim. ∎

With Observation 3.2 and Lemma 4.2 at hand, we are now ready to prove the main
result. Theorem 4.3 below provides a tool for determining the Stäckel class of
any non-degenerate 2D second-order superintegrable system.

The resulting technique is quite efficient (see examples below). Its proof,
however, is rather simple. In this respect it has to be acknowledged that the
second-order superintegrable systems in 2D are already classified [20] and
that we already know the complete list of Stäckel classes appearing for 2D
non-degenerate systems [23], too. As a result, we can simply check (3.4) for a
subset of the known normal forms of systems on flat space. Of course, this
simplicity of the proof is no impediment to the strength or scope of the
resulting method.

We are now going to see that two different Stäckel classes give rise to
different spaces $\mathcal{F}$. Therefore, from the knowledge of
$\mathcal{F}$, we can infer the Stäckel class of a non-degenerate second-order
superintegrable system. As an explicit example, take the Hamiltonian
$H_{8}=p_{x}p_{y}+V[a_{3},a_{2},a_{1},a_{0}]$ from [20] (see the list (4.1)
above). Here, $V^{\mathcal{U}}=V[a_{3},a_{2},a_{1},a_{0}]$ is a concrete
parametrisation. Since we work projectively, the potential is to be considered
modulo multiplication by an irrelevant constant, which we denote as
$V[a_{3}:a_{2}:a_{1}:a_{0}]$. We find

$\displaystyle\mathcal{Q}$
$\displaystyle=\left\\{V[a_{3}:a_{2}:a_{1}:a_{0}]{\rm d}x{\rm
d}y\colon\frac{\big{(}a_{0}a_{3}y^{6}+3a_{2}a_{3}y^{4}-3a_{0}a_{1}y^{2}-a_{1}a_{2}\big{)}y^{3}}{a_{3}xy^{4}+a_{0}y^{3}+a_{1}x+a_{2}y}=0\right\\}$
and thus $\displaystyle\mathcal{F}$
$\displaystyle=\\{V[a_{3}:a_{2}:a_{1}:a_{0}]{\rm d}x{\rm d}y\colon
a_{0}a_{3}=0,\,a_{2}a_{3}=0,\,a_{0}a_{1}=0,\,a_{1}a_{2}=0\\},$

wherever $a_{3}xy^{4}+a_{0}y^{3}+a_{1}x+a_{2}y\neq 0$. One therefore obtains a
space with two distinct components,

$\mathcal{F}=\\{V[a_{3}:a_{2}:a_{1}:a_{0}]{\rm d}x{\rm d}y\colon
a_{2}=0=a_{0}\\}\cup\\{V[a_{3}:a_{2}:a_{1}:a_{0}]{\rm d}x{\rm d}y\colon
a_{1}=0=a_{3}\\}$

($x\neq 0$). Table 1 summarises the results of the analogous computations for
all metrics in the list (4.1).

Table 1: The table details the non-degenerate Stäckel equivalence classes in dimension 2, their flat realisations and the associated quadric $\mathcal{F}$. Each row stands for a Stäckel class $\mathcal{S}$, specified in the first column and labeled as in [23]. The second column specifies, within this class, its flat realisations $h\in\mathcal{S}$ up to isometries (i.e., up to coordinate changes). These realisations $h$ are denoted with labels as in [20], cf. also the above list. The third column describes the variety $\mathcal{F}$ for the respective Stäckel class; note that it can be parametrised in various equivalent ways, but the varieties themselves are invariant and characteristic to each class. Stäckel class $\mathcal{S}$ | Flat systems in $\mathcal{S}$ | Description of $\mathcal{F}(\mathcal{S})$
---|---|---
up to Stäckel transform | up to isometries | as a variety
(0,11) | E3, E11, E20 | quadric surface $uv=a^{2}+b^{2}$
(21,0) | E7, E8, E17, E19 | two projective lines
(3,11) | E9, E10 | one projective line
(21,2) | E1, E16 | two projective points
(3,2) | E2 | one projective point
(111,11) | none | empty

######  Theorem 4.3.

The variety $\mathcal{F}$ determines the Stäckel class of a second-order non-
degenerate superintegrable system in dimension $2$ unambiguously.

###### Proof.

Any second-order superintegrable system in 2D is Stäckel equivalent to a
superintegrable system on a constant curvature space [6, 14]. Therefore Table
1 covers all cases and we immediately infer the asserted statement. ∎

With the help of this theorem, we can easily determine the Stäckel class for a
2-dimensional non-degenerate second-order superintegrable system. We
illustrate this with two examples of non-zero curvature.

######  Example 4.4.

As a first example we consider the system [S4], defined on the 2-sphere with
the metric (3.1) and the Hamiltonian (3.2). Using (3.3), we find that

$\mathcal{F}=\left\\{\left(\frac{a_{1}}{y}-\frac{a_{3}}{\sqrt{xy}}\right)\frac{{\rm
d}x{\rm d}y}{y}\colon\text{either $a_{1}=0$ or $a_{3}=0$, but not
both}\right\\}/{\sim},$

where we require $xy+4\neq 0$, and either $y\neq 0$ or $x\neq 0$,
respectively. In other words, $\mathcal{F}$ consists of two distinct
projective points. According to Table 1, this is the Stäckel class (21,2).
Indeed, the system [S4] – defined on the 2-sphere – is Stäckel equivalent to
the systems [E1] and [E16], i.e., the Hamiltonians $H_{1}$ and $H_{16}$ from
(4.1), defined on flat 2-space [23].

######  Example 4.5.

In [38], see also [28, 29], it is proven that any non-degenerate second-order
superintegrable system that admits one unique, essential projective
symmetry444A vector field is called projective if its local flow maps
geodesics into geodesics if we disregard parametrisation. It is called
essential if it is not a homothety. The word “unique” here means that the
projective symmetry algebra is 1-dimensional. is projectively equivalent
to555Note that (4.3) itself has a 1-dimensional projective algebra, but its
projective vector fields are homothetic. Nonetheless, any 2D second-order
superintegrable metric with a unique, essential projective symmetry is
projectively equivalent to (4.3). the Hamiltonian

$H=\frac{p_{x}p_{y}}{\big{(}x+y^{2}\big{)}}-a_{3}\frac{y\big{(}y^{2}+3x\big{)}}{x+y^{2}}+a_{2}\frac{y}{x+y^{2}}+\frac{a_{1}}{x+y^{2}}+a_{0}.$
(4.2)

The underlying metric is

$g=\big{(}x+y^{2}\big{)}{\rm d}x{\rm d}y,$ (4.3)

see for instance [4, 10, 30]. We find

$\mathcal{F}=\\{(a_{1}+a_{2}y){\rm d}x{\rm d}y\neq 0\\}/{\sim},$ (4.4)

where $y\neq 0$. Therefore, we have confirmed that $\mathcal{F}$ is a
projective line, and thus $H=g^{ij}p_{i}p_{j}+V$ belongs to the Stäckel class
(3,11).

We conclude this section with a closer look at the inner structure of the
invariant variety $\mathcal{F}$. By definition, the invariant variety
$\mathcal{F}$ is the space of all flat realisations of a given Stäckel class.
While before we were using $\mathcal{F}$ in order to characterise the Stäckel
class, we shall now take a different stance: We aim to understand the internal
structure of $\mathcal{F}$ as a space of flat superintegrable systems. In
other words, for each row of Table 1, i.e., for each Stäckel class, we ask how
the individual flat systems listed in the second column are “placed” within
the variety $\mathcal{F}$.

The strategy for this goal is rather simple: We compute $\mathcal{U}g$ for one
realisation $H=g^{ij}p_{i}p_{j}+V^{\mathcal{U}}$ from each row in Table 1. Due
to the invariance of $\mathcal{F}$ it does not matter which actual realisation
we select for the computation. For convenience, we may chose coordinates
$(x,y)$ such that $g={\rm d}x{\rm d}y$. Then the isometry operations are
$x\to\lambda x+a_{1}$ and $y\to\frac{y}{\lambda}+a_{2}$; the constants
$a_{1}$, $a_{2}$ and $\lambda\neq 0$ have to be adjusted such that we arrive
at the suitable normal form from [20].

######  Example 4.6.

For the class (3,11) we can use (4.4). Since either $a_{1}$ or $a_{3}$ has to
be non-zero, we arrive at two distinct cases:

  1. 1.

Case 1: $a_{3}=0$, but $a_{1}\neq 0$, i.e., w.l.o.g. $a_{1}=1$. Together with
$a_{2}=0=a_{4}$ we arrive at the Stäckel transform with conformal factor
$U=\frac{1}{x+y^{2}}$. The transformed Hamiltonian is therefore

$\displaystyle H^{\prime}$
$\displaystyle=\frac{H}{U}=p_{x}p_{y}+a_{1}-\tfrac{a_{2}}{2}y\big{(}y^{2}-3x\big{)}+a_{3}y+a_{4}\big{(}y^{2}+x\big{)}$
$\displaystyle=p_{x^{\prime}}p_{y^{\prime}}+a_{1}^{\prime}y^{\prime}+a_{2}^{\prime}\big{(}x^{\prime}-\tfrac{3}{2}(y^{\prime})^{2}\big{)}+a_{3}y^{\prime}\big{(}x^{\prime}-\tfrac{1}{2}(y^{\prime})^{2}\big{)}+a_{4}^{\prime},$

after a suitable change of coordinates from $(x,y)$ to
$(x^{\prime},y^{\prime})$, and redefinition of the parameters from the $a_{i}$
to the $a_{i}^{\prime}$. This is the system [E10] of [20].

  2. 2.

Case 2: $a_{3}\neq 0$. In an analogous way, one arrives to the system [E9] of
[20].

Continuing the procedure as in Example 4.6 for the other Stäckel classes, we
find the following.

######  Proposition 4.7.

For the class $(111,11)$, $\mathcal{F}$ is empty, $\mathcal{F}=\varnothing$,
i.e., no flat realisations exist. For the other Stäckel classes, we have:

Class (3,2).

The variety $\mathcal{F}$ consists of one projective point, corresponding to
the system [E2] of [20].

Class (21,2).

The variety $\mathcal{F}$ consists of two projective points, one corresponding
to [E1], the other to [E16].

Class (3,11).

This is the first non-trivial case. $\mathcal{F}$ consists of one projective
line, which generically is of type [E9]. The system [E10] corresponds to a
projective point lying within this projective line.

Class (21,0).

Two disjoined projective lines are contained in $\mathcal{F}$. One line is
generically [E19], containing one point that is [E17]. The other line is [E7]
generically and contains a point that is [E8].

Class (0,11).

The most interesting variety contains the Harmonic Oscillator and is governed
by the position-independent quadric $a^{2}+b^{2}=uv$ for
$a,b,u,v\in\mathbb{C}$. In the quadric defining $\mathcal{Q}$ the position
dependent contributions factor out. Somewhat surprisingly, the system [E20] is
realised generically, when $\mathcal{F}$ is described by
$v=\frac{a^{2}+b^{2}}{u}$ $(u\neq 0)$. The system [E11] is realised if $u=0$,
$a\neq 0$ and $b\neq 0$. The quadric is $a^{2}+b^{2}=(a+{\rm i}b)(a-{\rm
i}b)=0$. The projective point with $u=a=b=0$ realises [E3].

Note that for class (111,11) there exist no flat realisations. There are,
however, three distinct non-degenerate systems with constant sectional
curvature, called [S7], [S8] and [S9] in [20], which realise (111,11) on the
complex 2-sphere [23]. Analogously to $\mathcal{F}$, one could now define a
variety $\mathcal{E}=\\{q\in\mathcal{W}\colon\mathrm{Riem}(g)=\text{const}\neq
0\\}$ and ask for its inner structure analogously to Proposition 4.7. For
instance, for the class (111,11), one obtains a variety $\mathcal{E}$ whose
complete primary decomposition leads to 11 components. A further study of
these components does however not appear to reveal any new, interesting facts
relating to the purposes of the present paper. Hence we refrain from any
further analysis of $\mathcal{E}$.

##  5 Discussion and generalisations

###  5.1 Comparison with established method

It is worthwhile contrasting Theorem 4.3 with the method established by [23]
that appears to be the only preexisting technique for this task available for
immediate use. Essentially, one has to complete the following steps:

  1. 1.

Compute functionally independent integrals $F_{1}$, $F_{2}$ explicitly for the
Hamiltonian $H$.

  2. 2.

Compute their Poisson bracket $R=\\{F_{1},F_{2}\\}$.

  3. 3.

Rewrite $R^{2}$ (a sextic polynomial w.r.t. momenta) as a cubic in $H$,
$F_{1}$, $F_{2}$.

  4. 4.

Bring this cubic into normal form, cf. [23].

Note that there exist different choices for $F_{1}$ and $F_{2}$, which lead to
different but proportional results for $R$. This ambiguity is resolved only in
the last step by reverting to the normal forms. It is instructive to look at
an example:

######  Example 5.1.

We reconsider the system of Example 4.5. Its Hamiltonian (4.2) is defined on a
manifold of non-constant sectional curvature and its potential is non-
degenerate as it has four linear parameters with independent functions as
coefficients. We follow the four aforementioned steps:

  1. (i)

The Hamiltonian (4.2) has the advantage that we do not have to integrate for
the integrals of motion. In fact, two metrics that are projectively equivalent
to (4.3) are given in [30], and from these we can obtain the following two
integrals of motion by using the formula from [34], see also [38],

$\displaystyle
F_{1}=\left(p_{x}^{2}-\frac{2yp_{x}p_{y}}{y^{2}+x}\right)-\frac{a_{3}y^{2}\big{(}y^{2}-3x\big{)}}{y^{2}+x}+\frac{2a_{2}y^{2}}{y^{2}+x}+\frac{2a_{1}y}{y^{2}+x}+a_{0},$
$\displaystyle
F_{2}=\left(12xp_{x}^{2}-4y\frac{y^{2}+9x}{y^{2}+x}p_{x}p_{y}+9p_{y}^{2}\right)$
$\displaystyle\hphantom{F_{2}=}{}+\frac{a_{3}\big{(}y^{2}-3x\big{)}^{3}}{y^{2}+x}+\frac{2a_{2}\big{(}y^{2}-3x\big{)}^{2}}{y^{2}+x}+\left(\frac{8y\big{(}y^{2}-3x\big{)}}{y^{2}+x}-8\right)a_{1}+a_{0}.$

  2. (ii)

We compute

$\displaystyle
R=-12p_{x}\left(2p_{x}^{2}+\frac{3p_{y}^{2}}{x+y^{2}}-\frac{6yp_{x}p_{y}}{x+y^{2}}\right)$
$\displaystyle\hphantom{R=}{}+36p_{x}\frac{a_{3}\big{(}y^{4}+3x^{2}\big{)}+a_{2}\big{(}y^{2}-x\big{)}+2a_{1}y}{x+y^{2}}-36p_{y}\frac{2a_{3}y^{3}+a_{2}y+a_{1}}{x+y^{2}}.$

  3. (iii)

Its square $R^{2}$ can be written as

$\displaystyle R^{2}=576F_{1}^{3}+1728(a_{0}+a_{2})F_{1}^{2}-432a3F_{1}F_{2}$
$\displaystyle\hphantom{R^{2}=}{}+\big{(}{-}1728a_{1}H+432(4a_{0}^{2}+4(a_{0}-2a_{3})a_{1}+8a_{0}a_{2}+3a_{2}^{2}+a_{0}a_{3})\big{)}F_{1}$
$\displaystyle\hphantom{R^{2}=}{}+\big{(}144H^{2}-288a_{0}H+144\big{(}a_{0}^{2}-3a_{0}a_{3}\big{)}\big{)}F_{2}-144(a_{0}-8a_{1})H^{2}$
$\displaystyle\hphantom{R^{2}=}{}+288(a_{0}^{2}-(14a_{0}+9a_{2})a_{1})H+288\big{(}10a_{0}^{2}+9a_{0}a_{2}-12a_{0}a_{3}\big{)}a_{1}$
$\displaystyle\hphantom{R^{2}=}{}+432a_{0}^{3}+1728a_{0}^{2}a_{2}+1296a_{0}a_{2}^{2}+432a_{0}^{2}a_{3}+3888a_{1}^{2}a_{3}.$

  4. (iv)

After a linear redefinition $F_{1}^{\prime}=\sqrt[3]{576}(F_{1}+a_{0}+a_{2})$
and a rescaling $F_{2}^{\prime}=-36\sqrt[3]{3}F_{2}$, we find

$R^{2}=(F_{1}^{\prime})^{3}+a_{3}F_{1}^{\prime}F_{2}^{\prime}+\mathcal{O},$

where $\mathcal{O}$ is at most linear in $(F_{1}^{\prime},F_{2}^{\prime})$.
Therefore, according to [23], the system is of type (3,11). This is consistent
with what we found in Example 4.5.

Note that we have needed the explicit expressions for the integrals, and for
their Poisson bracket. This makes the procedure typically computationally more
intensive than the one presented in Theorem 4.3.

###  5.2 Conformal superintegrability

In the remainder of this section we comment on two possible generalisations of
Theorem 4.3. The first is the extension of our results to conformal
superintegrability, the other one to dimensions higher than 2.

Let us begin with a remark on conformal superintegrability. In this case,
integrals of motion (2.1) are replaced by conformal integrals: A _second-order
conformal integral_ is a function $F(x,p)=K^{ij}(x)p_{i}p_{j}+W(x)$ such that

$\\{F,H\\}=\omega H$ (5.1)

holds, for some polynomial $\omega=\omega^{i}(x)p_{i}$ linear in momenta.
Obviously, every integral of $H$ – satisfying (2.1) – is also a conformal
integral for $H$, with $\omega=0$. With (5.1) instead of (2.1), the components
$K_{ij}$ are now the components of a conformal Killing tensor. As a
consequence, instead of (2.2), we obtain the equation [14, 19, 26]

$\nabla_{[i}K\indices{{}_{j]}^{a}}\nabla_{a}V=K\indices{{}^{a}_{[i}}\nabla^{2}_{j]a}V+\omega_{[i}V_{,j]}+\omega_{[i,j]}V.$

Conformal superintegrability can then be defined analogously to proper
superintegrability, but we require functionally independent _conformal_
integrals (5.1) instead of proper ones, i.e., instead of (2.1). Non-degeneracy
is then also defined analogously, but one thing should be noted: For non-
degenerate properly superintegrable systems, we have
$V^{\mathcal{U}}=a_{0}+\sum_{k=1}^{n+1}a_{k}f_{k}$ for $(n+1)$-many functions
$f_{k}$. In conformal systems, however, we usually have $n+2$ non-constant
functions $f_{k}$, $V^{\mathcal{U}}=\sum_{k=0}^{n+1}a_{k}f_{k}$. In both
cases, the admissible potentials form a $(n+2)$-dimensional linear space
$\mathcal{U}$. Finally, the _conformal equivalence_ of conformally
superintegrable systems is defined similar to Stäckel equivalence, see, e.g.,
[5, 14, 26]. For our purposes, it is enough to allow, in (2.3), arbitrary
functions $U$, which not necessarily are compatible potentials of $H$. In
other words, we allow $U\not\in\mathcal{U}$ in addition. With these
generalisations of superintegrability at hand, it is easy to extend Theorem
4.3 to non-degenerate second-order 2D _conformally_ superintegrable systems.

######  Theorem 5.2.

The variety $\mathcal{F}$ determines the Stäckel class of a second-order non-
degenerate conformally superintegrable system in dimension $2$ unambiguously.

###### Proof.

It is shown in [6] that any conformally superintegrable system is Stäckel
equivalent to a properly superintegrable system, i.e., no new Stäckel classes
can appear. The assertion is then easily confirmed by following the individual
steps in the proof of Theorem 4.3. The key point in this respect is that
Observation 3.2 continues to hold true if $U\not\in\mathcal{U}$. ∎

###  5.3 Higher dimension

The current paper focuses on dimension $n=2$. In dimensions $n>2$ it is still
possible, by the same reasoning, to construct the invariant
$\mathcal{Q}\subset\mathcal{W}$, but it is generally not described by a single
quadric. Also, in higher dimension it is not possible to use $\mathcal{F}$ in
the same way as in 2D for a characterisation of the Stäckel class. In this
context, it seems appropriate to mention that also the method of [23] is
restricted to dimension 2. Indeed, one quickly finds that $\mathcal{Q}$ cannot
be used to identify the Stäckel class in dimension 3 already. In the 3D case,
the explicit normal forms of second-order non-degenerate superintegrable
systems [6, 15, 17] facilitate the computation; however, in most cases,
$\mathcal{F}$ is just one projective point. Such ambiguities should be
expected in any higher dimension, pointing to fundamental structural
particularities of 2D second-order superintegrable systems compared to higher
dimensions [24, 25, 26].

### Acknowledgements

The author is indebted to Jonathan Kress, Joshua Capel and Konrad Schöbel for
many discussions on superintegrability, Stäckel transforms and numerous
related topics, and thanks the anonymous referees for contributing valuable
suggestions that lead to improvements of this paper. A special thank you goes
to Konrad Schöbel for helpful comments on the manuscript.

This paper was principally written when the author was a fellow of the German
Research Foundation – Deutsche Forschungsgemeinschaft (DFG): Andreas Vollmer
acknowledges funding from a DFG research fellowship with the project number
353063958 as well as through a subsequent return fellowship. Andreas Vollmer
thanks the University of Stuttgart and the University of New South Wales for
hospitality.

## References

  * [1] Bertrand J.M., Mémoire sur quelques-unes des forms les plus simples que puissent présenter les intégrales des équations différentielles du mouvement d’un point matériel, J. Math. Pures Appl. 2 (1857), 113–140.
  * [2] Błaszak M., Marciniak K., On reciprocal equivalence of Stäckel systems, Stud. Appl. Math. 129 (2012), 26–50, arXiv:1201.0446.
  * [3] Błaszak M., Marciniak K., Classical and quantum superintegrability of Stäckel systems, SIGMA 13 (2017), 008, 23 pages, arXiv:1608.04546.
  * [4] Bolsinov A.V., Matveev V.S., Pucacco G., Normal forms for pseudo-Riemannian 2-dimensional metrics whose geodesic flows admit integrals quadratic in momenta, J. Geom. Phys. 59 (2009), 1048–1062, arXiv:0803.0289.
  * [5] Boyer C.P., Kalnins E.G., Miller Jr. W., Stäckel-equivalent integrable Hamiltonian systems, SIAM J. Math. Anal. 17 (1986), 778–797.
  * [6] Capel J.J., Invariant classification of second-order conformally flat superintegrable systems, Ph.D. Thesis, University of New South Wales, 2014.
  * [7] Darboux G., Sur un probléme de mécanique, Arch. Néerl. 6 (1901), 371–376.
  * [8] Daskaloyannis C., Quadratic Poisson algebras of two-dimensional classical superintegrable systems and quadratic associative algebras of quantum superintegrable systems, J. Math. Phys. 42 (2001), 1100–1119, arXiv:math-ph/0003017.
  * [9] Daskaloyannis C., Ypsilantis K., Unified treatment and classification of superintegrable systems with integrals quadratic in momenta on a two-dimensional manifold, J. Math. Phys. 47 (2006), 042904, 38 pages, arXiv:math-ph/0412055.
  * [10] Dini U., Sopra un problema che si presenta nella teoria generale delle rappresentazioni geografiche di una superficie su di un’altra, Ann. di Matem. 3 (1869), 269–293.
  * [11] Genest V.X., Ismail M.E.H., Vinet L., Zhedanov A., The Dunkl oscillator in the plane: I. Superintegrability, separated wavefunctions and overlap coefficients, J. Phys. A: Math. Theor. 46 (2013), 145201, 21 pages, arXiv:1212.4459.
  * [12] Hietarinta J., Grammaticos B., Dorizzi B., Ramani A., Coupling-constant metamorphosis and duality between integrable Hamiltonian systems, Phys. Rev. Lett. 53 (1984), 1707–1710.
  * [13] Kalnins E.G., Kress J.M., Miller Jr. W., Second-order superintegrable systems in conformally flat spaces. I. Two-dimensional classical structure theory, J. Math. Phys. 46 (2005), 053509, 28 pages.
  * [14] Kalnins E.G., Kress J.M., Miller Jr. W., Second order superintegrable systems in conformally flat spaces. II. The classical two-dimensional Stäckel transform, J. Math. Phys. 46 (2005), 053510, 15 pages.
  * [15] Kalnins E.G., Kress J.M., Miller Jr. W., Second order superintegrable systems in conformally flat spaces. IV. The classical 3D Stäckel transform and 3D classification theory, J. Math. Phys. 47 (2006), 043514, 26 pages.
  * [16] Kalnins E.G., Kress J.M., Miller Jr. W., Nondegenerate 2D complex Euclidean superintegrable systems and algebraic varieties, J. Phys. A: Math. Theor. 40 (2007), 3399–3411.
  * [17] Kalnins E.G., Kress J.M., Miller Jr. W., Nondegenerate three-dimensional complex Euclidean superintegrable systems and algebraic varieties, J. Math. Phys. 48 (2007), 113518, 26 pages, arXiv:0708.3044.
  * [18] Kalnins E.G., Kress J.M., Miller Jr. W., Separation of variables and superintegrability. The symmetry of solvable systems, IOP Expanding Physics, IOP Publishing, Bristol, 2018.
  * [19] Kalnins E.G., Kress J.M., Miller Jr. W., Post S., Laplace-type equations as conformal superintegrable systems, Adv. in Appl. Math. 46 (2011), 396–416.
  * [20] Kalnins E.G., Kress J.M., Pogosyan G.S., Miller Jr. W., Completeness of superintegrability in two-dimensional constant-curvature spaces, J. Phys. A: Math. Gen. 34 (2001), 4705–4720, arXiv:math-ph/0102006.
  * [21] Kalnins E.G., Miller Jr. W., Post S., Contractions of 2D 2nd order quantum superintegrable systems and the Askey scheme for hypergeometric orthogonal polynomials, SIGMA 9 (2013), 057, 28 pages, arXiv:1212.4766.
  * [22] Kalnins E.G., Pogosyan G.S., Miller Jr. W., Completeness of multiseparable superintegrability in two dimensions, Phys. Atomic Nuclei 65 (2002), 1033–1035.
  * [23] Kress J.M., Equivalence of superintegrable systems in two dimensions, Phys. Atomic Nuclei 70 (2007), 560–566.
  * [24] Kress J., Schöbel K., An algebraic geometric classification of superintegrable systems in the Euclidean plane, J. Pure Appl. Algebra 223 (2019), 1728–1752, arXiv:1602.07890.
  * [25] Kress J., Schöbel K., Vollmer A., An algebraic geometric foundation for a classification of superintegrable systems in arbitrary dimension, arXiv:1911.11925.
  * [26] Kress J., Schöbel K., Vollmer A., Algebraic conditions for conformal superintegrability in arbitrary dimension, arXiv:2006.15696.
  * [27] Maciejewski A.J., Przybylska M., Tsiganov A.V., On algebraic construction of certain integrable and super-integrable systems, Phys. D 240 (2011), 1426–1448, arXiv:1011.3249.
  * [28] Manno G., Vollmer A., (Super-)integrable systems associated to 2-dimensional projective connections with one projective symmetry, J. Geom. Phys. 145 (2019), 103476, 22 pages, arXiv:1905.01396.
  * [29] Manno G., Vollmer A., Normal forms of two-dimensional metrics admitting exactly one essential projective vector field, J. Math. Pures Appl. 135 (2020), 26–82, arXiv:1705.06630.
  * [30] Matveev V.S., Two-dimensional metrics admitting precisely one projective vector field, Math. Ann. 352 (2012), 865–909.
  * [31] Post S., Coupling constant metamorphosis, the Stäckel transform and superintegrability, AIP Conf. Proc. 1323 (2011), 265–274.
  * [32] Post S., Models of quadratic algebras generated by superintegrable systems in 2D, SIGMA 7 (2011), 036, 20 pages, arXiv:1104.0734.
  * [33] Sergyeyev A., Błaszak M., Generalized Stäckel transform and reciprocal transformations for finite-dimensional integrable systems, J. Phys. A: Math. Theor. 41 (2008), 105205, 20 pages, arXiv:0706.1473.
  * [34] Topalov P., Matveev V.S., Geodesic equivalence via integrability, Geom. Dedicata 96 (2003), 91–115, arXiv:math.DG/9911062.
  * [35] Tsiganov A.V., Duality between integrable Stäckel systems, J. Phys. A: Math. Gen. 32 (1999), 7965–7982, arXiv:solv-int/9812001.
  * [36] Tsiganov A.V., Transformation of the Stäckel matrices preserving superintegrability, J. Math. Phys. 60 (2019), 042701, 13 pages, arXiv:1809.05824.
  * [37] Tsiganov A.V., Superintegrable systems and Riemann–Roch theorem, J. Math. Phys. 61 (2020), 012701, 14 pages, arXiv:1910.08269.
  * [38] Vollmer A., Projectively equivalent 2-dimensional superintegrable systems with projective symmetries, J. Phys. A: Math. Theor. 53 (2020), 095202, 25 pages, arXiv:1812.03591.


# Asymptotic and Finite-time Cluster Synchronization of Neural Networks via
Two Different Controllers

Juan Cao<EMAIL_ADDRESS>Fengli Ren<EMAIL_ADDRESS>Dacheng Zhou
<EMAIL_ADDRESS>Department of Mathematics, Nanjing University of
Aeronautics and Astronautics, Nanjing 210016, P.R. China  School of
Mechatronics Engineering and Automation, Shanghai University, Shanghai 200444,
P.R. China

###### Abstract

In this paper, by using pinning impulse controller and hybrid controller
respectively, the research difficulties of asymptotic synchronization and
finite time cluster synchronization of time-varying delayed neural networks
are studied. On the ground of Lyapunov stability theorem and Lyapunov-
Razumikhin method, a novel sufficient criterion on asymptotic cluster
synchronization of time-varying delayed neural networks is obtained. Utilizing
Finite time stability theorem and hybrid control technology, a sufficient
criterion on finite-time cluster synchronization is also obtained. In order to
deal with time-varying delay and save control cost, pinning pulse control is
introduced to promote the realization of asymptotic cluster synchronization.
Following the idea of pinning control scheme, we design a progressive hybrid
control to promote the realization of finite time cluster synchronization.
Finally, an example is given to illustrate the theoretical results.

###### keywords:

Lyapunov-Razumikhin method , time-varying delay , finite-time cluster
synchronization , pinning impulsive control , hybrid control

##  1 Introduction

Synchronization in complex networks refers to a evolutionary process where two
or more chaotic systems improve their behavior to achieve a common phenomenon
under certain coupling conditions or under certain forces. It’s an important
topic that have been captured by many experts and scholars from various fields
of research [1, 2, 3, 4, 5, 6], such as physical science, natural science,
mathematics, communication, and engineering. Because of that, different types
of synchronization protocols have been studied in detail. For example,
projective synchronization [7, 8, 9], cluster synchronization [10, 11, 12, 13,
14, 15], complete synchronization [16], generalized synchronization [17],
phase synchronization [18], lag synchronization [19], and so on.

Sometimes it takes a lot of people to achieve a goal, and different aims may
be implemented by different people in real life, such as the division of
research teams and the setting of government function departments. Because of
different tasks, they have to achieve different goals. The same is true in
neural networks. Each node in the same part wants to achieve the same goal,
nodes in the different clusters want to achieve different goals, this is the
so-called cluster synchronization. Cluster synchronization has important
applications in science [20] and Bioengineering [21]. During the past few
years, cluster synchronization has been widely studied by a large number of
researchers, see [10, 11, 12, 13, 14, 15, 22, 23, 24, 25] and the relevant
references. In an attempt to investigate cluster synchronization of complex
systems, there have been many results related to cluster synchronization of
coupled systems. This is not only because cluster synchronization has unusual
correlations between network subsystems, but also because many coupled
networks can exhibit excellent clustering characteristic. Additionally, As we
all know, time delay is an ubiquitous phenomenon in physical complex networks.
This is inevitable and may lead to a undesirable state of the system [26].
Therefore, the influence of time delay should be considered. As a result, it
is essential to investigate cluster synchronization of coupled neural networks
with time-varying delay.

Because of the complexity of the network structure, researchers have proposed
many effective methods to synchronize the complex networks, some effective
methods are pinning control [27, 28, 29], impulsive control [30, 31, 32],
intermittent control [33], adaptive control [34], feedback control [25, 35,
36], and so on. Impulse effect is a common phenomenon due to the instantaneous
disturbance at some time. These phenomena are described by impulsive
differential equation which have been used efficiently in modeling many
practical problems that arise in the fields of engineering, physics, and
science. Regarding to this, pinning impulsive control becomes a practically
external control approach which is famous for low cost and energy-saving.
Pinning impulsive control is characterized by adding impulsive controllers on
only a fraction of network nodes. [31] investigated the pinning
synchronization problem of stochastic impulsive networks using Lyapunov
stability theory and pinning method. However, it only examined asymptotical
synchronization and exponential synchronization of such dynamical networks in
mean square. By introducing stochastic impulsive analysis technique and the
selected nodes of the stochastic network are pinned at the impulse moment,
authors in [22] discussed cluster synchronization of delayed stochastic neural
networks under pinning impulsive control. A disadvantage of the above two
papers is that they only considers the asymptotic behavior of the system as
time approaches infinity.

In control theory, a considerable number of papers are focused on the
utilization of pinning control to help complex networks achieve cluster
synchronization. Based on the Lyapunov stability theory, the article [37] is
devoted to study the cluster synchronization for network of a complex
dynamical network of non-identical nodes with delayed and non-delayed coupling
under a pinning control scheme. The finite-time and fixed-time cluster
synchronization problem for complex networks with or without pinning control
are studied in [38]. However, the systems in this paper are special coupled
networks where use a concrete function $\rm sig^{p}(x)$. The major objective
of [23] is to deal with cluster synchronization of coupled fractional-order
neural networks with time delay, while the two control laws in this work is
not pinning control but the control of all nodes in these networks. Based on
it, we want to extend the idea of pinning control to control some of the key
nodes in each cluster, and exert no control over the remaining nodes in the
cluster, or only exert weak control over the remaining nodes, which is so-
called hybrid control.

Based on above analysis, we study cluster synchronization of coupled neural
networks with time-varying delay both on asymptotic cluster synchronization
and on finite-time cluster synchronization. Unlike the literature [23] which
is devoted to constant delay, we deal with time-varying delay, so the method
in the literature [23] can’t be used here. We utilize Lyapunov-Razumikhin
method to deal with time-vary delay. Two novel sufficient conditions are
obtained on asymptotic and finite-time cluster synchronization of the neural
networks by exerting pinning impulsive control and hybrid control,
respectively. The article is arranged as follows. In section 2, some
preliminaries on asymptotic and finite-time cluster synchronization are
provided. In section 3, we give some major results. In section 4, a numerical
example is given to check the effectiveness of the consequences. The final
section is the conclusion of this paper.

##  2 Preliminaries

###  2.1 Model Description

Consider the following system which consists of $N$ coupled neural networks
with time-varying delay, the $i$th neural network is described by

$\left\\{\begin{array}[]{c}\dot{x}_{i}(t)=-C_{i}x_{i}(t)+A_{i}f\left(x_{i}(t)\right)+B_{i}f\left(x_{i}(t-\tau(t))\right)+\sum\limits_{j=1}^{N}g_{ij}x_{j}(t)\\\
+I_{i}+u_{i}(t),\\\
x_{i}(t)=\phi_{i}(t),t\in\left[-\tau,0\right],\end{array}\right.$ (1)

where $x_{i}(t)=\left(x_{i1}(t),x_{i2}(t),\cdots,x_{in}(t)\right)^{T}$ is the
state vector of the $i$th neural network at time $t$, $C_{i}>0$(means a
symmetrical positive determined matrix) denotes the self-inhibition rate,
$A_{i}=\left(a_{lr}^{i}\right)_{n\times n}\in R^{n\times n}$ and
$B_{i}=\left(b_{lr}^{i}\right)_{n\times n}\in R^{n\times n}$ are connection
weight matrices and delayed connection weight matrices, respectively. $f(x)$
is the activation function at time $t$ , $\tau(t)$ is transmittal delay. We
suppose there exists two constants $\tau$ and $\sigma$, render
$0<\tau(t)<\tau$ and $\dot{\tau}(t)\leq\sigma<1$. $I_{i}\in R^{n}$ means a
input from the outside of the system, $u_{i}(t)$ is the controller that will
be designed below. The element $g_{ij}$ of a linear coupling matrix is
expressed as that if there is a connection from node $i$ to another node $j$ ,
then $g_{ij}\neq 0$, otherwise $g_{ij}=0$. The diagonal terms of the matrix
satisfy the following diffusion coupling conditions:
$g_{ii}=-\sum\limits_{j=1,j\neq i}^{N}g_{ji}$.

###  2.2 Cluster Synchronization

Let$\\{\mathscr{C}_{1},\cdots,\mathscr{C}_{M}\\}$ be a partition of the set
$\\{1,\cdots,N\\}$ , by which $\\{1,\cdots,\\\ N\\}$ is divided into $M$
nonempty subsets, i.e. $\mathscr{C}_{l}\neq\phi$ and
$\bigcup\limits_{l=1}^{M}{\mathscr{C}_{l}}=\\{1,\cdots,N\\}$. For each
$l\in\\{1,\cdots,N\\}$, let $\widetilde{l}$ denotes the subscript of the
subset where the number $l$ is, i.e. $l\in\mathscr{C}_{\widetilde{l}}$.

A network with $N$ nodes is said to reach asymptotic cluster synchronization
if the state variables of the network satisfies
$\lim\limits_{t\rightarrow\infty}\left\|x_{i}(t)-x_{j}(t)\right\|=0$ for
${\widetilde{i}=\widetilde{j}}$ and
$\lim\limits_{t\rightarrow\infty}\left\|x_{i}(t)-x_{j}(t)\right\|\neq 0$ for
${\widetilde{i}\neq\widetilde{j}}$. Furthermore, A network with $N$ nodes is
said to reach finite-time cluster synchronization if there exists a
$t^{*}>t_{0}$ such that for ${\widetilde{i}=\widetilde{j}}$
,$\lim\limits_{t\rightarrow t^{*}}\left\|x_{i}(t)-x_{j}(t)\right\|=0$ and
$\|x_{i}(t)-x_{j}(t)\|=0$ for $t\geq t^{*}$.

Definition 1[23]. The function $H=(h_{ij}\in R^{m\times m})$ is said to belong
to class $\mathscr{A}_{1}$, if it satisfies $h_{ij}\geq
0,h_{ii}=-\sum\limits_{j=1,j\neq i}^{m}h_{ij}$.

Definition 2 [23]. The function $H=(h_{ij}\in R^{m\times m})$ belongs to class
$\mathscr{A}_{2}$, if it satisfies $\sum\limits_{j=1}^{m}h_{ij}=0$.

Assumption 1 [23]. For $i\in\mathscr{C}_{p},p=\\{1,2,\cdots,M\\}$, there
exists $\xi_{p}>0$, for all $x,y\in R^{n}$, $\|f(x)-f(y)\|\leq\xi_{p}\|x-y\|$.

Assumption 2 [23]. If all $N$ nodes are divided into $M$ clusters, The Laplace
matrix can be defined in the following form

$\begin{array}[]{l}\left(\begin{array}[]{ccccc}\bar{A}_{11}&\bar{A}_{12}&\bar{A}_{13}&\cdots&\bar{A}_{1M}\\\
\bar{A}_{21}&\bar{A}_{22}&\bar{A}_{23}&\cdots&\bar{A}_{2M}\\\
\bar{A}_{31}&\bar{A}_{32}&\bar{A}_{33}&\cdots&\bar{A}_{3M}\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
\bar{A}_{M1}&\bar{A}_{M2}&\bar{A}_{M3}&\cdots&\bar{A}_{MM}\end{array}\right),\end{array}$

where $\bar{A}_{ij}\in
R^{\left(v_{i}-v_{i-1}\right)\times\left(v_{j}-v_{j-1}\right)},\bar{A}_{pp}\in\mathscr{A}_{1},q=1,2,\cdots,M$,
$\bar{A}_{pq}\in\mathscr{A}_{2},p,q=1,2,\cdots,M,p\neq q$.

###  2.3 Mathematical Preparation

Provided that the $N$ nodes can be divided into $M$ clusters, that is to say
$\\{1,2,\cdots,N\\}=\mathscr{C}_{1}\cup\mathscr{C}_{2}\cup\cdots\cup\mathscr{C}_{M}$,
where $\mathscr{C}_{p}=\left\\{v_{p-1}+1,v_{p-1}+2,\cdots,v_{p}\right\\}$,
$p=1,2,\cdots,M$ and $v_{0}=0$, $v_{M}=N$, $v_{p-1}<v_{p}$. It is necessary to
assume that the coefficients of system (1) meet
$C_{v_{p-1}+1}=C_{v_{p-1}+2}=\cdots=C_{r_{p}}=C_{p}$,
$A_{p_{p-1}+1}=A_{p_{p-1}+2}=\cdots=A_{r_{p}}=A_{p}$,
$B_{v_{p-1}+1}=B_{v_{p-1}+2}=\cdots=B_{v_{p}}=B_{p}$,
$I_{v_{p-1}+1}=I_{v_{p-1}+2}=\cdots=I_{v_{p}}=I_{p}$. Suppose we want to
control network (1) to achieve the following ideal state
$x_{v_{0}+1}(t),\cdots,x_{v_{1}}(t)\rightarrow s_{1}(t)$;
$x_{v_{1}+1}(t),\cdots,x_{v_{2}}(t)\rightarrow s_{2}(t)$;
$x_{v_{M}+1}(t),\cdots,x_{v_{M}}(t)\rightarrow s_{M}(t)$. That is,
$s=\left(s_{1}(t),\cdots,s_{1}(t),s_{2}(t),\cdots,s_{2}(t),\cdots,s_{M}(t),\cdots,s_{M}(t)\right)\subset
R^{n\times N}$ is the desired leader. $s_{p}(t)$can be defined as

$\left.\dot{s}_{p}(t)=-C_{p}s_{p}(t)+A_{p}f\left(s_{p}(t)\right)+B_{p}f\left(s_{p}(t-\tau(t))\right)\right)+I_{p},p=1,2,\cdots,M.$
(2)

Then there is the error dynamic system

$\begin{array}[]{*{20}{l}}{\dot{e_{i}}(t)=\dot{x_{i}}(t)-\dot{s_{p}}(t)}\\\
\begin{array}[]{l}\;\;\;\;\;\;=-{C_{p}}{e_{i}}(t)+{A_{p}}\left[{f\left({{x_{i}}(t)}\right)-f\left({{s_{p}}(t)}\right)}\right]\\\
\;\;\;\;\;\;\;\;\;\;+{B_{p}}\left[{f\left({{x_{i}}(t-\tau(t))}\right)-f\left({{s_{p}}(t-\tau(t))}\right)}\right]+\sum\limits_{j=1}^{N}{{g_{ij}}}{e_{j}}(t)+{u_{i}}(t).\end{array}\end{array}$
(3)

Definition 3 [39]. Let $V:R^{+}\times R^{n}\rightarrow R^{+}$, then $V$ is
said to belong to class $\nu_{0}$, if
1.

$\lim\limits_{(t,y)\rightarrow\left(\tau_{k}^{+},x\right)}V(t,y)=V\left(\tau_{k}^{+},x\right),$
(4)

2\. $V$ is locally Lipschitzian in $x$ and $V(t,0)=0$ for all $t\in R^{+}$.

Consider a general impulsive delayed differential equation(IDDE) in the form
of

$\left\\{\begin{array}[]{c}\dot{x}(t)=f\left(t,x_{t}\right),t\neq t_{k},t\geq
t_{0},\\\ \Delta x\left(t_{k}\right)=I_{k}\left(t_{k},x_{t_{k}}\right),k\in
N,\\\ x_{t_{0}}=\phi\in PC\left([-\tau,0],R^{n}\right),\end{array}\right.$ (5)

where $x(t)=\left[x_{1}(t),x_{2}(t),\cdots,x_{n}(t)\right]^{T}$,
$x_{t}=\\{x(t+\theta):-\tau\leq\theta\leq 0\\}$. $f:R^{+}\times R^{n}\times
PC\left([-\tau,0],R^{n}\right)\rightarrow R^{n}$ is continuous. $\Delta
x\left(t_{k}\right)=x(t_{k}^{+})-x(t_{k})$ denotes impulsive interval,
$I:R^{+}\times R^{n}\rightarrow R^{n}$ is also continuous. We shall assume
$f(t,0)=I_{k}(t,0)=0$ for all $t\in R^{+}$ and $k\in N$ so that system (5)
admits the trivial solution.

Lemma 1 [39]. We suppose there exist $V\in\nu_{0}$ and $p>0$, $c>0$,
$c_{1}>0$, $c_{2}>0$, $\sigma>0$, $\lambda>0$, $\gamma\geq 1$ and
$\sigma-\lambda\geq c$, make the following four conditions true
(i)$c_{1}\|x\|^{s}\leq V(t,x)\leq c_{2}\|x\|^{s}$ , $\forall t\in R^{+}$,
$x\in R^{n}$;
(ii)$D^{+}V(t,\varphi(0))\leq cV(t,\varphi(0))$, $\forall
t\in[t_{k-1},t_{k}),k\in N$ hold, no matter when
$qV(t,\varphi(0))\geq V(t+\theta$, $\varphi(\theta))$ for
$\theta\in[-\tau,0]$, where the constant $q\geq\gamma e^{\lambda\tau}$;
(iii)$D^{+}V(t_{k},\varphi(0)+I_{k}(t_{k},\varphi))\leq\eta_{k}V(t_{k}^{-},\varphi(0))$,
where $0<\eta_{k-1}\leq 1$, $k\in N$;
(iv)$\gamma\geq 1/\eta_{k-1}$ and
$ln\eta_{k-1}<-(\sigma+\lambda)(t_{k}-t_{k-1}),k\in N$;
then IDDE(5) is global stability exponentially to the zero solution. The rate
of convergence is $\frac{\lambda}{s}$.

Lemma 2 [22]. $\forall x,y\in R_{n}$, there exist $\varpi>0$, $E\in R^{n\times
n}>0$(positive definite metrix), such that $2x^{T}y\leq\varpi
x^{T}Ex+\varpi^{-1}y^{T}E^{-1}y$.

Lemma 3 [24]. Let $x_{1},x_{2},\cdots,x_{n}\in R^{n}$ are any vectors and
$0<q<2$ is a real number satisfying:

$\left(\left\|x_{1}\right\|^{2}+\left\|x_{2}\right\|^{2}+\cdots+\left\|x_{n}\right\|^{2}\right)^{q/2}\leq\left\|x_{1}\right\|^{q}+\left\|x_{2}\right\|^{q}+\cdots+\left\|x_{n}\right\|^{q}.$

Lemma 4 [24]. Suppose that a continuous, positive-definite function $V(t)$
satisfies $\dot{V}(t)\leq-\alpha V^{\eta}(t),\forall t\geq t_{0},V(t_{0})\geq
0,$ where $\alpha>0,0<\eta<1$ are two constants, then for any given $t_{0}$,
$V(t)$ satisfies $V^{1-\eta}(t)\leq
V^{1-\eta}(t_{0})-\alpha(1-\eta)(t-t_{0}),t_{0}\leq t\leq t_{1}$, and
$V(t)=0,\forall t\geq t_{1}$, with $t_{1}$ given by
$t_{1}=t_{0}+\frac{V^{1-\eta}(t_{0})}{\alpha(1-\eta)}$.

##  3 Main Results

###  3.1 Asymptotic Cluster Synchronization

In this part, we obtain a sufficient condition for asymptotic cluster
synchronization of a class of coupled neural networks (1) with time-varying
delays by means of pinning impulsive control.

In order to achieve the purpose of asymptotic cluster synchronization, pinning
impulsive control only applies to certain nodes. We set $\delta(\cdot)$ as
Dirac delta function, $d_{k}$ as impulse gain. If $-2<d_{k}<0$, we choose the
first $\rho_{p}$ node to pin in the cluster $\mathscr{C}_{p}$, the selection
method follows the form (a); if $d_{k}\geq 0$ or $d_{k}\leq-2$, we choose the
first $\rho_{p}$ node as the pinning node in the cluster $\mathscr{C}_{p}$,
the selection method follows the form (b). We define the set of all pinned
nodes as
$\chi_{p}\left(t_{k}\right)=\left\\{i_{p1},i_{p2},\cdots,i_{p\rho_{p}}\right\\}\subset\mathscr{C}_{p}$.
In order to add a pinning impulsive control, we set $t_{k}\geq 0$ are
impulsive moments, and assume that
$0=t_{0}<t_{1}<\cdots<t_{k}<t_{k+1}<\cdots$,
$\lim\limits_{k\to\infty}t_{k}=+\infty$, $\sup\limits_{k\geq
0}\left\\{t_{k+1}-t_{k}\right\\}<+\infty$. When $t=t_{k}$, the node errors in
each cluster $\mathscr{C}_{p}$ are divided into the following two forms:
(a)$\|e_{i_{p1}}(t_{k})\|\geq\|e_{i_{p2}}(t_{k})\|\geq\cdots\geq\|e_{i_{p_{s}}}(t_{k})\|\geq\|e_{i_{p_{p(s+1)}}}(t_{k})\|\geq\cdots\geq\\\
\|e_{i_{p_{k}(v_{p}-v_{p-1})}}(t_{k})\|;$
(b)$\|e_{i_{p1}}(t_{k})\|\leq\|e_{i_{p2}}(t_{k})\|\leq\cdots\leq\|e_{i_{p_{ss}}}(t_{k})\|\leq\|e_{i_{p_{p(s+1)}}(t_{k})}\|\leq\cdots\leq\\\
\|e_{i_{p_{k}(v_{p}-p_{p-1})}}(t_{k})\|;$
where $i_{ps}\in\mathscr{C}_{p},p=1,2,\cdots,M$, $i_{pu}\neq i_{pw}$, when
$u\neq w$. What’s more, providing
$|e_{i_{ps}}\left(t_{k}\right)\|=\|e_{i_{p,(s+1)}}\left(t_{k}\right)\|$, then
$i_{ps}<i_{p,s+1}$.

Design the following pinning impulsive controllers

$u_{i}(t)=\left\\{\begin{array}[]{c}\sum\limits_{k=1}^{+\infty}d_{k}e_{i}(t)\delta\left(t-t_{k}\right),i\in\chi_{p}\left(t_{k}\right),\\\
0,i\in\mathscr{C}_{p}\backslash\chi_{p}\left(t_{k}\right).\end{array}\right.$
(6)

The form of the error dynamic system under the pinning impulsive controllers
(6) is as follows

$\left\\{\begin{array}[]{c}\dot{e}_{i}(t)=-C_{p}e_{i}(t)+A_{p}F\left(e_{i}(t)\right)+B_{p}F\left(e_{i}(t-\tau(t))\right)+\sum\limits_{j=1}^{N}g_{ij}e_{j}(t),\\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;i\in\mathscr{C}_{p},p=1,2,\cdots,M,t\neq
t_{k},\\\
e_{i}\left(t_{k}^{+}\right)=\left(1+d_{k}\right)e_{i}\left(t_{k}\right),i\in\chi_{p}\left(t_{k}\right),\\\
e_{i}\left(t_{k}^{+}\right)=e_{i}\left(t_{k}\right),i\in\mathscr{C}_{p}\backslash\chi_{p}\left(t_{k}\right),\end{array}\right.$
(7)

where $F\left(e_{i}(t)\right)=f\left(x_{i}(t)\right)-f\left(s_{p}(t)\right)$,
$F\left(e_{i}(t-\tau(t))\right)=f\left(x_{i}(t-\tau(t))\right)-f\left(s_{p}(t-\tau(t))\right)$,
$e_{i}\left(t_{k}^{+}\right)=\lim\limits_{h\rightarrow
0^{+}}e_{i}\left(t_{k}+h\right)$,
$e_{i}\left(t_{k}^{-}\right)=\lim\limits_{h\rightarrow
0^{-}}e_{i}\left(t_{k}+h\right)$, $e_{i}(t)$ is left continuous when
$t=t_{k}$. The initial condition of $e_{i}(t)$ is set as $e_{i}(t)=\phi(t)\in
PC\left([-\tau,0],R^{n}\right)$, where $PC\left([-\tau,0],R^{n}\right)$ is the
sum of all piecewise continuous functions $\phi$, $\phi$ has a supremum norm
$\|\phi\|=\sup\limits_{-\tau\leq\theta\leq 0}\|\phi(\theta)\|$.

Theorem 1. According to the assumptions 1-2, if there have $E_{1}\in
R^{n\times n}>0$, $E_{2}\in R^{n\times n}>0$, and $Q\in R^{n\times n}>0$, and
$q\geq\gamma e^{\lambda\tau}$, $\alpha>0$, $\beta>0$, and $\mu,\upsilon$, so
as to the following conditions are satisfied:

$\left(\begin{array}[]{ccc}\gamma I_{N}&\sqrt{\alpha}I_{N}\otimes
QA_{p}&\sqrt{\beta}I_{N}\otimes QB_{p}\\\ &-I_{N}\otimes E_{1}&0\\\
&*&-I_{N}\otimes E_{2}\end{array}\right)<0,$ (8)
$\begin{array}[]{l}\beta^{-1}\xi^{2}E_{2}<vQ,\\\ \end{array}$ (9)
$\begin{array}[]{l}\ln\eta_{k}\leq-(\sigma+\lambda)\Delta_{k-1},\\\
\end{array}$ (10)
$\begin{array}[]{l}\mu+\varepsilon\upsilon-(\sigma+\lambda)<0,\end{array}$
(11)

then the system (1) can realize cluster synchronization with exponential
convergence rate $\frac{\lambda}{2}$ under the pinning impulsive controllers
(6).

Proof. Set up a Lyapunov function as

$V(t)=\sum\limits_{i=1}^{N}e_{i}^{T}(t)Qe_{i}(t),$ (12)

where $Q\in R^{n\times n}>0$.

It is easy to observe that $\lambda_{\min}(Q)\left\|e_{i}(t)\right\|^{2}\leq
V(t)\leq\lambda_{\max}(Q)\left\|e_{i}(t)\right\|^{2},\\\ t\in R^{+}$, so
condition (i) in Lemma 1 is satisfied and $s=2$.

For $t\in(t_{k-1},t_{k}]$, from the formulation (7), we can deduce

$\displaystyle\frac{dV(t)}{dt}=2$
$\displaystyle\sum_{p=1}^{M}\sum_{i=v_{p-1}}^{v_{p}}e_{i}^{T}(t)Q\left[-C_{p}e_{i}(t)+A_{p}F\left(e_{i}(t)\right)\right.$
(13)
$\displaystyle+B_{p}F\left(e_{i}(t-\tau(t))\right)+\sum_{j=1}^{N}g_{ij}e_{j}(t)].$

For $i\in\mathscr{C}_{p}$, according to the assumption 1 and lemma 2, exist
$\alpha>0,\beta>0$ and $E_{1}\in R^{n\times n}>0$, $E_{2}\in R^{n\times n}>0$,
such that

$\begin{array}[]{l}2e_{i}^{T}(t)QA_{p}F\left(e_{i}(t)\right)\leq\alpha
e_{i}^{T}(t)QA_{p}E_{1}^{-1}A_{p}^{T}Qe_{i}(t)+\alpha^{-1}F^{T}\left(e_{i}(t)\right)E_{1}F\left(e_{i}(t)\right)\\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\leq\alpha
e_{i}^{T}(t)QA_{p}E_{1}^{-1}A_{p}^{T}Qe_{i}(t)+\alpha^{-1}\xi_{p}^{2}e_{i}^{T}(t)E_{1}e_{i}(t),\end{array}$
(14)

together with

$\begin{array}[]{l}2e_{i}^{T}(t)QB_{p}F\left(e_{i}(t-\tau(t))\right)\leq\beta
e_{i}^{T}(t)QB_{p}E_{2}^{-1}B_{p}^{T}Qe_{i}(t)\\\
+\beta^{-1}F^{T}\left(e_{i}(t-\tau(t))\right)\times
E_{2}F\left(e_{i}(t-\tau(t))\right)\leq\beta
e_{i}^{T}(t)QB_{p}E_{2}^{-1}B_{p}^{T}Qe_{i}(t)\\\
+\beta^{-1}\xi_{p}^{2}e_{i}^{T}(t-\tau(t))E_{2}e_{i}(t-\tau(t)).\end{array}$
(15)

Let $e(t)=(e_{1}^{T}(t),e_{2}^{T}(t),\cdots,e_{N}^{T}(t))^{T}$, then we have

$2\sum\limits_{p=1}^{M}\sum\limits_{i=v_{p-1}+1}^{v_{p}}e_{i}^{T}(t)Q\sum\limits_{j=1}^{N}g_{ij}e_{j}(t)=2e^{T}(t)(G\otimes
Q)e(t).$ (16)

Substituting (14)-(16) into (13) and we can get

$\begin{array}[]{l}\frac{dV(t)}{dt}=\sum\limits_{i=1}^{N}e_{i}^{T}(t)\Phi
e_{i}(t)+\sum\limits_{p=1}^{M}\sum\limits_{i=v_{p-1}+1}^{v_{p}}e_{i}^{T}(t)\left(\alpha^{-1}\xi_{p}^{2}E_{1}\right)e_{i}(t)\\\
+2e^{T}(t)(G\otimes
Q)e(t)+\beta^{-1}\sum\limits_{p=1}^{M}\sum\limits_{i=v_{p-1}+1}^{v_{p}}e_{i}^{T}(t-\tau(t))\xi_{p}^{2}Qe_{i}(t-\tau(t)),\\\
\end{array}$ (17)

where $\Phi=-QC+C^{T}Q+\alpha QA_{p}E_{1}^{-1}A_{p}^{T}Q+\beta
QB_{p}E_{2}^{-1}B_{p}^{T}Q$. According to (8) and the theory of schur
complement, we can easily reason out

$\begin{array}[]{l}\frac{dV(t)}{dt}\leq\mu\sum\limits_{i=1}^{N}e_{i}^{T}(t)Qe_{i}(t)+\upsilon[\sum\limits_{i=1}^{N}e_{i}^{T}(t-\tau(t))Qe_{i}(t-\tau(t))]\\\
\;\;\;\;\;\;\;\;\leq\mu V(t)+\upsilon[\sup\limits_{-\tau\leq\theta\leq
0}V(t+\theta)].\\\ \end{array}$ (18)

Due to $q\geq\gamma e^{\lambda\tau}$ is a constant, from Razumikhin condition,
for all $t\in[t_{k-1},t_{k}),k\in N$, $qV(t,\varphi(0))\geq V(t+s,\varphi(s))$
for $s\in[-\tau,0]$, we have $e_{i}^{T}(t-\tau(t))Qe_{i}(t-\tau(t))\leq
qe_{i}^{T}(t)Qe_{i}(t),t\in[t_{k-1},t_{k}),k\in N$. So the formula (18) can be
further deduced as

$\begin{array}[]{l}\frac{dV(t)}{dt}\leq(\mu+\upsilon q)V(t).\end{array}$ (19)

Therefore, the parameter $c$ in condition (ii) in Lemma 1 can be chosen as
$\mu+\upsilon q$, so that condition (ii) can be realized.

Next, let us consider when $t=t_{k}$, i.e.
$e_{i}\left(t_{k}^{+}\right)=\left(1+d_{k}\right)e_{i}\left(t_{k}\right),i\in\chi_{p}\left(t_{k}\right);e_{i}\left(t_{k}^{+}\right)=e_{i}\left(t_{k}\right),i\in\mathscr{C}_{p}\backslash\chi_{p}\left(t_{k}\right)$.
We can find that

$\begin{array}[]{l}V\left(t_{k}^{+}\right)=\sum\limits_{i=1}^{N}e_{i}^{T}\left(t_{k}^{+}\right)Qe_{i}\left(t_{k}^{+}\right)\\\
\;\;\;\;\;\;\;\;\;\;\;=\sum\limits_{p=1}^{M}\sum\limits_{i\in\chi_{p}\left(t_{k}\right)}e_{i}^{T}\left(t_{k}^{+}\right)Qe_{i}\left(t_{k}^{+}\right)+\sum\limits_{p=1}^{M}\sum\limits_{i\in\mathscr{C}_{p}\backslash\chi_{p}\left(t_{k}\right)}e_{i}^{T}\left(t_{k}^{+}\right)Qe_{i}\left(t_{k}^{+}\right)\\\
\;\;\;\;\;\;\;\;\;\;\;=\left(1+d_{k}\right)^{2}\sum\limits_{p=1}^{M}\sum\limits_{i\in\chi_{p}\left(t_{k}\right)}e_{i}^{T}\left(t_{k}\right)Qe_{i}\left(t_{k}\right)-d_{k}\left(2+d_{k}\right)\\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\times\sum\limits_{p=1}^{M}\sum\limits_{i\in\mathscr{C}_{p}\backslash\chi_{p}\left(t_{k}\right)}e_{i}^{T}\left(t_{k}\right)Qe_{i}\left(t_{k}\right).\end{array}$
(20)

If $-2<d_{k}<0$, in the light of pinned nodes in $\chi_{p}\left(t_{k}\right)$,
we have

$\begin{array}[]{l}\frac{1}{v_{p}-v_{p-1}-\rho_{p}}\sum\limits_{i\in\mathscr{C}_{p}\backslash\chi_{p}\left(t_{k}\right)}e_{i}^{T}\left(t_{k}\right)Qe_{i}\left(t_{k}\right)\\\
\;\;\;\;\;\;\;\;\;\;\;\;\leq\frac{\lambda_{\max}(Q)}{v_{p}-v_{p-1}-\rho_{p}}\sum\limits_{i\in\mathscr{C}_{p}\backslash\chi_{p}\left(t_{k}\right)}e_{i}^{T}\left(t_{k}\right)e_{i}\left(t_{k}\right)\\\
\;\;\;\;\;\;\;\;\;\;\;\;\leq\frac{\lambda_{\max}(Q)}{v_{p}-v_{p-1}}\sum\limits_{i=v_{p-1}+1}^{v_{p}}e_{i}^{T}\left(t_{k}\right)e_{i}\left(t_{k}\right)\\\
\;\;\;\;\;\;\;\;\;\;\;\;\leq\frac{\lambda_{\max}(Q)}{\lambda_{\min}(Q)\left(v_{p}-v_{p-1}\right)}\sum\limits_{i=v_{p-1}+1}^{v_{p}}e_{i}^{T}\left(t_{k}\right)Qe_{i}\left(t_{k}\right).\end{array}$
(21)

So we can get

$\begin{array}[]{l}V\left(t_{k}^{+}\right)\leq\left(1+d_{k}\right)^{2}\sum\limits_{p=1}^{M}\sum\limits_{i=v_{p-1}+1}^{v_{p}}e_{i}^{T}\left(t_{k}\right)Qe_{i}\left(t_{k}\right)-d_{k}\left(2+d_{k}\right)\\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\times\sum\limits_{p=1}^{M}\frac{\lambda_{\max}(Q)\left(v_{p}-v_{p-1}-\rho_{p}\right)}{\lambda_{\min}(Q)\left(v_{p}-v_{p-1}\right)}\sum\limits_{i=v_{p-1}+1}^{v_{p}}e_{i}^{T}\left(t_{k}\right)Qe_{i}\left(t_{k}\right)\\\
\;\;\;\;\;\;\;\;\;\;\;\;=\left[\left(1+d_{k}\right)^{2}-d_{k}\left(2+d_{k}\right)\frac{\lambda_{\max}(Q)\left(v_{p}-v_{p-1}-\rho_{p}\right)}{\lambda_{\min}(Q)\left(v_{p}-v_{p-1}\right)}\right]\sum\limits_{i=1}^{N}e_{i}^{T}\left(t_{k}\right)Qe_{i}\left(t_{k}\right)\\\
\;\;\;\;\;\;\;\;\;\;\;\;\doteq\eta_{k}V\left(t_{k}\right).\end{array}$ (22)

The same can be found if $d_{k}\leq-2$ or $d_{k}\geq 0$, so we omit here.
Utilizing Lemma 1 and (9)-(11), the trivial solution of system (7) can realize
globally exponentially stable with convergence rate $\frac{\lambda}{2}$.
Hence, the neural network (1) achieves exponential cluster synchronization
asymptotically with exponent $\frac{\lambda}{2}$ under the pinning impulsive
controllers (6).

###  3.2 Finite-time Cluster Synchronization

Motivated by pinning control scheme, in this part, we obtain a sufficient
condition for finite-time cluster synchronization of a class of coupled neural
networks (1) with time-varying delays by means of hybrid control.

In order to achieve the goal of finite-time cluster synchronization, we
redefine the notation $\chi_{p}$ as the nodes in the $p$th cluster which is
directly connected with the nodes in other clusters. So the notation
$\mathscr{C}_{p}\backslash\chi_{p}$ means the remaining nodes in the $p$th
cluster which isn’t directly connected to nodes in other clusters.

Contrive the following hybrid controllers

$u_{i}(t)=\left\\{\begin{array}[]{c}-g_{1}e_{i}(t)-\phi_{i}(t),i\in\chi_{p},\\\
-\varphi_{i}(t),i\in\mathscr{C}_{p}\backslash\chi_{p},\end{array}\right.$ (23)

where $\phi_{i}(t)=k{\rm
sign}\left(e_{i}(t)\right)\left|e_{i}(t)\right|^{\mu}+k\left(k_{1}\int_{t-\tau(t)}^{t}e_{i}^{T}(s)e_{i}(s)ds\right)^{\frac{1+\mu}{2}}\times\\\
\psi\left(e_{i}(t),\|e(t)\|\right)$, $\varphi_{i}(t)=k{\rm
sign}\left(e_{i}(t)\right)\left|e_{i}(t)\right|^{\mu}+k\left(k_{1}\int_{t-\tau(t)}^{t}e_{i}^{T}(s)e_{i}(s)ds\right)^{\frac{1+\mu}{2}}\times\\\
\psi\left(e_{i}(t),\|e(t)\|\right)+2k\left(g_{1}\int_{t}^{t_{1}}e_{i}^{T}(s)e_{i}(s)ds\right)^{\frac{1+\mu}{2}}\psi\left(e_{i}(t),\|e(t)\|\right)$,
$i=1,2,\cdots,N$, $k$ is tunable, $g_{1}$ is a positive constant which has the
practical meaning of control strength, and $k_{1}>0$.
$\psi\left(e_{i}(t),\|e(t)\|\right)$ is denoted as follows: if $\|e(t)\|\neq
0,\psi\left(e_{i}(t),\|e(t)\|\right)=\frac{e_{i}(t)}{\|e(t)\|^{2}}$, or else
$\psi\left(e_{i}(t),\|e(t)\|\right)=0$.

The form of the error dynamic system under the hybrid controllers (23) is as
follows

$\left\\{\begin{array}[]{c}\dot{e}_{i}(t)=-C_{p}e_{i}(t)+A_{p}F\left(e_{i}(t)\right)+B_{p}F\left(e_{i}(t-\tau(t))\right)+\sum\limits_{j=1}^{N}g_{ij}e_{j}(t)\\\
-g_{1}e_{i}(t)-\phi_{i}(t),i\in\chi_{p},\\\
\dot{e}_{i}(t)=-C_{p}e_{i}(t)+A_{p}F\left(e_{i}(t)\right)+B_{p}F\left(e_{i}(t-\tau(t))\right)+\sum\limits_{j=1}^{N}g_{ij}e_{j}(t)\\\
-\varphi_{i}(t),i\in\mathscr{C}_{p}\backslash\chi_{p}.\end{array}\right.$ (24)

Our next goal is to drive the coupled neural network (1) to achieve
synchronization among the node states $x_{i}(t)$ and the objective state
$s_{p}(t)$ over a finite-time interval by designing the hybrid controllers
(23) and applying the finite-time stability theory.

Theorem 2. Suppose that Assumptions 1-2 hold, if the following conditions are
satisfied:
(1)$\beta^{-1}\xi_{p}^{2}\lambda_{\max}\left(E_{2}\right)+\sigma<1$,
(2)$-\lambda_{\min}\left(C_{p}\right)+\frac{1}{2}\alpha\lambda_{\max}\left(A_{p}E_{1}^{-1}A_{p}^{T}\right)+\frac{1}{2}\alpha^{-1}\xi_{p}^{2}\lambda_{\max}\left(E_{1}\right)+\frac{1}{2}\beta\lambda_{\max}\left(B_{p}E_{2}^{-1}B_{p}^{T}\right)\\\
+\lambda_{\max}(G\otimes I)+\frac{1}{2}k_{1}-g_{1}<0$,
then by the hybrid controllers (23), the system (1) can achieve cluster
synchronization in finite time and the settling time is
$t_{1}=t_{0}+\frac{1}{k(1-\mu)}(2V(t_{0}))^{\frac{1-\mu}{2}}$.

proof. Set up a Lyapunov functional as

$\displaystyle V(t)=\frac{1}{2}$
$\displaystyle\sum_{i=1}^{N}e_{i}^{T}(t)e_{i}(t)+\frac{1}{2}k_{1}\sum_{i=1}^{N}\int_{t-\tau(t)}^{t}e_{i}^{T}(s)e_{i}(s)ds$
(25)
$\displaystyle+g_{1}\sum_{p=1}^{M}\sum_{\mathscr{C}_{p}\backslash\chi_{p}}\int_{t}^{t_{1}}e_{i}^{T}(s)e_{i}(s)ds.$

Calculating the derivative of $V(t)$ along the solutions of system (24)
outputs the following:

$\displaystyle\dot{V}(t)=$
$\displaystyle\sum_{i=1}^{N}e_{i}^{T}(t)\left[-C_{p}e_{i}(t)+A_{p}F\left(e_{i}(t)\right)+B_{p}F\left(e_{i}(t-\tau(t))\right)\right.$
(26)
$\displaystyle+\sum_{j=1}^{N}g_{ij}e_{j}(t)+u_{i}(t)]+\frac{1}{2}k_{1}\sum_{i=1}^{N}e_{i}^{T}(t)e_{i}(t)$
$\displaystyle-\frac{1-i(t)}{2}k_{1}\sum_{i=1}^{N}e_{i}^{T}(t-\tau(t))e_{i}(t-\tau(t))$
$\displaystyle-
g_{1}\sum_{p=1}^{M}\sum_{i\in\mathscr{C}_{p}\backslash\chi_{p}}e_{i}^{T}(t)e_{i}(t).$

Inserting the hybrid controllers (23) into $\dot{V}(t)$, we have

$\displaystyle\dot{V}(t)=$
$\displaystyle\sum_{i=1}^{N}e_{i}^{T}(t)\left[-C_{p}e_{i}(t)+A_{p}F\left(e_{i}(t)\right)+B_{p}F\left(e_{i}(t-\tau(t))\right)\right.$
$\displaystyle+\sum_{j=1}^{N}g_{ij}e_{j}(t)]+\frac{1}{2}k_{1}\sum_{i=1}^{N}e_{i}^{T}(t)e_{i}(t)-\frac{1-\dot{\tau}(t)}{2}\times$
$\displaystyle
k_{1}\sum_{i=1}^{N}e_{i}^{T}(t-\tau(t))e_{i}(t-\tau(t))-g_{1}\sum_{p=1}^{M}\sum_{i\in\mathscr{C}_{p}\backslash\chi_{p}}e_{i}^{T}(t)e_{i}(t)$
$\displaystyle-\sum_{p=1}^{M}\sum_{i\in\chi_{p}}g_{1}e_{i}^{T}(t)e_{i}(t)-k\sum_{i=1}^{N}e_{i}^{T}(t)\left[\operatorname{sign}\left(e_{i}(t)\right)\left|e_{i}(t)\right|^{\mu}\right.$
$\displaystyle+(k_{1}\int_{t-\tau(t)}^{t}e_{i}^{T}(s)e_{i}(s)ds)^{\frac{1+\mu}{2}}\psi\left(e_{i}(t),\|e(t)\|\right)]$
$\displaystyle-2k\sum_{p=1}^{M}\sum_{i\in\mathscr{C}_{p}\backslash\chi_{p}}e_{i}^{T}(t)[(g_{1}\int_{t}^{t_{1}}e_{i}^{T}(s)e_{i}(s)ds)^{\frac{1+\mu}{2}}\times$
$\displaystyle\left.\psi\left(e_{i}(t),\|e(t)\|\right)\right],$

that is

$\displaystyle\dot{V}(t)=$
$\displaystyle-\sum_{i=1}^{N}e_{i}^{T}(t)C_{p}e_{i}(t)+\sum_{i=1}^{N}e_{i}^{T}(t)A_{p}F\left(e_{i}(t)\right)$
$\displaystyle+\sum_{i=1}^{N}e_{i}^{T}(t)B_{p}F\left(e_{i}(t-\tau(t))\right)+\sum_{i=1}^{N}e_{i}^{T}(t)\sum_{j=1}^{N}g_{ij}e_{j}(t)$
$\displaystyle+\frac{1}{2}k_{1}\sum_{i=1}^{N}e_{i}^{T}(t)e_{i}(t)-\frac{1-\dot{\tau}(t)}{2}\times$
$\displaystyle
k_{1}\sum_{i=1}^{N}e_{i}^{T}(t-\tau(t))e_{i}(t-\tau(t))-g_{1}\sum_{p=1}^{M}\sum_{i\in\mathscr{C}_{p}\backslash\chi_{p}}e_{i}^{T}(t)e_{i}(t)$
$\displaystyle-\sum_{p=1}^{M}\sum_{i\in\chi_{p}}g_{1}e_{i}^{T}(t)e_{i}(t)-k\sum_{i=1}^{N}e_{i}^{T}(t)\left[\operatorname{sign}\left(e_{i}(t)\right)\left|e_{i}(t)\right|^{\mu}\right.$
$\displaystyle+(k_{1}\int_{t-\tau(t)}^{t}e_{i}^{T}(s)e_{i}(s)ds)^{\frac{1+\mu}{2}}\psi\left(e_{i}(t),\|e(t)\|\right)]$
$\displaystyle-2k\sum_{p=1}^{M}\sum_{i\in\mathscr{C}_{p}\backslash\chi_{p}}e_{i}^{T}(t)[(g_{1}\int_{t}^{t_{1}}e_{i}^{T}(s)e_{i}(s)ds)^{\frac{1+\mu}{2}}\times$
$\displaystyle\psi\left(e_{i}(t),\|e(t)\|\right)].$

Utilizing the same technique as in the proof of Theorem 1 and
$\dot{\tau}(t)\leq\sigma$,

$\displaystyle\dot{V}(t)\leq$
$\displaystyle-\lambda_{\min}\left(C_{p}\right)e^{T}(t)e(t)+\frac{1}{2}\sum_{i=1}^{N}[\alpha
e_{i}^{T}(t)A_{p}E_{1}^{-1}A_{p}^{T}e_{i}(t)$
$\displaystyle+\alpha^{-1}\xi_{p}^{2}e_{i}^{T}(t)E_{1}e_{i}(t)]+\frac{1}{2}\sum_{i=1}^{N}[\beta
e_{i}^{T}(t)B_{p}E_{2}^{-1}B_{p}^{T}e_{i}(t)$
$\displaystyle+\beta^{-1}\xi_{p}^{2}e_{i}^{T}(t-\tau(t))E_{2}e_{i}(t-\tau(t))]+e^{T}(t)(G\otimes
I)e(t)$
$\displaystyle+\frac{1}{2}k_{1}\sum_{i=1}^{N}e_{i}^{T}(t)e_{i}(t)-\frac{1-\sigma}{2}k_{1}\sum_{i=1}^{N}e_{i}^{T}(t-\tau(t))e_{i}(t-\tau(t))$
$\displaystyle-
g_{1}e^{T}(t)e(t)-k\sum_{i=1}^{N}e_{i}^{T}(t)[\operatorname{sign}(e_{i}(t))\left|e_{i}(t)\right|^{\mu}$
$\displaystyle+(k_{1}\int_{t-\tau(t)}^{t}e_{i}^{T}(s)e_{i}(s)ds)^{\frac{1+\mu}{2}}\psi(e_{i}(t),\|e(t)\|)]$
$\displaystyle-2k\sum_{p=1}^{M}\sum_{i\in\mathscr{C}_{p}\backslash\chi_{p}}e_{i}^{T}(t)[(g_{1}\int_{t}^{t_{1}}e_{i}^{T}(s)e_{i}(s)ds)^{\frac{1+\mu}{2}}\times$
$\displaystyle\psi(e_{i}(t),\|e(t)\|)].$

Expanding this out yields

$\displaystyle\dot{V}(t)\leq$
$\displaystyle-\lambda_{\min}\left(C_{p}\right)e^{T}(t)e(t)+\frac{1}{2}\alpha\lambda_{\max}\left(A_{p}E_{1}^{-1}A_{p}^{T}\right)e^{T}(t)e(t)$
$\displaystyle+\frac{1}{2}\alpha^{-1}\xi_{p}^{2}\lambda_{\max}\left(E_{1}\right)e^{T}(t)e(t)+\frac{1}{2}\beta\lambda_{\max}\left(B_{p}E_{2}^{-1}B_{p}^{T}\right)e^{T}(t)e(t)$
$\displaystyle+\frac{1}{2}\beta^{-1}\xi_{p}^{2}\lambda_{\max}\left(E_{2}\right)e^{T}(t-\tau(t))e(t-\tau(t))$
$\displaystyle+\lambda_{\max}(G\otimes
I)e^{T}(t)e(t)+\frac{1}{2}k_{1}e^{T}(t)e(t)$
$\displaystyle-\frac{1-\sigma}{2}k_{1}e^{T}(t-\tau(t))e(t-\tau(t))-g_{1}e^{T}(t)e(t)$
$\displaystyle-k\sum_{i=1}^{N}e_{i}^{T}(t)[\operatorname{sign}(e_{i}(t))|e_{i}(t)|^{\mu}+(k_{1}\int_{t-\tau(t)}^{t}e_{i}^{T}(s)e_{i}(s)ds)^{\frac{1+\mu}{2}}$
$\displaystyle\left.\times\psi\left(e_{i}(t),\|e(t)\|\right)\right]-2k\sum_{p=1}^{M}\sum_{i\in\mathscr{C}_{p}\backslash\chi_{p}}e_{i}^{T}(t)[(g_{1}\int_{t}^{t_{1}}e_{i}^{T}(s)e_{i}(s)ds)^{\frac{1+\mu}{2}}$
$\displaystyle\times\psi(e_{i}(t),\|e(t)\|)].$

By sorting through it, we have

$\displaystyle\dot{V}(t)\leq$
$\displaystyle[-\lambda_{\min}\left(C_{p}\right)+\frac{1}{2}\alpha\lambda_{\max}\left(A_{p}E_{1}^{-1}A_{p}^{T}\right)+\frac{1}{2}\alpha^{-1}\xi_{p}^{2}\lambda_{\max}\left(E_{1}\right)$
$\displaystyle+\frac{1}{2}\beta\lambda_{\max}\left(B_{p}E_{2}^{-1}B_{p}^{T}\right)+\lambda_{\max}(G\otimes
I)+\frac{1}{2}k_{1}-g_{1}]e^{T}(t)e(t)$
$\displaystyle+(\frac{1}{2}\beta^{-1}\xi_{p}^{2}\lambda_{\max}\left(E_{2}\right)-\frac{1-\sigma}{2})e^{T}(t-\tau(t))e(t-\tau(t))$
$\displaystyle-k[\sum_{i=1}^{N}e_{i}^{T}(t)\left(\operatorname{sign}(e_{i}(t)\right)\left|e_{i}(t)\right|^{\mu}+k_{1}\int_{t-\tau(t)}^{t}e_{i}^{T}(s)e_{i}(s)ds)^{\frac{1+\mu}{2}}$
$\displaystyle\times\psi(e_{i}(t),\|e(t)\|)]-2k\sum_{p=1}^{M}\sum_{i\in\mathscr{C}_{p}}e_{i}^{T}(t)[(g_{1}\int_{t}^{t_{1}}e_{i}^{T}(s)e_{i}(s)ds)^{\frac{1+\mu}{2}}$
$\displaystyle\times\psi(e_{i}(t),\|e(t)\|)].$

From the conditions (1) and (2) of Theorem 2, we get that

$\displaystyle\dot{V}(t)\leq$
$\displaystyle-k\sum_{i=1}^{N}e_{i}^{T}(t)[\operatorname{sign}\left(e_{i}(t)\right)\left|e_{i}(t)\right|^{\mu}+(k_{1}\int_{t-\tau(t)}^{t}e_{i}^{T}(s)e_{i}(s)ds)^{\frac{1+\mu}{2}}$
$\displaystyle\left.\psi\left(e_{i}(t),\|e(t)\|\right)\right]-2k\sum_{p=1}^{M}\sum_{i\in\mathscr{C}_{p}\backslash\chi_{p}}e_{i}^{T}(t)[(g_{1}\int_{t}^{t_{1}}e_{i}^{T}(s)e_{i}(s)ds)^{\frac{1+\mu}{2}}$
$\displaystyle\left.\times\psi\left(e_{i}(t),\|e(t)\|\right)\right]$
$\displaystyle\leq-$ $\displaystyle
k\sum_{i=1}^{N}\left|e_{i}^{T}(t)e_{i}(t)\right|^{1+\mu}-k\sum_{i=1}^{N}(k_{1}\int_{t-\tau(t)}^{t}e_{i}^{T}(s)e_{i}(s)ds)^{\frac{1+\mu}{2}}$
$\displaystyle-2k\sum_{p=1}^{M}\sum_{\mathscr{C}_{p}\backslash\chi_{p}}(g_{1}\int_{t}^{t_{1}}e_{i}^{T}(s)e_{i}(s)ds)^{\frac{1+\mu}{2}}.$

Through Lemma 3, one gain

$\displaystyle\dot{V}(t)\leq-$ $\displaystyle
2^{\frac{1+\mu}{2}}k(\frac{1}{2}\sum_{i=1}^{N}e_{i}^{2}(t)+\frac{1}{2}\sum_{i=1}^{N}k_{1}\int_{t-\tau(t)}^{t}e_{i}^{T}(s)e_{i}(s)ds$
$\displaystyle+\sum_{p=1}^{M}\sum_{i\in\mathscr{C}_{p}\backslash\chi_{p}}g_{1}\int_{t}^{t_{1}}e_{i}^{T}(s)e_{i}(s)ds)^{\frac{1+\mu}{2}}$
$\displaystyle=$ $\displaystyle-2^{\frac{1+\mu}{2}}kV^{\frac{1+\mu}{2}}(t).$

Utilizing Lemma 4, the error system (24) can achieve finite-time
synchronization no matter what initial condition is, the settling time is

$t_{1}=t_{0}+\frac{1}{k(1-\mu)}(2V(t_{0}))^{\frac{1-\mu}{2}}.$

Therefore, the neural network (1) can achieve finite-time cluster
synchronization under the hybrid controllers (23).

##  4 Illustrative example

In this section, an example with three cases is presented to demonstrate the
effectiveness of our results.

Example: Consider two-neuron time-varying delayed coupled neural networks (2),
where $f(r)=(f_{1}(r),f_{2}(r))^{T}=(\rm arctan(r),\rm arctan(r))^{T}$,
$\tau(t)=\frac{1.7e^{t}}{1+e^{t}}$, $M=2$, and
$C_{1}=C_{2}=\left(\begin{array}[]{ll}1&0\\\ 0&1\end{array}\right)$,
$A_{1}=\left(\begin{array}[]{cc}1.95&-0.1\\\ -5&3\end{array}\right)$,
$I_{1}=I_{2}=\left(\begin{array}[]{l}0\\\ 0\end{array}\right)$,
$A_{2}=\left(\begin{array}[]{cc}2&-0.11\\\ -5.1&3\end{array}\right)$,
$B_{1}=\left(\begin{array}[]{cc}-1.5&-0.1\\\ -0.3&-2.41\end{array}\right)$,
$B_{2}=\left(\begin{array}[]{cc}-1.5&-0.1\\\ -0.2&-2.45\end{array}\right)$.
The initial conditions are $S_{p}(t)=(0.4,0.6)^{T}$, $t\in[-1,0]$ with
$p=1,2$.

Given a seven coupled time-varying delayed neural networks (1), the coupling
matrix of which is

$G=\left(\begin{array}[]{ccccccc}-4&2&2&0.5&-0.5\\\ 2&-3&1&0.2&-0.2\\\
1&1&-2&0.3&-0.3\\\ 0.5&0.5&-1&-1&1\\\ 0.4&-0.4&0&1&-1\\\ \end{array}\right).$

Then the system (1) can be divided into two clusters
$\mathscr{C}_{1}=\\{1,2\\},\mathscr{C}_{2}=\\{3,4,5\\}$.

Case 1: We set the error $\left\|e_{1}\right\|$, $\left\|e_{2}\right\|$ of
synchronization with initial value $(10,-5),i=1,2$, $(8,-6),i=3,4,5$,
respectively. The error dynamic system with $u_{i}(t)=0$ can’t realize
synchronization, which is depicted in Fig.1 and Fig.2.

Case 2: The error dynamic system under the pinning impulsive controllers (6)
with $p_{1}=1,p_{2}=3,t_{k}=0.03k,d_{k}=-0.8$ can realize asymptotic cluster
synchronization, which is depicted in Fig.3 and Fig.4.

Case 3: The error dynamic system under the hybrid controllers (23)
with$k=2,k_{1}=1.4,g_{1}=41.4$ can realize finite-time cluster
synchronization, which is depicted in Fig.5 and Fig.6.

Figure 1: The error dynamic trajectory when $i=1,2$ without controller.

Figure 2: The error dynamic trajectory when $i=3,4,5$ without controller.

Figure 3: The error dynamic trajectory when $i=1,2$ with the pinning impulsive
controllers (6).

Figure 4: The error dynamic trajectory when $i=3,4,5$ with the pinning
impulsive controllers (6).

Figure 5: The error dynamic trajectory $i=1,2$ with the hybrid controllers
(23).

Figure 6: The error dynamic trajectory when $i=3,4,5$ with the hybrid
controllers (23).

##  5 Conclusion

In this article, the problem concerning asymptotic and finite-time cluster
synchronization of coupled neural networks with time-varying delay via two
different control schemes is mainly concerned. First, we select some typical
nodes in the coupling neural networks, and apply the control of pinning
impulse [40] which can achieve the goal of saving control costs and is helpful
to obtain sufficient conditions for asymptotic cluster synchronization of the
systems. Second, motivated by the goal of pinning control method which is
operated by precedence controlling some key nodes, we introduce the hybrid
controllers to deal with finite-time synchronization of the error system.
Third, we show a numerical example by exhibiting three cases to show the
correctness and effectiveness.

Using the properties of fractional-order calculus and the research technique
of fractional-order energy function, the influence of two control schemes on
the asymptotic and finite-time cluster synchronization of the fractional-order
coupled neural networks with time-varying delays is studied. The result will
emerge in the near future. What’s more, complex-valued or quaternion-valued
coupled neural networks with time-varying delays under the two control schemes
in this paper have attracted the interest of many researchers. Hence, the
future works also include asymptotic and finite-time cluster synchronization
of complex-valued and quaternion-valued coupled neural networks with time-
varying delays by pinning impulsive control and hybrid control.

## References

  * [1] J. Zha and C. Li.  Synchronization of complex network based on the theory of gravitational field.  Acta Physica Polonica B, 50(1):87–114, 2019.
  * [2] J. Zhou, J. Chen, J. Lu, and J. Lu.  On applicability of auxiliary system approach to detect generalized synchronization in complex network.  IEEE Transactions on Automatic Control, 62(7):3468–3473, 2017.
  * [3] S. Zhu, J. Zhou, X. Yu, and J. Lu.  Bounded synchronization of heterogeneous complex dynamical networks: A unified approach.  IEEE Transactions on Automatic Control, 66(4):1756–1762, 2020.
  * [4] Z. Xu, P. Shi, H. Su, Z. Wu, and T. Huang.  Global h∞ pinning synchronization of complex networks with sampled-data communications.  IEEE Transactions on Neural Networks and Learning Systems, 29(5):1467–1476, 2018.
  * [5] Y. Wu and L. Guo.  Enhancement of intercellular electrical synchronization by conductive materials in cardiac tissue engineering.  IEEE Transactions on Biomedical Engineering, 65(2):264–272, 2018\.
  * [6] F. Ren, F. Cao, and J. Cao.  Mittag-leffler stability and generalized mittag-leffler stability of fractional-order gene regulatory networks.  Neurocomputing, 160:185–190, 2015.
  * [7] S. Kumar, R.and Sarkar, S. Das, and J. Cao.  Projective synchronization of delayed neural networks with mismatched parameters and impulsive effects.  IEEE Transactions on Neural Networks and Learning Systems, 31(4):1211–1221, 2020.
  * [8] A. Abdurahman and H. Jiang.  Improved control schemes for projective synchronization of delayed neural networks with unmatched coefficients.  International Journal of Pattern Recognition and Artificial Intelligence, 34(3), 2020.
  * [9] Adel O., X. Wang, V. Pham, and G. Giuseppe.  Coexistence of identical synchronization, antiphase synchronization and inverse full state hybrid projective synchronization in different dimensional fractional-order chaotic systems.  Advances in Difference Equations, 2018(1):1–16, 2018.
  * [10] Q. Yang, H. Wu, and J. Cao.  Global cluster synchronization in finite time for complex dynamical networks with hybrid couplings via aperiodically intermittent control.  Optimal Control Applications and Methods, 41(4):1097–1117, 2020\.
  * [11] L. Liu, K. Liu, H. Xiang, and Q Liu.  Cluster synchronization for directed complex dynamical networks via pinning control.  Physica A: Statistical Mechanics and its Applications, 545:123580, 2020.
  * [12] A. Hu, J. Cao, M. Hu, and L. Guo.  Cluster synchronization of complex networks via event-triggered strategy under stochastic sampling.  Complexity, 434:99–110, 2015.
  * [13] Y. Wang, Z. Ma, J. Cao, and A. Alsaedi.  Adaptive cluster synchronization in directed networks with nonidentical nonlinear dynamics.  Complexity, 21:380–387, 2016.
  * [14] W. Wu, W. Zhou, and T. Chen.  Cluster synchronization of linearly coupled complex networks under pinning control.  IEEE Transactions on Circuits and Systems, 56(4):829–839, 2009\.
  * [15] Q. Yang, H. Wu, and J. Cao.  Global cluster synchronization in finite time for complex dynamical networks with hybrid couplings via aperiodically intermittent control.  Optimal Control Applications and Methods, 41(4):1097–1117, 2020\.
  * [16] J. Cao, G. Chen, and P. Li.  Generalized analytical solutions and experimental confirmation of complete synchronization in a class of mutually-coupled simple nonlinear electronic circuits.  Nonlinear Sciences, 113:294–307, 2017.
  * [17] A. Koronovskii, O. Moskalenko, V. Ponomarenko, M. Prokhorov, and A. Hramov.  Binary generalized synchronization.  Chaos Splitions and Fractals, 83:133–139, 2016.
  * [18] D. Senthilkumar, M. Lakshmanan, and J. Kurths.  Phase synchronization in unidirectionally coupled ikeda time-delay systems.  Nonlinear Sciences, 164(11):35–44, 2008.
  * [19] K. Li, J. Zhao, H. Zhang, and X. Li.  On successive lag synchronization of a dynamical network with delayed couplings.  IEEE Transactions on Control of Network Systems, page 1, 2021.
  * [20] S. Yang, C. Hu, J. Yu, and H. Jiang.  Finite-time cluster synchronization in complex-variable networks with fractional-order and nonlinear coupling.  Neural networks: the official journal of the International Neural Network Society, 135:212–224, 2020.
  * [21] S. Liu, N. Jiang, A. Zhao, Y. Zhang, and K. Qiu.  Secure optical communication based on cluster chaos synchronization in semiconductor lasers network.  IEEE Access, 8:11872–11879, 2020.
  * [22] L. Pan, J. Cao, U. Al-Juboori, and M. Abdel-Aty.  Cluster synchronization of stochastic neural networks with delay via pinning impulsive control.  Neurocomputing, 366(13):109–117, 2019.
  * [23] P. Liu, Z. Zeng, and J. Wang.  Asymptotic and finite-time cluster synchronization of coupled fractional-order neural networks with time delay.  IEEE Transactions on Neural Networks and Learning Systems, PP(99), 2020.
  * [24] J. Mei, M. Jiang, W. Xu, and B. Wang.  Finite-time synchronization control of complex dynamical networks with time delay.  Communications in Nonlinear Science and Numerical Simulation, 18(9):2462–2478, 2013.
  * [25] P. He, S. Ma, and T. Fan.  Finite-time mixed outer synchronization of complex networks with coupling time-varying delay.  Chaos, 22(4):175–423, 2012.
  * [26] Z. Hou, Jun P., Z. Liu, J. Zou, and J. Luo.  Theory of functional differential equations.  Journal of Changsha Railway University, 2001.
  * [27] X. Jin, Z. Wang, H. Yang, Q. Song, and M. Xiao.  Synchronization of multiplex networks with stochastic perturbations via pinning adaptive control.  Journal of the Franklin Institute, 2021.
  * [28] S. Ding and Z. Wang.  Synchronization of coupled neural networks via an event-dependent intermittent pinning control.  IEEE Transactions on Systems, Man, and Cybernetics: Systems, pages 1–7, 2020.
  * [29] W. Li, J. Zhou, J. Li, T. Xie, and J. Lu.  Cluster synchronization of two-layer networks via aperiodically intermittent pinning control.  IEEE Transactions on Circuits and Systems II: Express Briefs, page 1, 2020.
  * [30] X. Li, D. Peng, and J. Cao.  Lyapunov stability for impulsive systems via event-triggered impulsive control.  IEEE Transactions on Automatic Control, 65(11):4908–4913, 2020\.
  * [31] X. Li, X. Yang, and J. Cao.  Event-triggered impulsive control for nonlinear delay systems.  Automatica, 117, 2020.
  * [32] W. He, F. Qian, and J. Cao.  Pinning-controlled synchronization of delayed neural networks with distributed-delay coupling via impulsive control.  Neural Networks, 85:1–9, 2017.
  * [33] X. Xiong, X. Yang, J. Cao, and R. Tang.  Finite-time control for a class of hybrid systems via quantized intermittent control.  Sci. China Inf. Sci., 63(9):192201, 2020.
  * [34] S. Liu, Wu H., and J. Cao.  Fixed‐time synchronization for discontinuous delayed complex‐valued networks with semi‐markovian switching and hybrid couplings via adaptive control.  International Journal of Adaptive Control and Signal Processing, 34(10):1359–1382, 2020.
  * [35] Z. Wu, Q. Ye, and D. Liu.  Finite-time synchronization of dynamical networks coupled with complex-variable chaotic systems.  2013\.
  * [36] H. Wang, Z. Han, Q. Xie, and W. Zhang.  Finite-time chaos control via nonsingular terminal sliding mode control.  Communications in Nonlinear Science and Numerical Simulation, 14(6):2728–2733, 2009.
  * [37] L. Liu and Q. Liu.  Cluster synchronization in complex dynamical network of nonidentical nodes with delayed and non-delayed coupling via pinning control.  Physica Scripta, 94(4), 2019.
  * [38] X. Liu and T. Chen.  Finite-time and fixed-time cluster synchronization with or without pinning control.  IEEE Transactions on Cybernetics, 48(99):240–252, 2018.
  * [39] Q. Wu, Z. Jin, and X. Lan.  Global exponential stability of impulsive differential equations with any time delays.  Applied Mathematics Letters, 23(2):143–147, 2010.
  * [40] C. Aouiti, M. Bessifi, and X. Li.  Finite-time and fixed-time synchronization of complex-valued recurrent neural networks with discontinuous activations and time-varying delays.  Circuits Systems and Signal Processing, pages 1–23, 2020.