# Long-term Averages of the Stochastic Logistic Map

Maricela Cruza and Austin Weib and Johanna Hardinc and Ami Radunskayac CONTACT
Ami Radunskaya. Email<EMAIL_ADDRESS>aKaiser Permanente Washington
Health Research Institute, Seattle WA; bThe Ohio State University, Columbus
OH; cDept. of Mathematics and Statistics, Pomona College, Claremont CA

###### Abstract

The logistic map is a nonlinear difference equation well studied in the
literature, used to model self-limiting growth in certain populations. It is
known that, under certain regularity conditions, the stochastic logistic map,
where the parameter is varied according to a specified distribution, has a
unique invariant distribution. In these cases we can compare the long-term
behavior of the stochastic system with that of the deterministic system
evaluated at the average parameter value. Here we examine the relationship
between the mean of the stochastic logistic equation and the mean of orbits of
the deterministic logistic equation at the expected value of the parameter. We
formally prove that, in some cases, the addition of noise is beneficial to the
populations, in the sense that it increases the mean, while for other ranges
of parameters it is detrimental. A conjecture based on numerical evidence is
presented at the end.

###### keywords:

difference equation; invariant distribution; random dynamical system; long-
term average

##  1 Introduction

The logistic map is a nonlinear first-order difference equation first used to
describe the evolution of biological populations. It was formulated by Pierre
Francois Verhulst in $1838$ [4] but became a phenomenon in the $1940$’s when
John Von Neumann and Stanislav Ulam suggested its use as a random number
generator [24]. Since the seminal work of Robert May in 1976 [15], the
logistic model has become a canonical equation describing self-limiting
growth, and in the past fifty years it has been used extensively in both its
discrete and continuous versions. Our understanding of the behavior of the
logistic map can be extended to more general models, and, despite its
simplicity, the logistic model gives rise to much of the qualitative behavior
observed in population data, capturing both the stable and chaotic aspects of
growth. For example, many protozoan populations such as yeast and grain
beetles illustrate logistic growth in laboratory studies, exhibiting both
stable and chaotic behavior [19]. In the 1980’s, economists used logistic
models and their relatives to show that, surprisingly at the time, chaotic
behavior could occur in models of capital accumulation [6, 7].

If the environment or growth rate is noisy, it is useful to use a logistic
model that includes either additive or parametric noise. For example, when
making decisions in noisy environments, we might choose to use a stochastic
model. Nishimura and Stachurski give a survey of discrete time decision
models, both deterministic and stochastic, in which the goal is to optimize
savings and consumption [18]. They note the importance of understanding the
existence and uniqueness of invariant measures in the stochastic case, a topic
which we address in this paper. Many other forecasting problems can be
formulated in terms of discrete stochastic maps [20]; for example, in [21],
Satoh uses discrete stochastic logistic and Gompertz models to predict when
software is reliable enough to be released. In related work, Erguler and
Stumpf used the stochastic logistic map to contrast the “true” dynamics of a
system and observational noise, thereby addressing the reliability of
inference from present data [11]. The stochastic logistic model fits
experimental data as well. For example, in [8] predictions of the abundance of
microbial communities using stochastic generalized Lotka-Volterra models
provide good fits to experimental models.

In the study of the deterministic logistic map the focus is on the existence
and stability of fixed points and periodic orbits. In the stochastic case,
invariant measures, or distributions, take the place of periodic points. If
these invariant measures are stable, then they can be used to predict long-
term behavior. Diaconis and Freeman [10] develop a “contraction in the log
average” criterion for the existence of a stable distribution of Markov
processes. The idea of a stable distribution, i.e., a stable fixed point of
the Perron-Froebenius operator described in Section 2.4, can be studied in
both deterministic and stochastic maps. One standard example is the tent map;
the two-dimensional tent map has been explored in this context by Nakamura and
Mackey in [17]. Arethreya and Dai have studied the stochastic logistic map [1]
and have found situations in which there are multiple invariant measures, so
that the long term behavior of the system depends on the initial state [2].
For another example of conditions that ensure a unique invariant measure for
stochastic one-dimensional maps, see Nakamura’s study of non-expanding
piecewise linear maps which can exhibit stable periodic behavior even in the
presence of noise [16]. For a compendium of recent results and some new
formulations, the reader is referred to [5, 14].

In making predictions using stochastic models, a natural question is: “How do
these predictions change as the amplitude of the noise increases?” In this
paper we present a novel result that describes changes in the long term
average of the logistic map with multiplicative noise. In other words, we
study the logistic map when the parameter randomly fluctuates in a prescribed
range, and answer the question: “Does the presence of noise increase or
decrease the long-term average?” It turns out that the answer to this question
depends on the periodicity of the stable orbits of the underlying
deterministic system. Our theoretical results can be applied to a range of
predictive stochastic logistic models.

The paper is organized as follows: in Section 2 we present notation and
general results on the stochastic logistic map. The main results of the paper
are found in Section 3, where we prove that the long term average increases in
the period-one regime and decreases in the period-two regime. The proof of the
period-two result is quite technical, and the details are given in Section 5.
A discussion section follows in Section 4, where we present a more general
“flip-flop” conjecture.

##  2 Background

###  2.1 The Map

The logistic map is a one-parameter family which gives the difference
equation:

$x_{n+1}=\lambda x_{n}(1-x_{n}),$

where $x_{n}\in[0,1]$ can be interpreted as a measure of the population in the
$n$th time interval, and $\lambda\geq 0$ represents the intrinsic growth rate.

In this paper, we discuss the logistic map under two different conditions,
when $\lambda$ is fixed and when it is random. In order to write out the map
under the two conditions some notation is needed. Define,

$\vec{\lambda}=(\lambda_{1},\lambda_{2},\ldots\,\,\,);\lambda_{i}\stackrel{{\scriptstyle
iid}}{{\sim}}U(a,b)\,\,\,\text{and}$

$\sigma(\vec{\lambda})$ to be the shift operator such that
$\sigma:(\lambda_{1},\lambda_{2},\ldots)\mapsto(\lambda_{2},\lambda_{3},\ldots),$.
Let $\pi(\vec{\lambda})=\lambda_{1}$ be the projection onto the first
coordinate of $\vec{\lambda}$. Then

$S_{\lambda}(x)=\lambda x(1-x)\,\,$ (2.1)

denotes the logistic map when $\lambda$ is fixed, and

$T(\vec{\lambda},x)=(\sigma(\vec{\lambda}),\pi(\vec{\lambda})x(1-x))\,\,$
(2.2)

denotes the stochastic version of the logistic map as a skew product. Note
that $S_{\lambda_{0}}(x)=T(\vec{\lambda},x)$ with
$\vec{\lambda}=(\lambda_{0},\lambda_{0},\lambda_{0},\dots)$. An example of the
output of the stochastic logistic map is shown in Figure 1, where
$\lambda_{i}\stackrel{{\scriptstyle iid}}{{\sim}}U(2,2.5).$

Figure 1: $T^{i}(\vec{\lambda},x_{0})$ for $i=1,2,3$ where
$\vec{\lambda}=(2.1,2.4,2.3,\dots)$, and $x_{0}=0.12$. The graphs of
$S_{2.1}(x)$, $S_{2.4}(x)$ and $S_{2.3}(x)$ are shown, with the line $y=x$
superimposed.

We denote iterates of $T(\vec{\lambda},x)$ by $T^{i}(\vec{\lambda},x),$ where

$T^{i}(\vec{\lambda},x)=T(T^{i-1}(\vec{\lambda},x))=(\sigma^{i}(\vec{\lambda}),\pi(\sigma^{i-1}(\vec{\lambda}))x(1-x)).$

Define $\Pi(\vec{\lambda},x)=x$ as the projection of the pair
$(\vec{\lambda},x)$ onto the state space, $[0,1]$. We call the sequence
$x_{0},x_{1}=\Pi(T(\vec{\lambda},x_{0}),\dots,x_{n}=\Pi(T^{n}(\vec{\lambda},x_{0})),\dots$
the orbit of $x_{0}$ under $T(\vec{\lambda},x)$. If the initial value is a
random variable independent of $\lambda$, then the set of orbits form a Markov
Chain with state space some subset of $[0,1]$, depending on the distribution
of $\lambda$. We will use capital letters to denote this Markov Chain:
$X_{1}=\Pi\left({T(\vec{\lambda},X_{0})}\right),\dots
X_{i+1}=\Pi\left({T(\sigma^{i}(\vec{\lambda}),X_{i})}\right)$.

Note that in order to keep $x$ in the interval $[0,1]$ for all iterations (to
keep the unit interval invariant) $\lambda$ must lie in the interval $[0,4]$.

###  2.2 The Deterministic Logistic Map

Here we recall a few properties of the deterministic logistic map,
$S_{\lambda}(x)$. For a more complete discussion see, for example, [15, 9,
23].

  * •

For $\lambda$ in the interval $[0,1]$, 0 is the unique attracting fixed point
of $S_{\lambda}(x)$, and, for every $x_{0}\in[0,1]$, the orbit
$\\{x_{n}\\}\rightarrow 0$ as $n\rightarrow\infty$.

  * •

For $\lambda$ in the interval $(1,3]$ there are two fixed points of
$S_{\lambda}(x)$ in $[0,1]$: $x^{0}=0$ and
$x^{1}=\displaystyle{\frac{\lambda-1}{\lambda}}$. In this regime, zero is an
unstable fixed point, and $x^{1}$ is attracting. For every $x_{0}\in(0,1)$,
the orbit $\\{x_{n}\\}\rightarrow x^{1}$, the non-zero steady state. In other
words, any initial distribution of $x$-values converges to a measure supported
on a single point.

  * •

For $\lambda$ in the interval $(3,1+\sqrt{6}]$, the two fixed points (at zero
and $\frac{\lambda-1}{\lambda}$) are unstable, and the logistic map has a
stable period-$2$ cycle. Almost all initial values in $(0,1)$ have orbits that
converge to the period two orbit [23, p. 359]:

$p(\lambda)=\frac{(\lambda+1)-\sqrt{(\lambda-3)(\lambda+1)}}{2\lambda}$ (2.3)
$q(\lambda)=\frac{(\lambda+1)+\sqrt{(\lambda-3)(\lambda+1)}}{2\lambda}.$ (2.4)

Note the average along the period two orbit is
$\displaystyle{\frac{\lambda+1}{2\lambda}}$, so that the long term average
along almost every orbit is also $\displaystyle{\frac{\lambda+1}{2\lambda}}$.

  * •

For $\lambda$ in the interval $(1+\sqrt{6},3.54409]$ (approximately) both the
fixed points and the period two cycle are unstable. However, the logistic map
has a stable attracting period-4 cycle for parameter values in this interval:
the population oscillates between $4$ values. Almost all initial values of
$x_{0}\in(0,1)$ have orbits that converge to this period-4 cycle. Therefore,
any initial continuous distribution of $x$-values will converge to a
distribution supported on these four points.

  * •

The parameter values at which the system transitions into the stable period
2-regime and into the stable period-4 regime are called bifurcation points.
Define the parameter value $\lambda_{c_{k}}$ to be the value at which
bifurcation into a stable period $k$ regime occurs. The first few bifurcation
points can be calculated as $\lambda_{c_{2}}=3$ and
$\lambda_{c_{4}}=1+\sqrt{6}$. Since the period of the attracting cycle doubles
at these points, the system is said to undergo a period-doubling bifurcation.

  * •

Period doublings occur as $\lambda$ increases. The bifurcation parameter
values get closer and closer together, yielding stable cycles of period 8, 16,
32 and so on. As the period increases, the $\lambda$-interval between period
doubling narrows. At
$\lambda_{c_{2^{\omega}}}=\lim_{n\rightarrow\infty}\lambda_{c_{2^{n}}}\approx
3.56995$, stable periodic orbits of odd period appear in quick succession,
according to Sarkovskii’s theorem [9], culminating in a stable period-3 orbit
at $\lambda_{c_{3}}\approx 3.8284$. After the period-3 region, the
deterministic logistic map exhibits chaotic behavior [23, 3].

The long term behavior of the logistic map $S_{\lambda}(x)$ as $\lambda$
varies between 0 and 4 can be depicted in a bifurcation diagram, as shown in
Figure 2.

Figure 2: Bifurcation diagram for $S_{\lambda}(x)$. For each value of the
parameter, $\lambda$ One hundred initial values of $x_{0}$ were generated
uniformly on [0,1]. For each $x_{0}$ value, the logistic map was iterated 1000
times for each $\lambda$ on [0,4], in increments of 0.001. The last iteration
for each $x_{0}$ at each $\lambda$ is plotted. The bifurcation values for the
period 2, period 4 and period 3 bifurcations are indicated as vertical lines.

###  2.3 Bifurcation Diagram for the Stochastic Logistic Map

We can numerically construct a bifurcation diagram for the stochastic logistic
map by plotting the long-term behavior of a sample path as a function of the
mean of a distribution of $\lambda$-values. Figure 3 shows an example of such
a diagram where the parameter is uniformly distributed in a small interval:
$\lambda\sim{\mathcal{U}}([\bar{\lambda}-\Delta\lambda,\bar{\lambda}+\Delta\lambda])$,
$\Delta\lambda=0.1$, and $\bar{\lambda}$ ranges from $0+\Delta\lambda=0.1$ to
$4-\Delta\lambda=3.9$. For $\bar{\lambda}<1-\Delta\lambda=0.9$,
$S_{\lambda}(x)$ has a unique attracting fixed point at $x=0$ for all
$\lambda\in(\bar{\lambda}-\Delta\lambda,\bar{\lambda}+\Delta\lambda)$.
Therefore, all orbits under $T(\vec{\lambda},x)$ converge to 0, and the
bifurcation diagram in Figure 3 shows this with a horizontal line at 0 for
$\bar{\lambda}\in[0+\Delta\lambda,1-\Delta\lambda]$. For
$\bar{\lambda}\in(1+\Delta\lambda=1.1,3-\Delta\lambda=2.9)$, $S_{\lambda}(x)$
has a non-zero attracting fixed point at
$\displaystyle{\frac{\lambda-1}{\lambda}}$ for all
$\lambda\in(\bar{\lambda}-\Delta\lambda,\bar{\lambda}+\Delta\lambda)$. For
these values of $\bar{\lambda}$, orbits of $T(\vec{\lambda},x)$ eventually
enter the interval
$\displaystyle{\left({\frac{\bar{\lambda}-\Delta\lambda-1}{\bar{\lambda}-\Delta\lambda},\frac{\bar{\lambda}+\Delta\lambda-1}{\bar{\lambda}+\Delta\lambda}}\right)}$
with probability 1 and, once in this interval, they never leave it. Figure 3
shows a single curved strip of width
$\frac{2\cdot\Delta\lambda}{(\bar{\lambda}+\Delta\lambda)(\bar{\lambda}-\Delta\lambda)}$
centered at
$\displaystyle{\frac{\bar{\lambda}^{2}-\bar{\lambda}-(\Delta\lambda)^{2}}{\bar{\lambda}^{2}-(\Delta\lambda)^{2}}}$.
Figure 3 also shows a distinct period 2 regime when
$\bar{\lambda}\in(3+\Delta\lambda,1+\sqrt{6}-\Delta\lambda)$.

Bifurcations of families of stochastic logistic maps have also been explored
for other types of distributions of the parameter values. In [22], for
example, the “dichotomous” case, where the distribution is supported on two
$\lambda$-values, is studied. Implicit in the exploration of a stochastic
system via these bifurcation diagrams (and, indeed, in the presentation of
most numerical simulations of random phenomena) is that they represent generic
behavior, i.e., that almost all orbits have the same limiting distribution: we
would get roughly the same bifurcation diagram regardless of which value we
chose for $x_{0}$. This characterizes ergodicity, which we define in the next
section.

In the rest of this paper we are interested in statistical characterizations
of orbits when $\lambda$-values are supported on small intervals that are
completely contained in the stable periodic regimes of the deterministic
logistic map. In particular, we consider the average behavior along orbits of
the map $T(\vec{\lambda},x)$, where $\vec{\lambda}$ is an i.i.d. sequence of
random variables $\lambda_{1},\lambda_{2},\dots$ with common distribution
$g(\lambda)$. In order to make statements about “the” average behavior, we
need to establish the existence of a well-defined measure for the distribution
of $x$-values after many iterations. This is the content of the next section.

Figure 3:  A bifurcation diagram for $T(\vec{\lambda},x)$ with
$\Delta\lambda=0.1$ and $\lambda_{i}\stackrel{{\scriptstyle\
i.i.d.}}{{\sim}}{\mathcal{U}}(\bar{\lambda}-\Delta\lambda,\bar{\lambda}+\Delta\lambda)$,
$\bar{\lambda}\in[0+\Delta\lambda,4-\Delta\lambda]$. For each value of
$\bar{\lambda}$, plotted on the horizontal axis, we begin with 100 values of
$x_{0}$ chosen from a uniform distribution on $[0,1]$, and plot their 1000th
iterate under $T(\vec{\lambda},x)$ on the vertical axis, where $\vec{\lambda}$
is an i.i.d. sequence that is different for each simulation. In other words,
we simulated 100 sample paths of the Markov Chain defined by the stochastic
logistic map for each starting value, $x_{0}$. The periodic structure from the
deterministic setting is still visible for $\bar{\lambda}$-values in the
period-2 regime, but the bifurcations are blurred.

###  2.4 Ergodicity and Stability

We now consider the stochastic logistic map as a map on distributions. This
requires some new definitions.

Let ${\mathcal{B}}$ denote the Borel sigma-algebra on the state space
$\Omega=[0,1]$ and ${\mathcal{M}}$ the space of probability measures defined
on $(\Omega,{\mathcal{B}})$. The stochastic map, $T(\vec{\lambda},x)$ induces
a map on ${\mathcal{M}}$ as follows. Assume that $\vec{\lambda}$ is a sequence
of i.i.d. random variables with common density $g(\lambda)$ supported on
$[a,b]\subset[0,4]$. For $\mu\in{\mathcal{M}}$ define the Perron-Froebenius
operator

$P^{*}\mu(A)=\int_{0}^{4}\mu\left({S_{\lambda}^{-1}(A)}\right)g(\lambda)\,d\lambda$
(2.5)

Fixed points of this map take the place of fixed points in the deterministic
setting.

######  Definition 2.1.

An invariant measure is a measure $\mu^{*}\in{\mathcal{M}}$, such that,
$P^{*}\mu^{*}(A)=\mu^{*}(A)$ for all $A\in{\mathcal{B}}$.

Thus, an invariant measure is a fixed point of the operator, $P^{*}$. We can
also define stability for stochastic dynamical systems via the operator
$P^{*}$:

######  Definition 2.2.

Suppose $\mu^{*}$ is the unique invariant measure for the operator $P^{*}$.
Then the system $T(\vec{\lambda},x)$ is stable in distribution  if, for any
$\mu\in{\mathcal{M}}$

$\left|{\lim_{n\rightarrow\infty}(P^{*})^{n}\mu(A)-\mu^{*}(A)}\right|=0,\quad\forall\,A\in{\mathcal{B}}$

where $P^{*}$ is the Perron-Froebenius operator associated with $T$.

A set $A\in{\mathcal{B}}$ is invariant under $T$ if
$T^{-1}(\vec{\lambda},A)=A$. A probability measure $\mu$ is ergodic if
$\mu(A)=0$ or $1$ for every invariant set $A$. Suppose $\mu^{*}$ is invariant
and ergodic. Then $\mu^{*}$ is the unique invariant probability, and the
ergodic theorem tells us, that for any $A\in{\mathcal{B}}$, for almost every
$x_{0}$, with respect to $\mu^{*}$:

$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=0}^{n-1}\chi_{A}(x_{i})=\mu^{*}(A),$
(2.6)

where $\chi_{A}(x)$ is the indicator function of the set $A$ and
$\\{x_{i}\\}_{i=0}^{\infty}$ is an orbit of $x_{0}$ under the stochastic map.
If, in addition, the invariant measure $\mu^{*}$ is stable, then

$\mu^{*}(A)=\lim_{n\rightarrow\infty}(P^{*})^{n}\mu(A),\quad\forall\mu\in{\mathcal{M}}.$

Thus, we can estimate the long-term behavior along a $\mu^{*}$-generic orbit
by iterating simulated estimates of arbitrary distributions. In fact, this is
the justification behind the algorithm used to generate the stochastic
bifurcation diagram shown in Figure 3.

The following result from [5, p. 343] shows that the stochastic logistic map
with continuous parameter distributions supported in a given periodic regime
is stable.

Assume that $S_{\lambda}(x)$ has a stable periodic orbit with period $m$ for
all $\lambda$ in the support of $g(\lambda)$.

######  Proposition 2.3.

[5] If

$E[\log\lambda]>0\text{ and }E[\left|{\log(4-\lambda)}\right|]<\infty$

then

  1. 1.

the stochastic dynamical system $T(\vec{\lambda},x)$ has a unique invariant
probability measure $\mu^{*}\in{\mathcal{M}}$, and

  2. 2.

$\displaystyle{\left|{\lim_{n\rightarrow\infty}(P^{*})^{mn}\mu(A)-\mu^{*}(A)}\right|=0}$
for all $A\in{\mathcal{B}},\mu\in{\mathcal{M}}$.

In this paper, we consider the situation when $g(\lambda)$ is the uniform
density on an interval $[a,b]$, where $[a,b]$ is entirely within either the
non-zero fixed point regime: (1,3), the period-2 regime: $(3,1+\sqrt{6})$, or
the period-4 regime: $(1+\sqrt{6},3.54409)$. In all cases, $a$ and $b$ are
bounded away from 1 and 4, so that $E[\log\lambda]>0$ and
$E[\left|({\log(4-\lambda)}\right|]<\infty$. Therefore, Proposition 2.3
applies and we know that the long-term distribution of almost all orbits is
well-defined. Thus, “the expected value” exists in the various periodic
regimes, and we are in a position to compare the means of the deterministic
and stochastic logistic maps in these regimes. Note that an explicit formula
for the unique invariant measure is not known, but the existence and stability
of the invariant measure ensure that numerical simulations should reflect the
generic behavior of sample paths of the system.

We can verify the theoretical results numerically by exploring more closely
“slices” of the stochastic bifurcation diagram shown in Figure 3.

The red-solid densities in Figure 4 represent three different slices (panel
(a), (b), and (c), respectively) of the bifurcation diagram of
$T(\vec{\lambda},x)$ shown in Figure 3 using $\Delta\lambda=0.024$, with
$\lambda$-values supported in different periodic regions. Plot (a) is a
vertical slice of a bifurcation diagram of $T(\vec{\lambda},x)$ with
$\lambda\sim{\mathcal{U}}[1.508-\Delta\lambda,1.508+\Delta\lambda]$, within
the non-zero fixed point regime. The figure shows the empirical distribution
of the $x$ values for different iterations, i.e., it gives a numerical
approximation of iterates of the Perron-Froebenius operator, $P^{*}$ given in
Equation (2.5), applied to the uniform distribution. To generate the numerical
approximation, we start with $1000$ uniformly drawn random values of $X_{0}$.
This gives a numerical approximation to $\mu$, the uniform distribution on
$[0,1]$ indicated by the purple-two-dash curve in Figure 4. The blue-long-dash
curve in panel (a) represents the empirical distribution of
$X_{1}=T(\vec{\lambda},X_{0})$ with
$\lambda\sim{\mathcal{U}}[1.508-\Delta\lambda,1.508+\Delta\lambda]$. The other
curves give the empirical distributions of $X_{10},X_{50},X_{100}$ and finally
$X_{1000}$, in red-solid. Plots (b) and (c) show the evolution of the uniform
distribution under $P^{*}$ when the parameter values are in the period-2 and
period-4 regions; the distributions were created using
$\lambda\sim{\mathcal{U}}[3.208-\Delta\lambda,3.208+\Delta\lambda]$ and
$\lambda\sim{\mathcal{U}}[3.508-\Delta\lambda,3.508+\Delta\lambda]$,
respectively. The periodicity of the deterministic systems is reflected in the
multi-modality of the limiting distributions, although we note that these
empirically derived distributions suggest asymmetries. The numerical results
demonstrate that the 50th iteration of $P^{*}$ gives a fairly close
approximation to the stable invariant distribution, $\mu^{*}$. The choice of
$\Delta\lambda=0.024$ was made so that the visual intuition becomes clear, but
the theoretical results do not depend on any specific value of
$\Delta\lambda$, so long as the support of $\lambda$ stays in a particular
periodic regime. (See Section 2.2.)

Figure 4:  The distribution of the state variable with a uniform initial
distribution. The initial, $1$st, $10$th, $50$th, $100$th, and $10000$th
iterations are shown in purple-two-dash, blue-long-dash, green-dog-dash,
brown-dot-dot, orange-dash, and red-solid respectively. The parameter values
in each of the three plots are distributed according to (a)
$\lambda\sim{\mathcal{U}}[1.508-\Delta\lambda,1.508+\Delta\lambda]$, (b)
$\lambda\sim{\mathcal{U}}[3.208-\Delta\lambda,3.208+\Delta\lambda]$, and (c)
$\lambda\sim{\mathcal{U}}[3.508-\Delta\lambda,3.508+\Delta\lambda]$. Note that
the red-solid densities in each plot give an empirical estimate of the
invariant distribution for the given distribution on $\lambda$.

##  3 Stochastically-induced changes in the average behavior.

###  3.1 Mean versus Mean

If $g(\lambda)$ has support in one of the periodic regimes, then Proposition
2.3 gives (let $A=(0,1)$):

$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=0}^{n-1}T^{i}(\vec{\lambda},x)=\int_{0}^{1}x\,d\mu^{*}(x)=E_{\mu^{*}}[X],$

for almost all initial values of $x.$

The result in this paper is a compariosn of the expected value of $X$ under
$T(\vec{\lambda},\cdot)$ with the mean along the periodic orbit under
$S_{\lambda}(x)=\lambda x(1-x)$, the deterministic map, using the expected
value of $\lambda$. Similar investigations have been discussed in [12] in the
context of the Beverton-Holt equation, and in [13], in the context of self-
limiting exponential growth. In the next section we compare the average
behavior of the stochastic logistic map with the deterministic map in several
parameter regimes, uncovering a pattern of alternating larger and smaller
expected values under stochasticity.

###  3.2 Period One

Suppose $\lambda\sim{\mathcal{U}}([a,b])$. In Figure 5 we compare the
“deterministic mean”, i.e., the long term average of orbits under
$S_{\bar{\lambda}}$ where $\bar{\lambda}=\displaystyle{\frac{a+b}{2}}$, with
the long term behavior under the stochastic dynamical system,
$T(\vec{\lambda},\cdot)$. The left panel of Figure 5 shows the stable fixed
point at $\bar{\lambda}=1.508$ (red line), the limit set for almost all orbits
under $S_{\bar{\lambda}}$. In the same panel we plot the empirical
distribution of $\left({P^{*}}\right)^{10000}(\mu)$ (black histogram), where
$\mu$ is the uniform distribution on $(0,1)$, $\Delta\lambda=0.024$, and
$\lambda\sim{\mathcal{U}}([1.508-\Delta\lambda,1.508+\Delta\lambda])$; the
average of the empirical distribution is given by the blue line. Because the
system is stable, we can interpret $\left({P^{*}}\right)^{10000}(\mu)$ as an
estimate of the invariant distribution, $\mu^{*}$. The right panel of Figure 5
shows a magnification of the left panel. Observe that the blue line is to the
left of the red line, hence the (empirically estimated) average of $\mu^{*}$
is less than the stable fixed point of the deterministic map when
$\bar{\lambda}=1.508$. Since the stable fixed point is the average along
almost every orbit of the deterministic map, Figure 5 gives a visual
comparison of the mean of the stochastic system, where the parameter values
have mean $\bar{\lambda}$ (blue line), with the mean of the deterministic
system with parameter value equal to $\bar{\lambda}$ (red line).

Figure 5: Numerically estimated histogram of the $10000^{th}$ iterate of
$P^{*}$, the Perron-Froebenius operator associated with $T(\vec{\lambda},x)$.
In this simulation, the parameter value at each iteration is given by
$\lambda\sim U[1.508-\Delta\lambda,1.508+\Delta\lambda]$ with
$\Delta\lambda=0.024$; 20,000 initial $x$-values were drawn from a uniform
distribution on $[0,1]$. Since the system has a unique stable invariant
measure, the histogram is an estimate of the distribution of the limiting
invariant measure, $\mu^{*}$. The right panel shows a magnification of the
central portion of the histogram, highlighting that the mean of the empirical
histogram (blue line) is less than the stable fixed point of the logistic map
with the average parameter value, $\bar{\lambda}=1.508$ (red line).

The following theorem, an application of Jensen’s inequality, proves that the
“stochastic mean” will always be less than the deterministic mean. In other
words, the presence of randomness lowers the long term average in the period-
one regime.

Define $x^{*}(\lambda)=\displaystyle{\frac{\lambda-1}{\lambda}}$ to be the
non-zero fixed point of the map $S_{\lambda}(x)$. Recall that this fixed point
is attracting for $\lambda\in(1,3)$ and, therefore, the Césaro average along
almost any orbit converges to $x^{*}(\lambda)$:

$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=0}^{n-1}S^{i}_{\lambda}(x)=\frac{\lambda-1}{\lambda}=x^{*}(\lambda)\quad\hbox{if
}\;\lambda\in(1,3).$

######  Theorem 3.1.

Let $g(\lambda)$ be a continuous density whose support is in the period one
region, $(1,3)$. Then the mean of $X$ under the stochastic logistic map
$T(\vec{\lambda},\cdot)$ is less than the stable fixed point of the
deterministic map $S_{\bar{\lambda}}(x)$, where
$\bar{\lambda}=\int_{0}^{4}\lambda g(\lambda)d\lambda$.

###### Proof.

Let $\mu^{*}$ be the unique invariant distribution of the Perron-Froebenius
operator associated with $T(\vec{\lambda},\cdot)$. Since $\lambda$ is
independent of $X$:

$\displaystyle E_{\mu^{*}}(X)$ $\displaystyle=$
$\displaystyle\int_{0}^{1}xd\mu^{*}(x)$ $\displaystyle=$
$\displaystyle\int_{0}^{1}xdP^{*}(\mu^{*})(x)$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\int_{0}^{4}\lambda x(1-x)g(\lambda)d\lambda
d\mu^{*}(x)$ $\displaystyle=$
$\displaystyle\bar{\lambda}\int_{0}^{1}x-x^{2}d\lambda d\mu^{*}(x)$
$\displaystyle=$
$\displaystyle\bar{\lambda}\left({E_{\mu^{*}}(X)-E_{\mu^{*}}(X^{2})}\right)$
$\displaystyle\leq$
$\displaystyle\bar{\lambda}\left({E_{\mu^{*}}(X)-E_{\mu^{*}}(X)^{2}}\right)\quad\hbox{since,
by Jensen's inequality }E_{\mu^{*}}(X)^{2}\leq E_{\mu^{*}}(X^{2})$
$\displaystyle\Rightarrow(\bar{\lambda}-1)E_{\mu^{*}}(X)-\bar{\lambda}E_{\mu^{*}}^{2}(X)\geq
0$ $\displaystyle\Rightarrow
E_{\mu^{*}}(X)\left({\bar{\lambda}-1-\bar{\lambda}E_{\mu^{*}}(X)}\right)\geq
0$ $\displaystyle\Rightarrow
E_{\mu^{*}}(X)\leq\frac{\bar{\lambda}-1}{\bar{\lambda}}=x^{*}(\bar{\lambda})$

since $E_{\mu^{*}}(X)\geq 0$. Furthermore, the inequality is strict unless
$E_{\mu^{*}}(X^{2})=0$. ∎

Note: the final inequality in the proof of Theorem 3.1 holds for $\lambda$ in
any region. However, $\displaystyle{\frac{\bar{\lambda}-1}{\bar{\lambda}}}$ is
the average of the deterministic system only in the non-zero stable period-one
regime, $\lambda\in(1,3)$. Therefore, the proof of theorem 3.1 does not tell
us anything about the relationship between the stochastic and deterministic
means when $g(\lambda)$ is supported in other regions.

###  3.3 Period Two

Figure 6 shows a histogram of
$\left({P^{*}}\right)^{10000}(\mu)\approx\mu^{*}$ where the parameter values
have a uniform distribution:
$\lambda\sim{\mathcal{U}}([3.208-\Delta\lambda,3.208+\Delta\lambda])$,
$\Delta\lambda=0.024$. As in Figure 5, the empirical stochastic and
deterministic means with $\bar{\lambda}=3.208$ are indicated by green and
orange vertical lines, respectively. The right panel of Figure 6 shows a
magnified region of the left panel, showing that the orange line is to the
left of the green line, so the empirical estimate of the stochastic mean is
greater than the average along the entire stable period-2 orbit of the average
parameter value. Note that this is opposite of the situation depicted in
Figure 5. The numerical results, along with other simulated histograms of
$(P^{*})^{10000}(\mu)$ (not shown) when $\lambda$ is in the period two region,
demonstrate that the expected value of $\pi(T(\vec{\lambda},X))$ is greater
than the mean along the stable period-2 orbit of the deterministic map
evaluated at $\bar{\lambda}$. In other words, the relationship between the
stochastic and deterministic means in the period two region is the reverse of
the relationship in the period one region. We prove that this is the case in
Theorem 3.2.

We can gain some intuition about this reversal from the following observation,
also depicted in Figure 6. For
$\lambda\in[3.208-\Delta\lambda,3.208+\Delta\lambda]$, the map $S_{\lambda}$
has a stable period 2 orbit: $\\{p(\lambda),q(\lambda)\\}$ where
$p(\lambda)<q(\lambda)$ are given by Equations (2.3) and (2.4), respectively.
Thus, almost all orbits of the stochastic system will eventually enter two
disjoint intervals, $I_{p}$ and $I_{q}$, where $I_{q}$ is the image of the
interval: $[p(3.208+\Delta\lambda),p(3.208-\Delta\lambda)]$ and $I_{p}$ is the
image of the interval $[q(3.208-\Delta\lambda),q(3.208+\Delta\lambda)]$. A
generic orbit will alternately visit $I_{p}$ and $I_{q}$ and thus the
invariant measure, $\mu^{*}$, can be written as a sum of two measures,
$\mu_{p}^{*}$ and $\mu_{q}^{*}$, supported on $I_{p}$ and $I_{q}$,
respectively, and $E_{\mu^{*}}(X)$ will be the average of $E_{\mu_{p}^{*}}(X)$
and $E_{\mu_{q}^{*}}(X)$. Figure 6 compares $E_{\mu_{p}^{*}}(X)$ and
$E_{\mu_{q}^{*}}(X)$ with the stable period-2 points of the deterministic
system with the average parameter value, $\bar{\lambda}=3.208$. We see that
$E_{\mu_{p}^{*}}(X)>p(\bar{\lambda})$ and
$E_{\mu_{q}^{*}}(X)<q(\bar{\lambda})$, but we formally prove in the Appendix
that the first inequality is larger than the second, so that
$E_{\mu^{*}}(X)>\frac{p(\bar{\lambda})+q(\bar{\lambda})}{2}$. This is the
conclusion of Theorem 3.2.

Figure 6:  Numerically estimated histogram of the $10000^{th}$ iterate of
$P^{*}$, the Perron-Froebenius operator associated with $T(\vec{\lambda},x)$.
In this simulation, the parameter value at each iteration is given by
$\lambda\sim U[3.208-\Delta\lambda,3.208+\Delta\lambda]$ with
$\Delta\lambda=0.024$; 20,000 initial $x$-values were drawn from a uniform
distribution on $[0,1]$. Since the system has a unique stable invariant
measure, this histogram is an estimate of the distribution of the limiting
invariant measure, $\mu^{*}$. The right panel shows a magnification of the
central portion of the bimodal histogram, highlighting that the mean of the
empirical histogram (green line, $(E_{\mu_{p}^{*}}(X)+E_{\mu_{q}^{*}}(X))/2$)
is greater than the average of the stable fixed points of the invariant
distribution at the average parameter value, $\bar{\lambda}=3.208$ (orange
line, $(p(\bar{\lambda})+q(\bar{\lambda}))/2$).

The previous motivation leads to the following theorem. Rather than a
convexity argument similar to the one used to prove Theorem 3.1, the proof
presented here relies on several rather technical lemmas which are proved in
the Appendix.

######  Theorem 3.2.

In the period two region, $\lambda\in(3,1+\sqrt{6})$, the expected value of X
under $\mu^{*}$, the stochastic logistic map (i.e., $\Delta\lambda>0$), is
greater than the average of the stable period-2 points of the deterministic
map with the average parameter value, $\bar{\lambda}$.

$E_{\mu^{*}}(X)>\frac{p(\overline{\lambda})+q(\overline{\lambda})}{2}=\frac{\overline{\lambda}+1}{2\overline{\lambda}}$

Suppose
$\lambda\sim{\mathcal{U}}(\bar{\lambda}-\Delta\lambda,\bar{\lambda}+\Delta\lambda)$
where
$(\bar{\lambda}-\Delta\lambda,\bar{\lambda}+\Delta\lambda)\subset(3,1+\sqrt{6})$
and $\Delta\lambda$ is small enough so that the support of the invariant
distribution, $\mu^{*}$, consists of two disjoint intervals (see Lemma 5.1 in
the Appendix). Then define the measures $\mu_{p}^{*}$ and $\mu_{q}^{*}$ as
follows:

$\mu_{p}^{*}(A)=2\mu^{*}\Bigg{(}A\cap\bigg{[}0,\frac{\bar{\lambda}-1}{\bar{\lambda}}\bigg{]}\Bigg{)}\quad\quad\mu_{q}^{*}(A)=2\mu^{*}\Bigg{(}A\cap\bigg{(}\frac{\bar{\lambda}-1}{\bar{\lambda}},1\bigg{]}\Bigg{)}$
(3.1)

Thus, $\mu_{p}^{*}$ gives the left peak of the invariant distribution, and
$\mu_{q}^{*}$ gives the right peak.

The proof of Theorem 3.2 involves showing that, for small $\Delta\lambda>0$,
the expected value of the lower peak, $E_{\mu_{p}^{*}}[X]$, is farther from
$p(\bar{\lambda})$ than the distance of the expected value of the upper peak,
$E_{\mu_{q}^{*}}[X]$, to $q(\bar{\lambda})$. The entire proof is in the
Appendix.

###  3.4 Period Four

Figure 7 shows the average of the four stable fixed points for
$\bar{\lambda}=3.508$, corresponding to the four points on the vertical slice
of the bifurcation diagram (see Figure 2) of $S_{\lambda}(x)$ with
$\lambda=3.508$ (orange line). In the same figure, we plot the average of the
points that make up the empirical histogram (green line). Observe in the right
panel of Figure 7, the zoomed in version of the left panel, the $x$-coordinate
of the green line is less than that of the orange line, so the average of the
empirical estimation of $\mu^{*}$ is strictly less than the average of the
stable period-four points, just as in the period one region. Figure 7, and
other histograms of the last four iterations (not shown) of
$T(\vec{\lambda},x)$ for $\lambda$ in the period four region, demonstrate that
the expected value of $\mu^{*}$ under the stochastic logistic map is less than
or equal to the long term average under the corresponding deterministic map.

Figure 7: Numerically estimated histogram of the $10000^{th}$ iterate of
$P^{*}$, the Perron-Froebenius operator associated with $T(\vec{\lambda},x)$.
In this simulation, the parameter value at each iteration is given by
$\lambda\sim U[3.508-\Delta\lambda,3.508+\Delta\lambda]$ with
$\Delta\lambda=0.024$; 20,000 initial $x$-values were drawn from a uniform
distribution on $[0,1]$. Since the system has a unique stable invariant
measure, this histogram is an estimate of the distribution of the limiting
invariant measure, $\mu^{*}$. The right panel shows a magnification of the
central portion of the histogram, highlighting that the mean of the empirical
histogram (green line) is less than the average of the stable fixed points of
the invariant distribution at the average parameter value,
$\bar{\lambda}=3.508$ (orange line).

##  4 Discussion

Previous studies have explored the effect of randomness on the average
behavior of families of one-dimensional maps with stable fixed points [12,
13]. In this paper we compared the expected value of the deterministic and
stochastic logistic map in different periodic regimes. We showed that a unique
stable invariant measure existed under the Perron-Froebenius map associated
with the stochastic logistic map in the parameter regions of interest. This
allowed us to interpret numerical experiments, and allowed us to compare the
expected value of the stochastic logistic map with that of the deterministic
map.

In the period one region, the average of the deterministic map, which is the
fixed point of the map at the average parameter values, is greater than the
expected value of $X$ under the unique stable invariant distribution of the
stochastic map. Since this distribution is stable, the long term average of
almost all orbits under $T(\vec{\lambda},x)$ is strictly less than the long-
term average under the related deterministic map. In terms of a growth model,
if an environmental parameter that affects the birth rate of the population
fluctuates randomly around a mean value $\bar{\lambda}$, then the average
population will be smaller than in the deterministic model with parameter
fixed at $\bar{\lambda}$.

These results for the period one region are similar to those in [12], which
apply to the stochastic Beverton-Holt model. This is not surprising, since the
proof of Theorem 3.1 uses only the fact that the logistic map is unimodal and
concave down, features shared by the Beverton-Holt model. In [13] a similar
result is proved in greater generality, allowing for arbitrary distributions
of the parameter.

In contrast to the period one region, in the period two region the average of
the deterministic map is less than the expected value of $X$ under the unique
stable invariant distribution of the stochastic map. The period two result is
given in Theorem 3.2 and formally proved in the Appendix. The switch in
behavior led us to explore what happens after further period doubling
bifurcations.

Our empirical simulations (shown in Figure 7) demonstrate that the behavior in
the period four region reverts to that of the period one region. In
particular, it seems as though the average of the deterministic map is again
greater than the expected value of $X$ under the unique stable invariant
distribution of the stochastic map. The oscillating behavior across the
different regions leads us to the following “flip-flop” conjecture.

######  Conjecture 4.1.

Flip-flop conjecture.

  1. 1.

For $\rho$ odd, consider regions with period $2^{\rho}$. Let
$p_{i}(\bar{\lambda})$ be the stable period-$2^{\rho}$ points of the
deterministic map with the average parameter value, $\bar{\lambda}$.

The expected value of X under the stochastic logistic map is greater than the
average of the stable period-$2^{\rho}$ points of the deterministic map with
the average parameter value, $\bar{\lambda}$.

$E_{\mu^{*}}(X)>\frac{1}{2^{\rho}}\sum_{i=1}^{2^{\rho}}p_{i}(\bar{\lambda})$

  2. 2.

For $\rho$ even, consider regions with period $2^{\rho}$. Let
$p_{i}(\bar{\lambda})$ be the stable period-$2^{\rho}$ points of the
deterministic map with the average parameter value, $\bar{\lambda}$.

The expected value of X under the stochastic logistic map is less than the
average of the stable period-$2^{\rho}$ points of the deterministic map with
the average parameter value, $\bar{\lambda}$.

$E_{\mu^{*}}(X)<\frac{1}{2^{\rho}}\sum_{i=1}^{2^{\rho}}p_{i}(\bar{\lambda})$

We note that a renormalization framework can be used to explore the
conjecture, since the convexity of the iterates: $S_{\lambda}(x)$,
$S^{2}_{\lambda}(x)$, $S^{4}_{\lambda}(x),\dots$ at the fixed point switches
as the number of iterates is doubled. Thus, it makes sense to conjecture that
as we increase the period, the direction of the inequality switches back and
forth, at least through the period doubling range of $\lambda$.

##  5 Appendix

This Appendix contains the formal proof of the main result of the paper,
Theorem 3.2. The proof consists of a series of lemmas. To guide the reader and
to highlight the steps in the argument, we first provide an outline of the
lemmas.

Recall Theorem 3.2:
In the period two region, $\lambda\in(3,1+\sqrt{6})$, the expected value of X
under $\mu^{*}$ (the stochastic logistic map) is greater than the average of
the stable period-2 points of the deterministic map with the average parameter
value, $\bar{\lambda}$.

$E_{\mu^{*}}(X)>\frac{p(\overline{\lambda})+q(\overline{\lambda})}{2}=\frac{\overline{\lambda}+1}{2\overline{\lambda}}$

  1. 1.

First, we establish that, for small amplitudes of noise, the support of the
invariant measure is contained in two disjoint intervals (Lemma 5.1).

  2. 2.

The expected value of X with respect to the right hand measure (as defined in
Equation (3.1)) is the same as the expected value of the first iterate of the
deterministic map at $\overline{\lambda}$ with respect to the left hand
measure (Lemma 5.2).

$E_{\mu_{q}^{*}}[X]=E_{\mu_{p}^{*}}[S_{\bar{\lambda}}(X)]$

  3. 3.

The left peak of the invariant measure, i.e., the expected value with respect
to $\mu_{p}^{*}$, is larger than the smaller point on the period-2 cycle of
$S_{\bar{\lambda}}$ (Lemma 5.3).

$E_{\mu_{p}^{*}}[X]>p(\bar{\lambda})$

To prove Lemma 5.3: create functions $H$ and $F$, apply Lemma 5.2 twice, then
Lemma 5.1 and use Jensen’s inequality.

  4. 4.

We define a function of $\Delta\lambda$ giving the variance of X with respect
to the measure $\mu_{q}^{*}$. We show that the right hand derivative of the
function $V(\Delta\lambda)$ is zero when $\Delta\lambda=0$ (Lemma 5.4).

$V_{+}^{\prime}(0)=\lim_{h\rightarrow 0^{+}}\frac{V(h+0)-V(0)}{h}=0$

We use this result to prove the next lemma.

  5. 5.

The change in the expected value of the left peak of the invariant
distribution is greater in magnitude than the change in the expected value of
the right peak at $\Delta\lambda=0$ (Lemma 5.5).

$\frac{dE_{\mu_{p}^{*}}[X]}{d\Delta\lambda}\geq-\frac{dE_{\mu_{q}^{*}}[X]}{d\Delta\lambda}$

  6. 6.

Which allows us to prove Theorem 3.2. By Lemma 5.5, for very small
$\Delta\lambda\geq 0$, $E_{\mu_{p}^{*}}[X]$ increases faster than
$E_{\mu_{q}^{*}}[X]$ decreases in $\Delta\lambda$, implying that their mean,
$E[X]$, increases with $\Delta\lambda$.

We provide some definitions of notation that will be used throughout the
proof.

$\displaystyle\mbox{for }\lambda\sim
U[\bar{\lambda}-\Delta\lambda,\bar{\lambda}+\Delta\lambda],\ \ \ \ \ \
g(\lambda)=\begin{cases}\frac{1}{\bar{\lambda}+\Delta\lambda-(\bar{\lambda}-\Delta\lambda)}=\frac{1}{2\Delta\lambda},&\bar{\lambda}-\Delta\lambda\leq\lambda\leq\bar{\lambda}+\Delta\lambda,\\\
0&\hbox{otherwise}\end{cases}.$ (5.1)

For values of $\lambda$ greater than $3$, the logistic map has two unstable
fixed points at 0 and $x_{\lambda}^{*}=1-\frac{1}{\lambda}$, and two period-2
points, $p_{\lambda}<q_{\lambda}$ at

$p(\lambda)=\frac{\lambda+1-\sqrt{\lambda^{2}-2\lambda-3}}{2\lambda},\quad\quad
q(\lambda)=\frac{\lambda+1+\sqrt{\lambda^{2}-2\lambda-3}}{2\lambda}.$ (5.2)

Note that the fixed points are solutions of the quadratic:

$\lambda x(1-x)-x=0$

while the period-2 points are solutions to the quartic:

$\lambda^{2}x(1-x)(1-\lambda x(1-x))-x=0$

that are not fixed points.

$\displaystyle p_{\pm}=p(\bar{\lambda}\pm\Delta\lambda)\quad
q_{\pm}=q(\bar{\lambda}\pm\Delta\lambda)$ (5.3)

The period-two points for $\bar{\lambda}=3.2$, $\Delta\lambda=0.1$, are shown
in Figure 8. Note that $p_{+}<p_{-}$ while $q_{-}<q_{+}$.

###  5.1 Proof of Theorem 3.2

The proof of Theorem 3.2 is divided up into several steps. First, we establish
that, for small amplitudes of noise, the support of the invariant measure is
contained in two disjoint intervals. This was discussed in Section 3.3.

Suppose the values of $\lambda$ are in some interval
$[\bar{\lambda}-\Delta\lambda,\bar{\lambda}+\Delta\lambda]$. Let $I_{p}$ be
the image under $T(\vec{\lambda},\cdot)$ of the interval of period-2 points,
$[q(\bar{\lambda}-\Delta\lambda),q(\bar{\lambda}+\Delta\lambda)]$, and let
$I_{q}$ be the image of the other interval of period-2 points:
$[p(\bar{\lambda}+\Delta\lambda),p(\bar{\lambda}-\Delta\lambda)]$. We will see
that $I_{p}\cap I_{q}=\emptyset$.

######  Lemma 5.1.

For $\bar{\lambda}\in(3,1+\sqrt{6})$ with $\Delta\lambda$ such that
$[\bar{\lambda}-\Delta\lambda,\bar{\lambda}+\Delta\lambda]\subset(3,1+\sqrt{6})$,
the support of the invariant distribution, $I$, is contained in the disjoint
union $I_{p}\cup I_{q}$.

###### Proof.

If $\lambda\in(\bar{\lambda}-\Delta\lambda,\bar{\lambda}+\Delta\lambda)$, with

$3<\bar{\lambda}-\Delta\lambda<\bar{\lambda}+\Delta\lambda<1+\sqrt{6}$ (5.4)

then the dynamical system given by

$x_{t+1}=S_{\lambda}(x_{t})=\lambda x_{t}(1-x_{t})$

has an asymptotically stable period-2 orbit [9] that attracts almost all
orbits in $[0,1]$. In other words, for a fixed value of $\lambda$, for almost
all $x_{0}\in(0,1)$ with respect to Lebesgue measure, the limit set of
$\\{x_{t}\\}_{t=0}^{\infty}$ is $\\{p(\lambda),q(\lambda)\\}$ as given in
Equation (5.2). This implies that almost all orbits of the stochastic map will
approach a neighborhood of the set of period-2 points:
$[p_{+},p_{-}]\cup[q_{-},q_{+}]$.

Let $T(\vec{\lambda},x)$ be the stochastic logistic map with
$\lambda\sim{\mathcal{U}}(\bar{\lambda}-\Delta\lambda,\bar{\lambda}+\Delta\lambda)$,
and let $\mu^{*}$ be the unique stable invariant measure under $T$, as defined
in Section 2.4. Denote the support of $\mu^{*}$ by $I$. We will show that
$I\subset I_{p}\cup I_{q}$, where are given by $I_{p}=[x_{p,\min},x_{p,\max}]$
and $I_{q}=[x_{q,\min},x_{q,\max}]$ as seen in Figure 9. We give an explicit
description of $I_{p}$ and $I_{q}$ below.

For a fixed $\bar{\lambda}$ and $\Delta\lambda$ satisfying Equation (5.4), let
$p_{\pm}$ and $q_{\pm}$ be defined as in Equation (5.3). Remember that, if a
process is ergodic, the support of the unique invariant measure contains all
invariant sets of positive measure. The stable period-2 points of
$S_{\lambda}(x)$ for all
$\lambda\in(\bar{\lambda}-\Delta\lambda,\bar{\lambda}+\Delta\lambda)$ must be
contained in the support of $\mu^{*}$ since these points are invariant under
$S_{\lambda}(x)$ for each
$\lambda\in(\bar{\lambda}-\Delta\lambda,\bar{\lambda}+\Delta\lambda)$. Hence
$I\supset[p_{+},p_{-}]\cup[q_{-},q_{+}]$. Figure 8 shows that $I$ can, in
fact, contain more than the set of stable period-2 points. This follows from
the observation that the function $S_{\lambda}(x)$ can be non-monotonic in the
interval $[p_{+},p_{-}]$, and the functions are decreasing on the interval
$[q_{-},q_{+}]$ for
$\lambda\in(\bar{\lambda}-\Delta\lambda,\bar{\lambda}+\Delta\lambda)$, so that
$S_{\bar{\lambda}_{+}}(q_{-})>S_{\bar{\lambda}_{-}}(q_{-})=p_{-}$, and
$S_{\bar{\lambda}_{-}}(q_{+})<S_{\bar{\lambda}_{+}}(q_{+})=p_{+}$

Figure 8: Two iterates of the stochastic logistic map, $T(\vec{\lambda},x)$
with initial value $x_{0}=.7755$ and parameter distribution
$\lambda\sim{\mathcal{U}}(3.1,3.3)$, i.e.,
$\bar{\lambda}=3.2,\Delta\lambda=.1$.  Figure 9: The disjoint intervals
$[x_{p,\min},x_{p,\max}]$ and $[x_{q,\min},x_{q,\max}]$ are the images of
$[q_{-},q_{+}]$ and $[p_{+},p_{-}]$, respectively, under
$S_{\lambda_{2}}(S_{\lambda_{1}})(x)$ for all
$\lambda_{1,2}\in[\bar{\lambda}-\Delta\lambda,\bar{\lambda}+\Delta\lambda]$.

For illustration, consider the case when
$1+\sqrt{5}\in(\bar{\lambda}-\Delta\lambda,\bar{\lambda}+\Delta\lambda)$, so
that $p_{+}\leq\frac{1}{2}\leq p_{-}$ and the logistic map is non-monotonic on
the interval $[p_{+},p_{-}]$. Let $x_{t}\in[p_{+},p_{-}]$ and consider
$x_{t+1}=T(\vec{\lambda},x_{t})$. We define $x_{q,\min}$ to be the minimum
value of $x_{t+1}$, so that $x_{q,\min}$ is the smaller of $q_{-}$ and
$S_{\bar{\lambda}-\Delta\lambda}(p_{+})$, noting that it is possible that
$x_{q,\min}$ can be smaller than $q_{-}$. Similarly, in this case the maximum
value of $x_{t+1}$ is
$x_{q,\max}=S_{\bar{\lambda}+\Delta\lambda}\left({\frac{1}{2}}\right)=\frac{(\bar{\lambda}+\Delta\lambda)}{4}.$

In general, define:

$\displaystyle x_{p,\min}$ $\displaystyle=$ $\displaystyle
S_{\bar{\lambda}-\Delta\lambda}(q_{+}),\quad\quad\quad\quad\quad
x_{p,\max}=S_{\bar{\lambda}+\Delta\lambda}(q_{-})$ (5.5) $\displaystyle\quad
x_{q,\min}$ $\displaystyle=$
$\displaystyle\min\\{q_{-},S_{\bar{\lambda}-\Delta\lambda}(p_{+})\\},\quad
x_{q,\max}=\max\\{S_{\bar{\lambda}+\Delta\lambda}(1/2),S_{\bar{\lambda}+\Delta\lambda}(p_{+}),S_{\bar{\lambda}+\Delta\lambda}(p_{-})\\}$

These intervals are shown in Figure 9.

By writing $x_{p,\max}$ and $x_{q,\min}$ explicitly as functions of $\lambda$
and $\Delta\lambda$, we see that the intervals $I_{p}$ and $I_{q}$ lie on
either side of the non-zero fixed points corresponding to
$\bar{\lambda}+\Delta\lambda$ and $\bar{\lambda}-\Delta\lambda$, as follows:

$\displaystyle p_{+}<p_{-}=p(\bar{\lambda}-\Delta\lambda)\leq
x_{p,\max}<\frac{\bar{\lambda}-\Delta\lambda-1}{\bar{\lambda}-\Delta\lambda}<\frac{\bar{\lambda}+\Delta\lambda-1}{\bar{\lambda}+\Delta\lambda}<x_{q,\min}\leq
q_{-}=q(\bar{\lambda}-\Delta\lambda)<q_{+}$ (5.6)

so the intervals $I_{p}$ and $I_{q}$ are indeed disjoint. See Figure 9 which
displays the ordered values given in Equation (5.6). ∎

Remark: the supports of $\mu_{p}^{*}$ and $\mu_{q}^{*}$ are contained in
$I_{p}$ and $I_{q}$, respectively. Lemma 5.1 therefore implies that all points
in the support of $\mu_{p}^{*}$ are mapped to the support of $\mu_{q}^{*}$
under one iterate of $T(\vec{\lambda},x)$, so that, by the definition of the
Perron-Frobenius operator, $P^{*}:\mu_{p}^{*}\mapsto\mu_{q}^{*}$ and vice-
versa. Therefore, by construction,
$\mu^{*}(support(\mu_{q}^{*}))=\mu^{*}(support(\mu_{p}^{*}))=\frac{1}{2}$.

######  Lemma 5.2.

$E_{\mu_{q}^{*}}[X]=E_{\mu_{p}^{*}}[S_{\bar{\lambda}}(X)].$

when $\lambda\sim U[\bar{\lambda}-\Delta\lambda,\bar{\lambda}+\Delta\lambda]$
with density $g(\lambda)$ as given in Equation (5.1).

###### Proof.

We note that $P^{*}\mu_{p}^{*}(x)=\mu_{q}^{*}(x)$, where $P^{*}$ is the
Perron-Froebenius operator given in Equation (2.5). Therefore, we have:

$E_{\mu_{q}^{*}}[X]=\int_{0}^{1}x\mu_{q}^{*}(x)\,dx=\int_{0}^{1}xP^{*}\mu_{p}^{*}(x)\,dx=\int_{0}^{1}x\int_{V_{x}}\mu_{p}^{*}(y)g\left({\frac{x}{y(1-y)}}\right)\frac{1}{y(1-y)}\,dy\,dx,$

where the set $V_{x}$ is $\\{(\lambda,y):\lambda y(1-y)=x\\}$. Changing the
order of integration, we have that:

$E_{\mu_{q}^{*}}[X]=\int_{0}^{1}\left[{\int_{0}^{1}xg\left({\frac{x}{y(1-y)}}\right)\,dx}\right]\frac{1}{y(1-y)}\mu_{p}^{*}(y)\,dy.$

The $\lambda$’s are uniformly distributed on
$[\bar{\lambda}-\Delta\lambda,\bar{\lambda}+\Delta\lambda]$ with $g(\lambda)$
is as given in Equation (5.1). Using $g(\lambda)$ in the expression for
$E_{\mu_{q}^{*}}[X]$ and evaluating the integral we get:

$\displaystyle E_{\mu_{q}^{*}}[X]$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\left[{\frac{x^{2}}{2}}\right|_{(\bar{\lambda}-\Delta\lambda)y(1-y)}^{(\bar{\lambda}+\Delta\lambda)y(1-y)}\,\frac{\mu_{p^{*}}(y)}{y(1-y)(2\Delta\lambda)}\,dy$
$\displaystyle=$
$\displaystyle\int_{0}^{1}\frac{\left({(\bar{\lambda}+\Delta\lambda)^{2}-(\bar{\lambda}-\Delta\lambda)^{2}}\right)\left({(y(1-y))^{2}}\right)}{2}\,\frac{\mu_{p^{*}}(y)}{y(1-y)(2\Delta\lambda)}\,dy$
$\displaystyle=$
$\displaystyle\bar{\lambda}\int_{0}^{1}y(1-y)\mu_{p^{*}}(y)\,dy$

We can rewrite $E_{\mu_{q}^{*}}[X]$ as:

$E_{\mu_{q}^{*}}[X]=\bar{\lambda}\left({E_{\mu_{p^{*}}}[X]-E_{\mu_{p}^{*}}[X^{2}]}\right)=\bar{\lambda}E_{\mu_{p^{*}}}[S_{1}(X)].$

∎

######  Lemma 5.3.

The left peak of the invariant measure, i.e., the expected value with respect
to $\mu_{p}^{*}$, is larger than the smaller point on the period-2 cycle of
$S_{\bar{\lambda}}$.

$E_{\mu_{p}^{*}}[X]>p(\bar{\lambda}).$

###### Proof.

Note that we are assuming that $\Delta\lambda>0$ in order to get a strict
inequality. To prove Lemma 5.3, we introduce a new function,
$H_{\bar{\lambda}}(x)$, which helps us put bounds on the second iterate of
$S_{\bar{\lambda}}(x)$.

Fix values of $\bar{\lambda}$ and $\Delta\lambda$ in the period-2 regime:
$\bar{\lambda}\pm\Delta\lambda\in(3,1+\sqrt{6})$.

Consider the functions $F_{\bar{\lambda}}(x)=(S_{\bar{\lambda}}\circ
S_{\bar{\lambda}})(x)-x$, and
$H_{\bar{\lambda}}(x)=\bar{\lambda}h_{\bar{\lambda},\epsilon}(x)-x$ where

$h_{\bar{\lambda},\epsilon}(x)=\bar{\lambda}x(1-x)-\bar{\lambda}^{2}\left({x(1-x)}\right)^{2}+\epsilon$
(5.7)

for some $\epsilon>0$. From here on we drop the subscripts on $h$ for clarity.

The function $F_{\bar{\lambda}}(x)$ has four zeros. Two of the zeros are the
two fixed points of $S_{\bar{\lambda}}(x)$: $0$ and
$\frac{\bar{\lambda}-1}{\bar{\lambda}}=x^{*}(\bar{\lambda})$; two zeros are
given by the period-2 orbit: $p(\bar{\lambda})$ and $q(\bar{\lambda})$, as
defined in Equation (5.2). We see that the function $H$ is just a shift upward
of the function $F$ by $\bar{\lambda}\epsilon$:
$H(x)=F(x)+\bar{\lambda}\epsilon$. If $\epsilon$ is small enough, $H$ will
have the same number of zeros as $F$. If we label the zeros of H as:
$z_{H}<p_{H}<x^{*}_{H}<q_{H}$, we have (see Figure 10):

$z_{H}<0<p(\bar{\lambda})<p_{H}<x^{*}_{H}<x^{*}(\bar{\lambda})=\frac{\bar{\lambda}-1}{\bar{\lambda}}<q(\bar{\lambda})<q_{H}<1.$

Now, fix $\bar{\lambda}$, and we will drop the subscript $\bar{\lambda}$ on
all the functions for readability.

Consider the region $I_{p}$ which contains the support for $\mu_{p}^{*}(x)$,
the left peak of the invariant measure. We note that $h(I_{p})$ is concave up,
i.e., $h^{\prime\prime}(x)>0$ for all $x\in I_{p}$. This implies by Jensen’s
Inequality that

$E_{\mu_{p}^{*}}(h(X))>h(E_{\mu_{p}^{*}}[X])\Rightarrow
E_{\mu_{p}^{*}}(h(X))=h(E_{\mu_{p}^{*}}[X])+\delta\quad\hbox{for some
}\delta>0.$ (5.8)

Note that $\delta$ is strictly greater than zero if $h^{\prime\prime}(x)$ is
strictly positive. The rest of the proof of Lemma 5.3 hinges on this fact.

######  Claim 5.1.

The second derivative of $h(x)$ is strictly positive for $x\in I_{p}$.

###### Proof.

Let $x^{*}_{+}$ and $x^{*}_{4}$ be the non-zero fixed points of the function
when $\lambda=\bar{\lambda}+\Delta\lambda$ and $\lambda=4$, respectively;
$q_{4}$ is the value of $q(\lambda)$ given in Equation (5.2) evaluated at
$\lambda=4$. Note that
$I_{p}\subset[S_{\bar{\lambda}-\Delta\lambda}(q_{+}),x^{*}_{+}]\subset[S_{3}(q_{4}),x^{*}_{4}]$
since $3<\bar{\lambda}-\Delta\lambda<\bar{\lambda}+\Delta\lambda\leq 4$. So it
is sufficient to show that $h^{\prime\prime}(x)>0$ for all
$x\in[S_{3}(q_{4}),x^{*}_{4}]=[3q_{4}(1-q_{4}),\frac{3}{4}]$. A direct
calculation gives:

$h^{\prime\prime}(x)=-2(\bar{\lambda}+\bar{\lambda}^{2})+12\bar{\lambda}^{2}(x-x^{2})>-2(4+4^{2})+12\cdot
3^{2}(x-x^{2})=-40+108(x-x^{2}).$

Multiplying by $-\frac{1}{4}$, we see that $h^{\prime\prime}(x)>0$ between the
two roots of the quadratic $27x^{2}-27x+10$. These roots are:

$r_{1,2}=\frac{27\pm\sqrt{27^{2}-108}}{54}.$

By direct calculation,

$[r_{1},r_{2}]\supset\bigg{[}3q_{4}(1-q_{4}),\frac{3}{4}\bigg{]}\supset
I_{p}.$

Therefore, $h^{\prime\prime}(x)>0$ for all $x\in I_{p}$, which establishes
Equation (5.8) with $\delta>0$. ∎

Now we return to the proof of Lemma 5.3. By applying Lemma 5.2 twice and using
Equation (5.8) we get:

$E_{\mu_{q}^{*}}[X]=E_{\mu_{p}^{*}}[S_{\bar{\lambda}}(X)]=E_{\mu_{q}^{*}}[S^{2}_{\bar{\lambda}}(X)]=\bar{\lambda}E_{\mu_{q}^{*}}[h(X)]-\bar{\lambda}\epsilon=\bar{\lambda}h(E_{\mu_{q}^{*}}[X])+\bar{\lambda}(\delta-\epsilon).$

Rearranging the equation, we have

$\bar{\lambda}h(E_{\mu_{q}^{*}}[X])-E_{\mu_{q}^{*}}[X]=\bar{\lambda}(\epsilon-\delta)=H(E_{\mu_{q}^{*}}[X]).$
(5.9)

Since $\epsilon$ can be arbitrarily small, we can make sure that
$0<\epsilon<\delta$, so that $H(E_{\mu_{q}^{*}}[X])<0$. This means that
$E_{\mu_{q}^{*}}[X]$ is either less than $z_{H}<0$, greater than $q_{H}$, or
in the interval $(p_{H},x^{*}_{H})$ (see Figure 10). By Lemma 5.1, it is not
possible for $E_{\mu_{p}^{*}}[X]$ to be in the support of $\mu_{q}^{*}$, which
implies that $p_{H}<E_{\mu_{q}^{*}}[X]<x^{*}_{H}$. Because $H(x)>F(x)$,
$p(\bar{\lambda})<p_{H}$, from which we conclude that

$E_{\mu_{p}^{*}}[X]>p(\bar{\lambda})$ (5.10)

and, as promised, the “left peak” shifts with noise added, i.e., when
$\Delta\lambda>0$. ∎

Figure 10: $H(x)$ (dashed blue line) is the function which shifts $F(x)$
(solid orange line) up by a small amount, $\epsilon>0$. The curvature of the
two functions is identical, but the zeros of $H(x)$ are all shifted. In
particular, the zero at $p_{H}$ is to the right of $p(\bar{\lambda})$, which
is the zero of $F$ corresponding to the smaller period-2 point of
$S_{\bar{\lambda}}$.

In this discussion, we fix $\bar{\lambda}$ and consider values of the
parameter, $\lambda$, uniformly distributed between
$\bar{\lambda}-\Delta\lambda$ and $\bar{\lambda}+\Delta\lambda$. We want to
emphasize here and in what follows that the measures $\mu_{p}^{*}$ and
$\mu_{q}^{*}$ are functions of $\Delta\lambda$, defined in Equation (3.1). In
particular, we define a function:

$\displaystyle
V(\Delta\lambda)=E_{\mu_{q}^{*}(\Delta\lambda)}[X^{2}]-E_{\mu_{q}^{*}(\Delta\lambda)}[X]^{2}.$
(5.11)

That is, $V(\Delta\lambda)$ gives the variance of $X$ with respect to the
measure $\mu_{q}^{*}$, which is a function of $\Delta\lambda$. The variance
function, $V(\Delta\lambda)$ measures the difference between
$E_{\mu_{q}^{*}(\Delta\lambda)}[X^{2}]$ and
$E_{\mu_{q}^{*}(\Delta\lambda)}[X]^{2}$ as a function of $\Delta\lambda$.

The technical lemma, Lemma 5.4, is used to prove Lemma 5.5. We show directly
that the right-hand derivative of the function $V(\Delta\lambda)$ is zero when
$\Delta\lambda=0$. To show this, we us a very rough bound of the variance as
the squared difference between the maximum and minimum values of the support
of $\mu^{*}_{q}$.

######  Lemma 5.4.

$V_{+}^{\prime}(0)=\lim_{h\rightarrow 0^{+}}\frac{V(h+0)-V(0)}{h}=0$

###### Proof.

We first note that $V(0)=0$.

We consider the support of
$\mu_{q}^{*}(\cdot,\Delta\lambda,\bar{\lambda})=(x_{q,\min},x_{q,\max})$, the
right peak of the invariant distribution, $\mu^{*}$, in three distinct cases,
and prove the lemma for each case separately.

  * Case 1

If $\bar{\lambda}>1+\sqrt{5}$ and $\Delta\lambda$ is sufficiently small, in
which case $x_{q,\min}=S_{\bar{\lambda}-\Delta\lambda}(p_{+})$ and
$x_{q.\max}=S_{\bar{\lambda}+\Delta\lambda}(p_{-})$.

  * Case 2

If $\bar{\lambda}<1+\sqrt{5}$ and $\Delta\lambda$ is sufficiently small, in
which case $x_{q,\min}=q_{-}$ and $x_{q,\max}=q_{+}$.

  * Case 3

If $\bar{\lambda}=1+\sqrt{5}$ in which case $x_{q,\min}=q_{-}$ and
$x_{q,\max}=S_{\bar{\lambda}+\Delta\lambda}(\frac{1}{2})$.

  * Case 1

We start with $\bar{\lambda}>1+\sqrt{5}$, which means that
$p(\bar{\lambda})<\frac{1}{2}$. For small enough $\Delta\lambda$, it follows
that $p_{+}<p_{-}<\frac{1}{2}$, i.e., $S_{\lambda}$ is monotonically
increasing on the interval $[p_{+},p_{-}]$ for all
$\lambda\in[\bar{\lambda}-\Delta\lambda,\bar{\lambda}+\Delta\lambda]$. This
means that, for $x_{t}\in[p_{+},p_{1}]$, the smallest possible value of
$x_{t+1}$ is $S_{\bar{\lambda}-\Delta\lambda}(p_{+})\triangleq x_{q,\min}$ and
the largest possible value of $x_{t+1}$ is
$S_{\bar{\lambda}+\Delta\lambda}(p_{-})\triangleq x_{q,\max}$.

It follows that the squared difference of of the length of the interval from
$x_{q,min}$ to $x_{q,max}$ gives a (rather crude) upper bound for the
variance, i.e., $V(h)\leq(x_{q,\max}-x_{q,\min})^{2}$. Thus:

$\lim_{h\rightarrow 0^{+}}\frac{V(h)}{h}\leq\lim_{h\rightarrow
0^{+}}\frac{\left({S_{\bar{\lambda}-\Delta\lambda}(p_{+})-S_{\bar{\lambda}+\Delta\lambda}(p_{-})}\right)^{2}}{h}$

Writing $\Delta\lambda=h$, and expressing everything in terms of $q_{+}$ and
$q_{-}$:

$\displaystyle\left({S_{\bar{\lambda}-\Delta\lambda}(p_{+})-S_{\bar{\lambda}+\Delta\lambda}(p_{-})}\right)^{2}$
$\displaystyle=$
$\displaystyle\left({(\bar{\lambda}+h)p_{-}(1-p_{-})-(\bar{\lambda}-h)p_{+}(1-p_{+})}\right)^{2}$
(5.12) $\displaystyle=$
$\displaystyle\left({(\bar{\lambda}-h)p_{-}(1-p_{-})+2hp_{-}(1-p_{-})-(\bar{\lambda}+h)p_{+}(1-p_{+})+2hp_{+}(1-p_{+})}\right)^{2}$
$\displaystyle=$
$\displaystyle\left({(q_{+}-q_{-})-2h\left[{\frac{q_{-}}{\bar{\lambda}-h}+\frac{q_{+}}{\bar{\lambda}+h}}\right]}\right)^{2}$
$\displaystyle=$
$\displaystyle(q_{+}-q_{-})^{2}-4h(q_{+}-q_{-})\left[{\frac{q_{-}}{\bar{\lambda}-h}+\frac{q_{+}}{\bar{\lambda}+h}}\right]+$
$\displaystyle\hskip
59.75095pt4(h)^{2}\left[{\frac{q_{-}}{\bar{\lambda}-h}+\frac{q_{+}}{\bar{\lambda}+h}}\right]^{2}$

We now divide both sides of Equation (5.12) by $h$ and take the limit as $h$
goes to zero:

$\lim_{h\rightarrow 0^{+}}\frac{V(h)}{h}\leq\lim_{h\rightarrow
0^{+}}\left({\frac{(q_{+}-q_{-})^{2}}{h}-4(q_{+}-q_{-})\left[{\frac{q_{-}}{\bar{\lambda}-h}+\frac{q_{+}}{\bar{\lambda}+h}}\right]+4(h)\left[{\frac{q_{-}}{\bar{\lambda}-h}+\frac{q_{+}}{\bar{\lambda}+h}}\right]^{2}}\right)$
(5.13)

Since $\lim_{h\rightarrow 0^{+}}q_{-}=\lim_{h\rightarrow 0^{+}}q_{+}$, we see
that the middle term of Equation (5.13) goes to zero, and the last term goes
to zero because $h\rightarrow 0$. Expanding the first term using Equation
(5.2) gives:

$q_{+}-q_{-}=\frac{h}{(\bar{\lambda}+h)(\bar{\lambda}-h)}-\frac{(\bar{\lambda}+h)\sqrt{(\bar{\lambda}-h)^{2}-2(\bar{\lambda}-h)-3}}{2(\bar{\lambda}+h)(\bar{\lambda}-h)}+\frac{(\bar{\lambda}-h)\sqrt{(\bar{\lambda}+h)^{2}-2(\bar{\lambda}+h)-3}}{2(\bar{\lambda}+h)(\bar{\lambda}-h)}$

Squaring and dividing by $h$:

$\frac{(q_{+}-q_{-})^{2}}{h}=\mbox{(A)}+\mbox{(B)}+\frac{\mbox{(C)}}{\mbox{(D)}}$

where

(A) $\displaystyle=$
$\displaystyle\frac{h}{(\bar{\lambda}+h)^{2}(\bar{\lambda}-h)^{2}}$ (B)
$\displaystyle=$
$\displaystyle\frac{1}{(\bar{\lambda}+h)(\bar{\lambda}-h)}\left[{\frac{(\bar{\lambda}-h)\sqrt{(\bar{\lambda}+h)^{2}-2(\bar{\lambda}+h)-3}}{(\bar{\lambda}+h)(\bar{\lambda}-h)}-\frac{(\bar{\lambda}+h)\sqrt{(\bar{\lambda}-h)^{2}-2(\bar{\lambda}-h)-3}}{(\bar{\lambda}+h)(\bar{\lambda}-h)}}\right]$
(C) $\displaystyle=$
$\displaystyle(\bar{\lambda}-h)^{2}((\bar{\lambda}+h)^{2}-2(\bar{\lambda}+h)-3)+(\bar{\lambda}+h)^{2}((\bar{\lambda}-h)^{2}-2(\bar{\lambda}-h)-3)-$
$\displaystyle
2(\bar{\lambda}+h)(\bar{\lambda}-h)\sqrt{((\bar{\lambda}+h)^{2}-2(\bar{\lambda}+h)-3)((\bar{\lambda}-h)^{2}-2(\bar{\lambda}-h)-3)}$
(D) $\displaystyle=$ $\displaystyle
4h(\bar{\lambda}+h)^{2}(\bar{\lambda}-h)^{2}$

Taking the limit as $h$ goes to zero on both sides, we see that (A) is zero
because of the $h$ in the numerator; (B) is zero, since
$\bar{\lambda}-h\rightarrow\bar{\lambda}+h\rightarrow\bar{\lambda}$ as
$h\rightarrow 0^{+}$. We can see that $\frac{\mbox{(C)}}{\mbox{(D)}}$ also
goes to zero as $h\rightarrow 0^{+}$ by applying l’Hopital’s rule. Therefore,
$\displaystyle{\lim_{h\rightarrow
0^{+}}\frac{(q_{+}-q_{-})^{2}}{h}=\lim_{h\rightarrow 0^{+}}\frac{V(h)}{h}}=0$.

Since the variance is always positive, we have:

$0\leq\lim_{h\rightarrow 0^{+}}\frac{V(h)}{h}\leq\lim_{h\rightarrow
0^{+}}\frac{(x_{q,max}-x_{q,min})^{2}}{h}=0\Rightarrow
V^{\prime}_{+}(0)=\lim_{h\rightarrow 0^{+}}\frac{V(h)}{h}=0$

when $\bar{\lambda}>1+\sqrt{5}$.

  * Case 2

Next, when $\bar{\lambda}<1+\sqrt{5}$, for small $\Delta\lambda$, the function
$S_{\lambda}(x)$ is increasing on the interval $[p_{+},p_{-}]$ for all
$\lambda\in[\bar{\lambda}-\Delta\lambda,\bar{\lambda}+\Delta\lambda]$. It
follows that $x_{q,\min}=q_{-}$ and $x_{q,\max}=q_{+}$ so the proof
simplifies:

$\lim_{h\rightarrow 0^{+}}\frac{V(h)}{h}\leq\lim_{h\rightarrow
0^{+}}\frac{(q_{+}-q_{-})^{2}}{h}.$

It has already been shown that

$\lim_{h\rightarrow 0^{+}}\frac{(q_{+}-q_{-})^{2}}{h}=0,$

therefore

$0\leq\lim_{h\rightarrow 0^{+}}\frac{V(h)}{h}\leq\lim_{h\rightarrow
0^{+}}\frac{(q_{+}-q_{-})^{2}}{h}=0\Rightarrow
V^{\prime}_{+}(0)=\lim_{h\rightarrow 0^{+}}\frac{V(h)}{h}=0$

when $\bar{\lambda}<1+\sqrt{5}$.

  * Case 3

Suppose $\bar{\lambda}=1+\sqrt{5}$, so that the maximum of the function
coincides with the lower period-2 point when the parameter is $\bar{\lambda}$,
i.e., $p(\bar{\lambda})=\frac{1}{2}$. We claim that in this case

$\lim_{h\rightarrow 0^{+}}\frac{V(h)}{h}\leq\lim_{h\rightarrow
0^{+}}\frac{(S_{\bar{\lambda}+h}(1/2)-q_{-})^{2}}{h}.$ (5.14)

To see why this is true, we let $f(x)=p(x)=\frac{x+1-\sqrt{x^{2}-2x-3}}{2x}$
be the the smaller period-2 point when $\lambda=x$. We are interested in
values of $x$ near $1+\sqrt{5}$ that are in the stable period-2 regime, i.e.,
$3<x<1+\sqrt{6}$. We see that

$f^{\prime}(x)=\frac{-x-3-\sqrt{x^{2}-2x-3}}{2x^{2}\sqrt{x^{2}-2x-3}}<0.$

Furthermore,

$f^{\prime\prime}(x)=\frac{x^{3}+(\sqrt{x^{2}-2x-3}+3)x^{2}-(2\sqrt{x^{2}-2x-3}+9)x-3(\sqrt{x^{2}-2x-3}+3)}{x^{3}(x^{2}-2x-3)^{3/2}}>0\quad\hbox{
when }x>3.$

We conclude that $f(x)$ has negative slope and is concave up for $x$ close to
$1+\sqrt{5}$, which means that $f(1+\sqrt{5}-\Delta\lambda)$ is farther from
$f(1+\sqrt{5})$ than the distance from $f(1+\sqrt{5}+\Delta\lambda)$ to
$f(1+\sqrt{5})$. In other words, $p_{-}-\frac{1}{2}>\frac{1}{2}-p_{+}$. This
implies that

$x_{q,\min}=S_{\bar{\lambda}-\Delta\lambda}\left({p(\bar{\lambda}-\Delta\lambda)}\right)=q_{-}\quad.$

We know that $x_{q,\max}$ is the image of $\frac{1}{2}$ under
$S_{\bar{\lambda}+\Delta\lambda}$. Letting $\Delta\lambda=h$ we have:

$V(h)\leq(S_{\bar{\lambda}+h}(1/2)-q_{-})^{2}$

and Equation (5.14) is verified.

We now calculate the limit:

$\lim_{h\rightarrow 0^{+}}\frac{V(h)}{h}\leq\lim_{h\rightarrow
0^{+}}\frac{(S_{\bar{\lambda}+h}(1/2)-q_{-})^{2}}{h}.$

Notice that the numerator in the second limit is zero, since, as $h\rightarrow
0,p_{-}\rightarrow\frac{1}{2}$

$\Rightarrow q_{-}=S_{\bar{\lambda}-h}(p_{-})\rightarrow
S_{\bar{\lambda}}(1/2)=\lim_{h\rightarrow 0}S_{\bar{\lambda}+h}(1/2).$

Therefore, we apply l’Hopital’s rule to find:

$\lim_{h\rightarrow
0^{+}}\frac{(S_{\bar{\lambda}+h}(1/2)-q_{-})^{2}}{h}=\lim_{h\rightarrow
0^{+}}2\left({S_{\bar{\lambda}+h}(1/2)-q_{-}}\right)\frac{d}{dh}\left({S_{\bar{\lambda}+h}(1/2)-q_{-}}\right)$

This limit is zero as long as the last derivative term is finite. We verify
this:

$\frac{d}{dh}\left({S_{\bar{\lambda}+h}(1/2)-q_{-}}\right)=\frac{d}{dh}\left[{\frac{\bar{\lambda}+h}{4}-\frac{\bar{\lambda}-h+1-\sqrt{(\bar{\lambda}-h)^{2}-2(\bar{\lambda}-h)-3}}{2(\bar{\lambda}-h)}}\right]$
$=\frac{1}{4}+\frac{1}{4(\bar{\lambda}-h)^{2}}\left[{1+\frac{h-\bar{\lambda}+1}{2\sqrt{(\bar{\lambda}-h)^{2}-2(\bar{\lambda}-h)-3}}2(\bar{\lambda}-h)+2\left({\bar{\lambda}-h+1-\sqrt{(\bar{\lambda}-h)^{2}-2(\bar{\lambda}-h)-3}}\right)}\right]$

We now take the limit as $h\rightarrow 0$, remembering that
$\bar{\lambda}=1+\sqrt{5}$, so that
$\sqrt{\bar{\lambda}^{2}-3\bar{\lambda}=3}=1$, which gives:

$\displaystyle\lim_{h\rightarrow
0}\frac{d}{dh}\left({S_{\bar{\lambda}+h}(1/2)-q_{-}}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{4}+\frac{1}{4\bar{\lambda}^{2}}\left[{1+\bar{\lambda}(1-\bar{\lambda})+2\bar{\lambda}}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{4}+\frac{1}{4\bar{\lambda}^{2}}\left[{1-\bar{\lambda}^{2}+3\bar{\lambda}}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{4}\left({\frac{1+3\bar{\lambda}}{\bar{\lambda}^{2}}}\right)<\infty$

Therefore, by l’Hopital’s rule, the limit is zero, and

$\lim_{h\rightarrow 0^{+}}\frac{V(h)}{h}\leq\lim_{h\rightarrow
0^{+}}\frac{(S_{\bar{\lambda}+h}(1/2)-q_{-})^{2}}{h}=0$

∎

######  Lemma 5.5.

The change in the expected value of the left peak of the invariant
distribution is greater in magnitude than the change in the expected value of
the right peak.

$\frac{dE_{\mu_{p}^{*}}[X]}{d\Delta\lambda}\geq-\frac{dE_{\mu_{q}^{*}}[X]}{d\Delta\lambda}$

###### Proof.

From Lemma 5.4 we know that

$\frac{d}{d\Delta\lambda}\bigg{[}E_{\mu_{q}^{*}}[X]^{2}-E_{\mu_{q}^{*}}[X^{2}]\bigg{]}\bigg{|}_{\Delta\lambda=0}=V_{+}^{\prime}(0)=0$

Because $V^{\prime}(\Delta\lambda)$ is continuous from above at zero 111
$\lim_{\Delta\lambda\rightarrow
0}V^{\prime}(\Delta\lambda)=\lim_{\Delta\lambda\rightarrow
0}\lim_{h\rightarrow 0}\frac{V(\Delta\lambda+h)}{h}=\lim_{h\rightarrow
0}\frac{1}{h}\lim_{\Delta\lambda\rightarrow 0}V(\Delta\lambda+h)=0$ we know
that there exists a $\gamma$ such that
$|\gamma|<\frac{3-\bar{\lambda}}{\bar{\lambda}}\frac{d}{d\Delta\lambda}E_{\mu_{q}^{*}}[X]$
(note that $3-\bar{\lambda}$ and $\frac{d}{d\Delta\lambda}E_{\mu_{q}^{*}}[X]$
are both negative, so their product is positive) and

$V^{\prime}(\Delta\lambda)=\frac{d}{d\Delta\lambda}\bigg{[}E_{\mu_{q}^{*}}[X]^{2}-E_{\mu_{q}^{*}}[X^{2}]\bigg{]}=\gamma,$

for $\Delta\lambda$ small enough. Note that $\gamma$ might be positive or
negative, but for a small value of $\Delta\lambda$ we can find a small enough
$\gamma$ in magnitude such that the two expected values are sufficiently
close.

$\displaystyle\frac{d}{d\Delta\lambda}E_{\mu_{p}^{*}}[X]$ $\displaystyle=$
$\displaystyle\bar{\lambda}\cdot\frac{d}{d\Delta\lambda}\bigg{[}E_{\mu_{q}^{*}}[X]-E_{\mu_{q}^{*}}[X^{2}]\bigg{]},\mbox{
from Lemma }\ref{Lemma1}$ (5.15) $\displaystyle=$
$\displaystyle\bar{\lambda}\cdot\frac{d}{d\Delta\lambda}\bigg{[}E_{\mu_{q}^{*}}[X]-E_{\mu_{q}^{*}}[X]^{2}\bigg{]}+\bar{\lambda}\gamma$
$\displaystyle=$
$\displaystyle\bar{\lambda}\cdot\bigg{[}\frac{d}{d\Delta\lambda}E_{\mu_{q}^{*}}[X]+2E_{\mu_{q}^{*}}[X]\bigg{(}-\frac{d}{d\Delta\lambda}E_{\mu_{q}^{*}}[X]\bigg{)}\bigg{]}+\bar{\lambda}\gamma,\mbox{
note that }E_{\mu_{q}^{*}}[X]<0$ $\displaystyle>$
$\displaystyle\bar{\lambda}\cdot\bigg{[}\frac{d}{d\Delta\lambda}E_{\mu_{q}^{*}}[X]+2\bigg{(}\frac{\bar{\lambda}-1}{\bar{\lambda}}\bigg{)}\bigg{(}-\frac{d}{d\Delta\lambda}E_{\mu_{q}^{*}}[X]\bigg{)}\bigg{]}+\bar{\lambda}\gamma,\mbox{
by defn of
}\mu_{q}^{*},E_{\mu_{q}^{*}}[X]>\frac{\bar{\lambda}-1}{\bar{\lambda}}$
$\displaystyle=$
$\displaystyle(2-\bar{\lambda})\cdot\frac{d}{d\Delta\lambda}E_{\mu_{q}^{*}}[X]+\bar{\lambda}\gamma$
$\displaystyle>$
$\displaystyle(2-\bar{\lambda})\cdot\frac{d}{d\Delta\lambda}E_{\mu_{q}^{*}}[X]+\bar{\lambda}\bigg{(}\frac{\bar{\lambda}-3}{\bar{\lambda}}\frac{d}{d\Delta\lambda}E_{\mu_{q}^{*}}[X]\bigg{)},\mbox{
because
}\gamma>\frac{\bar{\lambda}-3}{\bar{\lambda}}\frac{d}{d\Delta\lambda}E_{\mu_{q}^{*}}[X]$
$\displaystyle=$ $\displaystyle-\frac{d}{d\Delta\lambda}E_{\mu_{q}^{*}}[X]$

∎

###### Proof.

of Theorem 3.2. From Lemma 5.3 we know that
$\displaystyle{\frac{dE_{\mu_{q}^{*}}[X]}{d\Delta\lambda}}<0$. Combining this
with Lemma 5.5, it follows that
$\displaystyle{\frac{dE_{\mu_{p}^{*}}[X]}{d\Delta\lambda}}$ is both positive
and greater in magnitude than
$\displaystyle{\frac{dE_{\mu_{q}^{*}}[X]}{d\Delta\lambda}}$. Thus, for some
interval around $\Delta\lambda=0$, $E_{\mu_{p}^{*}}[X]$ is farther from
$p(\bar{\lambda})$ than $E_{\mu_{q}^{*}}[X]$ is from $q(\bar{\lambda})$,
implying that their mean, $E[X]$, is larger than the expected value of the
deterministic map for $\lambda=\bar{\lambda}$. ∎

## Disclosure statement

The authors report there are no competing interests to declare.

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