1. Introduction
Coupled map lattices were introduced many years ago as a model of identical (nonlinear) dynamical systems interacting by means of some coupling (see, e.g., [Reference Bunimovich and Sinaĭ3, Reference Keller, Bandt, Graf and Zähle10, Reference Keller and Liverani11] and references therein). Many copies of the same dynamical system $f \colon X \to X$ are distributed over the points of some lattice Σ (that can be finite or infinite), and an interaction is used to couple the maps on different points. Among some phenomena that can be studied in this context, we have the propagation of some signals through the lattice, solitons, certain kinds of phase transitions when invariant measures are considered (see [Reference Keller, Künzle and Nowicki12]), the entropy of the map and its dependence on the coupling (see [Reference Bunimovich, Ming-Chia and Lyu4]), and the so-called synchronization where the orbits of distinct points of the lattice have the same asymptotic behavior (see, e.g., [Reference Jost and Joy9]; for an approach using the ideas of extreme value theory, see [Reference Faranda, Ghoudi, Guiraud and Vaienti7]).
The purpose of this work is to find some general conditions for ensuring synchronization when the usual linear coupling is replaced by a general Markov kernel, extending part of what appears in [Reference Jost and Joy9] to less symmetrical situations and also to understand what may happen when those conditions are relaxed. The dynamics on the base is assumed to be $C^1-$smooth, allowing also the use of the concept of Lyapunov exponent. Some numerical experiments suggest that part of the results about synchronization can remain valid for a Lipschitz map, but the infinitesimal argument used in the $C^1-$smooth context cannot be reproduced in this more general case.
We also relax some conditions on the lattice Σ, replacing this by a more general set that is assumed to be a Polish space in what we call a coupled map system; a similar idea appeared in [Reference Poignard13], where the lattice is replaced by a Cantor set, and a possible time discretization of the Amari neural field equation (see [Reference Amari1, Reference Bressloff2]) used in biomathematics leads to a case where the lattice is replaced by the real line. It is also interesting to notice that some classical analytical objects, like the Hardy–Littlewood operator acting on real functions, can be seen as limits of the action of a certain coupled map system where the lattice is replaced, again, by the real line. In these new situations, it is also possible to adapt some of the arguments used to get synchronization.
This paper is organized as follows: in $\S$ 2, we fix the notation and the main definitions. In $\S$ 3, we have the results about synchronization of $C^1-$smooth transformations for coupled map lattices and coupled map systems. In $\S$ 4, we present the proofs. In $\S$ 5, we introduce several classes of coupled map lattices and systems to which our results apply. In $\S$ 6, we provide some counter-examples which illustrate the need for the hypothesis of the theorems as well as the limitations in their conclusions. Some problems and future directions, part of them based on numerical experiments, are discussed in $\S$ 7.
2. Definitions
A topological space Σ is called a Polish space if it is separable, and there exists a metric d on Σ which defines the topology of Σ such that $(\Sigma,d)$ is complete. Every Polish space is a Radon space. In particular, every Borel measure on Σ is regular (see [Reference Schwartz15]). From now on Σ will denote a Polish space and $\mathscr{F}$ will be the Borel σ-algebra on Σ.
A stochastic kernel on Σ is any function $K \colon \Sigma \times\mathscr{F}\to [0,1]$ such that
(1) the function $B\mapsto K(x,B)$, from $\mathscr{F}$ to $[0,1]$, is a probability measure for any $x\in \Sigma$;
(2) the function $x\mapsto K(x,B)$, from Σ to $[0,1]$, is $\mathscr{F}$-measurable for any $B\in\mathscr{F}$.
The convolution of two kernels K 1 and K 2 on Σ is
The space of kernels on Σ is a convolution semigroup with identity, where the identity is the kernel $K(x,.)=\delta_x$, and δ x stands for the Dirac measure supported on x. The iterated kernels are defined recursively, setting $K^1=K$, and for $n \geq 1$,
A probability measure µ on $(\Sigma,\mathscr{F})$ is called K-stationary if for all $B\in\mathscr{F}$,
A set $B\in\mathscr{F}$ is said to be K-invariant when $K(x,B)=1$ for all $x\in B$ and $K(x,B)=0$ for all $x\in X\setminus B$. A K-stationary measure µ is called ergodic when there is no K-invariant set $B\in \mathscr{F}$ such that $0\lt\mu(B)\lt1$. As usual, ergodic measures are the extremal points in the convex set of K-stationary measures.
We say that a stochastic kernel K is strongly mixing if it admits a unique stationary probability measure µ with $\operatorname{{supp}}(\mu)=\Sigma$ and there are constants C > 0 and $0\lt\rho\lt1$ such that for every $\psi \in L^\infty(\Sigma)$, all $x\in\Sigma$ and $n\in\mathbb{N}$,
We say that K satisfies the Doeblin condition if there is a positive finite measure ρ on $(\Sigma,\mathscr{F})$ and some ɛ > 0 such that for all $x\in\Sigma$ and $B\in\mathscr{F}$,
Theorem 2.1 Let K be a stochastic kernel on $(\Sigma,\mathscr{F})$. If K satisfies the Doeblin condition, then there are sets $\Sigma_1,\ldots, \Sigma_m$ in $\mathscr{F}$ and probability measures $\nu_1,\ldots, \nu_m$ on Σ such that for all $i,j=1,\ldots, m$,
(1) $\Sigma_i\cap \Sigma_j=\emptyset$ when i ≠ j,
(2) $\Sigma_i$ is K-forward invariant, that is, $K(x,\Sigma_i)=1$ for $x\in\Sigma_i$,
(3) ν i is K-stationary and ergodic with $\nu_i(\Sigma_j)=\delta_{ij}$,
(4) $\lim_{n\to+\infty} K^{n}(x,\Sigma_1\cup \cdots \cup\Sigma_m)=1$, with geometric uniform speed of convergence, for all $x\in\Sigma$,
(5) $\nu(\Sigma_1\cup \cdots \cup\Sigma_m)=1$, for every K-stationary probability ν.
Moreover, for every $1\leq i\leq m$, there is an integer $p_i\in\mathbb{N}$ and measurable sets $\Sigma_{i,1},\ldots, \Sigma_{i,p_i}\in\mathscr{F}$ such that
(1) $\{\Sigma_{i,1},\ldots, \Sigma_{i,p_i}\}$ is a partition of $\Sigma_i$,
(2) $K(x,\Sigma_{i,j+1})=1$ for $x\in \Sigma_{i,j}$ and $1\leq j \leq p_i$, with $\Sigma_{i,p_i+1}=\Sigma_{i,1}$,
(3) the stochastic kernel $K^{p_i}$ on $ \Sigma_{i,j}$ is strongly mixing for all $1\leq j \leq p_i$.
Proof. See [Reference Doob5, section V-5].
In the context of Doeblin condition, we call the sets $\Sigma_i$, $1 \leq i \leq m$, the ergodic components of K and the integers p i their periods. We call the sets $ \Sigma_{i,j}$, $1 \leq j \leq p_i$, the mixing subcomponents of the ergodic component $\Sigma_i$. We shall say that an ergodic component is aperiodic when its period is equal to 1.
A stochastic kernel K determines an operator $K:L^\infty(\Sigma)\to L^\infty(\Sigma)$ defined by
This operator satisfies the following.
Proposition 2.2. For any $\psi\in L^\infty(\Sigma)$,
(a) $K\mathbb{1}=\mathbb{1}$, where $\mathbb{1}$ denotes the constant function 1.
(b) $\int_\Sigma K \psi\, {\rm d}\mu= \int_\Sigma \psi\,{\rm d}\mu$,
(c) $\left\|{K \psi}\right\|_\infty \leq \left\|{\psi}\right\|_\infty$.
In particular, if $H_0=\{\psi\in L^\infty(\Sigma)\colon \int_\Sigma \psi\,{\rm d}\mu=0\}$, then $L^\infty(\Sigma) =\mathbb{R}\mathbb{1}\oplus H_0$ is a K-invariant decomposition and $\rho(K\vert_{H_0})\leq 1$.
Proof. Since $\int_\Sigma K(x,dy) =1$, items (a) and (c) follow. Item (b) is a consequence of µ being a K-stationary probability measure.
If the kernel K on Σ is strongly mixing, we define its mixing rate to be the spectral radius of the restriction of K to the invariant subspace H 0 in Proposition 2.2, that is,
In general, if K satisfies the Doeblin condition, using the notation of Theorem 2.1, we define its escape rate $\beta^*(K)$ to be the spectral radius of the operator $r_\Lambda\circ K\vert_{L^\infty(\Lambda)}$, where $\Lambda=\Sigma\setminus (\Sigma_1\cup \cdots \cup \Sigma_m)$ and $r_\Lambda:L^\infty(\Sigma)\to L^\infty(\Lambda)$ is the restriction operator. Notice that the operator $K_\Lambda:L^\infty(\Lambda)\to L^\infty(\Lambda)$, $K_\Lambda= r_\Lambda\circ K\vert_{L^\infty(\Lambda)}$ is sub-stochastic and hence has spectral radius less than 1. We also define the mixing rate of the component $\Sigma_i$ as $\tau^*(K,\Sigma_i):=\sqrt[p_i]{\tau^\ast(K^{p_i},\Sigma_{i,j})} $, where $\tau^\ast(K^{p_i},\Sigma_{i,j})$ denotes the mixing rate of the strongly mixing stochastic kernel obtained by restriction of the kernel $K^{p_i}$ to any of the mixing subcomponents $\Sigma_{i,j}$, $1 \leq j \leq p_i$.
Let Σ be a discrete space, that we call a lattice. Consider a convex and compact set $X \subset \mathbb{R}^d$, a map $f \colon X \to X$, the base dynamical system and a probability kernel $K \colon \Sigma \times \Sigma \to [0,1]$. If Σ is finite with k elements, then the stochastic kernel K can be identified with a k × k stochastic matrix. We recall that a (row) stochastic matrix on a finite set Σ is any square matrix $A=[a_{ij}] \in \mathbb{R}^{k \times k}$ such that $a_{ij} \geq 0$ for all $i,j \in \Sigma$ and $\sum_{j \in \Sigma} a_{ij}=1$ for all $i \in \Sigma$. A stochastic matrix A is called primitive if there exists $n \geq 1$ for which the power matrix $A^n = [a_{ij}^{n}]$ has all entries strictly positive, that is, $a_{ij}^{n} \gt 0$ for $i, j \in \Sigma$. Notice that if A is primitive, then the kernel K on Σ determined by A is strongly mixing. In particular, Σ is an aperiodic ergodic component. We define the mixing rate of a primitive matrix A, denoted by $\tau^*(A)$, as the mixing rate of the strongly mixing kernel defined by A on Σ.
The coupled map lattice is the dynamical system $F:X^{\Sigma}\to X^\Sigma$ defined by
where $X^{\Sigma}$ denotes the space of all functions from Σ to X. Since K is a probability kernel, we get that for the constant function $\varphi = c$, the image $K (f\circ \varphi )= K f(c) = f(c)$. Hence, the set of constant functions is invariant under the dynamics defined by F.
More generally, given a Polish space Σ, consider the space $L^\infty(\Sigma,X)$ of all measurable functions $\varphi:\Sigma\to X$. Notice that since X is compact, all functions in $L^\infty(\Sigma,X)$ are bounded. We endow this space with the uniform convergence topology, which is determined by the uniform distance
As before, we can define a transformation $F:L^\infty(\Sigma,X)\to L^\infty(\Sigma,X)$ by
which we will refer to as a coupled map system.
We say that the coupled map system $F:L^\infty(\Sigma,X)\to L^\infty(\Sigma,X)$ has global synchronization over a subset $\Lambda\subset \Sigma$ if for every δ > 0, there exists $n_0 \in \mathbb{N}$ such that for $n \geq n_0$ and for every $\varphi \in L^\infty(\Sigma,X)$ and all $x,y \in \Lambda$, one has
When $\Lambda=\Sigma$, we simply talk about global synchronization of F. Let $\Delta \subset L^\infty(\Sigma,X)$ denote the compact and convex subset of all constant functions. Notice that F has global synchronization if and only if Δ is a global attractor of F on $L^\infty(\Sigma,X)$.
We say that F has local synchronization over a subset $\Lambda\subset \Sigma$ if there exists a neighbourhood $\mathscr{U}$ of Δ in $L^\infty(\Sigma,X)$ such that for every δ > 0, there exists $n_0 \in \mathbb{N}$ such that for $n \geq n_0$ and for every $\varphi \in \mathscr{U}$ and all $x,y \in \Lambda$, one has
The map F has local synchronization (over Σ) if and only if Δ is a local attractor of F on $L^\infty(\Sigma,X)$.
Clearly, global synchronization implies local synchronization, but the converse is not true.
3. Main results
Throughout this section, we assume that $X\subset \mathbb{R}^d$ is a compact convex set and $f:X\to X$ is a C 1-smooth map whose Lipschitz constant is denoted by $\operatorname{{Lip}}(f)$.
We define
This limit exists by Fekete’s Lemma. As usual, the derivative of the map f at a point $x\in X$ is denoted by $Df_x:\mathbb{R}^d\to \mathbb{R}^d$. The top Lyapunov exponent of a periodic point $x=f^n(x)$ with period n is defined to be
By the mean value theorem, $\left\|{D f_x}\right\|\leq \operatorname{{Lip}}(f)$. Hence, for all periodic points $x\in X$ of f, one has
We define the oscillation semi-norm $\left\|{{.} }\right\|_{\mathrm{o}}$ of a function $\varphi \in L^\infty(\Sigma,\mathbb{R}^d)$ by
Proposition 3.1. The oscillation semi-norm satisfies
(1) $\left\|{{g \circ \varphi} }\right\|_{\mathrm{o}} \leq \operatorname{{Lip}}(g) \left\|{{\varphi} }\right\|_{\mathrm{o}}$ for any Lipschitz function $g \colon \mathbb{R}^d \to \mathbb{R}^d$, and
(2) $\left\|{{\varphi} }\right\|_{\mathrm{o}}=0 \Leftrightarrow \varphi \,\, \mathrm{is constant}$.
Proof. Given any Lipschitz function $g \colon \mathbb{R}^d \to \mathbb{R}^d$,
and, clearly, $\left\|{{\varphi} }\right\|_{\mathrm{o}}=0$ if and only if φ is constant.
We define the norm of a stochastic kernel K, regarded as an operator, by
Theorem 3.2 If $\vert\!\vert\!\vert {K} \vert\!\vert\!\vert_{\mathrm{o}}\, \operatorname{{Lip}}(f) \lt 1$, then F has global synchronization.
Proof. Given $\varphi\in L^\infty(\Sigma,X)$,
Hence,
converges to 0 as $n\to +\infty$. This proves that F has global synchronization.
Theorem 3.3 Given a finite set Σ and a stochastic primitive matrix A on Σ, if there exists a periodic orbit $x\in X$ with Lyapunov exponent $\lambda=\lambda(f,x)$ such that
then F does not have local synchronization over Σ.
Theorem 3.4 Given a Polish space Σ and a stochastic kernel K on Σ satisfying the Doeblin condition with a unique aperiodic ergodic component Σ0, if
then F has local synchronization over Σ.
The next theorem provides a sufficient condition for local synchronization over the mixing subcomponents of an ergodic component.
Theorem 3.5 Given a Polish space Σ and a stochastic kernel K on Σ satisfying the Doeblin condition with an ergodic component
where the sets $\Sigma_{i,j}$ stand for the mixing subcomponents of $\Sigma_i$, if
then F has local synchronization over all mixing subcomponents $\Sigma_{i,j}$ of $\Sigma_{i}$.
4. Proofs
Proof of Theorem 3.3
Let $\Delta=\{u\in X^\Sigma\colon u_i=u_j\ \forall\, i,j\in\Sigma \}$. The transformation F has local synchronization over Σ iff Δ is a local attractor whose basin of attraction is a neighbourhood of Δ. To see that F does not have local synchronization over Σ, it is enough to find the F-periodic point $u\in \Delta$ with period p such that the derivative $D F^p_u$ has an eigenvalue $\alpha\in\mathbb{C}$ with $\left\vert {\alpha}\right\vert\gt1$ associated to some (complex) eigenvector, which does not lie in the complexification of the tangent space $T_u\Delta$.
Consider the periodic point $x=f^p(x)$ whose existence is hypothesized in this theorem, and let $u=(x,x,\ldots, x)\in X^\Sigma$. Then $u\in\Delta$ is a periodic point of F with the same period p. Let $w\in\mathbb{C}^\Sigma$ be an eigenvector of A such that $\sum_{i\in\Sigma} w_i=0$ and $Aw=\beta w$, where $\left\vert {\beta}\right\vert=\tau^\ast(A)$. Writing $A=[a_{ij}]\in \mathbb{R}^{\Sigma\times \Sigma}$, let $\hat A\in (\mathbb{R}^{d\times d})^{\Sigma\times\Sigma}$ be the matrix with entries $\hat a_{ij}:= a_{ij} I_d$, where I d is the identity matrix in $\mathbb{R}^{d\times d}$. Let $M_j= Df_{f^j(x)}$ and denote by $D_{M_j}$ the block diagonal matrix
Finally, let $v\in\mathbb{C}^d$ be a non-zero vector such that $D f^p_x\, v= M_{p-1}\cdots M_1\, M_0\, v= \alpha\, v$ with $\alpha\in\mathbb{C}$ and $\frac{1}{p}\,\log \left\vert {\alpha}\right\vert=\lambda(x,f)$. The Jacobian matrix of F at each point $F^j(u)$ is the matrix
Since the matrices $\hat A$ and $D_{M_j}$ commute, we have
where $M= M_{p-1}\cdots M_1\, M_0=D f^p_x$. Now the complex vector $V=(w_i v)_{i\in \Sigma}\in (\mathbb{C}^d)^\Sigma$ satisfies
Since the vector V does not lie in the complexification, $\Delta_{\mathbb{C}}=\{(w,w,\ldots, w)\colon w\in\mathbb{C}\}$ of the tangent space $T_u\Delta$ and
the F-invariant set Δ cannot be a local attractor.
Proof of Theorem 3.4
Assume first that the stochastic kernel K is strongly mixing, and consider the space
introduced in Proposition 2.2. Let $\tau^*=\tau^*(K)\lt1$ be the spectral radius of $K\vert_{H_0}$.
In the end, we explain how to proceed in the general case. Taking κ > 0 with $\tau^*(K)\, {\rm e}^{\ell(f)}\lt\kappa \lt1$, there exists an integer $m\in\mathbb{N}$ such that $\vert\!\vert\!\vert {K^m} \vert\!\vert\!\vert_{\mathrm{o}}\,\operatorname{{Lip}}(f^m)\lt\kappa^m $.
Because f is $C^1-$smooth, then so is F. On the diagonal Δ, the derivatives of the linear action of K and f commute. Consequently, for all $\varphi\in\Delta$
with $0\lt\kappa\lt1$. Hence, there is a neighbourhood $\mathscr{U}$ of Δ where F n is a contraction. This implies that for all $\varphi\in \mathscr{U}$,
Thus, for all $0\leq j\lt n$,
and therefore
The case where K is not strongly mixing but there is only one ergodic acyclic component Σ0 is the same as the previous one because the operator $K\vert_{H_0}$ has spectral radius equal to $\max\{\beta^*(K),\tau^*(K,\Sigma_0)\}$.
Proof of Theorem 3.5
Same argument as before working in the mixing subcomponent.
5. Coupled map classes
In this section, we introduce several classes of coupled map systems and lattices to which our results apply. In all examples below, $X\subset \mathbb{R}^d$ is any compact convex set and $f:X\to X$ is a $C^1-$smooth function.
Finite coupled map lattices
Consider a finite set $\Sigma=\{1,\ldots, k\}$ and a stochastic k × k matrix $A=[ a_{ij} ]$. The associated coupled map lattice is the transformation $F:X^k\to X^k$ with components
In this setting, Theorem 2.1 is a simple consequence of classical Markov Chain’s Theory. Let $\Sigma_1,\ldots, \Sigma_m$ be the ergodic components of the stochastic kernel K defined by A on Σ, and let $\Lambda=\Sigma\setminus (\Sigma_1\cup \ldots \cup \Sigma_m)$. The escape rate $\beta^*(K)$ and the mixing rate $\tau^*(K,\Sigma_i)$ can be estimated as described below. See [Reference Doob5, section V-2] and also [Reference Duarte and Torres6, section 5]Footnote *. Recall that $\tau^*(K,\Sigma_i):=\sqrt[p_i]{\tau^\ast(K^{p_i},\Sigma_{i,j})} $, where $\tau^\ast(K^{p_i},\Sigma_{i,j})$ denotes the mixing rate of the strongly mixing stochastic kernel obtained by restriction of the kernel $K^{p_i}$ to any of the subcomponents $\Sigma_{i,j}$. Let $A^{p_i}=[a_{ij}^{p_i}]$. We have that
where
and
where
This class contains, for example, coupled map lattices with translation-invariant coupling and arbitrary Lipschitz continuous individual map on the periodic lattice $\mathbb{Z}_L:=\{s \in \mathbb{Z} \, \mbox{mod} \, L\}$ $(L\gt1)$.
Infinite coupled map lattices
Assume now that Σ is an infinite (countable) lattice, say $\Sigma=\mathbb{N}=\{0,1,\ldots\}$. Consider an infinite (row) stochastic matrix $A=[ a_{ij} ]$ such that for some α > 0, we have $a_{i0}\geq \alpha$ for all $i\in \mathbb{N}$. This assumptions ensure that the stochastic kernel defined by A satisfies the Doeblin condition.
The associated coupled map lattice is the transformation $F:X^{\mathbb{N}}\to X^{\mathbb{N}}$ with components
The previous finite state formulas also hold in this countable case.
Coupled map systems
Assume that Σ is a compact metric space, say $\Sigma=[0,1]$, and consider a bounded measurable function $k: [0,1]\times [0,1] \to [0,+\infty)$ such that
This function determines a kernel K on $[0,1]$ defined by
In this setting, the stochastic kernel K satisfies the Doeblin condition.
The associated coupled map system is the transformation $F:L^\infty([0,1],X)\to L^\infty([0,1],X)$
Remark 5.1. It is interesting to mention that the uncoupled map, say, the case where $F(\varphi)(x) = f(\varphi(x))$, corresponds to a dynamical system already considered in [Reference Huang and Feng8].
Remark 5.2. We note that the class of coupled map systems may contain the mean field limit of all-to-all coupled finite systems. In [Reference Sélley and Tanzi16], the authors study models of this type, with very similar dynamics to our current setup of coupled map systems.
The previous given formulas for the escape rate and the mixing rate of the ergodic components $\Sigma_i$, $1 \leq i \leq m$, extend to stochastic kernels satisfying the Doeblin condition (see [Reference Doob5, section V-5] and also [Reference Duarte and Torres6, section 5])Footnote *. We have that
where
and
where
Example 5.3. The Amari neural field equation used in biomathematics is the integro-differential equation
where u represents the average neural activity, ${\mathcal K}$ is the connectivity kernel (modelling the interaction between neurons at distinct places) and S is the firing rate function. A first-order time discretization, considering a time step of size 1, consists in replacing the partial derivative by the difference $u(t+1, x) - u(t, x)$; we then get the map
which is an example of a coupled map system where Σ is the real line (for another approximation of the Amari’s model, the reader can see [Reference Salasnich14]).
Example 5.4. Consider an integrable function $\varphi \colon \mathbb{R} \to \mathbb{R}$. The Hardy–Littlewood maximal operator maps φ to the function that is defined for any $x \in \mathbb{R}$ as
where, for any given r > 0, F r is given by
in this expression, ${\rm d}y$ is the Lebesgue measure on $\mathbb{R}$, $B(x, r)$ is the open interval $(x-r, x+r)$ and $|B|$ denotes its Lebesgue measure. Notice that F r corresponds to the coupled map system, where $X = \Sigma = \mathbb{R}$, $f(\cdot) = | \cdot |$ and $k(x, y) = \frac{dy}{|B(x, r)|}$ is the normalized Lebesgue measure restricted to the ball $B(x, r)$.
6. Counter-examples
In this section, we provide two counter-examples which illustrate the need for the hypothesis of the theorems as well as the limitations in their conclusions.
In the first example, the hypothesis of Theorem 3.4 is satisfied but the coupled lattice map has no global synchronization.
Example 6.1. Let $X=[0,3]$ and $f:X\to X$ be a $C^1-$smooth map such that $f(1)=2/3$, $f(8/5)=2$, $f(2)=16/9$ and
An example (see Figure 1) is given by
Indeed, given $(t_0, x_0, v_0), (t_1, x_1, v_1)\in\mathbb{R}^3$ with $t_0\lt t_1$, there exists a unique cubic polynomial p(t) that interpolates these tuples in the sense that
The above function is the unique piecewise cubic polynomial of class C 1 that interpolates the sequence of tuples
One can easily check that, in each branch, the maximum absolute value of $f^\prime(t)$ is less than or equal to $20/9$.
Consider the stochastic matrix
and the coupled lattice map $F:[0,3]^3 \to [0,3]^3$ defined by
The point $p=(8/5, 1, 2)$ is a fixed point of F. Because this fixed point is off the diagonal $\Delta=\{(x,x,x)\colon x\in [0,3] \}$, the map F cannot have global synchronization.
The matrix A has eigenvalues 1 and $\frac{1}{10}(2 \pm \sqrt{5}\,i)$ (with absolute value $3/10$). The stochastic kernel K determined by A has a unique aperiodic ergodic component $\Sigma_0=\Sigma$. Hence, $\tau^*(K,\Sigma)=3/10$ and $\beta^*(K)=0$. Consequently,
The second example illustrates a transition from local synchronization over Σ to local synchronization over an ergodic component.
Example 6.2. Consider the following family of stochastic matrices
with $0\leq t\leq 1$, whose associated family of kernels K t has a unique aperiodic ergodic component $\Sigma_0=\{1,2\}$. Applying the formulas (2)–(3), we get $\beta^*(K_t) = \frac{11-10\,t}{18}$ and $\tau^*(K_t,\Sigma_0)=\frac{5t}{18}$.
Let $f:[0,1]\to [0,1]$ be the map $f(x):= 4x(1-x)$, which has Lipschitz constant $\operatorname{{Lip}}(f)=4$. We have ${\rm e}^{\ell(f)}= 4$ because 0 is a fixed point with derivative $f^\prime(0)=4$.
In this example, the threshold conditions of Theorems 3.4 and 3.5 are, respectively,
An analog of the threshold condition in Theorem 3.3 with Lyapunov exponent $\lambda(f)=\log 2$ w.r.t. Lebesgue measure is
See Figure 2.
(1) in region ( $\mathscr{R}_1$), (local) synchronization over Σ occurs;
(2) in region $(\mathscr{R}_2)$, (local) synchronization over Σ0 occurs.
For $t\lt0.65=13/20$, an easy calculation shows that $p_t=(0,0,\frac{20 t-13}{20 t-22})$ is a fixed point of the coupled map F t off the diagonal. This shows that there is no global synchronization for F t on this range.
We run several numerical experiments, randomly choosing an initial condition in $[0,1]^3$, computing n = 200 iterates and then plotting the last 100 iterates. These experiments indicate that (see Figures 3 and 4):
(3) in region ( $\mathscr{R}_1$), global synchronization over Σ occurs;
(4) in region ( $\mathscr{R}_3$), local synchronization over Σ occurs;
(5) outside region ( $\mathscr{R}_3$), synchronization over Σ never occurs.
7. Conclusions
All examples that we have analyzed indicate that Theorems 3.4 and 3.5 should hold for any Lipschitz map $f:X\to X$. We note that the infinitesimal argument used in the $C^1-$smooth context cannot be reproduced in the Lipschitz case.
This motivates the following problem:
Problem 1.
Prove Theorems 3.4 and 3.5 for Lipschitz maps $f:X\to X$, or at least for piecewise smooth maps.
The two (Lipschitz) examples below provide numerical evidence that there is no local synchronization whenever the main hypothesis of Theorems 3.4 and 3.5 fail.
Example 7.1. Consider the following family of stochastic matrices
with $0\leq t\leq 1$, whose associated family of kernels K t has a unique aperiodic ergodic component $\Sigma_0=\{1,2\}$. Applying the formulas (2)–(3), we have that $\beta^*(K_t) = \frac{9-5t}{12}$ and $\tau^*(K_t,\Sigma_0)=\frac{3t}{5}$.
Let $f:[0,1]\to [0,1]$ be the piecewise linear map
which has Lipschitz constant $\operatorname{{Lip}}(f)=2$. A simple calculation shows that ${\rm e}^{\ell(f)}= 2$.
In this example, the threshold condition
of Theorem 3.4 is equivalent to $0.6=\frac{3}{5} \lt t \lt \frac{5}{6} \sim 0.833$ (see Figure 5). Experiments we run indicate that in this example, local synchronization never occurs whenever the threshold condition fails, that is, $t\lt\frac{3}{5}$ or $t\gt\frac{5}{6}$ (see Figure 6).
In the simulations, we randomly chose initial conditions near the diagonal Δ in $[0,1]^3$ and computed n = 200 iterates, plotting the last 100 iterates.
Example 7.2. Consider the following family of stochastic matrices
with $0\leq t\leq 1$. Each associated kernel K t has a unique ergodic component $\Sigma_1 = \Sigma=\{1,2,3,4\}$ with period 2. The mixing subcomponents are $\Sigma_{1,1}=\{1,2\}$ and $\Sigma_{1,2}=\{3,4\}$. Since the eigenvalues of A t are 1, −1, $\frac{5\,t}{8}$ and $-\frac{5\,t}{8}$, we have that $\tau^*(K_t,\Sigma_1)=\frac{5\,t}{8}$.
Let $f:[0,1]\to [0,1]$ be the piecewise linear map
which has Lipschitz constant $\operatorname{{Lip}}(f)=2$. A simple calculation shows that ${\rm e}^{\ell(f)}= 2$.
In this example, the threshold condition
of Theorem 3.5 is equivalent to t < 0.8. Experiments we run have shown that in this example, local synchronization over the mixing subcomponents never occurs whenever the threshold condition fails, that is, t > 0.8.
In the simulations, we randomly chose initial conditions near the diagonal Δ in $[0,1]^4$ and computed n = 200 iterates, plotting the last 100 iterates. Figures 7 and 8 illustrate the projections of these orbits in $[0,1]^4$ to all six coordinate 2-planes.
Example 6.2 (in the smooth setting) and Examples 7.1 and 7.2 (in the Lipschitz case) suggest a weaker concept of synchronization.
Let $f:X\to X$ be a piecewise C 1-diffeomorphism preserving a probability measure µ on the Borel σ-algebra of X. Consider on $L^\infty(\Sigma,X)\subset X^\Sigma$ the product measure $\mu^\Sigma$. We say that a coupled map system has almost sure global synchronization over $\Lambda\subset \Sigma$ if for $\mu^\Sigma$ almost every $\varphi\in L^\infty(\Sigma,X)$, given δ > 0, there exists $n_0 \in \mathbb{N}$ such that for $n \geq n_0$ and all $x,y \in \Lambda$, one has
In Example 6.2, the simulations indicate that almost sure global synchronization over $\Sigma=\{1,2,3\}$ occurs for t > 0.2, in spite of the fact that the existence of a fixed point off the diagonal implies no global synchronization for t < 0.65.
This motivates the following problems:
Problem 2.
Under the assumptions of Theorem 3.4, denoting by $\lambda(f)=\lambda(f,\mu)$ the top Lyapunov exponent of f w.r.t. µ, if
can one prove that almost sure global synchronization (over Σ) occurs?
Problem 3.
Under the assumptions of Theorem 3.5, denoting by $\lambda(f)=\lambda(f,\mu)$ the top Lyapunov exponent of f w.r.t. µ, if
can one prove that almost sure global synchronization over all mixing subcomponents occur?
In other words, we wonder if almost sure global synchronization occurs when we replace, in the threshold conditions of Theorems 3.4 and 3.5, the Lipschitz term $\ell(f)$ by the Lyapunov exponent $\lambda(f)$.
Acknowledgements
The authors would like to thank the referee for his useful comments and suggestions.
Funding Statement
All authors were partially supported by the Project ‘New trends in Lyapunov exponents’ (PTDC/MAT-PUR/29126/ 2017). A.B. would like to thank the hospitality of Universidade de Lisboa. P.D. was partially supported by CMAFCIO through FCT project UIDB/04561/2020. M.J.T. was partially financed by Portuguese Funds through FCT (Fundação para a Ciência e a Tecnologia) within the Projects UIDB/00013/2020 and UIDP/00013/2020.
Competing Interests
The authors declare none.