Lifted lattices, hyperbolic structures, and topological disorders in coupled map lattices.

*(English)*Zbl 0868.58059A coupled map lattice (CML) of \(d\)-dimensional and \(p\)-component unbounded media is a system of the following form:
\[
u_j(n+1)=F(\{u_j(n)\}^s)\tag{1}
\]
where \(n\in\mathbb{Z}^+\) is a discrete-time step, \(j\in \mathbb{Z}^d\), \(\mathbb{Z}^d\) is the \(d\)-dimensional integer lattice provided with the norm
\[
|j_1|:=\max(|j|,\dots,|j_d|),\{u_j(n)\}^s=\{u_i(n): i\in\mathbb{Z}^d,\;|i-j|\geq s\},\quad s\in \mathbb{Z}^+,\quad u_j(n)\in\mathbb{R}^p.
\]
There are two important dynamical systems associated with (1): the evolutional dynamical system and the translational one. One investigates solutions of (1) of various types: standing waves, travelling waves, spatial (temporal, spatiotemporal) disorder or chaos. The paper presents a general approach to characterize the stability and hyperbolicity of various solution structures and to study of the existence of solutions of the types mentioned above by introducing a so-called lifted lattice of the form
\[
w_{j,k}(n+1)=F(\{w_{j,k-1}(n)\}^s)\tag{2}
\]

\[ \text{where }k\in\mathbb{Z}, \quad j\in\mathbb{Z}^d,\quad \{w_{j,k-1}(n)\}^s=\{w_{i,k-1}(n):i\in \mathbb{Z}^d,|i-j|\leq s\}. \] The method is based on the fact that there is a one-to-one correspondence between stationary solutions of (2) and solutions of (1) having backward extensions (e.g. standing solutions, periodic solutions, travelling waves etc.). Types of hyperbolicity of stationary solutions of (2) give information about stability and hyperbolicity of solutions of (1). Section 4 gives theorems on existence of spatial, temporal and spatiotemporal (topological) disorders, Section 5 presents some discussion on time-almost-periodic CLMs; the existence of almost-periodic solutions in a discrete-time almost-periodic Nagumo equation (being a “discrete version” of the equation \(u_t=\Delta u+f(u))\) is proved.

\[ \text{where }k\in\mathbb{Z}, \quad j\in\mathbb{Z}^d,\quad \{w_{j,k-1}(n)\}^s=\{w_{i,k-1}(n):i\in \mathbb{Z}^d,|i-j|\leq s\}. \] The method is based on the fact that there is a one-to-one correspondence between stationary solutions of (2) and solutions of (1) having backward extensions (e.g. standing solutions, periodic solutions, travelling waves etc.). Types of hyperbolicity of stationary solutions of (2) give information about stability and hyperbolicity of solutions of (1). Section 4 gives theorems on existence of spatial, temporal and spatiotemporal (topological) disorders, Section 5 presents some discussion on time-almost-periodic CLMs; the existence of almost-periodic solutions in a discrete-time almost-periodic Nagumo equation (being a “discrete version” of the equation \(u_t=\Delta u+f(u))\) is proved.

Reviewer: A.Pelczar (Kraków)

##### MSC:

37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |

54H20 | Topological dynamics (MSC2010) |

37J40 | Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion |

37D99 | Dynamical systems with hyperbolic behavior |