Narrow Results

Stability of steady-state solutions of 1-dim coupled map lattices is studied. The stability is determined by the spectrum of linear operators on two-sided sequences of vectors in arising as a linearization of the corresponding nonlinear evolution operators. Theoretical results are applied to several examples.

Coupled map lattices with multidimensional lattice are considered. A method for the determination of the stability of spatially homogeneous and spatially periodic steady-state solutions is derived. This method is based on the determination of the spectrum of the linearized operator by means of Gelfand transformation of some appropriate Banach algebra. The results are applied to several examples.

This paper deals with the existence of monotonic traveling and standing wave solutions for a certain class of lattice differential equations. Employing the techniques of monotone iteration coupled with the concept of upper and lower solutions in the theory of monotone dynamical systems, we can classify the monotonic traveling wave solutions with various asymptotic boundary conditions. For the case of zero wave speed, a novel discrete monotone iteration scheme is established for proving the existence of monotonic standing wave solutions. Applications are made to several models including cellular neural networks, original and modified RTD-based cellular neural networks. Numerical simulations of the monotone iteration schemes are also given.

Individual sites in spatially extended systems of coupled identical maps may exhibit chaotic behavior even if their intrinsic (local) dynamics is regular and stable. For this to happen it is imperative that the spatial interactions are sufficiently strong. So far, this scenario of generating chaos from simple local dynamics has been established rigorously only for special, very narrow classes of local maps. The present article largely extends previous results by showing that the corresponding mechanism of peak-crossing is in fact very general and robust: whenever the local map is sufficiently expanding and exhibits a horseshoe then the emergence of spatial intermittency will be observed in the form of chaotically oscillating sites surrounded by quasi-regular clusters. The results firmly establish peak-crossing as a natural scenario on the route to spacetime chaos.

The density of the measure-theoretic directional entropy for the lattice dynamical system is introduced and it is shown that for a lattice dynamical transformation commuting with the shift transformation the density coincides with the entropy of the ℤ²-action generated by these two transformations, i.e. it is a constant function with respect to the direction. It is also proved that in the noncommutative case this result fails to be true.

We determine the essential spectrum of certain types of linear operators which arise in the study of the stability of steady state or traveling wave solutions in coupled map lattices. The basic tool is the Gelfand transformation which enables us to determine the essential spectrum completely.

This work investigates three-dimensional pattern generation problems and their applications to three-dimensional Cellular Neural Networks (3DCNN). An ordering matrix for the set of all local patterns is established to derive a recursive formula for the ordering matrix of a larger finite lattice. For a given admissible set of local patterns, the transition matrix is defined and the recursive formula of high order transition matrix is presented. Then, the spatial entropy is obtained by computing the maximum eigenvalues of a sequence of transition matrices. The connecting operators are used to verify the positivity of the spatial entropy, which is important in determining the complexity of the set of admissible global patterns. The results are useful in studying a set of global stationary solutions in various Lattice Dynamical Systems and Cellular Neural Networks.

This work investigates the diversity of traveling wave solutions for a class of delayed cellular neural networks on the one-dimensional integer lattice ℤ¹. The dynamics of a given cell is characterized by instantaneous self-feedback and neighborhood interaction with distributed delay due to, for example, finite switching speed and finite velocity of signal transmission. Applying the monotone iteration scheme, we can deduce the existence of monotonic traveling wave solutions provided the templates satisfy the so-called quasi-monotonicity condition. We then consider two special cases of the delayed cellular neural network in which each cell interacts only with either the nearest m left neighbors or the nearest m right neighbors. For the former case, we can directly figure out the analytic solution in an explicit form by the method of step with the help of the characteristic function and then prove that, in addition to the existence of monotonic traveling wave solutions, for certain templates there exist nonmonotonic traveling wave solutions such as camel-like waves with many critical points. For the latter case, employing the comparison arguments repeatedly, we can clarify the deformation of traveling wave solutions with respect to the wave speed. More specifically, we can describe the transition of profiles from monotonicity, damped oscillation, periodicity, unboundedness and back to monotonicity as the wave speed is varied. Some numerical results are also given to demonstrate the theoretical analysis.

In this paper, we prove the existence of solutions for first order lattice dynamical systems with continuous nonlinear term obtained via discretization of a reaction–diffusion system. Since the uniqueness of the Cauchy problem is not guaranteed, we define a multivalued semiflow and prove the existence of a global compact attractor.

In this paper we consider a lattice dynamical system generated by a parabolic equation modeling suspension flows. We prove the existence of a global compact connected attractor for this system and the upper semicontinuity of this attractor with respect to finite-dimensional approximations. Also, we obtain a sequence of approximating discrete dynamical systems by the implementation of the implicit Euler method, proving the existence and the upper semicontinuous convergence of their global attractors.

The existence of a pullback attractor for the nonautonomous $p$-Laplacian type equations on infinite lattices is established under certain natural dissipative conditions. In particular, there is no restriction on the power index $q$ of the nonlinearity relative to the index $p$. The forward limiting behavior is also discussed and, under suitable assumptions on the time dependent terms, the lattice system is shown to be asymptotically autonomous with its pullback attractor component sets converging upper semi-continuously to the autonomous global attractor of the limiting autonomous system.

This paper is devoted to the long-term behavior of nonautonomous random lattice dynamical systems with nonlinear diffusion terms. The nonlinear drift and diffusion terms are not expected to be Lipschitz continuous but satisfy the continuity and growth conditions. We first prove the existence of solutions, and establish the existence of a multi-valued nonautonomous cocycle. We then show the existence and uniqueness of pullback attractors parameterized by sample parameters. Finally, we establish the measurability of this pullback attractor by the method based on the weak upper semicontinuity of the solutions.