In this paper, the exponential stability of travelling waves solutions for nonlinear cellular neural networks with distribute delays in the lattice is studied. The weighted energy method and comparison principle are employed to derive the sufficient conditions under which the networks proposed are exponentially stable. Following the study  on the existence of the travelling wave solutions in nonlinear delayed cellular neural networks, this paper is focused on the exponential stability of these travelling wave solutions.,In this paper, the exponential stability of travelling waves solutions for nonlinear cellular neural networks with distribute delays in the lattice is studied. The weighted energy method and comparison principle are employed to derive the sufficient conditions under which the networks proposed are exponentially stable. Following the study [ 13 ] on the existence of the travelling wave solutions in nonlinear delayed cellular neural networks, this paper is focused on the exponential stability of these travelling wave solutions.
In this article we prove the stability of some mean field systems similar to the Winfree model in the synchronized state. The model is governed by the coupling strength parameter κ and the natural frequency of each oscillator. The stability is proved independently of the number of oscillators and the distribution of the natural frequencies. The main result is proved using the positive invariant cone method for the linearized system. This method can be applied to other mean field models as in the Kuramoto model.
Anosov families are non-stationary dynamical systems with hyperbolic behaviour. Non-trivial examples of Anosov families will be given in this paper. We show the existence of invariant manifolds, the structrural stability and a characterization for a certain class of Anosov families.
In this note we describe the maximal and the minimal values of Lyapunov exponents for second-order discrete time-invariant linear system perturbed by time-varying bounded perturbations. An interpretation of the results in terms of generalized spectral radius is given. An application of obtained formulas to the robust stability problem is demonstrated on a numerical example.
In this note we give a statistical approximation of the unstable manifold of a connected isolated unstable attractor of a smooth flow using coherent measures relative to it. In the main result we show that almost all orbits in the support of a coherent measure relative to an isolated unstable attractor are contained in its unstable manifold.
Let be a dynamical system, where is a compact metric space and is a continuous map. Using the concepts of g-almost product property and uniform separation property introduced by Pfister and Sullivan in Pfister and Sullivan [On the topological entropy of saturated sets, Ergodic Theory Dyn. Syst. 27 (2007), pp. 929-956], we give a variational principle for certain non-compact with relation to the asymptotically additive topological pressure. We also study the set of points that are irregular for a collection finite or infinite of asymptotically additive sequences and we show that carried the full asymptotically additive topological pressure. These results are suitable for systems such as mixing shifts of finite type, β-shifts, repellers and uniformly hyperbolic diffeomorphisms.
In this paper, we consider the following real analytic Hamiltonian system where A is a constant Hamiltonian matrix with the different eigenvalues , where for are real, and is quasi-periodic with frequencies . Without any non-degeneracy condition with respect to ϵ, we prove that by a quasi-periodic symplectic mapping, then for most of the sufficiently small parameter ϵ, the Hamiltonian system is reducible.
In this paper, we consider the effective reducibility of the following quasi-periodic nonlinear system where A is a constant matrix with the different and nonzero eigenvalues, , and , and are analytic quasi-periodic on with respect to t. Under non-resonance conditions, without any non-degeneracy condition, by a quasi-periodic transformation, the system can be reducible to a quasi-periodic system where and are exponentially small in ϵ.
A class of autonomous Kolmogorov systems that are dissipative and competitive with the origin as a repellor are considered when each nullcline surface is either concave or convex. Geometric method is developed by using the relative positions of the upper and lower planes of the nullcline surfaces for global asymptotic stability of an interior or a boundary equilibrium point. Criteria are also established for global repulsion of an interior or a boundary equilibrium point on the carrying simplex. This method and the theorems can be viewed as a natural extension of those results for Lotka-Volterra systems in the literature.
We consider a two-sided sequence of bounded operators in a Banach space which are not necessarily injective and satisfy two properties (SVG) and (FI). The singular value gap (SVG) property says that two successive singular values of the cocycle at some index d admit a uniform exponential gap; the fast invertibility (FI) property says that the cocycle is uniformly invertible on the fastest d-dimensional direction. We prove the existence of a uniform equivariant splitting of the Banach space into a fast space of dimension d and a slow space of codimension d. We compute an explicit constant lower bound on the angle between these two spaces using solely the constants defining the properties (SVG) and (FI). We extend the results obtained by Bochi and Gourmelon in the finite-dimensional case for bijective operators and the results obtained by Blumenthal and Morris in the infinite dimensional case for injective norm-continuous cocycles, in the direction that the operators are not required to be globally injective, that no dynamical system is involved and no compactness of the underlying system or smoothness of the cocycle is required. Moreover we give quantitative estimates of the angle between the fast and slow spaces that are new even in the case of finite-dimensional bijective operators in Hilbert spaces.
We show that any neighbourhood of a non-degenerate reversible bifocal homoclinic orbit contains chaotic suspended invariant sets on N-symbols for all . This will be achieved by showing switching associated with networks of secondary homoclinic orbits. We also prove the existence of super-homoclinic orbits (trajectories homoclinic to a network of homoclinic orbits), whose presence leads to a particularly rich structure.
We prove the saturation of a generalized partially hyperbolic attractor of a map. As a consequence, we show that any generalized partially hyperbolic horseshoe-like attractor of a -generic diffeomorphism has zero volume. In contrast, by modification of the Poincaré cross section of Lorenz geometric model, we build a -diffeomorphism with a partially hyperbolic horseshoe-like attractor of positive volume.