Journal of Dynamics and Differential Equations最新文献

We study a family of skew-products of smooth functions having a unique critical point of degree (Dge 2) over a strongly expanding map of the circle and prove that these systems admit two positive Lyapunov exponents. This extends an analogous result of Viana who considered, in the seminal paper (Viana in Inst Hautes Études Sci Publ Math 85:63–96, 1997), the quadratic case (D=2).

The main aim of this paper is to study the limit cycle bifurcations near the homoclinic cycle in the discontinuous systems. Based on the impoved Lin’s method, we establish the bifurcation equation, which presents the existence of limit cycles bifurcated from nonsmooth homoclinic cycles under perturbation. Furthermore, we give an example to support our conclusions. After solving a boundary value problem with numerical tools, we provide the exact parameter values for the system having a limit cycle.

Abstract

In this paper, we study the existence of global attractors for a class of discrete dynamical systems naturally originated from impulsive dynamical systems. We establish sufficient conditions for the existence of a discrete global attractor. Moreover, we investigate the relationship among different types of global attractors, i.e., the attractor ({mathcal {A}}) of a continuous dynamical system, the attractor (tilde{{mathcal {A}}}) of an impulsive dynamical system and the attractor (hat{{mathcal {A}}}) of a discrete dynamical system. Two applications are presented, one involving an integrate-and-fire neuron model, and the other involving a nonlinear reaction-diffusion initial boundary value problem.

We show that a Kupka–Smale riemannian metric on a closed surface contains a finite primary set of closed geodesics, i.e. they intersect any other geodesic and divide the surface into simply connected regions. From them we obtain a finite set of disjoint surfaces of section of genera 0 or 1, which intersect any orbit of the geodesic flow. As an application we obtain that the geodesic flow of a Kupka–Smale riemannian metric on a closed surface has homoclinic orbits for all branches of all of its hyperbolic closed geodesics.

Affine flows on vector bundles with chain transitive base flow are lifted to linear flows and the decomposition into exponentially separated subbundles provided by Selgrade’s theorem is determined. The results are illustrated by an application to affine control systems with bounded control range.

The aim of this paper is to construct invariant manifolds for a coupled system, consisting of a parabolic equation and a second-order ordinary differential equation, set on (mathbb {T}^3) and subject to periodic boundary conditions. Notably, the “spectral gap condition" does not hold for the system under consideration, leading to the use of the spatial averaging principle, together with the graph transform method. This approach facilitates the construction of the relevant invariant manifold, characterized by attributes such as Lipschitz continuity, local invariance, infinite dimensionality, and exponential tracking, thus mirroring the properties traditionally associated with a classical global manifold.

We found a dichotomy involving rigidity and measure of maximal entropy of a (C^{infty })-special Anosov endomorphism of the 2-torus. Considering (widetilde{m} ) the measure of maximal entropy of a (C^{infty })-special Anosov endomorphism of the 2-torus, either (widetilde{m}) satisfies the Pesin formula (in this case we get smooth conjugacy with the linearization) or there is a set Z, such that (widetilde{m}(Z) = 1,) but Z intersects every unstable leaf on a set of zero measure of the leaf. Also, we can characterize the absolute continuity of the intermediate foliation for a class of volume-preserving special Anosov endomorphisms of (mathbb {T}^3).

Abstract

The collective dynamics of interacting dynamical units on a network crucially depends on the properties of the network structure. Rather than considering large but finite graphs to capture the network, one often resorts to graph limits and the dynamics thereon. We elucidate the symmetry properties of dynamical systems on graph limits—including graphons and graphops—and analyze how the symmetry shapes the dynamics, for example through invariant subspaces. In addition to traditional symmetries, dynamics on graph limits can support generalized noninvertible symmetries. Moreover, as asymmetric networks can have symmetric limits, we note that one can expect to see ghosts of symmetries in the dynamics of large but finite asymmetric networks.

A new type of random attractors is introduced to study dynamics of a stochastic modified Swift–Hohenberg equation with a general delay. A compact, pullback attracting and dividedly invariant set is called a backward attractor, while the criteria for its existence are established in terms of increasing dissipation and backward asymptotic compactness of a cocycle. If the delay term in the equation is Lipschitz continuous such that the Lipschitz bound and the external force are backward limitable, then we prove the existence of a backward attractor, which further leads to the longtime stability as well as the existence of a pullback attractor, where the pullback attractor and the backward attractor are shown to be random and dividedly random, respectively. Two examples of the delay term are provided to illustrate variable and distributed delays without restricting the upper bound of Lipschitz bounds.

The long time behavior of Wong–Zakai approximations of 2D as well as 3D non-autonomous stochastic convective Brinkman–Forchheimer (CBF) equations with non-linear diffusion terms on some bounded and unbounded domains is discussed in this work. To establish the existence of pullback random attractors, the concept of asymptotic compactness (AC) is used. In bounded domains, AC is proved via compact Sobolev embeddings. In unbounded domains, due to the lack of compact embeddings, the ideas of energy equations and uniform tail-estimates are exploited to prove AC. In the literature, CBF equations are also known as Navier–Stokes equations (NSE) with damping, and it is interesting to see that the modification in NSE by linear and nonlinear damping provides better results than that available for NSE in 2D and 3D. The presence of linear damping term helps to establish the results in the whole space (mathbb {R}^d). The nonlinear damping term supports to obtain the results in 3D and to cover a large class of nonlinear diffusion terms also. In addition, we prove the existence of a unique pullback random attractor for stochastic CBF equations driven by additive white noise. Finally, for additive as well as multiplicative white noise cases, we establish the convergence of solutions and upper semicontinuity of pullback random attractors for Wong–Zakai approximations of stochastic CBF equations towards the pullback random attractors for stochastic CBF equations when the correlation time of colored noise converges to zero.