Journal of Dynamics and Differential Equations最新文献

Abstract

This paper is concerned with the existence of quasi-periodic response solutions (i.e., solutions that are quasi-periodic with the same frequencies as forcing term) for a class of forced reversible wave equations with derivative nonlinearity. The forcing frequency (omega in mathbb {R}^2) would be Liouvillean which is weaker than the usual Diophantine and Brjuno conditions. The derivative nonlinearity in the equation also leads to some difficulty in measure estimate. To overcome it, we also use the Töplitz–Lipschitz property of vector field. The proof is based on an infinite dimensional Kolmogorov–Arnold–Moser theorem for reversible systems.

Abstract

We find an explicit form of entropy solution to a Riemann problem for a degenerate nonlinear parabolic equation with piecewise constant velocity and diffusion coefficients. It is demonstrated that this solution corresponds to the minimum point of some strictly convex function of a finite number of variables. We also discuss the limit when piecewise constant coefficients approximate the arbitrary ones.

We study the existence of non-collision orbits for a class of singular Hamiltonian systems

where (q:{mathbb {R}} longrightarrow {mathbb {R}}^2) and (Vin C^2({mathbb {R}}^2 {setminus } {e},, {mathbb {R}})) is a potential with a singularity at a point (enot =0). We consider V which behaves like (displaystyle -1/|q-e|^alpha ) as ( qrightarrow e ) with (alpha in ]0,2[.) Under the assumption that 0 is a strict global maximum for V, we establish the existence of a homoclinic orbit emanating from 0. Moreover, in case (displaystyle V(q) longrightarrow 0) as (|q|rightarrow +infty ), we prove the existence of a heteroclinic orbit “at infinity" i.e. a solution q such that

Abstract

The aim of this paper is to extend the results associated with periodic orbits from two-dimensions to higher-dimensions. Because of the one-to-one correspondence between solutions for the monotone recurrence relation and orbits for the induced high-dimensional cylinder twist map, we consider the system of solutions for monotone recurrence relations instead. By introducing intersections of type (k, l), we propose the definition of generalized Birkhoff solutions, generalizing the concept of Birkhoff solutions. We show that if there is a (p, q)-periodic solution which is not a generalized Birkhoff solution, then the system has positive topological entropy and the Farey interval of p/q is contained in the rotation set.

In this work we analyze the boundedness properties of the solutions of a nonautonomous parabolic degenerate logistic equation in a bounded domain. The equation is degenerate in the sense that the logistic nonlinearity vanishes in a moving region, K(t), inside the domain. The boundedness character of the solutions depends not only on, roughly speaking, the first eigenvalue of the Laplace operator in K(t) but also on the way this moving set evolves inside the domain and in particular on the speed at which it moves.

In this paper we construct a new type of noise of fractional nature that has a strong regularizing effect on differential equations. We consider an equation driven by a highly irregular vector field and study the effect of this noise on such dynamical systems. We employ a new method to prove existence and uniqueness of global strong solutions, where classical methods fail because of the “roughness” and non-Markovianity of the driving process. In addition, we prove the rather remarkable property that such solutions are infinitely many times classically differentiable with respect to the initial condition in spite of the vector field being discontinuous. The technique used in this article corresponds, in a certain sense, to the Nash–Moser iterative scheme in combination with a new concept of “higher order averaging operators along highly fractal stochastic curves”. This approach may provide a general principle for the study of regularization by noise effects in connection with important classes of partial differential equations.

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.