Journal of Dynamics and Differential Equations最新文献

We study a nonlocal balance equation that describes the evolution of a system consisting of infinitely many identical particles those move along a deterministic dynamics and can also either disappear or give a spring. In this case, the solution of the balance equation is considered in the space of nonnegative measures. We prove the superposition principle for the examined nonlocal balance equation. Furthermore, we interpret the source/sink term as a probability rate of jumps from/to a remote point. Using this idea and replacing the deterministic dynamics of each particle by a nonlinear Markov chain, we approximate the solution of the balance equation by a solution of a system of ODEs and evaluate the corresponding approximation rate. This result can be used for construction of numerical solutions of the nonlocal balance equation.

We investigate mixing properties of piecewise affine non-Markovian maps acting on ([0,1]^2) or ([0,1]^3) and preserving the Lebesgue measure, which are natural generalizations of the heterochaos baker maps introduced in Saiki et al. (Nonlinearity 34:5744–5761, 2021). These maps are skew products over uniformly expanding or hyperbolic bases, and the fiber direction is a center in which both contracting and expanding behaviors coexist. We prove that these maps are mixing of all orders. For maps with a mostly expanding or contracting center, we establish exponential mixing for Hölder functions. Using this result, for the Dyck system originating in the theory of formal languages, we establish exponential mixing for Hölder functions with respect to its two coexisting ergodic measures of maximal entropy.

We study the well-posedness of two systems modeling the non-equilibrium dynamics of pumped decaying Bose–Einstein condensates. In particular, we present the local theory for rough initial data using the Fourier restricted norm method introduced by Bourgain. We extend the result globally for initial data in (L^{2}).

We introduce first-time sensitivity for a homeomorphism of a compact metric space, that is a condition on the first increasing times of open balls of the space. Continuum-wise expansive homeomorphisms, the shift map on the Hilbert cube, and also some partially hyperbolic diffeomorphisms satisfy this condition. We prove the existence of local unstable continua satisfying similar properties with the local unstable continua of continuum-wise expansive homeomorphisms, but assuming first-time sensitivity. As a consequence we prove that first-time sensitivity (with some additional technical assumptions) implies positive topological entropy.

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.