Journal of Dynamics and Differential Equations最新文献

This work explores a synchronization-like phenomenon induced by common noise for continuous-time Markov jump processes given by chemical reaction networks. Based on Gillespie’s stochastic simulation algorithm, a corresponding random dynamical system is formulated in a two-step procedure, at first for the states of the embedded discrete-time Markov chain and then for the augmented Markov chain including random jump times. We uncover a time-shifted synchronization in the sense that—after some initial waiting time—one trajectory exactly replicates another one with a certain time delay. Whether or not such a synchronization behavior occurs depends on the combination of the initial states. We prove this partial time-shifted synchronization for the special setting of a birth-death process by analyzing the corresponding two-point motion of the embedded Markov chain and determine the structure of the associated random attractor. In this context, we also provide general results on existence and form of random attractors for discrete-time, discrete-space random dynamical systems.

Abstract

Let N be an n-dimensional compact riemannian manifold, with (nge 2) . In this paper, we prove that for any (alpha in [0,n]) , the set consisting of homeomorphisms on N with lower and upper metric mean dimensions equal to (alpha ) is dense in (text {Hom}(N)) . More generally, given (alpha ,beta in [0,n]) , with (alpha le beta ) , we show the set consisting of homeomorphisms on N with lower metric mean dimension equal to (alpha ) and upper metric mean dimension equal to (beta ) is dense in (text {Hom}(N)) . Furthermore, we also give a proof that the set of homeomorphisms with upper metric mean dimension equal to n is residual in (text {Hom}(N)) .

The principle of linearized stability and instability is established for a classical model describing the spatial movement of an age-structured population with nonlinear vital rates. It is shown that the real parts of the eigenvalues of the corresponding linearization at an equilibrium determine the latter’s stability or instability. The key ingredient of the proof is the eventual compactness of the semigroup associated with the linearized problem, which is derived by a perturbation argument. The results are illustrated with examples.

We are concerned with the global well-posedness to the compressible Oldroyd-B model without a damping term in the stress tensor equation. By exploiting the intrinsic structure of the equations and introducing several new quantities for the density, the velocity and the divergence of the stress tensor, we overcome the difficulty of the lack of dissipation for the density and the stress tensor, and construct unique global solutions to this system with initial data in critical Besov spaces. As a byproduct, we obtain the optimal time decay rates of the solutions by using the pure energy argument. A similar result can be also proved for the compressible viscoelastic system without “div–curl" structure.

In this paper, we develop the basic theory of the shadowing property for local homeomorphisms of metric locally compact spaces, with a focus on applications to edge shift spaces connected with C*-algebra theory. For the local homeomorphism (the Deaconu–Renault system) associated with a directed graph, we completely characterize the shadowing property in terms of conditions on sets of paths. Using these results, we single out classes of graphs for which the associated system presents the shadowing property, fully characterize the shadowing property for systems associated with certain graphs, and show that the system associated with the rose of infinite petals presents the shadowing property and that the Renewal shift system does not present the shadowing property.

In this work we study the continuity (both upper and lower semicontinuity), defined using the Hausdorff semidistance, of the unbounded attractors for a family of fractional perturbations of a scalar reaction-diffusion equation with a non-dissipative nonlinear term.

Every volume-preserving accessible centre-bunched fibred partially hyperbolic system with 2-dimensional centre either (a) has two distinct centre Lyapunov exponents, or (b) exhibits an invariant continuous line field (or pair of line fields) tangent to the centre leaves, or (c) admits a continuous conformal structure on the centre leaves invariant under both the dynamics and the stable and unstable holonomies. The last two alternatives carry strong restrictions on the topology of the centre leaves: (b) can only occur on tori, and for (c) the centre leaves must be either tori or spheres. Moreover, under some additional conditions, such maps are rigid, in the sense that they are topologically conjugate to specific algebraic models. When the system is symplectic (a) implies that the centre Lyapunov exponents are non-zero, and thus the system is (non-uniformly) hyperbolic.

In the article ’Criteria for Strong and Weak Random Attractors’ necessary and sufficient conditions for strong attractors and weak attractors are studied. In this note we correct two of its theorems on strong attractors.

A Riemann problem for the conservation law (u_{t}+[phi (u)]_{x}=kH(x-vt)), where x, t, k, v and (u=u(x,t)) are real numbers, is studied with the goal of getting singular solutions in a convenient space of distributions that contains delta shock waves. Here (phi ) stands for an entire function taking real values on the real axis and H represents the Heaviside function. When u is seen as a density of matter some surprises may appear such as the creation of matter from a vacuum state. In a particular case, as the time goes on, such a matter grows continuously, running away from any spatial bounded region, what can be viewed as a unidimensional model of universe.

We prove stochastic stability of the three-dimensional Rayleigh–Bénard convection in the infinite Prandtl number regime for any pair of temperatures maintained on the top and the bottom. Assuming that the non-degenerate random perturbation acts in a thin layer adjacent to the bottom of the domain, we prove that the law of the random flow periodic in the two infinite directions stabilises to a unique stationary measure, provided that there is at least one point accessible from any initial state. We also prove that the latter property is satisfied if the amplitude of the noise is sufficiently large.