Discrete & Continuous Dynamical Systems - S最新文献

We consider the large time behavior of solutions of compressible viscoelastic system around a motionless state in a three-dimensional whole space. We show that if the initial data belongs to begin{document}$ W^{2,1} $end{document}, and is sufficiently small in begin{document}$ H^4cap L^1 $end{document}, the solutions grow in time at the same rate as begin{document}$ t^{frac{1}{2}} $end{document} in begin{document}$ L^1 $end{document} due to diffusion wave phenomena of the system caused by interaction between sound wave, viscous diffusion and elastic wave.

We consider the low regularity well-posedness problem for the Maxwell-Dirac system in 3+1 dimensions:

begin{document}$ begin{align*} partial^{mu} F_{mu nu} & = - langle psi, alpha_{nu} psi rangle -i alpha^{mu} partial_{mu} psi & = A_{mu} alpha^{mu} psi , , end{align*} $end{document}

where begin{document}$ F_{mu nu} = partial^{mu} A_{nu} - partial^{nu} A_{mu} $end{document}, and begin{document}$ alpha^{mu} $end{document} are the 4x4 Dirac matrices. We assume the temporal gauge begin{document}$ A_0 = 0 $end{document} and make use of the fact that some of the nonlinearities fulfill a null condition. Because we work in the temporal gauge we also apply a method, which was used by Tao for the Yang-Mills system.

One considers a system on begin{document}$ mathbb{C}^2 $end{document} close to an invariant curve which can be viewed as a generalization of the semi-standard map to a trigonometric polynomial with many Fourier modes. The radius of convergence of an analytic linearization of the system around the invariant curve is bounded by the exponential of the negative Brjuno sum of begin{document}$ dalpha $end{document}, where begin{document}$ din mathbb{N}^* $end{document} and begin{document}$ alpha $end{document} is the frequency of the linear part, and the error function is non decreasing with respect to the smallest coefficient of the trigonometric polynomial.

In the paper we study what sets can be obtained as begin{document}$ alpha $end{document}-limit sets of backward trajectories in graph maps. We show that in the case of mixing maps, all those begin{document}$ alpha $end{document}-limit sets are begin{document}$ omega $end{document}-limit sets and for all but finitely many points begin{document}$ x $end{document}, we can obtain every begin{document}$ omega $end{document}-limits set as the begin{document}$ alpha $end{document}-limit set of a backward trajectory starting in begin{document}$ x $end{document}. For zero entropy maps, every begin{document}$ alpha $end{document}-limit set of a backward trajectory is a minimal set. In the case of maps with positive entropy, we obtain a partial characterization which is very close to complete picture of the possible situations.

The main purpose of this paper is to study the global propagation of singularities of the viscosity solution to discounted Hamilton-Jacobi equation

begin{document}$ begin{align} lambda v(x)+H( x, Dv(x) ) = 0 , quad xin mathbb{R}^n. quadquadquad (mathrm{HJ}_{lambda})end{align} $end{document}

with fixed constant begin{document}$ lambdain mathbb{R}^+ $end{document}. We reduce the problem for equation begin{document}$(mathrm{HJ}_{lambda})$end{document} into that for a time-dependent evolutionary Hamilton-Jacobi equation. We prove that the singularities of the viscosity solution of begin{document}$(mathrm{HJ}_{lambda})$end{document} propagate along locally Lipschitz singular characteristics begin{document}$ {{bf{x}}}(s):[0,t]to mathbb{R}^n $end{document} and time begin{document}$ t $end{document} can extend to begin{document}$ +infty $end{document}. Essentially, we use begin{document}$ sigma $end{document}-compactness of the Euclidean space which is different from the original construction in [4]. The local Lipschitz issue is a key technical difficulty to study the global result. As a application, we also obtain the homotopy equivalence between the singular locus of begin{document}$ u $end{document} and the complement of Aubry set using the basic idea from [9].

We study the approximation of the nonlocal-interaction equation restricted to a compact manifold begin{document}$ {mathcal{M}} $end{document} embedded in begin{document}$ {mathbb{R}}^d $end{document}, and more generally compact sets with positive reach (i.e. prox-regular sets). We show that the equation on begin{document}$ {mathcal{M}} $end{document} can be approximated by the classical nonlocal-interaction equation on begin{document}$ {mathbb{R}}^d $end{document} by adding an external potential which strongly attracts to begin{document}$ {mathcal{M}} $end{document}. The proof relies on the Sandier–Serfaty approach [23,24] to the begin{document}$ Gamma $end{document}-convergence of gradient flows. As a by-product, we recover well-posedness for the nonlocal-interaction equation on begin{document}$ {mathcal{M}} $end{document}, which was shown [10]. We also provide an another approximation to the interaction equation on begin{document}$ {mathcal{M}} $end{document}, based on iterating approximately solving an interaction equation on begin{document}$ {mathbb{R}}^d $end{document} and projecting to begin{document}$ {mathcal{M}} $end{document}. We show convergence of this scheme, together with an estimate on the rate of convergence. Finally, we conduct numerical experiments, for both the attractive-potential-based and the projection-based approaches, that highlight the effects of the geometry on the dynamics.

In this paper we study generalized Poincaré-Andronov-Hopf bifurcations of discrete dynamical systems. We prove a general result for attractors in begin{document}$ n $end{document}-dimensional manifolds satisfying some suitable conditions. This result allows us to obtain sharper Hopf bifurcation theorems for fixed points in the general case and other attractors in low dimensional manifolds. Topological techniques based on the notion of concentricity of manifolds play a substantial role in the paper.

A zero-dimensional (resp. symbolic) flow is a suspension flow over a zero-dimensional system (resp. a subshift). We show that any topological flow admits a principal extension by a zero-dimensional flow. Following [6] we deduce that any topological flow admits an extension by a symbolic flow if and only if its time-begin{document}$ t $end{document} map admits an extension by a subshift for any begin{document}$ tneq 0 $end{document}. Moreover the existence of such an extension is preserved under orbit equivalence for regular topological flows, but this property does not hold for singular flows. Finally we investigate symbolic extensions for singular suspension flows. In particular, the suspension flow over the full shift on begin{document}$ {0,1}^{mathbb Z} $end{document} with a roof function begin{document}$ f $end{document} vanishing at the zero sequence begin{document}$ 0^infty $end{document} admits a principal symbolic extension or not depending on the smoothness of begin{document}$ f $end{document} at begin{document}$ 0^infty $end{document}.