Journal of Dynamics and Differential Equations最新文献

This paper aims to investigate the global existence and uniqueness of weak solutions for a class of heat equations with logarithmic nonlinearity in ({mathbb {R}}^{N}), as well as to examine some of their qualitative properties. Additionally, we analyze the boundedness and higher integrability of these solutions.

This paper provides a novel analysis of the rich and complex propagation dynamics to a model of precursor and differentiated cells, with the appearance of non-isolated equilibria on a line in the phase space. We established the existence of traveling waves in the monostable monotone case by means of continuation argument via perturbation in a weighted functional space, by applying the abstract implicit function theorem. We proved necessary and sufficient conditions of the minimal wave speed selection and showed the existence of the transition (turning point) (k^*) for the minimal wave speed when the parameters (lambda ) and (gamma ) are fixed. Two explicit estimates about (k^*) were given by the easy-to-apply theorem we derived. We investigated the decay rate of the minimal traveling wave as (zrightarrow infty ) in terms of the value of k. We further proved the existence of non-negative wavefronts in the monostable non-monotone case and found that the minimal wave speed is always linearly selected. Finally in the bistable monotone case, the existence and uniqueness of bistable traveling waves were proved via constructing an auxiliary parabolic non-local equation.

In this paper, we show that for any (C^1) three-dimensional vector fields with positive topological entropy, the topological entropy can be approximated by horseshoes. Precisely, for any (C^1) three-dimensional vector field X with positive topological entropy, there exists a vector field Y arbitrarily close (in the (C^1) topology) to X exhibiting a horseshoe (Lambda ) such that the topological entropy of Y restricted on (Lambda ) can arbitrarily approximate the topological entropy of X. This extends a classical result (Katok in Inst Hautes Études Sci Publ Math 51:137–173, 1980) of Katok for (C^{1+alpha }(alpha >0)) surface diffeomorphisms and a result (Wu and Liu in Proc Am Math Soc 148(1):223–233, 2020) for (C^1) surface diffeomorphisms.

Let G be an infinite countable discrete amenable group. For any G-action on a compact metric space X, it is proved that for any sequence ((G_n)_{nge 1}) consisting of non-empty finite subsets of G with (lim _{nrightarrow infty }|G_n|=infty ), Pinsker (sigma )-algebra is a characteristic factor for ((G_n)_{nge 1}). As a consequence, for a class of G-topological dynamical systems, positive topological entropy implies mean Li–Yorke chaos along a class of sequences consisting of non-empty finite subsets of G.

In this paper, we study the size of the level sets of all Lyapunov exponents. For typical cocycles, we establish a variational relation between the topological entropy of the level sets of Lyapunov exponents and the topological pressure of the generalized singular value function.

We prove the nonstationary bounded distortion property for (C^{1 + varepsilon }) smooth dynamical systems on multidimensional spaces. The results we obtain are motivated by potential application to study of spectral properties of discrete Schrödinger operators with potentials generated by Sturmian sequences.

In this paper, we address the existence and multiplicity of 2-periodic solutions to differential equations with a distributed delay of the form

Combining Kaplan–Yorke’s method with pseudoindex theory, we estimate the number of periodic solutions when the equations are both resonant and nonresonant. More specifically, we define two indices using asymptotic linear coefficient matrices at the origin and at infinity. Then the lower bound on the number of periodic solutions to the equations is estimated by the indices. Finally, two examples are given to illustrate our results.

Iterated function systems (IFSs) and their attractors have been central to the theory of fractal geometry almost from its inception. Moreover, contractivity of the functions in the IFS has been central to the theory of iterated functions systems. If the functions in the IFS are contractions, then the IFS is guaranteed to have a unique attractor. The converse question, does the existence of an attractor imply that the IFS is contractive, originates in a 1959 work by Bessaga which proves a converse to the contraction mapping theorem. Although a converse is true in that case, it is known that it does not always hold for an IFS. In general, there do exist IFSs with attractors and which are not contractive. However, in the context of IFSs in Euclidean space, this question has been open. In this paper we show that a highly non-contractive iterated function system in Euclidean space can have an attractor. In order to do that, we introduce the concept of an L-expansive map, i.e., a map that has Lipschitz constant strictly greater than one under any remetrization. This is necessitated by the absence of positively expansive maps on the interval.

We consider a smooth semiflow strongly focusing monotone with respect to a cone of rank k on a Banach space. We obtain its generic dynamics, that is, semiorbits with initial data from an open and dense subset of any open bounded set either are pseudo-ordered or convergent to an equilibrium. For the case (k=1), it is the celebrated Hirsch’s Generic Convergence Theorem. For the case (k=2), we obtain the generic Poincaré-Bendixson Theorem.