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Register a new userAsymptotic behavior of solutions of nonautonomous neutral dynamical systems
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This paper studies the dynamics of families of monotone nonautonomous neutral functional differential equations with nonautonomous operator, of great importance for their applications to the study of the long-term behavior of the trajectories of problems described by this kind of equations, such us compartmental systems and neural networks among many others. Precisely, more general admissible initial conditions are included in the study to show that the solutions are asymptotically of the same type as the coefficients of the neutral and non-neutral part.
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Let $mathcal{M}(X)$ be the space of Borel probability measures on a compact metric space $X$ endowed with the weak$^ast$-topology. In this paper, we prove that if the topological entropy of a nonautonomous dynamical system $(X,{f_n}_{n=1}^{+infty})$ vanishes, then so does that of its induced system $(mathcal{M}(X),{f_n}_{n=1}^{+infty})$; moreover, once the topological entropy of $(X,{f_n}_{n=1}^{+infty})$ is positive, that of its induced system $(mathcal{M}(X),{f_n}_{n=1}^{+infty})$ jumps to infinity. In contrast to Bowens inequality, we construct a nonautonomous dynamical system whose topological entropy is not preserved under a finite-to-one extension.
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Recently, the synchronization of coupled dynamical systems has been widely studied. Synchronization is referred to as a process wherein two (or many) dynamical systems are adjusted to a common behavior as time goes to infinity, due to coupling or forcing. Therefore, before discussing synchronization, a basic problem on continuation of the solution must be solved: For given initial conditions, can the solution of coupled dynamical systems be extended to the infinite interval $[0,+infty)$? In this paper, we propose a general model of coupled dynamical systems, which includes previously studied systems as special cases, and prove that under the assumption of QUAD, the solution of the general model exists on $[0,+infty)$.
We introduce a new concept of finite-time entropy which is a local version of the classical concept of metric entropy. Based on that, a finite-time version of Pesins entropy formula and also an explicit formula of finite-time entropy for $2$-D systems are derived. We also discuss about how to apply the finite-time entropy field to detect special dynamical structures such as Lagrangian coherent structures.
Monomial mappings, $xmapsto x^n$, are topologically transitive and ergodic with respect to Haar measure on the unit circle in the complex plane. In this paper we obtain an anologous result for monomial dynamical systems over $p-$adic numbers. The process is, however, not straightforward. The result will depend on the natural number $n$. Moreover, in the $p-$adic case we never have ergodicity on the unit circle, but on the circles around the point 1.
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