A Note on Variational Principle of Subsets for Nonautonomous Dynamical Systems

In this paper, we introduce measure-theoretic for Borel probability measures to characterize upper and lower Katok measure-theoretic entropies in discrete type and the measure-theoretic entropy for arbitrary Borel probability measure in nonautonomous case. Then we establish new variational principles for Bowen topological entropy for nonautonomous dynamical systems. JEL classification numbers: 37A35. Keywords: Nonautonomous, Measure-theoretical entropies, Variational principles.

Reduction principle at work

The purpose of this informal paper is three-fold: First, filling a gap in the literature, we provide a (necessary and sufficient) principle of linearized stability for nonautonomous difference equations in Banach spaces based on the dichotomy spectrum. Second, complementing the above, we survey and exemplify an ambient nonautonomous and infinite-dimensional center manifold reduction, that is Pliss’s reduction principle suitable for critical stability situations. Third, these results are applied to integrodifference equations of Hammerstein- and Urysohn-type both in C- and Lp -spaces. Specific features of the nonautonomous case are underlined. Yet, for the simpler situation of periodic time-dependence even explicit computations are feasible.

Dynamic equation on time scale with almost periodic coefficients

In this paper, we discuss a nonautonomous dynamical equation on time scale in a Banach space. The nonautonomous case is particularly important and needs to be studied because it is frequently met in the mathematical models of evolutionary processes. We give sufficient condition for equation to have an exponentially stable almost periodic solution in terms of the accretiveness of an operator. At the end, examples are given to illustrate the analytical findings.

Existence of Positive Ground State Solutions for Choquard Systems

We study the existence of positive ground state solution for Choquard systems. In the autonomous case, we prove the existence of at least one positive ground state solution by the Pohozaev manifold method and symmetric-decreasing rearrangement arguments. Moreover, we show that each positive ground state solution is radial symmetric. While, in the nonautonomous case, a positive ground state solution is obtained by using a monotonicity trick and a global compactness lemma. We remark that, under our assumptions of the nonlinearity {W_{u}}, the search of ground state solutions cannot be reduced to the study of critical points of a functional restricted to a Nehari manifold.

Dynamic Behavior of Positive Solutions for a Leslie Predator-Prey System with Mutual Interference and Feedback Controls

We consider a Leslie predator-prey system with mutual interference and feedback controls. For general nonautonomous case, by using differential inequality theory and constructing a suitable Lyapunov functional, we obtain some sufficient conditions which guarantee the permanence and the global attractivity of the system. For the periodic case, we obtain some sufficient conditions which guarantee the existence, uniqueness, and stability of a positive periodic solution.

Positive Periodic Solutions of an Epidemic Model with Seasonality

An SEI autonomous model with logistic growth rate and its corresponding nonautonomous model are investigated. For the autonomous case, we give the attractive regions of equilibria and perform some numerical simulations. Basic demographic reproduction numberRdis obtained. Moreover, only the basic reproduction numberR0cannot ensure the existence of the positive equilibrium, which needs additional conditionRd>R1. For the nonautonomous case, by introducing the basic reproduction number defined by the spectral radius, we study the uniform persistence and extinction of the disease. The results show that for the periodic system the basic reproduction number is more accurate than the average reproduction number.

Inertial manifolds for parabolic differential equations: The fully nonautonomous case

<p style='text-indent:20px;'>We study the existence of an inertial manifold for the solutions to fully non-autonomous parabolic differential equation of the form</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \dfrac{du}{dt} + A(t)u(t) = f(t,u),\, t&gt; s. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>We prove the existence of such an inertial manifold in the case that the family of linear partial differential operators <inline-formula><tex-math id="M1">\begin{document}$ (A(t))_{t\in { \mathbb {R}}} $\end{document}</tex-math></inline-formula> generates an evolution family <inline-formula><tex-math id="M2">\begin{document}$ (U(t,s))_{t\ge s} $\end{document}</tex-math></inline-formula> satisfying certain dichotomy estimates, and the nonlinear forcing term <inline-formula><tex-math id="M3">\begin{document}$ f(t,x) $\end{document}</tex-math></inline-formula> satisfies the Lipschitz condition such that certain dichotomy gap condition holds.</p>