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In this paper, we are concerned with a problem of periodic solution for a second order nonautonomous differential equation with a discontinuity line. Under two types of assumptions, we analyze the geometric properties of solutions respectively. Then by using a fixed point theorem, we obtain several existence criteria of periodic solutions, where the periodic solutions are crossing harmonic and subharmonic solutions. [ABSTRACT FROM AUTHOR]

We prove that smoothness of nonautonomous linearization is of class C 2. Our approach admits the existence of stable and unstable manifolds determined by a family of nonautonomous hyperbolicities, including the non uniform exponential case, while for the classic exponential dichotomy we obtain the same class of differentiability except for a zero Lebesgue measure set. Moreover, our goal is reached without spectral conditions. [ABSTRACT FROM AUTHOR]

The bank run phenomenon, mostly due to rumor spread about the financial health of given financial institutions, is prejudicious to the stability of financial systems. In this paper, by using the epidemiological approach, we propose a nonlinear model for describing the impact of rumor on the banking crisis spread. We establish conditions under which the crisis dies out or remains permanent. We also solve an optimal control problem focusing on the minimization, at the lowest cost, of the number of stressed banks, as well as the number of banks undergoing the restructuring process. Numerical simulations are performed to illustrate theoretical results obtained.

In this work, we implement multiplicative noise to the Duffing oscillator with variable coefficients. The stochastic differential equations are solved using the fourth-order Runge–Kutta method with the Box-Müller algorithm and the corresponding integral of motion is obtained. Some numerical experiments are performed and the results show that the integral of motion is highly unaffected by the multiplicative noise. [ABSTRACT FROM AUTHOR]

Neuroprotection seeks to halt cell death after brain ischemia and has been shown to be possible in laboratory studies. However, neuroprotection has not been successfully translated into clinical practice, despite voluminous research and controlled clinical trials. We suggested these failures may be due, at least in part, to the lack of a general theory of cell injury to guide research into specific injuries. The nonlinear dynamical theory of acute cell injury was introduced to ameliorate this situation. Here we present a revised nonautonomous nonlinear theory of acute cell injury and show how to interpret its solutions in terms of acute biomedical injuries. The theory solutions demonstrate the complexity of possible outcomes following an idealized acute injury and indicate that a "one size fits all" therapy is unlikely to be successful. This conclusion is offset by the fact that the theory can (1) determine if a cell has the possibility to survive given a specific acute injury, and (2) calculate the degree of therapy needed to cause survival. To appreciate these conclusions, it is necessary to idealize and abstract complex physical systems to identify the fundamental mechanism governing the injury dynamics. The path of abstraction and idealization in biomedical research opens the possibility for medical treatments that may achieve engineering levels of precision. [ABSTRACT FROM AUTHOR]

In this paper we consider the global and local stability and instability of solutions of a scalar nonlinear differential equation with non-negative solutions. The differential equation is a perturbed version of a globally stable autonomous equation with unique zero equilibrium where the perturbation is additive and independent of the state. It is assumed that the restoring force is asymptotically negligible as the solution becomes large, and that the perturbation tends to zero as time becomes indefinitely large. It is shown that solutions are always locally stable, and that solutions either tend to zero or to infinity as time tends to infinity. In the case when the perturbation is integrable, the zero solution is globally asymptotically stable. If the perturbation is non-integrable, and tends to zero faster than a critical rate which depends on the strength of the restoring force, then solutions are globally stable. However, if the perturbation tends to zero more slowly than this critical rate, and the initial condition is sufficiently large, the solution tends to infinity. Moreover, for every initial condition, there exists a perturbation which tends to zero more slowly than the critical rate, for which the solution once again escapes to infinity. Some extensions to general scalar equations as well as to finite-dimensional systems are also presented, as well as global convergence results using Liapunov techniques.

A pullback incremental attraction, a nonautonomous version of incremental stability, is introduced for nonautonomous systems that may have unbounded limiting solutions. Its characterisation by a Lyapunov function is indicated

Abstract: In this article, new concepts of (exponential) attractivity for nonautonomous differential equations on a finite time interval are introduced. Due to nonuniqueness of finite-time objects, areas of attraction are studied rather than solutions which are attractive. A sufficient and necessary condition for the existence of such areas is presented, which makes use of a time-varying Riemannian metric. Several examples illustrate the theoretical results and definitions. [Copyright &y& Elsevier]

The aim of this article is to introduce nonautonomous and finite-time versions of central concepts from the theory of dynamical systems such as attractivity and bifurcation. The discussion includes an appropriate spectral theory for linear systems as well as finite-time analogues of the well-known transcritical and pitchfork bifurcation. [ABSTRACT FROM AUTHOR]

Abstract: Although, bifurcation theory of ordinary differential equations with autonomous and periodic time dependence is a major object of research in the study of dynamical systems since decades, the notion of a nonautonomous bifurcation is not yet established. In this article, two different approaches are discussed which are based on special notions of attractivity and repulsivity. Generalizations of the well-known one-dimensional transcritical and pitchfork bifurcation are obtained. [Copyright &y& Elsevier]