Planar nonautonomous polynomial equations: The Riccati equation

We give a few sufficient conditions for the existence of two periodic solutions of the Riccati ordinary differential equation in the plane. We give also examples of the equation without periodic solutions.     

Coexistence of domains of attraction of nonautonomous neural networks with time-varying delays

This paper presents new dynamical behavior, i.e., the coexistence of 2^N domains of attraction of N-dimensional nonautonomous neural networks with time-varying delays. By imposing some new assumptions on activation functions and system parameters, we construct 2^N invariant basins for neural system and derive some criteria on the boundedness and exponential attractivity for each invariant basin. Particularly, when neural system degenerates into periodic case, we not only attain the coexistence of 2^N periodic orbits in bounded invariant basins but also give their domains of attraction. Moreover, our results are suitable for autonomous neural systems. Our new results improve and generalize former ones. Finally, computer simulation is performed to illustrate the feasibility of our results.     

Research on the properties of some planar polynomial differential equations

In this article, we use the new method of reflecting function to study the behavior of solutions of nonlinear time-vary differential equations, and give the sufficient conditions for these equations which have the reflecting function in the form of linear and fractional. We applied the obtained results to discuss the qualitative behavior of solutions of the higher degree polynomial differential systems and derive the sufficient conditions for a critical point to be a center.     

The Conley index for fast-slow systems II: Multidimensional slow variable

We use the Conley index theory to develop a general method to prove existence of periodic and heteroclinic orbits in a singularly perturbed system of ODEs. This is a continuation of the authors' earlier work [T. Gedeon, H. Kokubu, K. Mischaikow, H. Oka, J. Reineck, The Conley index for fast-slow systems I: One-dimensional slow variable, J. Dynam. Differential Equations 11 (1999) 427-470] which is now extended to systems with multidimensional slow variables. The key new idea is the observation that the Conley index in fast-slow systems has a cohomological product structure. The factors in this product are the slow index, which captures information about the flow in the slow direction transverse to the slow flow, and the fast index, which is analogous to the Conley index for fast-slow systems with one-dimensional slow flow [T. Gedeon, H. Kokubu, K. Mischaikow, H. Oka, J. Reineck, The Conley index for fast-slow systems I: One-dimensional slow variable, J. Dynam. Differential Equations 11 (1999) 427-470].     

An extension of Sharkovsky's theorem to periodic difference equations

We present an extension of Sharkovsky's theorem and its converse to periodic difference equations. In addition, we provide a simple method for constructing a p-periodic difference equation having an r-periodic geometric cycle with or without stability properties.     

A proof of Culter’s theorem on the existence of periodic orbits in polygonal outer billiards

We prove a recent theorem by C. Culter every polygonal outer billiard in the affine plane has a periodic trajectory.      

Forward attraction in nonautonomous difference equations

Two types of attractors consisting of families of sets that are mapped into each other under the dynamics have been defined for nonautonomous difference equations, one using pullback convergence with information about the system in the past and the other using forward convergence with information about the system in the future. In both cases, the component sets are constructed using a pullback argument within a positively invariant family of sets. The forward attractor so constructed also uses information about the past, which is very restrictive and not essential for determining future behaviour. Here an alternative is investigated, essentially the omega-limit set of the system, which Chepyzhov and Vishik called the uniform attractor. It is shown here that this set is asymptotically positively invariant, thus providing it with an hitherto missing form of invariance, if in somewhat weaker than usual, that one expects an attractor to possess. As a consequence this set provides useful information about the behaviour in current time during the approach to the limit.     

Selfdual Variational Principles for Periodic Solutions of Hamiltonian and Other Dynamical Systems

Selfdual variational principles are introduced in order to construct solutions for Hamiltonian and other dynamical systems which satisfy a variety of linear and nonlinear boundary conditions including many of the standard ones. These principles lead to new variational proofs of the existence of parabolic flows with prescribed initial conditions, as well as periodic, anti-periodic and skew-periodic orbits of Hamiltonian systems. They are based on the theory of anti-selfdual Lagrangians developed recently in Ghoussoub (2007a Ghoussoub , N. ( 2007a ). Anti-selfdual Lagrangians: Variational resolutions of non self-adjoint equations and dissipative evolutions . AIHP-Analyse Non Linéaire 24 : 171 – 205 . [Google Scholar] b Ghoussoub , N. ( 2007b ). Anti-selfdual Hamiltonians: Variational resolution for Navier-Stokes equations and other nonlinear evolutions . Comm. Pure & Applied Math. 60 ( 5 ): 619 – 653 .[Crossref], [Web of Science ®] , [Google Scholar] c Ghoussoub , N. ( 2007c ). Selfdual partial differential systems and their variational principals . Submitted for publication . [Google Scholar]).     

Pullback exponential attractors for nonautonomous equations Part II: Applications to reaction-diffusion systems

The existence of a pullback exponential attractor being a family of compact and positively invariant sets with a uniform bound on their fractal dimension which at a uniform exponential rate pullback attract bounded subsets of the phase space under the evolution process is proved for the nonautonomous logistic equation and a system of reaction-diffusion equations with time-dependent external forces including the case of the FitzHugh-Nagumo system.     

Pullback attractors in nonautonomous difference equations

Nonautonomous difference equations are formulated as cocycles which generalize semigroups corresponding to autonomous difference equations. Pullback attractors are the appropriate generalization of autonomous attractors to cocycles. The existence of a pullback attractor follows when the difference equation cocycle has a pullback absorbing set. Results from the literature are outlined, including the construction of a Lyapunov function characterizing pullback attraction, and illustrated with several examples.     

Integrability conditions for nonautonomous quad-graph equations

This paper presents a systematic investigation of the integrability conditions for nonautonomous quad-graph maps, using the Lax pair approach, the ultra-local singularity confinement criterion and direct construction of conservation laws. We show that the integrability conditions derived from each of the methods are the one and the same, suggesting that there exists a deep connection between these techniques for partial difference equations.     

Topological simplification of nonautonomous difference equations

In this paper, we continue the study of geometric properties of nonautonomous difference equations in arbitrary Banach spaces which was begun in [2 Aulbach, B. 1998. The fundamental existence theorem on invariant fiber bundles. Journal of Difference Equations and Applications, 3(5–6): 501–537.  [Google Scholar],3 Aulbach, B. and Wanner, T. 2003. Invariant foliations and decoupling of non-autonomous difference equations. Journal of Difference Equations and Applications, 9(5): 459–472. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]]. Building on previous results on invariant fiber bundles and foliations, this paper addresses the problem of topological simplifications via continuous conjugacies and semiconjugacies. In particular, we establish a reduction principle for not necessarily invertible difference equations, as well as a generalized Hartman–Grobman theorem for systems with not necessarily invertible linear part.     

Pullback attractors of nonautonomous reaction-diffusion equations

In this paper, firstly we introduce the concept of norm-to-weak continuous cocycle in Banach space and give a technical method to verify this kind of continuity, then we obtain some abstract results for the existence of pullback attractors about this kind of cocycle, using the measure of noncompactness. As an application, we prove the existence of pullback attractors in of the cocycle associated with the solutions for some nonlinear nonautonomous reaction-diffusion equations. The attractor pullback attracts all bounded subsets of in the norm of .     

Periodic orbits of a one-dimensional non-autonomous Hamiltonian system

In this paper we study the properties of the periodic orbits of with x∈S¹ and a T₀ periodic potential. Called the frequency of windings of an orbit in S¹ we show that exists an infinite number of periodic solutions with a given ρ. We give a lower bound on the number of periodic orbits with a given period and ρ by means of the Morse theory.     

Continuous symmetric perturbations of planar power law forces

We show the existence of periodic solutions for continuous symmetric perturbations of certain planar power law problems.     

Uniformly attracting solutions of nonautonomous differential equations

Understanding the structure of attractors is fundamental in nonautonomous stability and bifurcation theory. By means of clarifying theorems and carefully designed examples we highlight the potential complexity of attractors for nonautonomous differential equations that are as close to autonomous equations as possible. We introduce and study bounded uniform attractors and repellors for nonautonomous scalar differential equations, in particular for asymptotically autonomous, polynomial, and periodic equations. Our results suggest that uniformly attracting or repelling solutions are the true analogues of attracting or repelling fixed points of autonomous systems. We provide sharp conditions for the autonomous structure to break up and give way to a bewildering diversity of nonautonomous bifurcations.