Fixed points for planar maps with multiple twists, with application to nonlinear equations with indefinite weight

In this paper, we investigate the dynamical properties associated with planar maps which can be represented as a composition of twist maps together with expansive–contractive homeomorphisms. The class of maps we consider present some common features both with those arising in the context of the Poincaré–Birkhoff theorem and those studied in the theory of topological horseshoes. In our main theorems, we show that the multiplicity results of fixed points and periodic points typical of the Poincaré–Birkhoff theorem can be recovered and improved in our setting. In particular, we can avoid assuming area-preserving conditions and we also obtain higher multiplicity results in the case of multiple twists. Applications are given to periodic solutions for planar systems of non-autonomous ODEs with sign-indefinite weights, including the non-Hamiltonian case. The presence of complex dynamics is also discussed. This article is part of the theme issue ‘Topological degree and fixed point theories in differential and difference equations’.

How to Construct Complex Dynamics? A Note on a Topological Approach

We investigate the presence of complex behaviors for the solutions of two different dynamical systems: one is of discrete type and the other is continuous. We give evidence of “chaos” in the framework of topological horseshoes and show how different problems can be analyzed by the same procedure.

Complex Dynamics in One-Dimensional Nonlinear Schrödinger Equations with Stepwise Potential

We prove the existence and multiplicity of periodic solutions as well as solutions presenting a complex behavior for the one-dimensional nonlinear Schrödinger equation -ε2u′′+V(x)u=f(u), where the potential V(x) approximates a two-step function. The term f(u) generalizes the typical p-power nonlinearity considered by several authors in this context. Our approach is based on some recent developments of the theory of topological horseshoes, in connection with a linked twist maps geometry, which are applied to the discrete dynamics of the Poincaré map. We discuss the periodic and the Neumann boundary conditions. The value of the term ε>0, although small, can be explicitly estimated.

A New 3D Autonomous Continuous System with Two Isolated Chaotic Attractors and Its Topological Horseshoes

Based on the 3D autonomous continuous Lü chaotic system, a new 3D autonomous continuous chaotic system is proposed in this paper, and there are coexisting chaotic attractors in the 3D autonomous continuous chaotic system. Moreover, there are no overlaps between the coexisting chaotic attractors; that is, there are two isolated chaotic attractors (in this paper, named “positive attractor” and “negative attractor,” resp.). The “positive attractor” and “negative attractor” depend on the distance between the initial points (initial conditions) and the unstable equilibrium points. Furthermore, by means of topological horseshoes theory and numerical computation, the topological horseshoes in this 3D autonomous continuous system is found, and the topological entropy is obtained. These results indicate that the chaotic attractor emerges in the new 3D autonomous continuous system.

A Chaotic System with an Infinite Number of Equilibrium Points: Dynamics, Horseshoe, and Synchronization

Discovering systems with hidden attractors is a challenging topic which has received considerable interest of the scientific community recently. This work introduces a new chaotic system having hidden chaotic attractors with an infinite number of equilibrium points. We have studied dynamical properties of such special system via equilibrium analysis, bifurcation diagram, and maximal Lyapunov exponents. In order to confirm the system’s chaotic behavior, the findings of topological horseshoes for the system are presented. In addition, the possibility of synchronization of two new chaotic systems with infinite equilibria is investigated by using adaptive control.