Properties of Dynamical Systems on Dendrites and Graphs

Dynamical systems generated by continuous maps on compact metric spaces can have various properties, e.g. the existence of an arc horseshoe, the positivity of topological entropy, the existence of a homoclinic trajectory, the existence of an omega-limit set containing two minimal sets and other. In [Kočan et al., 2014] we consider six such properties and survey the relations among them for the cases of graph maps, dendrite maps and maps on compact metric spaces. In this paper, we consider fourteen such properties, provide new results and survey all the relations among the properties for the case of graph maps and all known relations for the case of dendrite maps. We formulate some open problems at the end of the paper.

Horseshoes, Entropy, Homoclinic Trajectories, and Lyapunov Stability

We consider six properties of continuous maps, such as the existence of an arc horseshoe, the positivity of topological entropy, the existence of a homoclinic trajectory, or Lyapunov instability on the set of periodic points. The relations between the considered properties are provided in the case of graph maps, dendrite maps and maps on compact metric spaces. For example, by [Llibre & Misiurewicz, 1993] in the case of graph maps, the existence of an arc horseshoe implies the positivity of topological entropy, but we construct a continuous map on a Peano continuum with an arc horseshoe and zero topological entropy. We also formulate one open problem.

CHAOS ON ONE-DIMENSIONAL COMPACT METRIC SPACES

We consider various kinds of chaotic behavior of continuous maps on compact metric spaces: the positivity of topological entropy, the existence of a horseshoe, the existence of a homoclinic trajectory (or perhaps, an eventually periodic homoclinic trajectory), three levels of Li–Yorke chaos, three levels of ω-chaos and distributional chaos of type 1. The relations between these properties are known when the space is an interval. We survey the known results in the case of trees, graphs and dendrites.

Comparison of the Hyperbolic Perturbation Method and the Hyperbolic Lindstedt-Poincaré Method for Homoclinic Solutions of Self-Excited Systems

The comparison of the hyperbolic perturbation method and the hyperbolic Lindstedt-Poincaré method for homoclinic solutions of self-excited systems is studied in this paper. The homoclinic solution of a generalized Van del Pol system with strongly quadratic nonlinearity is analytically derived by both of the methods. The critical value of the bifurcation parameter under which homoclinic trajectory forms can be determined by the both of the perturbation procedures. Typical numerical examples are studied in detail and compared to illustrate the accuracy and the efficiency.

Entropy, horseshoes and homoclinic trajectories on trees, graphs and dendrites

It is known that the positiveness of topological entropy, the existence of a horseshoe and the existence of a homoclinic trajectory are mutually equivalent, for interval maps. The aim of the paper is to investigate the relations between the properties for continuous maps of trees, graphs and dendrites. We consider three different definitions of a horseshoe and two different definitions of a homoclinic trajectory. All the properties are mutually equivalent for tree maps, whereas not for maps on graphs and dendrites. For example, positive topological entropy and the existence of a homoclinic trajectory are independent and neither of them implies the existence of any horseshoe in the case of dendrites. Unfortunately, there is still an open problem, and we formulate it at the end of the paper.

The interaction of two spatially resonant patterns in thermal convection. Part 1. Exact 1:2 resonance

The linear stability of two superimposed layers of fluid, heated from below and separated by a thin conducting plate, is investigated. It is shown that when the ratio of the depths of the layers is close to 1:2, two distinct modes of convection can occur with preferred horizontal wavenumber in the ratio 1:2. The nonlinear evolution of a disturbance consisting of both modes is considered, and it is shown that travelling waves are the preferred mode of nonlinear convection for a wide range of parameter values. Other possible types of behaviour, including modulated waves and an attracting homoclinic trajectory, are also described in detail.