Orbits and chaos in dynamical systems

Paul Cleary
1988 Bulletin of the Australian Mathematical Society  
Chaos is a widespread phenomenon occuring in dynamical systems in such diverse fields as fluid dynamics, astrophysics, electronics, plasma physics and biology. It is the aim of this thesis to study the global properties of systems, such as their integrability, and their local properties, such as the topology of their orbit families, for several difFerent dynamical problems. An n -dimensional system is integrable if it possesses n isolating integrals of the motion. Integrable systems are the
more » ... systems are the exception and not the rule. One tool currently being used to examine their integrability is Painleve Analysis. This is an analytic technique based on singularity analysis in the complex time plane. In this thesis the effects of generalising the analysis by enlarging the class of admissible singuarities we have examined, and a selection of two dimensional quartic polynomial potentials which were either integrable or close to integrable, found. The integrability of some of these potentials was confirmed by explicit analytic calculation of the second integrals of the motion and numerical Surface of Section calculations. A classification scheme for families of periodic and quasi-periodic orbits based on both the geometry of the orbits and their topological structure in the Surface of Section was developed. This classification scheme was then used to describe and classify all the orbits occuring in the previously located quartic potentials. There appears to be a strong relationship between the periodic orbits existing in a potential and its integrability. The KAM theorem gives sufficient conditions on the existence of invariant curves in the Surfaces of Section of non-integrable potentials. There is no converse theorem about which invariant curves definitely do break up under perturbation. However it was found that some rational invariant curves definitely do exist in non-integrable potentials. This leads to the idea of a potential being locally integrable about a periodic orbit if all the rational invariant curves, (corresponding to tori in phase space with rational winding numbers), surrounding a periodic orbit in the Surface of Section exist. There is also a strong relationship between the topology of families of regular orbits, (a local property), and the integrability of the potential,
doi:10.1017/s000497270002712x fatcat:dutnfdtb6bdiddhh7btfxlsbpe