Dynamical Systems On Three Manifolds Part II: 3-Manifolds,Heegaard Splittings and Three-Dimensional Systems

TL;DR

This paper explores the complex global behavior of nonlinear systems on 2- and 3-manifolds, demonstrating the possibility of arbitrarily intricate chaotic regimes through topological methods like Heegaard splittings.

Contribution

It introduces a topological approach to constructing nonlinear systems with highly complex chaotic behaviors on 3-manifolds, expanding understanding of global dynamics.

Findings

Systems can have arbitrarily many knotted and linked chaotic regimes.

Heegaard splittings enable the design of systems with complex global behavior.

Topological methods are effective in analyzing nonlinear systems on manifolds.

Abstract

The global behaviour of nonlinear systems is extremely important in control and systems theory since the usual local theories will only give information about a system in some neighbourhood of an operating point. Away from that point, the system may have totally different behaviour and so the theory developed for the local system will be useless for the global one. In this paper we shall consider the analytical and topological structure of systems on 2- and 3- manifolds and show that it is possible to obtain systems with 'arbitrarily strange' behaviour, i.e., arbitrary numbers of chaotic regimes which are knotted and linked in arbitrary ways. We shall do this by considering Heegaard Splittings of these manifolds and the resulting systems defined on the boundaries.