Title: Essential dynamics in chaotic attractors
Abstract:
Assume we have a three-dimensional flow whose dynamics are hyperbolic on some attracting invariant set. Whenever this is the case the dynamics on the attractor will be "chaotic" – for example, the attractor will include infinitely many periodic orbits belonging to infinitely many different knot types (i.e. the Birman-Williams Theorem). Due to such striking qualitative properties, it is often useful to think of hyperbolic flows as toy models for chaotic behavior - and since such non-linear phenomena are all around us we are led to the following natural question: just how much of this rich and beautiful theory of hyperbolic dynamics can be applied to analytically study real-life chaotic attractors - i.e., chaotic attractors discovered through science and engineering? At present, the answer is “not much”. In fact, of all the real-life attractors known to us, to date only one was proven to generate hyperbolic dynamics - and to make things worse, in many other interesting models there are good indications the dynamics will not be hyperbolic.
That being said, by most numerical studies real-life chaotic attractors often behave as if they were hyperbolic. This observation led Giovanni Gallavotti to conjecture the famous “Chaotic Hypothesis” - namely, that the dynamics on chaotic attractors are orbitally equivalent to some hyperbolic dynamical system. Inspired by the Thurston-Nielsen Classification Theorem and the Orbit Index Theory (the latter originally due to K. Aligood, J.A. Yorke and J. Mallet-Paret), we give a partial answer to the Chaotic Hypothesis. Namely, we prove that whenever a certain heteroclinic condition is satisfied by a given flow, the topology of the resulting phase space will force certain chaotic behavior – which shares many qualitative properties one often associates with hyperbolic systems. Following that, we apply these ideas to study two famous examples of "real-life" chaotic dynamical systems – the Lorenz and the Rössler attractors. Time permitting, we will also conjecture how these results can be generalized to derive a possible proof for the Chaotic Hypothesis, as well as extend the notion of topological forcing to three-dimensional flows.