Essential dynamics in chaotic attractors
组织者
演讲者
Eran Igra
时间
2024年12月10日 09:00 至 10:30
地点
Online
线上
Zoom 518 868 7656
(BIMSA)
摘要
Assume we have a smooth vector field of $S^3$ whose fixed points are all connected by unstable heteroclinic orbits. It is well-known from numerical studies that such heteroclinic knots can trap between them chaotic attractors - which raises the following question: can we prove analytically these chaotic attractors exist? One approach to answer this question is to prove the existence of chaotic dynamics by adding some extra assumptions - like, say, some form of a Hyperbolicity condition. This begs another question - since we usually cannot prove the original flow to be hyperbolic, just how much the results we prove under hyperbolicity assumptions actually hold for the original flow?
In this talk we give a partial answer to this question. Inspired by the Thurston-Nielsen Classification Theorem we prove that in certain heteroclinic scenarios one can define a class of periodic orbits for the flow which persist (without changing their knot type) under a certain class of smooth homotopies of the vector field which keep the heteroclinic condition fixed. This has the following meaning - assume we can smoothly deform the dynamics trapped between the heteroclinic knot into a hyperbolic (or more precisely, singular hyperbolic) dynamical system, then all the periodic orbits for the singular hyperbolic system are also generated by the original flow. Following that will show how our results can be applied to study the dynamics of the Lorenz and Rössler attractors, and time permitting, conjecture how they can be generalized to derive a forcing theory for three-dimensional flows.
In this talk we give a partial answer to this question. Inspired by the Thurston-Nielsen Classification Theorem we prove that in certain heteroclinic scenarios one can define a class of periodic orbits for the flow which persist (without changing their knot type) under a certain class of smooth homotopies of the vector field which keep the heteroclinic condition fixed. This has the following meaning - assume we can smoothly deform the dynamics trapped between the heteroclinic knot into a hyperbolic (or more precisely, singular hyperbolic) dynamical system, then all the periodic orbits for the singular hyperbolic system are also generated by the original flow. Following that will show how our results can be applied to study the dynamics of the Lorenz and Rössler attractors, and time permitting, conjecture how they can be generalized to derive a forcing theory for three-dimensional flows.