Resumen de Aspects of global dynamics in nonautonomous dynamical systems
Carlos Lopesino Jiménez de Zadava Lissón
Lagrangian descriptors (LDs) are a recent tool that has been used in multiple applications as a methodto uncover the phase space of time-dependent dynamical systems. The main goal of this work is to providerigorous results about this technique in the discrete and continuous time setting.First we extend the definition of LD to apply it in two dimensional, area-preserving, autonomous andnonautonomous discrete time dynamical systems. We then proceeded to prove rigorous results by consider-ing four different model problems: a hyperbolic saddle point for a linear, area-preserving autonomous map,a hyperbolic saddle point for a nonlinear, area-preserving autonomous map, a hyperbolic saddle point for alinear, area-preserving nonautonomous map, and a hyperbolic saddle point for a nonlinear, area-preservingnonautonomous map. The choice of a specific norm allows us to provide a rigorous setting for the notion of”singular sets” that correspond to invariant manifolds of hyperbolic points. From the computational pointof view, we also analyze the performance of LDs to reveal chaotic invariant sets.We then extend these results to the continuous setting by also considering analogous particular cases: ahyperbolic saddle point for linear autonomous systems, a hyperbolic saddle point for nonlinear autonomoussystems, a hyperbolic saddle point for linear nonautonomous systems and a hyperbolic saddle point for non-linear nonautonomous systems. Additionally, we discuss further rigorous results which show the ability of LDsto highlight other invariant sets, such as n-tori. These results are an extension of the ergodic partition theorywhich we illustrate by applying LDs to some examples, such as the planar field of the harmonic oscillatorand the 3D ABC flow. We also provide a discussion on the requirement of the objectivity (frame-invariance)property for tools designed to reveal phase space structures.Finally, we address the challenge of rigorously proving the presence of chaotic invariant sets in aperiodicallytime-dependent systems. In the context of discrete dynamical systems, we prove the existence of a chaoticsaddle for a well-known piecewise-linear map of the plane, named the Lozi map. This is studied in itsorientation and area-preserving version. We apply the first and second Conley-Moser conditions to obtainthe proof of the existence of a chaotic saddle in the autonomous setting. Then we generalize the Lozi map toits nonautonomous version and prove that the first and the third Conley-Moser conditions are satisfied, thusimplying the existence of a chaotic saddle. Lastly, equipped with a discrete LD, we numerically demonstratehow the structure of this nonautonomous chaotic saddle varies as parameters are varied.
