New Theorem on Chaos Transitions in Second-Order Dynamical Systems with Tikhonov Regularization

TL;DR

This paper investigates how Tikhonov regularization affects chaos and bifurcations in second-order dynamical systems, providing a theoretical framework for understanding and controlling complex behaviors in various scientific fields.

Contribution

It introduces new stability conditions and methods for managing nonlinear instabilities in second-order systems with Tikhonov regularization, advancing the understanding of chaos transitions.

Findings

Identification of critical bifurcation points leading to chaos

Characterization of strange attractors in regularized systems

Proposed methods for controlling nonlinear instabilities

Abstract

This study examines second-order dynamical systems incorporating Tikhonov regularization. It focuses on how nonlinearities induce bifurcations and chaotic dynamics. By using Lyapunov functions, bifurcation theory, and numerical simulations, we identify critical transitions that lead to complex behaviors like strange attractors and chaos. The findings provide a theoretical framework for applications in optimization, machine learning, and biological modeling. Key contributions include stability conditions, characterization of chaotic regimes, and methods for managing nonlinear instabilities in interdisciplinary systems.