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

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.