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

The paper asserts a new theorem on stability-to-chaos transitions for a second-order system with Tikhonov regularization but provides only a high-level, internally inconsistent sketch with several mathematical errors (e.g., incorrect energy-derivative algebra, misuse of Hopf bifurcation in a nonautonomous setting, and inconsistent model definitions) and no rigorous proof of the claimed critical parameter set or chaos criteria. The candidate model’s solution is conceptually stronger and uses a standard dissipative/variational/Floquet framework, including a correct energy-balance identity and a plausible route to a positive largest Lyapunov exponent; however, it still leaves gaps (e.g., it does not rigorously prove that the zero-crossings of the Lyapunov exponent and of the average energy derivative coincide, nor does it prove the existence of a chaotic invariant set).