Comparative Study of Oscillator Dynamics Under Deterministic and Stochastic Influences with Soliton Robustness Darboux Transformations and Chaos Transition

This paper presents a comprehensive study of nonlinear wave and oscillator dynamics under both deterministic and stochastic influences. By comparing soliton-like and dispersive waveforms, we employ spectral solvers, Darboux transformations, and nonlinear diagnostics, including Lyapunov exponents, power spectral analysis, and multidimensional phase-space reconstructions, to examine transitions from quasiperiodic motion to chaotic and stochastic regimes. The results highlight the robustness of soliton solutions in preserving energy and structure, in contrast to the degradation observed in dispersive waves under noise and damping. We also show that spectral broadening, entropy growth, and ergodic phase-space patterns are caused by the critical influence of initial conditions and noise intensity on system behavior. Incorporating control strategies such as OGY chaos control, this work provides a flexible framework for analyzing, modeling, and stabilizing nonlinear systems. Applications span nonlinear optics, fluid flows, and electrical lattices, offering insight into the interplay of nonlinearity and noise with implications for both theoretical understanding and practical system design.