Dynamics of Soliton Solutions, Dynamical Transitions, and Statistical Features of Noise‐Induced Multistable Nonlinear Systems

ABSTRACT In this paper, we will give a comprehensive theoretical and computational discussion of the concept of the fractional nonlinear dispersive systems, and how deterministic and stochastic driving forces influence complex oscillatory dynamics. Special attention is given to the overlap of soliton solutions which throws light on the sensitivity and stability of nonlinear waveforms that are of importance in plasma physics and optics. Despite bifurcation structures indicating ways to chaos and simultaneous multi‐stable states, dynamical geometry in analyzing bifurcation structures has shown fractal basin geometries, multistability transitions and attractor sensitivity. Periodically forced and noise‐driven oscillators are compared and an analysis of methods of controlling chaos, including OGY‐based stabilization, are also analyzed. Statistical indicators of stability, the coherence, and attractor geometry are measured in both deterministic and stochastic regimes by the methods of spectral analysis, complexity based on entropy, stochastic synchronization, and Monte Carlo simulation. When combined, these analyzes provide a cohesive method for identifying multistability, assessing basin resilience, and achieving efficient control in a system with nonlinear oscillations.