# Multistability, Chaos, and Control in the Deterministic and Stochastic Dynamics of Noise-Driven Nonlinear Oscillators

## TL;DR

This paper studies the complex behavior of a nonlinear oscillator influenced by noise and periodic forces, revealing chaotic dynamics and transitions under different conditions.

## Contribution

The study introduces a novel overlap-mapping approach to analyze and compare traveling-wave solutions in nonlinear oscillators.

## Key findings

- The system exhibits multistability and chaotic behavior under varying parameters and initial conditions.
- High-dimensional chaos is confirmed using Lyapunov exponents, Poincaré sections, and return-map analysis.
- Sensitivity to parameter perturbations and initial conditions is systematically quantified.

## Abstract

This paper presents a detailed investigation of the deterministic and stochastic dynamics of a noise-driven forced nonlinear oscillator in a periodically driven framework. An overlap-mapping approach is used to compare multiple traveling-wave solutions and verify the structural consistency among distinct solution families. The qualitative behavior of the system is further characterized through geometric and stability-based analysis, supported by two- and three-dimensional phase portraits, time-series responses, and reconstructed three-dimensional attractors to examine periodic and chaotic regimes under varying parameters and initial conditions. The sensitivity to parameter perturbations is quantified and the distribution of final states is analyzed to identify chaotic regions in the phase space. The high-dimensional chaotic nature of the dynamics is rigorously confirmed through Lyapunov exponent estimation, Poincaré sections, and return-map analysis, collectively demonstrating strong sensitivity to initial conditions and systematic transitions induced by parameter variations. These results provide a comprehensive dynamical description of the nonlinear oscillator and contribute to a deeper understanding of noise-influenced nonlinear driven systems.