Chaotic and quasi-periodic dynamics in nonlinear micro-strain wave systems: Exact symbolic solutions and symmetry-based analysis

• First use of Khat III and GRat methods on the MSW nonlinear wave model. • Exact symbolic traveling waves validated by He’s numerical iteration. • Hamiltonian bifurcation reveals quasi-periodic and chaotic dynamics. • Results clarify symmetry structures and parameter-driven wave behavior. • Framework links symbolic derivation, computation, and physical modeling. This work presents a systematic theoretical and computational investigation of the micro-strain wave ( MSW ) model, a fundamental nonlinear evolution equation governing propagation phenomena in media with microscale deformation effects. The model encapsulates the coupling between nonlinear dispersion, material heterogeneity, and microscale elasticity, providing a versatile framework for studying wave modulation and self-sustaining dynamics in structured media. The MSW framework captures the intricate interplay between nonlinear dispersive structures and material microgeometry, with far-reaching implications in materials science, continuum mechanics, and nonlinear optics. By employing the Khater III ( K hat III) and generalized rational ( GR at) approaches, we construct closed-form solutions that describe permanent-form traveling waves intrinsic to the system. These explicit analytical forms serve as benchmarks for validating computational models and reveal parameter-dependent regimes of stability and modulation. Numerical validation is achieved through He’s Variational Iteration ( HVI ) scheme, providing a quantitative benchmark that demonstrates excellent agreement between symbolic solutions and their computational counterparts. This dual verification confirms both the mathematical rigor and the computational robustness of the applied symbolic methodologies. In addition, a bifurcation analysis based on the Hamiltonian function H ( ψ , υ ) is carried out to classify equilibrium configurations. The results reveal the onset of chaotic dynamics as well as quasi-periodic behaviors, thereby enriching the dynamical portrait of the governing model. Such findings illuminate the transition mechanisms between ordered and irregular wave states, offering insight into energy localization, stability loss, and multiscale interactions. Collectively, these findings introduce a robust framework for studying nonlinear wave phenomena and contribute substantively to ongoing developments in applied mathematics and theoretical physics.