Bifurcation, quasi-periodic dynamics, chaos, and soliton waves in the van der Waals normal form for fluidized granular matter

The fluidized granular media is a complex phenomenon with sharp regime changes, pattern formation, and coherent density-wave propagation that is usually treated by nonlinear evolution equations, which are reduced. Of these, the van der Waals normal form, a fourth-order nonlinear oscillator with viscous/frictional dissipation and cubic nonlinearity, offers a general model with which to investigate such phenomena. Although there has been a considerable amount of effort on the construction of precise travelling-wave solutions to van der Waals-type models, in past work, systematic comparisons of the auxiliary-equation choices and their effects on admissible parameter regimes, as well as combining solution construction with an extensive dynamical-systems viewpoint are commonly not made. This study closes such gaps by creating a unified travelling-wave solution, based on Bernoulli-type and Riccati-type auxiliary equations, and an explicit parameterized family of wave solutions, consisting of kink-type, bright-dark soliton, and complex-valued solitons. They are then connected to a normalized nonlinear oscillator to analyze phase-space structures, stability, bifurcations, and sensitivity to perturbations. Symmetry-driven pitchfork bifurcations, dissipation-driven transitions, and an abundance of periodic, quasi-periodic, and chaotic behavior become apparent as key results through equilibrium analysis, 2D/3D phase portraits, time series, and Poincaré diagnostics. The high-dimensional parameter space and model idealism are drawbacks. However, analytical wave solutions coupled with dynamical diagnostics offer strong benchmarks to the numerical dynamics of the propagation of density waves in fluidized granular systems and have been used in augmenting predictive insights about their propagation. Both the theoretical and practical research in the areas of granular flow, material processing, and pattern formation have important implications for these findings. In contrast to previous works, which use one framework for auxiliary equations, a systematic comparison of four ansatz schemes within one common Bernoulli-Riccati framework. The analytical solutions are additionally connected to the dynamics of nonlinear oscillators via the tools of equilibria, bifurcation, and chaos diagnostics, and a direct link is established between exact travelling waves and the structures of phase space.