Soliton formation and chaos dynamics in a one-dimensional pre-compressed granular crystal

In this study, the dynamical behavior and wave profiles of a pre-compressed one-dimensional granular crystal are investigated using transformation schemes and the one-dimensional optimal system. Initially, we perform a dynamical analysis using bifurcation analysis. Next, we apply an external force to the dynamical system and examine its chaotic behavior. The presence of chaos is confirmed through 2D phase portraits, return maps, fractal dimensions, bifurcation diagrams, time series, basins of attraction, permutation entropy, Lyapunov exponents, and multistability under various values of frequency and amplitude of the external force. Subsequently, soliton solutions are derived using the generalized Riccati equation mapping method, yielding trigonometric, hyperbolic, and exponential forms. To illustrate the physical significance of the model, 2D and 3D graphical results are presented in Mathematica under suitable parameter selections. Finally, the stability analysis of the considered model is discussed. All simulations are performed in Python.