Exploring the non-classical symmetry, bifurcation with sensitivity analysis of a (3 + 1)-dimensional nonlinear evolution equation

Abstract

In this research, we systematically examined the special solutions of the $(3+1)$-dimensional evolution equation corresponding to nonclassical symmetries. By employing the identified symmetries, we developed invariant solutions that reveal the underlying structure of the equation and its solutions. Additionally, bifurcation analysis was conducted to understand the qualitative shifts in the system's behavior. We investigated sensitivity to initial conditions and the presence of unusual attractors to better comprehend the system's chaotic dynamics. While invariant solutions provide exact representations of certain dynamical states, bifurcation and chaos analyses offer insights into the system's intrinsic transitions and complexity. This study not only enhances the theoretical understanding of high-dimensional nonlinear evolution equations but also provides a foundation for future applications in various fields where complex processes are modeled by such equations.