Exploring memory effects and chaotic transitions in the fractional–order Bogoyavlenskii dynamical system

We explore the fractional form of the Bogoyavlenskii equation in this work using the modified version of the Riemann-Liouville fractional derivative operator introduced by Jumarie to reveal the memory and hereditary properties of the fractional dynamical systems. This analytical treatment is done as an Addendum to the method proposed by Kudryashov which allows one to put together an exact solitary wave solution in a systematic way. This leads to the derivation of a number of bright, dark, and singular soliton solutions. The dynamical properties of these solutions are depicted in graphical simulations, 3D plots are used to represent the integer-order case, and the 2D profiles are used to show the changes related to the fractional-order parameter. In addition, the research looks at various nonlinear dynamical measures such as phase portrait, energy density distribution, maximum Lyapunov exponent, and time-series analysis to gain a deeper insight into the system dynamics. These studies offer further understanding on the stability, complexity, and dynamical progression of the fractional model, in terms of the efficacy of the suggested analytical framework to examine nonlinear fractional differential equations.