Comparative Study of Constant- and Variable-Order Fractional Dynamics with RBFNN Approximation

The analysis of the complexity of dynamical systems generally presents a major dilemma between including the complexity of memory effects and being computationally efficient. For this purpose, this study proposes an unusual three-dimensional variable-order fractional chaotic system with absolute-value nonlinearity, which yields hidden chaotic attractors. The basic feature of novelty in this research is not about any particular mathematical or computational technique but rather the integration of such techniques. In particular, a major breakthrough has been made by integrating the recently developed chaotic system, which has no equilibrium point, with the memory property of the variable-order Liouville–Caputo derivative operator and overcoming all computational obstacles associated with such a system using the RBFNN surrogate model. The Liouville–Caputo fractional derivative is used for embedding these realistic memory effects, and the system is thoroughly studied via phase plots, bifurcations, Lyapunov exponents, and time series solutions. The comparison between the variable-order and equivalent constant-order cases clearly shows that the variable-order case exhibits more diverse dynamical behaviors compared to the constant-order case in terms of better dynamical transitions from periodic, quasi-periodic, and chaotic states, thus revealing the unique merits of the time-varying fractional order derivatives. In order to overcome the excessive computational costs usually involved in calculating the variable-order derivatives based on memory effects, a Radial Basis Function Neural Network (RBFNN) is proposed as a data-driven modeling approach. By training the RBFNN using the numerical data computed from the Lagrange polynomial interpolation method, the RBFNN is able to simulate the system responses. In summary, the findings reveal that the novel framework not only reveals hidden chaotic dynamics and rich dynamical behaviors but does so efficiently in terms of faster convergence and smaller mean-square error values at steady state.