Dynamics of Hidden Attractors in Four-Dimensional Dynamical Systems with Power Law
Zareen A. Khan, Javed Khan, Sayed Saifullah, Amir Ali
Abstract
Four-dimensional continuous chaotic models with Caputo fractional derivative are presented. Fixed point theory is used to investigate the existence and uniqueness of complex systems. The dynamical properties are studied, including the Lyapunov exponent, phase portrait, and time series analysis. The hyperchaotic nature of each system is revealed by the positive exponents. The numerical method is introduced to describe the influence of the order of the Caputo fractional derivative. The phase portraits are presented to investigate the behavior and effect of some key parameters and fractional orders on model dynamics. The systems approach fixed point attractors for fractional-order and increase the visibility of the attractor by decreasing fractional order. This means that a change in fractional order has a significant impact on the dynamics of the models. When the order of the derivative is equal to one, both systems oscillate frequently. However, as the fractional order is reduced, the system oscillations decrease as compared to the integer-order, and the system moves towards its fixed point, reveals the hidden attractors inherent in the system, and enabling it to develop to a stable state more efficiently.