Rich complex dynamics in new fractional-order hyperchaotic systems using a modified Caputo operator based on the extended Gamma function

Abstract

In this work, rich complex dynamics are observed in new fractional-order hyperchaotic systems when employing a modified Caputo-operator based on the extended Gamma function. Thus, the new parameter of the extended Gamma function provides higher degree of freedom that plays a vital role in the appearance (or disappearance) of chaotic attractors. Two numerical examples are provided to validate the proposed method. The numerical examples show that various chaotic, hyperchaotic and non-chaotic attractors are obtained when varying the new parameter of the modified Caputo-operator.In conclusion, this work shows that the modified Caputo-operator possesses higher degree of freedom than the classic one. This interesting phenomena result in obtaining rich complex dynamics in the systems that involve this new operator including phase transition from hyperchaotic to chaotic states and also phase transition from chaotic to non-chaotic states. Moreover, we show in the second numerical example that chaotic attractors exist only with the modified Caputo-operator using a specific choice of the system’s parameters, while the system’s counterpart does not show chaotic behaviors when using the classic Caputo-operator. Thus, the fractional systems with the modified Caputo-operator are shown to exhibit stronger chaotic dynamics than their counterparts with the old (classic) Caputo-operator.