Abstract
A fractional-order system with complex variables is proposed. Firstly, the dynamics of the system including symmetry, equilibrium points, chaotic attractors, and bifurcations with variation of system parameters and derivative order are studied. The routes leading to chaos including the period-doubling and tangent bifurcations are obtained. Then, based on the stability theory of fractional-order systems, the scheme of synchronization for the fractional-order complex system is presented. By designing appropriate controllers, the synchronization for the system is realized. Numerical simulations are carried out to demonstrate the effectiveness of the proposed scheme.
Highlights
Fractional calculus considered as the generalization of the integer-order calculus can be dated back to the 17th century. It has attracted many researchers’ interest for the ability to describe practical problems. Rich dynamics such as chaos and bifurcation exist in many fractionalorder systems [1,2,3]
It should be noted that fractional-order chaotic systems with complex variables can be used to increase the content of transmitting information signals and enhance their security further
It is well known that the initial conditions for the fractional differential equations with Caputo derivatives take on the same form as those for the integer-order ones, which is very suitable for practical problems [15]
Summary
Fractional calculus considered as the generalization of the integer-order calculus can be dated back to the 17th century. It has attracted many researchers’ interest for the ability to describe practical problems. It is an interesting and meaningful topic for researchers to study the dynamics and synchronization for fractional-order complex nonlinear systems. The chaos and synchronization for discrete fractional-order systems have been investigated in detail [10,11,12,13]. Based on the stability theory of fractional-order systems, the scheme of synchronization for the fractionalorder complex system is presented, and numerical simulations are used to demonstrate the effectiveness and feasibility of the proposed scheme.
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