On fractional and distributed order hyperchaotic systems with line and parabola of equilibrium points and their synchronization

Abstract

In this article, we introduced fractional and distributed order hyperchaotic Lü, Chen and Lorenz systems with both line and parabola of equilibrium points (EPs). Their dynamics which include invariance, dissipation, EPs and their stability, chaotic and hyperchaotic solutions are studied. Numerically we calculated the values of the systems parameters and the fractional order at which these systems have chaotic, hyperchaotic attractors and solutions that approach EPs. Those systems with no line and parabola of EPs have no hyperchaotic attractors of order 2 and 3. We discussed the difference between fractional order hyperchaotic (FOH) systems with line and parabola of EPs and distributed order hyperchaotic (DOH) systems have line and parabola of EPs. Finally we presented a scheme to investigate the generalization of combination-combination synchronization (GCCS) between two FOH systems and two DOH systems. A theorem is stated and proved to provide us with the analytical formula of the control functions to achieve this kind of synchronization. These analytical results are confirmed via numerical computations.