The Synchronization Of Different Fractional Order Chaotic Systems With Different Dimensions

Chaos is a special complex form of dynamic motion，it is determined by itself but cannot be predicted, and it is extremely sensitive to initial values, the random and complexbehavior widely exist in nonlinear dynamic systems. A few hundred years ago, people onlyregard chaos as a harmful phenomenon, but now people conduct research actively usingchaotic phenomenon, and chaos has been widely used in many fields. Currently, chaos hasbecome one of the focus of attention.Chaos is considered to be the third major breakthrough in physics, and fractionalcalculus theory is an important part of the theory of the calculus, the research of the fractionalcalculus has also undergone a long period of time. So far, people can describe the dynamiccharacteristics of the actual system using fractional calculus operator more accurately thanbefore. Therefore, the study of the fractional order chaotic system has gradually attracted theattention of many scholars.As a chaotic application of pivotal technologies, chaos synchronization is one of researchhotspot in recent years, and so far, there are a lot of great achievements on the research of thechaotic synchronization, and most of these methods applied to the self-synchronization ofchaotic systems with the same dimension and the synchronization of different chaotic systems,but there are only a few researches about the synchronization of different chaotic systemswith different orders. According to the current situation, in this paper, the fractional orderchaotic systems are regarded as the research object, the problem of the synchronization ofdifferent chaotic systems with different orders is studied using the theoretical derivation andnumerical simulation, and the research results as follows:First, based on sliding mode theory and adaptive control theory, the synchronization oftwo different fractional order chaotic systems is investigated, a fractional sliding surface withstrong robustness is designed and a suitable adaptive sliding controller is constructed, themovement trajectories of the system tend to the sliding surface using the controller, and canbe controlled to the sliding surface, then reach the default state along the sliding mode toguarantee the generalized synchronized behaviors between two chaotic systems with differentdimensions. Secondly, Numerical simulations on two pairs of chaotic systems(the hyper Chenchaotic system and Chen chaotic system,hyper Chen chaotic system and Lu chaotic system) are also carried out respectively to realize the synchronization control of chaotic systems. Thenumerical simulation results in good effect, and the generalized synchronization is achievedand the general applicability of this theorem is proved. Finally, based on nonlinear systemstability theorem and the active control principle, a new kind of nonlinear feedbacksynchronization controller is designed for fractional order chaotic system with differentdimensions. The structure of controller is simple and the selection is convenient. Numericalsimulations on the Lu chaotic system and a new hyper chaotic system are also carried outrespectively to guarantee the synchronized behaviors between two fractional order chaoticsystems with different dimensions. Simulation results show that two fractional order chaoticsystems with different dimensions can be realized and the effectiveness and feasibility of thecontroller is proved.