Translated Abstract
Fractional calculus, which is the generalization of the classical calculus, is one of important mathematical tools. Fractional-order nonlinear systems can be described by fractional differential equations. Compared with integer-order systems, fractional-order ones are more accurate to represent the physical characteristic of real systems. Therefore, fractional-order nonlinear systems play an irreplaceable role in modeling nonlinear dynamical systems. Nowadays, because the reports about the dynamics and synchronization for fractional-order nonlinear systems are few, the investigations about them have the practical significance for the application of this kind of systems.
In this dissertation, the dynamics and synchronization for fractional-order systems are studied. Two deterministic fractional-order systems are proposed. The fundamental properties and dynamics starting from one initial condition for these systems are investigated. The synchronization is realized via the adaptive method when the systems are chaotic, respectively. Due to the lack of numerical algorithm for global dynamics of fractional-order nonlinear systems, an extended generalized cell mapping method is proposed. The global dynamics, bifurcations, and crises for this kind of systems are discussed. Meanwhile, two special systems, that is, a fractional-order system with a random parameter and one with delay, are presented. The dynamics and synchronization for them are studied. The main works of this dissertation are listed as follows:
Firstly, a fractional-order Genesio-Tesi system with fifth order nonlinearity is proposed. The basic properties including the stability of equilibriums, numerical solutions and the minimum order to remain chaotic are discussed. The minimum order for the system to remain chaotic is obtained by computing the instability measure in the case of commensurate-order and incommensurate-order. The bifurcation and an interior crisis from single-scroll to double-scroll attractors are found with the variation of a derivative order. Furthermore, based on the stability theory of fractional-order systems, the synchronization of the system with unknown parameters is realized by designing appropriate controllers. Numerical simulations are carried out to demonstrate the effectiveness and flexibility of the controllers.
Secondly, a fractional-order complex T system is presented. The numerical solutions of the system are obtained by the improved predictor-corrector algorithm. The dynamics of the system including symmetry, the stability of equilibrium points, bifurcations with variation of system parameters and derivative orders are investigated. Period-doubling and tangent bifurcations with appropriate derivative orders and system parameters are observed. Besides, the control problem of the system is examined by using feedback control technique. Furthermore, based on the stability theory of fractional-order systems, the scheme of function projective synchronization for the fractional-order complex T system is presented. And function projective synchronization for the system is realized by designing appropriate synchronization controller. Numerical simulations are used to demonstrate the effectiveness and feasibility of the proposed scheme.
Thirdly, due to the absence of effective numerical algorithms, an extended generalized cell mapping method (EGCM) is developed for analysis of global dynamics for fractional-order systems. The operator of fractional calculus is non-local which causes the evolution of a fractional-order system cannot be described by the Markov chain. Taking advantage of the short memory principle, one can investigate the global dynamics of fractional-order systems by using the EGCM which combines the generalized cell mapping with the improved predictor-corrector algorithm. On the basis of the characteristics of the cell state space, the bound of the truncation error is defined to ensure that the truncation error is less than a half of a cell size. Four typical fractional-order systems, that is, a Logistic equation, a autonomous Duffing system, a Van der Pol oscillator, and a non-autonomous Duffing system are taken as examples. Periodic and chaotic attractors, boundaries, basins of attraction, and saddles are obtained by the EGCM. Our examples confirm the previous results, and furthermore demonstrate the accuracy and effectiveness of the proposed EGCM method for fractional-order systems.
Fourthly, by means of the extended generalized cell mapping (EGCM), crises of the non-autonomous fractional-order Duffing system are studied as a system parameter or derivative order is varied. A crisis is a sudden discontinuous change in chaotic sets involving three different chaotic basic sets: a chaotic attractor, a chaotic set on a fractal basin boundary, and a chaotic set in the interior of a basin and disjoint from the attractor. A boundary crisis results from the collision of a periodic (or chaotic) attractor with a chaotic (or regular) saddle in the fractal (or smooth) boundary. In such a case the attractor, together with its basin of attraction, is suddenly destroyed as the derivative order or system parameter passes through a critical value, leaving behind a chaotic saddle in the place of the original attractor and saddle after the crisis. An interior crisis happens when an unstable chaotic set in the basin of attraction collides with a periodic attractor, which causes the appearance of a chaotic attractor. In the same time the attractor and the unstable chaotic set are converted to the part of the chaotic attractor after the crisis. As a result, a crisis is generally defined as a collision between a chaotic basic set and a basic set, either periodic or chaotic. These results further reveal that the EGCM is a powerful tool to determine the mechanism of crises in fractional-order systems. To our knowledge, the boundary and interior crises in fractional-order systems are represented for the first time from the global perspective.
Fifthly, dynamics and synchronization of a stochastic fractional-order Lorenz system with complex variables are studied. The stochastic system is reduced into the equivalent deterministic one by using the Laguerre polynomial approximation method. Two important responses, that is, the ensemble mean responses (EMRs) and the responses of the mean parameter system (SRMs), are determined. In the analysis of dynamics, we suggest that the EMRs of the stochastic system have an extremely long transient with regular geometrical structures converging to a fixed point as . Besides, based on the stability theory of fractional-order systems, the synchronization for the stochastic fractional-order system with fully unknown parameters is realized in the sense of ensemble mean.
Finally, a fractional-order system with time delay is proposed. The predictor-corrector method for delayed fractional differential equations is used to solve the system. Then, the dynamics of the system with variation of the time delay is studied. Different periodic and chaotic attractors are obtained by numerical simulations. Based on the stability theory of fractional-order systems, the scheme of synchronization for the time-delayed fractional-order system is presented. By designing appropriate controllers, the synchronization for the proposed system is achieved.
Approaches developed in this dissertation could be used to analyzing fractional-order nonlinear systems and the results may facilitate a new method for this kind of systems.
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