Stability And Nonlinear Dynamics Of Several Kinds Of Fractional-order System

The fractional calculus is the generalization of classical differential and inte-gral calculus. The advantages of fractional-order models comparing with classical onesshow that, on one hand, these models has less variables and simpler mathematical form,on the other hand, the fractional calculus closely related to the fractal, has specialproperties such as non-local, memory and power laws. For example, the fractional-orderconstitutive equation in viscoelastic materials and non-Newtonian fluid mechanics.The fractional-order models can describe many anomalous behaviors and phenomena incomplex systems naturally and effectively. While the theory and applications offractional calculus are still in a developmental stage. The aim of this paper is toimprove the theoretical basis of fractional-order differential equation and its system,and to apply such new system to science and engineering. Specifically, we perfect thealgorithm and the stability testing of fractional-order system theoretically. In theapplication, we construct specific fractional-order dynamical systems according tosome phenomena and study the complex nonlinear dynamical behaviors of such system inthe sense of dynamics.The properties of solution and accurate, efficient algorithems are well known inclassical differential equations analytically or numerically. However, it is diffi-cult to obtain similar results in fractional-order case. Although some classicalalgorithms for solving ordinary differential equations are applied to fractional-order case, we still could not obtain an algorithm exactly and efficiently similar tofourth order Runge-Kutta method. Hence, this problem is one of the concerned issuesin solving fractional-order differential equations. In this paper, we try to extendthe classical multi-step differential transform method to the fractional-order caseand point out the obvious error when neglecting the effect of the memory term by usingthis generalized method in relevant reference. To overcome this fault, we will proposethe modified multi-step differential transform algorithm to solve linear and nonlinearfractional-order differential equations as the order varies between0and2.To the stability testing of fractional-order system, some authors proposed thefirst and second Lyapunov method and the generalized Mittag-Leffler stability, butthe strict mathematical proof of such methods are needed to improve. Some classical criterions based on the well known characteristic roots analysis method have alreadyextended to the fractional-order case, however, they are often limited or unpractical.In this paper, we will apply the classical Hassard stability criterion to a class oflinear fractional-order time-delay system.In the application, we will concern some phenomena in complex network and bio-engineering, and obtain the fractional-order dynamic models. Further, the Hopf bifur-cation and complex nonlinear dynamic behavior such as chaos are investigated. Thefractal structure, the inner property of many complex networks contributes to againstinterference or attack outside and improves the robustness of the network system. Inthis paper, we construct the nonlinear fractional-order time-delay small world networkbased on integer case and then study the nonlinear dynamical behaviors of such newmodel. In order to enhance the stability of the network, we use the effective controlstrategy to delay the occurring of the Hopf bifurcation.Similarly, fractional cal-culus is an adequate tool to describe the dynamics of biological cell and tissues. Weconstruct the fractional-order Morris-Lecar （M-L） neuron system based on the cla-ssical M-L neuron model due to the real dielectric behavior of the cell membranedescribed well by the fractional-order capacitor. Then, we use the classic bifurcationtheory of fast-slow dynamic system to reveal the abundant bursting pattern of this newneuron model.