Research On Dynamics Of Several Dynamical Systems Based On Logistic Models And Parameter Identification

Chaos and bifurcation phenomena are the main forms of complex dynamics in non-linear dynamical systems. At present, chaos and bifurcation can be found in discretesystems,continuous systems (ordinary diferential systems,fractional-order diferentialsystems, partial diferential systems and so on). So far, there is no strict unifed cri-terion to defne chaos. Some most popular defnitions of chaos are as follows: Li-Yorkechaos,Devaney chaos,Wiggins chaos and so on. But, chaos has some distinctive fea-tures: certainty,boundedness,sensitive dependence on initial conditions,ergodicity,positive largest lyapunov exponent, etc. However, proving strictly the existence of chaos,proving mathematically the bifurcation and identifying the parameters of chaotic systemsare all technically difcult problems.The discrete or continuous Logistic models play an important place on the study ofchaos and bifurcation. Based on the Logistic models, we study the dynamics of a fewhigher-dimensional discrete systems and fractional-order systems and parameter identif-cation. When one-dimension discrete Logistic map is extended to two-dimension discretepredator-prey system, this thesis proves rigorously that there exists bifurcation and showsthe approximate expression of invariant curve which is produced by Neimark-Sacker bi-furcation. When the higher-dimensional discrete systems are coupled with Logistic map,it is called a coupled map lattice system. With open boundary conditions, the thesisproves strictly the existence of chaos in this system by using the theory of Marotto. Atthe same time, when the improved continuous Logistic equation is extended to fractional-order case, homotopy analysis method and optimal homotopy analysis are used to solveapproximate analytical solutions for a fractional-order Logistic equation with Allee ef-fect. Moreover, we prove theoretically existence and uniqueness of solution of a kind ofgeneralized fractional-order Logistic equation. Finally, we frstly study the parameteridentifcation for fractional-order chaotic systems by using Particle Swarm Optimization.This article is made up of fve chapters. The main contents are as follows.Chapter1introduces the background, signifcance of the research and the majorworks. It recalls the history of chaos and bifurcation and introduces local bifurcationtheory of discrete systems,defnitions of chaos,common methods for chaos. It briefydescribes the knowledge of chaos synchronization and chaos control,basic concepts andpropositions of fractional-order diferential and integral,numerical method for fractional-order diferential equation,i.e., predictor-corrector algorithm.Chapter2studies bifurcation in a two species discrete two-dimensional predator-prey system which is derived from the single population of one dimensional discrete Logisticequation. Here, stability of the fxed points is discussed in detail. Using the theoryof center manifold theorem and bifurcation, we mathematically prove the existence ofFlip bifurcation and Neimark-Sacker bifurcation and obtain the approximate expressionof invariant curve which is derived from the Neimark-Sacker bifurcation. Numericalanalysis indicates that the period doubling bifurcation is an important route to chaosand there exists various periodic windows in chaotic regions. What's more, we controlthe occurrence of bifurcation using hybrid control method which can advance (or delay)Flip bifurcation and Neimark-Sacker bifurcation. Numerical simulations are consistentwith the theoretical results.Chapter3considers the complex dynamics in coupled map lattice system with openboundary conditions. The local function is Logistic mapping. With the help of Marottotheorem, we frst strictly prove that there exists Li-Yorke chaos in this discrete high-dimension system and obtain the chaotic phase diagrams and bifurcation fgures. More-over,0-1chaos test method is applied to analyze chaos, which is consistent with thetheoretical analysis of Li-Yorke chaos. Finally, we control the chaos to fxed point bydelayed feedback control method.Chapter4studies a fractional-order Logistic equation with Allee efect and a gener-alized fractional-order Logistic equation. By the homotopy analysis method and optimalhomotopy analysis method, we obtain the approximate analytic solution of a fractional-order Logistic equation with Allee efect and prove that the convergence of solution. It isconsistent with numerical solutions by predictor-corrector algorithm. At the same time,with the Banach fxed point theorem, we prove strictly the existence and uniqueness ofsolutions of a class of generalized fractional-order Logistic equation (Das model). Also,we analyze the stability of equilibrium points and carry out numerical simulations whichare consistent with theoretical results.Chapter5analyzes the parameter identifcation of fractional-order chaotic systems.Parameter (or order) identifcation is converted to optimization problem. Using PSOalgorithm for identifying parameters avoid the difculties of design for parameter up-date rule by using chaotic synchronization. With the identifcation results, we study thesynchronization between diferent (identical) fractional-order chaotic systems with com-mensurate (or incommensurate) orders. At the same time, we analyze basic dynamicsof a new three-dimension fractional-order chaotic system with four attractor, such as:the stability of equilibrium points, phase diagrams, chaotic attractor, existence anduniqueness of solutions.