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PrefaceThis book collects the recent development on the theory of continuous dynamicalsystems from a different point of view.From the author's 20 years researches andteaching experience,the author places terials in five chapters to provide a betterunderstanding of stability,stability switching,bifurcation,complexity and chaosin nonlinear continuous dynamical systems.For completeness,the theory for dis-crete and switching systems with transports is presented in a different volume.The stability theory of linear continuous dynamical systems is comprehen-sively discussed in Chapter I from the author's teaching experience.The terialspresented in this book are a foundation to understand stability and bifurcation the-ory in nonlinear dynamical systems.In Chapter 2,the author used a different pointof view to present the stability,stability switching and bifurcation of equilibriumsfor nonlinear continuous systems.Such presentation kes the stability and bifur-cation theory much simpler,readable and doable.In Chapter 3,an yticalmethod is presented to obtain the ytical solutions of periodic flows and chaosin nonlinear dynamical systems.The ytical solution of chaos ends the historyof chaos numerically simulated only.In Chapter 4,the global transversality of aflow to the generic separatrix in nonlinear dynamical systems is presented,and thetheory of global transversality is the base to understand the complexity of flows innonlinear dynamical systems.How to determine the separatrix and all possiblefirst integral nifolds in n-dimensional dynamical systems is still unsolved.InChapter 5,chaos in stochastic and resonant layers in nonlinear Hamiltonian sys-tems is discussed.The physical mechani of chaos in stochastic layers is theresonance interaction to form the hyperbolic characteristics.Finally,I would like to appreciate my student,Jianzhe Huang for completingnumerical computations in Chapter 3.Herein,I thank my wife (Sherry X.Huang)and my children (Yanyi Luo,Robin Ruo-Bing Luo,and Robert Zong-Yuan Luo)for tolerance,patience,understanding and support.I hope this book will be a goodgift for them.Albert C.J.LuoEdwardsville,IllinoisContentsPrefaceChapter I Linear Systems and Stability.1.I Linear systems with distinct eigenvalues............I1.2Operator exponentials..1.3 Linear systems with repeated eigenvalues...............81.5 Linear systems with periodic coefficients…21.6 Stability and boundary......221.7 Lower-dimensional linear systems......…331.7.1 One-dimensional linear systems....................…331.7.2 Planar linear systems......................…351.7.3 Three-dimensional linear systemsReferences............…53Chapter 2 Stability Switching and Bifurcation…552.I Continuous dynamical systems...........…552.2 Equilibriums and stability.582.3 Bifurcation and stability switching...…702.3.】Stability and switching…76References…l06Chapter 3 Analytical Periodic Flows and Chaos.1093.1 Analytical periodic flows...1093.1.1 Autonomous nonlinear systems.....1093.1.2 Periodically forced nonlinear systems....1173.2 Nonlinear vibration systems.1213.2.1 Free vibration systems......…1213.2.2 Periodically forced vibration systems.............1313.3 A periodically forced Duffing oscillator......135
