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This unique book presents the discretization of continuous systems and implicit mapping dynamics of periodic motions to chaos in continuous nonlinear systems. The stability and bifurcation theory of fixed points in discrete nonlinear dynamical systems is reviewed, and the explicit and implicit maps of continuous dynamical systems are developed through the single-step and multi-step discretizations. The implicit dynamics of period-m solutions in discrete nonlinear systems are discussed. The book also offers a generalized approach to finding analytical and numerical solutions of stable and unstable periodic flows to chaos in nonlinear systems with/without time-delay. The bifurcation trees of periodic motions to chaos in the Duffing oscillator are shown as a sample problem, while the discrete Fourier series of periodic motions and chaos are also presented. The book offers a valuable resource for university students, professors, researchers and engineers in the fields of applied mathematics, physics, mechanics, control systems, and engineering.

Physics. --- Nonlinear Dynamics. --- Difference and Functional Equations. --- Vibration, Dynamical Systems, Control. --- Functional equations. --- Vibration. --- Physique --- Equations fonctionnelles --- Vibration --- Discretization (Mathematics) --- Nonlinear systems. --- Physics --- Physical Sciences & Mathematics --- Atomic Physics --- Differential equations. --- Systems, Nonlinear --- 517.91 Differential equations --- Differential equations --- Difference equations. --- Statistical physics. --- Dynamical systems. --- Dynamics. --- System theory --- Applications of Nonlinear Dynamics and Chaos Theory. --- Cycles --- Mechanics --- Sound --- Equations, Functional --- Functional analysis --- Dynamical systems --- Kinetics --- Mathematics --- Mechanics, Analytic --- Force and energy --- Statics --- Calculus of differences --- Differences, Calculus of --- Equations, Difference --- Mathematical statistics --- Statistical methods

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“Discontinuous Dynamical Systems” presents a theory of dynamics and flow switchability in discontinuous dynamical systems, which can be as the mathematical foundation for a new dynamics of dynamical system networks. The book includes a theory for flow barriers and passability to boundaries in discontinuous dynamical systems that will completely change traditional concepts and ideas in the field of dynamical systems. Edge dynamics and switching complexity of flows in discontinuous dynamical systems are explored in the book and provide the mathematical basis for developing the attractive network channels in dynamical systems. The theory of bouncing flows to boundaries, edges and vertexes in discontinuous dynamical systems with multi-valued vector fields is described in the book as a “billiard” theory of dynamical system networks. The theory of dynamical system interactions in discontinued dynamical systems can be used as a general principle in dynamical system networks, which is applied to dynamical system synchronization.　 The book represents a valuable reference work for university professors and researchers in applied mathematics, physics, mechanics, and control.   Dr. Albert C.J. Luo is an internationally respected professor in nonlinear dynamics and mechanics, and he works at Southern Illinois University Edwardsville, USA.

Differentiable dynamical systems. --- Discontinuous functions. --- Nonlinear oscillations. --- Differentiable dynamical systems --- System theory --- Civil & Environmental Engineering --- Mathematics --- Engineering & Applied Sciences --- Physical Sciences & Mathematics --- Applied Mathematics --- Operations Research --- Geometry --- System theory. --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Systems, Theory of --- Systems science --- Mathematics. --- Statistical physics. --- Vibration. --- Dynamical systems. --- Dynamics. --- Systems Theory, Control. --- Nonlinear Dynamics. --- Vibration, Dynamical Systems, Control. --- Science --- Philosophy --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Systems theory. --- Applications of Nonlinear Dynamics and Chaos Theory. --- Cycles --- Mechanics --- Sound --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Physics --- Statics --- Mathematical statistics --- Statistical methods

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"Nonlinear Deformable-body Dynamics" mainly consists in a mathematical treatise of approximate theories for thin deformable bodies, including cables, beams, rods, webs, membranes, plates, and shells. The intent of the book is to stimulate more research in the area of nonlinear deformable-body dynamics not only because of the unsolved theoretical puzzles it presents but also because of its wide spectrum of applications. For instance, the theories for soft webs and rod-reinforced soft structures can be applied to biomechanics for DNA and living tissues, and the nonlinear theory of deformable bodies, based on the Kirchhoff assumptions, is a special case discussed. This book can serve as a reference work for researchers and a textbook for senior and postgraduate students in physics, mathematics, engineering and biophysics. Dr. Albert C.J. Luo is a Professor of Mechanical Engineering at Southern Illinois University, Edwardsville, IL, USA. Professor Luo is an internationally recognized scientist in the field of nonlinear dynamics in dynamical systems and deformable solids.

Differentiable dynamical systems. --- Nonlinear theories --- Deformations (Mechanics) --- Engineering & Applied Sciences --- Physics --- Physical Sciences & Mathematics --- Applied Physics --- Physics - General --- Nonlinear theories. --- Physics. --- Dynamics. --- Ergodic theory. --- Continuum physics. --- Mechanics. --- Statistical physics. --- Dynamical systems. --- Continuum mechanics. --- Dynamical Systems and Ergodic Theory. --- Statistical Physics, Dynamical Systems and Complexity. --- Classical Continuum Physics. --- Continuum Mechanics and Mechanics of Materials. --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Nonlinear problems --- Nonlinearity (Mathematics) --- Calculus --- Mathematical analysis --- Mathematical physics --- Mechanics, Applied. --- Classical Mechanics. --- Complex Systems. --- Classical and Continuum Physics. --- Solid Mechanics. --- Statistical Physics and Dynamical Systems. --- Applied mechanics --- Engineering, Mechanical --- Engineering mathematics --- Classical mechanics --- Newtonian mechanics --- Dynamics --- Quantum theory --- Mathematical statistics --- Statistical methods --- Classical field theory --- Continuum physics --- Continuum mechanics --- Dynamical systems --- Kinetics --- Mathematics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Statics --- Ergodic transformations --- Continuous groups --- Measure theory --- Transformations (Mathematics)

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Regularity and Complexity in Dynamical Systems describes periodic and chaotic behaviors in dynamical systems, including continuous, discrete, impulsive,discontinuous, and switching systems. In traditional analysis, the periodic and chaotic behaviors in continuous, nonlinear dynamical systems were extensively discussed even if unsolved. In recent years, there has been an increasing amount of interest in periodic and chaotic behaviors in discontinuous dynamical systems because such dynamical systems are prevalent in engineering. Usually,the smoothening of discontinuous dynamical system is adopted in order to use the theory of continuous dynamical systems. However, such technique cannot provide suitable results in such discontinuous systems. In this book, an alternative way is presented to discuss the periodic and chaotic behaviors in discontinuous dynamical systems. This book also: Illustrates new concepts and methodology in discontinuous dynamical systems Uses different ideas to describe complicated dynamical systems in real worlds Discuss the complete dynamics and the corresponding Ying-Yang theory as well as  complexity and factuality of chaos in dynamical systems Discusses the mechanism of chaos and diffusion in impulsive systems Discusses strange attractor fragmentation and hidden mathematical structures Contains intuitive illustrations and systematical description as well as complete example demonstrations Regularity and Complexity in Dynamical Systems is an ideal book for those interested in better understanding complexity and chaos caused by nonlinearity, discontinuity, switching, and impulses.

Chaotic behavior in systems. --- Dynamics -- Mathematics. --- Engineering mathematics. --- System theory. --- Engineering mathematics --- System theory --- Chaotic behavior in systems --- Dynamics --- Civil & Environmental Engineering --- Engineering & Applied Sciences --- Mathematics --- Physical Sciences & Mathematics --- Civil Engineering --- Geometry --- Applied Mathematics --- Dynamics. --- Dynamical systems --- Kinetics --- Mathematical and Computational Engineering. --- Complex Systems. --- Numerical and Computational Physics, Simulation. --- Applications of Nonlinear Dynamics and Chaos Theory. --- Engineering --- Engineering analysis --- Mathematical analysis --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Applied mathematics. --- Physics. --- Statistical physics. --- Mathematical statistics --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Systems, Theory of --- Systems science --- Science --- Statistical methods --- Philosophy

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This book examines discrete dynamical systems with memory—nonlinear systems that exist extensively in biological organisms and financial and economic organizations, and time-delay systems that can be discretized into the memorized, discrete dynamical systems. It book further discusses stability and bifurcations of time-delay dynamical systems that can be investigated through memorized dynamical systems as well as bifurcations of memorized nonlinear dynamical systems, discretization methods of time-delay systems, and periodic motions to chaos in nonlinear time-delay systems. The book helps readers find analytical solutions of MDS, change traditional perturbation analysis in time-delay systems, detect motion complexity and singularity in MDS; and determine stability, bifurcation, and chaos in any time-delay system.

Complexity, Computational. --- Applications of Nonlinear Dynamics and Chaos Theory. --- Engineering. --- Biological models. --- Econometrics. --- Complexity. --- Complex Systems. --- Systems Biology. --- Discrete-time systems. --- DES (System analysis) --- Discrete event systems --- Sampled-data systems --- Digital control systems --- System analysis --- Linear time invariant systems --- Construction --- Industrial arts --- Technology --- Economics, Mathematical --- Statistics --- Models, Biological --- Computational complexity. --- Statistical physics. --- System theory. --- Systems biology. --- Computational biology --- Bioinformatics --- Biological systems --- Molecular biology --- Systems, Theory of --- Systems science --- Science --- Physics --- Mathematical statistics --- Complexity, Computational --- Electronic data processing --- Machine theory --- Philosophy --- Statistical methods