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Lyapunov exponents lie at the heart of chaos theory, and are widely used in studies of complex dynamics. Utilising a pragmatic, physical approach, this self-contained book provides a comprehensive description of the concept. Beginning with the basic properties and numerical methods, it then guides readers through to the most recent advances in applications to complex systems. Practical algorithms are thoroughly reviewed and their performance is discussed, while a broad set of examples illustrate the wide range of potential applications. The description of various numerical and analytical techniques for the computation of Lyapunov exponents offers an extensive array of tools for the characterization of phenomena such as synchronization, weak and global chaos in low and high-dimensional set-ups, and localization. This text equips readers with all the investigative expertise needed to fully explore the dynamical properties of complex systems, making it ideal for both graduate students and experienced researchers.

Lyapunov exponents. --- Differential equations.

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This heavily illustrated book collects in one source most of the mathematically simple systems of differential equations whose solutions are chaotic. It includes the historically important systems of van der Pol, Duffing, Ueda, Lorenz, Rossler, and many others, but it goes on to show that there are many other systems that are simpler and more elegant. Many of these systems have been only recently discovered and are not widely known. Most cases include plots of the attractor and calculations of the spectra of Lyapunov exponents. Some important cases include graphs showing the route to chaos.The

Lyapunov exponents. --- Flows (Differentiable dynamical systems) --- Chaotic behavior in systems --- Mathematics.

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Mathematical control systems --- Ordinary differential equations --- Differentiable dynamical systems --- Lyapunov exponents --- Stochastic systems --- Dynamique différentiable --- Systèmes stochastiques --- Congresses --- Congrès --- Congresses. --- 51 --- -Lyapunov exponents --- -Stochastic systems --- -Systems, Stochastic --- Stochastic processes --- System analysis --- Liapunov exponents --- Lyapunov characteristic exponents --- Differential equations --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Global analysis (Mathematics) --- Topological dynamics --- Mathematics --- -Mathematics --- 51 Mathematics --- -51 Mathematics --- Systems, Stochastic --- Dynamique différentiable --- Systèmes stochastiques --- Congrès --- Stochastic systems - Congresses. --- Lyapunov exponents - Congresses. --- Differentiable dynamical systems - Congresses.

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Mathematical control systems --- Analytical spaces --- Ordinary differential equations --- Random dynamical systems. --- Lyapunov exponents. --- Ergodic theory. --- Invariant manifolds. --- Banach spaces. --- Systèmes dynamiques aléatoires --- Liapounov, Exposants de --- Théorie ergodique --- Variétés invariantes --- Banach, Espaces de --- 51 <082.1> --- Functions of complex variables --- Generalized spaces --- Topology --- Invariants --- Manifolds (Mathematics) --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Liapunov exponents --- Lyapunov characteristic exponents --- Differential equations --- Dynamical systems, Random --- Differentiable dynamical systems --- Mathematics--Series --- Systèmes dynamiques aléatoires --- Théorie ergodique --- Variétés invariantes --- Banach spaces --- Ergodic theory --- Invariant manifolds --- Lyapunov exponents --- Random dynamical systems --- Systèmes dynamiques

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Since the publication of the first edition of the present volume in 1980, the stochastic stability of differential equations has become a very popular subject of research in mathematics and engineering. To date exact formulas for the Lyapunov exponent, the criteria for the moment and almost sure stability, and for the existence of stationary and periodic solutions of stochastic differential equations have been widely used in the literature. In this updated volume readers will find important new results on the moment Lyapunov exponent, stability index and some other fields, obtained after publication of the first edition, and a significantly expanded bibliography. This volume provides a solid foundation for students in graduate courses in mathematics and its applications. It is also useful for those researchers who would like to learn more about this subject, to start their research in this area or to study the properties of concrete mechanical systems subjected to random perturbations.

Mathematics. --- Probabilities. --- Mechanics. --- Probability Theory and Stochastic Processes. --- Classical mechanics --- Newtonian mechanics --- Physics --- Dynamics --- Quantum theory --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk --- Math --- Science --- Stochastic differential equations. --- Differential equations. --- Stochastic systems. --- Systems, Stochastic --- Stochastic processes --- System analysis --- 517.91 Differential equations --- Differential equations --- Fokker-Planck equation --- Lyapunov exponents. --- Distribution (Probability theory). --- Qualitative theory. --- Distribution (Probability theory. --- Classical Mechanics. --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities