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ISBN:
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Languages : en
Pages : 2
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Book Description
Pinsky's research is concerned with the exponential growth rate (= Lyapunov exponent) of solutions of stochastic differential equations. In a paper to appear in the Annals of Applied Probability, a formula is obtained for the quadratic Lyapunov exponent of the simple harmonic oscillator in the presence of a finite-state Markov noise process. In case the noise process is reversible, the quadratic Lyapunov exponent is strictly less than for the corresponding white noise process obtained from the central limit theorem. An example is presented of a non-reversible Markov noise process for which this inequality is reversed. In another article, to appear in the volume 'Stochastic Partial Differential Equations and their Applications' in the Springer Verlag Lecture Notes in Control and Information Sciences (Proceedings of the 1991 Charlotte NC Conference on SPDE, ed. B. Rozovskii), the Lyapunov exponent is computed for the, solution of a hyperbolic partial differential equation with damping. In this case, one studies the exponential growth rate of the energy of the solution with Dirichlet boundary conditions. The detailed results depend on the size of the damping constant (overdamped vs. underdamped case). To our knowledge, this is the first study ever of the Lyapunov exponent for a partial differential equation. Lyapunov exponent, Stochastic oscillator, Fourier transform, Heat kernel.
Author:
Publisher:
ISBN:
Category :
Languages : en
Pages : 2
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Book Description
Pinsky's research is concerned with the exponential growth rate (= Lyapunov exponent) of solutions of stochastic differential equations. In a paper to appear in the Annals of Applied Probability, a formula is obtained for the quadratic Lyapunov exponent of the simple harmonic oscillator in the presence of a finite-state Markov noise process. In case the noise process is reversible, the quadratic Lyapunov exponent is strictly less than for the corresponding white noise process obtained from the central limit theorem. An example is presented of a non-reversible Markov noise process for which this inequality is reversed. In another article, to appear in the volume 'Stochastic Partial Differential Equations and their Applications' in the Springer Verlag Lecture Notes in Control and Information Sciences (Proceedings of the 1991 Charlotte NC Conference on SPDE, ed. B. Rozovskii), the Lyapunov exponent is computed for the, solution of a hyperbolic partial differential equation with damping. In this case, one studies the exponential growth rate of the energy of the solution with Dirichlet boundary conditions. The detailed results depend on the size of the damping constant (overdamped vs. underdamped case). To our knowledge, this is the first study ever of the Lyapunov exponent for a partial differential equation. Lyapunov exponent, Stochastic oscillator, Fourier transform, Heat kernel.
Author: Yuri Kifer
Publisher: Birkhäuser
ISBN: 9781461581833
Category : Mathematics
Languages : en
Pages : 294
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Book Description
Mathematicians often face the question to which extent mathematical models describe processes of the real world. These models are derived from experimental data, hence they describe real phenomena only approximately. Thus a mathematical approach must begin with choosing properties which are not very sensitive to small changes in the model, and so may be viewed as properties of the real process. In particular, this concerns real processes which can be described by means of ordinary differential equations. By this reason different notions of stability played an important role in the qualitative theory of ordinary differential equations commonly known nowdays as the theory of dynamical systems. Since physical processes are usually affected by an enormous number of small external fluctuations whose resulting action would be natural to consider as random, the stability of dynamical systems with respect to random perturbations comes into the picture. There are differences between the study of stability properties of single trajectories, i. e. , the Lyapunov stability, and the global stability of dynamical systems. The stochastic Lyapunov stability was dealt with in Hasminskii [Has]. In this book we are concerned mainly with questions of global stability in the presence of noise which can be described as recovering parameters of dynamical systems from the study of their random perturbations. The parameters which is possible to obtain in this way can be considered as stable under random perturbations, and so having physical sense. -1- Our set up is the following.
Author: Frank Moss
Publisher: Cambridge University Press
ISBN: 0521352290
Category : Mathematics
Languages : en
Pages : 410
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Book Description
A specially written review of all areas of noise and nonlinear in natural environments.
Author: Xiaoxin Liao
Publisher: Elsevier
ISBN: 0080550614
Category : Mathematics
Languages : en
Pages : 719
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Book Description
The main purpose of developing stability theory is to examine dynamic responses of a system to disturbances as the time approaches infinity. It has been and still is the object of intense investigations due to its intrinsic interest and its relevance to all practical systems in engineering, finance, natural science and social science. This monograph provides some state-of-the-art expositions of major advances in fundamental stability theories and methods for dynamic systems of ODE and DDE types and in limit cycle, normal form and Hopf bifurcation control of nonlinear dynamic systems. Presents comprehensive theory and methodology of stability analysis Can be used as textbook for graduate students in applied mathematics, mechanics, control theory, theoretical physics, mathematical biology, information theory, scientific computation Serves as a comprehensive handbook of stability theory for practicing aerospace, control, mechanical, structural, naval and civil engineers
Author: Yuri Kifer
Publisher: Springer Science & Business Media
ISBN: 1461581818
Category : Mathematics
Languages : en
Pages : 301
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Book Description
Mathematicians often face the question to which extent mathematical models describe processes of the real world. These models are derived from experimental data, hence they describe real phenomena only approximately. Thus a mathematical approach must begin with choosing properties which are not very sensitive to small changes in the model, and so may be viewed as properties of the real process. In particular, this concerns real processes which can be described by means of ordinary differential equations. By this reason different notions of stability played an important role in the qualitative theory of ordinary differential equations commonly known nowdays as the theory of dynamical systems. Since physical processes are usually affected by an enormous number of small external fluctuations whose resulting action would be natural to consider as random, the stability of dynamical systems with respect to random perturbations comes into the picture. There are differences between the study of stability properties of single trajectories, i. e. , the Lyapunov stability, and the global stability of dynamical systems. The stochastic Lyapunov stability was dealt with in Hasminskii [Has]. In this book we are concerned mainly with questions of global stability in the presence of noise which can be described as recovering parameters of dynamical systems from the study of their random perturbations. The parameters which is possible to obtain in this way can be considered as stable under random perturbations, and so having physical sense. -1- Our set up is the following.
Author:
Publisher: Springer Science & Business Media
ISBN: 0817644865
Category : Differentiable dynamical systems
Languages : en
Pages : 516
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Book Description
In the analysis and synthesis of contemporary systems, engineers and scientists are frequently confronted with increasingly complex models that may simultaneously include components whose states evolve along continuous time and discrete instants; components whose descriptions may exhibit nonlinearities, time lags, transportation delays, hysteresis effects, and uncertainties in parameters; and components that cannot be described by various classical equations, as in the case of discrete-event systems, logic commands, and Petri nets. The qualitative analysis of such systems requires results for finite-dimensional and infinite-dimensional systems; continuous-time and discrete-time systems; continuous continuous-time and discontinuous continuous-time systems; and hybrid systems involving a mixture of continuous and discrete dynamics. Filling a gap in the literature, this textbook presents the first comprehensive stability analysis of all the major types of system models described above. Throughout the book, the applicability of the developed theory is demonstrated by means of many specific examples and applications to important classes of systems, including digital control systems, nonlinear regulator systems, pulse-width-modulated feedback control systems, artificial neural networks (with and without time delays), digital signal processing, a class of discrete-event systems (with applications to manufacturing and computer load balancing problems) and a multicore nuclear reactor model. The book covers the following four general topics: * Representation and modeling of dynamical systems of the types described above * Presentation of Lyapunov and Lagrange stability theory for dynamical systems defined on general metric spaces * Specialization of this stability theory to finite-dimensional dynamical systems * Specialization of this stability theory to infinite-dimensional dynamical systems Replete with exercises and requiring basic knowledge of linear algebra, analysis, and differential equations, the work may be used as a textbook for graduate courses in stability theory of dynamical systems. The book may also serve as a self-study reference for graduate students, researchers, and practitioners in applied mathematics, engineering, computer science, physics, chemistry, biology, and economics.
Author: Vadim Semenovich Anishchenko
Publisher: World Scientific
ISBN: 9789810221423
Category : Science
Languages : en
Pages : 410
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Book Description
In this book, bifurcational mechanisms of the development, structure and properties of chaotic attractors are investigated by numerical and physical experiments based on the methods of the modern theory of nonlinear oscillations. The typical bifurcations of regular and chaotic attractors which are due to parameter variations are analyzed.Regularities of the transition to chaos via the collapse of quasiperiodic oscillations with two and three frequencies are investigated in detail. The book deals with the problems of chaotic synchronization, interaction of attractors and the phenomenon of stochastic resonance. The problems of fluctuation influence on the bifurcations and properties of chaotic attractors are investigated more closely.All principal problems are investigated by the comparison of theoretical and numerical results and data from physical experiments.
Author: Tomasz Kapitaniak
Publisher: World Scientific
ISBN: 9789810204105
Category : Science
Languages : en
Pages : 256
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Book Description
As in the first edition, the influence of random noise on the chaotic behavior of dissipative dynamical systems is investigated. Problems are illustrated by mechanical examples. This revised and updated edition contains new sections on the summary of probability theory, homoclinic chaos, Melnikov method, routes to chaos, stabilization of period-doubling, and Hopf bifurcation by noise. Some chapters have been rewritten and new examples have been added.
Author:
Publisher:
ISBN:
Category : Aeronautics
Languages : en
Pages : 880
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Book Description
Lists citations with abstracts for aerospace related reports obtained from world wide sources and announces documents that have recently been entered into the NASA Scientific and Technical Information Database.
Author: Sergey M. Bezrukov
Publisher: Springer Science & Business Media
ISBN: 9780735401273
Category : Science
Languages : en
Pages : 644
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Book Description
All papers in this proceedings volume were peer reviewed. The purview of this third conference was shifted toward biology and medicine. Among the topics covered were: the constructive role of noise in the central nervous system, neuronal networks, and sensory transduction (hearing in humans, photo- and electroreception in marine animals), encoding of information into nerve pulse trains, single molecules and noise (including single molecule detection and characterization by nanopores - molecular "Coulter counting"), concepts of noise in neurophysiology (randomness and order in brain and heart electrical activities under normal conditions and in pathology), the role of noise in genetic regulation and gene expression, biosensors, etc.
