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
Reprint of classic reference work. Over 400 books have been published in the series Classics in Mathematics, many remain standard references for their subject. All books in this series are reissued in a new, inexpensive softcover edition to make them easily accessible to younger generations of students and researchers. "... The book has many good points: clear organization, historical notes and references at the end of every chapter, and an excellent bibliography. The text is well-written, at a level appropriate for the intended audience, and it represents a very good introduction to the basic theory of dynamical systems."
Filling a gap in the literature, this volume offers the first comprehensive analysis of all the major types of system models. Throughout the text, there are many examples and applications to important classes of systems in areas such as power and energy, feedback control, artificial neural networks, digital signal processing and control, manufacturing, computer networks, and socio-economics. 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 a huge variety of fields.
An introduction to aspects of the theory of dynamial systems based on extensions of Liapunov's direct method. The main ideas and structure for the theory are presented for difference equations and for the analogous theory for ordinary differential equations and retarded functional differential equations. The latest results on invariance properties for non-autonomous time-varying systems processes are presented for difference and differential equations.
An introduction to aspects of the theory of dynamical systems based on extensions of Liapunov's direct method. The main ideas and structure for the theory are presented for difference equations and for the analogous theory for ordinary differential equations and retarded functional differential equations.
This monograph presents new insights into the perturbation theory of dynamical systems based on the Gromov-Hausdorff distance. In the first part, the authors introduce the notion of Gromov-Hausdorff distance between compact metric spaces, along with the corresponding distance for continuous maps, flows, and group actions on these spaces. They also focus on the stability of certain dynamical objects like shifts, global attractors, and inertial manifolds. Applications to dissipative PDEs, such as the reaction-diffusion and Chafee-Infante equations, are explored in the second part. This text will be of interest to graduates students and researchers working in the areas of topological dynamics and PDEs.
There are plenty of challenging and interesting problems open for investigation in the field of switched systems. Stability issues help to generate many complex nonlinear dynamic behaviors within switched systems. The authors present a thorough investigation of stability effects on three broad classes of switching mechanism: arbitrary switching where stability represents robustness to unpredictable and undesirable perturbation, constrained switching, including random (within a known stochastic distribution), dwell-time (with a known minimum duration for each subsystem) and autonomously-generated (with a pre-assigned mechanism) switching; and designed switching in which a measurable and freely-assigned switching mechanism contributes to stability by acting as a control input. For each of these classes this book propounds: detailed stability analysis and/or design, related robustness and performance issues, connections to other control problems and many motivating and illustrative examples.
This book systematically presents a fundamental theory for the local analysis of bifurcation and stability of equilibriums in nonlinear dynamical systems. Until now, one does not have any efficient way to investigate stability and bifurcation of dynamical systems with higher-order singularity equilibriums. For instance, infinite-equilibrium dynamical systems have higher-order singularity, which dramatically changes dynamical behaviors and possesses the similar characteristics of discontinuous dynamical systems. The stability and bifurcation of equilibriums on the specific eigenvector are presented, and the spiral stability and Hopf bifurcation of equilibriums in nonlinear systems are presented through the Fourier series transformation. The bifurcation and stability of higher-order singularity equilibriums are presented through the (2m)th and (2m+1)th -degree polynomial systems. From local analysis, dynamics of infinite-equilibrium systems is discussed. The research on infinite-equilibrium systems will bring us to the new era of dynamical systems and control. Presents an efficient way to investigate stability and bifurcation of dynamical systems with higher-order singularity equilibriums; Discusses dynamics of infinite-equilibrium systems; Demonstrates higher-order singularity.
The communication of knowledge on nonlinear dynamical systems, between the mathematicians working on the analytic approach and the scientists working mostly on the applications and numerical simulations has been less than ideal. This volume hopes to bridge the gap between books written on the subject by mathematicians and those written by scientists. The second objective of this volume is to draw attention to the need for cross-fertilization of knowledge between the physical and biological scientists. The third aim is to provide the reader with a personal guide on the study of global nonlinear dynamical systems. Contents:IntroductionTopics in Topology and Differential GeometryIntroduction to Global Analysis and Infinite Dimensional ManifoldsGeneral Theory of Dynamical SystemsStability Theory and Liapunov's Direct MethodIntroduction to the General Theory of Structural StabilityApplications Readership: Applied mathematicians and engineers. Keywords:Asymptotically Stable;Bifurcation;Dispersive Systems;Global Analysis;Integral Flow;Liapunov Functions;Linearization;Nonlinear Dynamical Systems;Stable Manifolds;Structural StabilityReview:“The author's style is clear, formulations of mathematical results are precise … The book helps the reader to create a good global picture of the theory of dynamical systems. The author has gathered a considerable number of facts about dynamical systems in this book, including almost 1000 references, so that it can also serve as a handbook for mathematicians beginning to work in this area … The book is useful not only for technicians, but also for mathematicians, and we recommend it to anyone working in dynamical systems.”Alois Klíc Mathematical Reviews
