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Bifurcation And Stability In Nonlinear Discrete Systems
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Book Synopsis Bifurcation and Stability in Nonlinear Discrete Systems by : Albert C. J. Luo
Download or read book Bifurcation and Stability in Nonlinear Discrete Systems written by Albert C. J. Luo and published by Springer Nature. This book was released on 2020-08-13 with total page 313 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book focuses on bifurcation and stability in nonlinear discrete systems, including monotonic and oscillatory stability. It presents the local monotonic and oscillatory stability and bifurcation of period-1 fixed-points on a specific eigenvector direction, and discusses the corresponding higher-order singularity of fixed-points. Further, it explores the global analysis of monotonic and oscillatory stability of fixed-points in 1-dimensional discrete systems through 1-dimensional polynomial discrete systems. Based on the Yin-Yang theory of nonlinear discrete systems, the book also addresses the dynamics of forward and backward nonlinear discrete systems, and the existence conditions of fixed-points in said systems. Lastly, in the context of local analysis, it describes the normal forms of nonlinear discrete systems and infinite-fixed-point discrete systems. Examining nonlinear discrete systems from various perspectives, the book helps readers gain a better understanding of the nonlinear dynamics of such systems.
Book Synopsis Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields by : John Guckenheimer
Download or read book Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields written by John Guckenheimer and published by Springer Science & Business Media. This book was released on 2013-11-21 with total page 475 pages. Available in PDF, EPUB and Kindle. Book excerpt: An application of the techniques of dynamical systems and bifurcation theories to the study of nonlinear oscillations. Taking their cue from Poincare, the authors stress the geometrical and topological properties of solutions of differential equations and iterated maps. Numerous exercises, some of which require nontrivial algebraic manipulations and computer work, convey the important analytical underpinnings of problems in dynamical systems and help readers develop an intuitive feel for the properties involved.
Book Synopsis Nonlinear Stability and Bifurcation Theory by : Hans Troger
Download or read book Nonlinear Stability and Bifurcation Theory written by Hans Troger and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 419 pages. Available in PDF, EPUB and Kindle. Book excerpt: Every student in engineering or in other fields of the applied sciences who has passed through his curriculum knows that the treatment of nonlin ear problems has been either avoided completely or is confined to special courses where a great number of different ad-hoc methods are presented. The wide-spread believe that no straightforward solution procedures for nonlinear problems are available prevails even today in engineering cir cles. Though in some courses it is indicated that in principle nonlinear problems are solveable by numerical methods the treatment of nonlinear problems, more or less, is considered to be an art or an intellectual game. A good example for this statement was the search for Ljapunov functions for nonlinear stability problems in the seventies. However things have changed. At the beginning of the seventies, start ing with the work of V.1. Arnold, R. Thom and many others, new ideas which, however, have their origin in the work of H. Poincare and A. A. Andronov, in the treatment of nonlinear problems appeared. These ideas gave birth to the term Bifurcation Theory. Bifurcation theory allows to solve a great class of nonlinear problems under variation of parameters in a straightforward manner.
Book Synopsis Bifurcation and Stability in Nonlinear Dynamical Systems by : Albert C. J. Luo
Download or read book Bifurcation and Stability in Nonlinear Dynamical Systems written by Albert C. J. Luo and published by Springer Nature. This book was released on 2020-01-30 with total page 418 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Nonlinear Physical Systems by : Oleg N. Kirillov
Download or read book Nonlinear Physical Systems written by Oleg N. Kirillov and published by John Wiley & Sons. This book was released on 2013-12-11 with total page 328 pages. Available in PDF, EPUB and Kindle. Book excerpt: Bringing together 18 chapters written by leading experts in dynamical systems, operator theory, partial differential equations, and solid and fluid mechanics, this book presents state-of-the-art approaches to a wide spectrum of new and challenging stability problems. Nonlinear Physical Systems: Spectral Analysis, Stability and Bifurcations focuses on problems of spectral analysis, stability and bifurcations arising in the nonlinear partial differential equations of modern physics. Bifurcations and stability of solitary waves, geometrical optics stability analysis in hydro- and magnetohydrodynamics, and dissipation-induced instabilities are treated with the use of the theory of Krein and Pontryagin space, index theory, the theory of multi-parameter eigenvalue problems and modern asymptotic and perturbative approaches. Each chapter contains mechanical and physical examples, and the combination of advanced material and more tutorial elements makes this book attractive for both experts and non-specialists keen to expand their knowledge on modern methods and trends in stability theory. Contents 1. Surprising Instabilities of Simple Elastic Structures, Davide Bigoni, Diego Misseroni, Giovanni Noselli and Daniele Zaccaria. 2. WKB Solutions Near an Unstable Equilibrium and Applications, Jean-François Bony, Setsuro Fujiié, Thierry Ramond and Maher Zerzeri, partially supported by French ANR project NOSEVOL. 3. The Sign Exchange Bifurcation in a Family of Linear Hamiltonian Systems, Richard Cushman, Johnathan Robbins and Dimitrii Sadovskii. 4. Dissipation Effect on Local and Global Fluid-Elastic Instabilities, Olivier Doaré. 5. Tunneling, Librations and Normal Forms in a Quantum Double Well with a Magnetic Field, Sergey Yu. Dobrokhotov and Anatoly Yu. Anikin. 6. Stability of Dipole Gap Solitons in Two-Dimensional Lattice Potentials, Nir Dror and Boris A. Malomed. 7. Representation of Wave Energy of a Rotating Flow in Terms of the Dispersion Relation, Yasuhide Fukumoto, Makoto Hirota and Youichi Mie. 8. Determining the Stability Domain of Perturbed Four-Dimensional Systems in 1:1 Resonance, Igor Hoveijn and Oleg N. Kirillov. 9. Index Theorems for Polynomial Pencils, Richard Kollár and Radomír Bosák. 10. Investigating Stability and Finding New Solutions in Conservative Fluid Flows Through Bifurcation Approaches, Paolo Luzzatto-Fegiz and Charles H.K. Williamson. 11. Evolution Equations for Finite Amplitude Waves in Parallel Shear Flows, Sherwin A. Maslowe. 12. Continuum Hamiltonian Hopf Bifurcation I, Philip J. Morrison and George I. Hagstrom. 13. Continuum Hamiltonian Hopf Bifurcation II, George I. Hagstrom and Philip J. Morrison. 14. Energy Stability Analysis for a Hybrid Fluid-Kinetic Plasma Model, Philip J. Morrison, Emanuele Tassi and Cesare Tronci. 15. Accurate Estimates for the Exponential Decay of Semigroups with Non-Self-Adjoint Generators, Francis Nier. 16. Stability Optimization for Polynomials and Matrices, Michael L. Overton. 17. Spectral Stability of Nonlinear Waves in KdV-Type Evolution Equations, Dmitry E. Pelinovsky. 18. Unfreezing Casimir Invariants: Singular Perturbations Giving Rise to Forbidden Instabilities, Zensho Yoshida and Philip J. Morrison. About the Authors Oleg N. Kirillov has been a Research Fellow at the Magneto-Hydrodynamics Division of the Helmholtz-Zentrum Dresden-Rossendorf in Germany since 2011. His research interests include non-conservative stability problems of structural mechanics and physics, perturbation theory of non-self-adjoint boundary eigenvalue problems, magnetohydrodynamics, friction-induced oscillations, dissipation-induced instabilities and non-Hermitian problems of optics and microwave physics. Since 2013 he has served as an Associate Editor for the journal Frontiers in Mathematical Physics. Dmitry E. Pelinovsky has been Professor at McMaster University in Canada since 2000. His research profile includes work with nonlinear partial differential equations, discrete dynamical systems, spectral theory, integrable systems, and numerical analysis. He served as the guest editor of the special issue of the journals Chaos in 2005 and Applicable Analysis in 2010. He is an Associate Editor of the journal Communications in Nonlinear Science and Numerical Simulations. This book is devoted to the problems of spectral analysis, stability and bifurcations arising from the nonlinear partial differential equations of modern physics. Leading experts in dynamical systems, operator theory, partial differential equations, and solid and fluid mechanics present state-of-the-art approaches to a wide spectrum of new challenging stability problems. Bifurcations and stability of solitary waves, geometrical optics stability analysis in hydro- and magnetohydrodynamics and dissipation-induced instabilities will be treated with the use of the theory of Krein and Pontryagin space, index theory, the theory of multi-parameter eigenvalue problems and modern asymptotic and perturbative approaches. All chapters contain mechanical and physical examples and combine both tutorial and advanced sections, making them attractive both to experts in the field and non-specialists interested in knowing more about modern methods and trends in stability theory.
Book Synopsis New Methods for Chaotic Dynamics by : Nikolai Aleksandrovich Magnitskii
Download or read book New Methods for Chaotic Dynamics written by Nikolai Aleksandrovich Magnitskii and published by World Scientific. This book was released on 2006 with total page 382 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents a new theory on the transition to dynamical chaos for two-dimensional nonautonomous, and three-dimensional, many-dimensional and infinitely-dimensional autonomous nonlinear dissipative systems of differential equations including nonlinear partial differential equations and differential equations with delay arguments. The transition is described from the Feigenbaum cascade of period doubling bifurcations of the original singular cycle to the complete or incomplete Sharkovskii subharmonic cascade of bifurcations of stable limit cycles with arbitrary period and finally to the complete or incomplete homoclinic cascade of bifurcations. The book presents a distinct view point on the principles of formation, scenarios of occurrence and ways of control of chaotic motion in nonlinear dissipative dynamical systems. All theoretical results and conclusions of the theory are strictly proved and confirmed by numerous examples, illustrations and numerical calculations. Sample Chapter(s). Chapter 1: Systems of Ordinary Differential Equations (1,736 KB). Contents: Systems of Ordinary Differential Equations; Bifurcations in Nonlinear Systems of Ordinary Differential Equations; Chaotic Systems of Ordinary Differential Equations; Principles of the Theory of Dynamical Chaos in Dissipative Systems of Ordinary Differential Equations; Dynamical Chaos in Infinitely-Dimensional Systems of Differential Equations; Chaos Control in Systems of Differential Equations. Readership: Graduate students and researchers in complex and chaotic dynamical systems.
Book Synopsis Practical Bifurcation and Stability Analysis by : Rüdiger U. Seydel
Download or read book Practical Bifurcation and Stability Analysis written by Rüdiger U. Seydel and published by Springer Science & Business Media. This book was released on 2009-11-27 with total page 493 pages. Available in PDF, EPUB and Kindle. Book excerpt: Probably the first book to describe computational methods for numerically computing steady state and Hopf bifurcations. Requiring only a basic knowledge of calculus, and using detailed examples, problems, and figures, this is an ideal textbook for graduate students.
Book Synopsis Stability and Stabilization of Nonlinear Systems by : Dirk Aeyels
Download or read book Stability and Stabilization of Nonlinear Systems written by Dirk Aeyels and published by Springer. This book was released on 2008-01-26 with total page 394 pages. Available in PDF, EPUB and Kindle. Book excerpt: These papers were presented at the first EC-TMR Nonlinear Control Network Workshop, on Stability and Stabilization of Nonlinear Systems, that took place in March 1999, Ghent, Belgium. The TMR programme offers a unique opportunity for the academic community to expand their knowledge, share their experience and identify and discuss strategic issues in aspects of nonlinear control engineering. The aim is to create a resource centre of available expertise and research interests. This outstanding reference volume presents current and emerging research directions, including: Stability analysis of nonlinear dynamical systems and converse Lyapunov theorems; Stabilization and regulation of nonlinear dynamical control systems; Control of physical systems using physics-based Lyapunov functions and passivity, as well as bifurcation analysis and optimal control. This collection of peer-reviewed papers provides a comprehensive overview of this field of research for graduate students and researchers in engineering and applied mathematics.
Book Synopsis Bifurcation Dynamics in Polynomial Discrete Systems by : Albert C. J. Luo
Download or read book Bifurcation Dynamics in Polynomial Discrete Systems written by Albert C. J. Luo and published by Springer Nature. This book was released on 2020-11-09 with total page 440 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is the first book focusing on bifurcation dynamics in 1-dimensional polynomial nonlinear discrete systems. It comprehensively discusses the general mathematical conditions of bifurcations in polynomial nonlinear discrete systems, as well as appearing and switching bifurcations for simple and higher-order singularity period-1 fixed-points in the 1-dimensional polynomial discrete systems. Further, it analyzes the bifurcation trees of period-1 to chaos generated by period-doubling, and monotonic saddle-node bifurcations. Lastly, the book presents methods for period-2 and period-doubling renormalization for polynomial discrete systems, and describes the appearing mechanism and period-doublization of period-n fixed-points on bifurcation trees for the first time, offering readers fascinating insights into recent research results in nonlinear discrete systems.
Book Synopsis Nonlinear Stability of Structures by : A.N. Kounadis
Download or read book Nonlinear Stability of Structures written by A.N. Kounadis and published by Springer. This book was released on 2014-05-04 with total page 418 pages. Available in PDF, EPUB and Kindle. Book excerpt: The present volume gives a very modern treatment of all theoretical as well as computational aspects of nonlinear structural stability. The theoretical part starts with the basic concepts of nonlinear static stability and classical dynamics and proceeds subsequently with recent progress in nonlinear dynamic stability and dynamic buckling of structures including an introduction to chaos. The first paper overviews theory and modelling of various structural instability problems. In the second section, nonlinear dynamic buckling and stability of autonomous discrete dissipative structural systems, gradient and non-gradient are discussed. The third paper handles stability and bifurcation phenomena in dynamical systems. The fourth paper contains an introduction to nonlinear dynamics and chaos. Special attention is devoted to the direct computation of critical points and path-switching strategies. A variety of numerical simulations for complicated nonlinear unstable responses also illustrate this part.
Book Synopsis Nonlinear Systems: Stability, Dynamics And Control by : Guanrong Chen
Download or read book Nonlinear Systems: Stability, Dynamics And Control written by Guanrong Chen and published by World Scientific. This book was released on 2023-06-28 with total page 234 pages. Available in PDF, EPUB and Kindle. Book excerpt: The topic of nonlinear systems is fundamental to the study of systems engineering. So extensive investigations have been carried out by both the nonlinear control and nonlinear dynamics communities, but the focus can be different — on controllers design and dynamics analysis, respectively. The last two decades have witnessed the gradual merging of control theory and dynamics analysis, but not yet to the extent of controlling nonlinear dynamics such as bifurcations and chaos. This monograph is an attempt to fill that gap while presenting a rather comprehensive coverage of the fundamental nonlinear systems theory in a self-contained and approachable manner.This introductory treatise is written for self-study and, in particular, as an elementary textbook that can be taught in a one-semester course to advanced undergraduates or entrance level graduates with curricula focusing on nonlinear systems, both on control theory and dynamics analysis.
Book Synopsis Elements of Applied Bifurcation Theory by : Yuri Kuznetsov
Download or read book Elements of Applied Bifurcation Theory written by Yuri Kuznetsov and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 648 pages. Available in PDF, EPUB and Kindle. Book excerpt: Providing readers with a solid basis in dynamical systems theory, as well as explicit procedures for application of general mathematical results to particular problems, the focus here is on efficient numerical implementations of the developed techniques. The book is designed for advanced undergraduates or graduates in applied mathematics, as well as for Ph.D. students and researchers in physics, biology, engineering, and economics who use dynamical systems as model tools in their studies. A moderate mathematical background is assumed, and, whenever possible, only elementary mathematical tools are used. This new edition preserves the structure of the first while updating the context to incorporate recent theoretical developments, in particular new and improved numerical methods for bifurcation analysis.
Book Synopsis Nonlinear Differential Equations by : Piero de Mottoni
Download or read book Nonlinear Differential Equations written by Piero de Mottoni and published by Academic Press. This book was released on 2014-05-10 with total page 370 pages. Available in PDF, EPUB and Kindle. Book excerpt: Nonlinear Differential Equations: Invariance, Stability, and Bifurcation presents the developments in the qualitative theory of nonlinear differential equations. This book discusses the exchange of mathematical ideas in stability and bifurcation theory. Organized into 26 chapters, this book begins with an overview of the initial value problem for a nonlinear wave equation. This text then focuses on the interplay between stability exchange for a stationary solution and the appearance of bifurcating periodic orbits. Other chapters consider the development of methods for ascertaining stability and boundedness and explore the development of bifurcation and stability analysis in nonlinear models of applied sciences. This book discusses as well nonlinear hyperbolic equations in further contributions, featuring stability properties of periodic and almost periodic solutions. The reader is also introduced to the stability problem of the equilibrium of a chemical network. The final chapter deals with suitable spaces for studying functional equations. This book is a valuable resource for mathematicians.
Download or read book Nonlinear Systems written by P. G. Drazin and published by Cambridge University Press. This book was released on 1992-06-26 with total page 354 pages. Available in PDF, EPUB and Kindle. Book excerpt: The theories of bifurcation, chaos and fractals as well as equilibrium, stability and nonlinear oscillations, are part of the theory of the evolution of solutions of nonlinear equations. A wide range of mathematical tools and ideas are drawn together in the study of these solutions, and the results applied to diverse and countless problems in the natural and social sciences, even philosophy. The text evolves from courses given by the author in the UK and the United States. It introduces the mathematical properties of nonlinear systems, mostly difference and differential equations, as an integrated theory, rather than presenting isolated fashionable topics. Topics are discussed in as concrete a way as possible and worked examples and problems are used to explain, motivate and illustrate the general principles. The essence of these principles, rather than proof or rigour, is emphasized. More advanced parts of the text are denoted by asterisks, and the mathematical prerequisites are limited to knowledge of linear algebra and advanced calculus, thus making it ideally suited to both senior undergraduates and postgraduates from physics, engineering, chemistry, meteorology etc. as well as mathematics.
Book Synopsis Memorized Discrete Systems and Time-delay by : Albert C. J. Luo
Download or read book Memorized Discrete Systems and Time-delay written by Albert C. J. Luo and published by Springer. This book was released on 2016-11-18 with total page 307 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Discretization and Implicit Mapping Dynamics by : Albert C. J. Luo
Download or read book Discretization and Implicit Mapping Dynamics written by Albert C. J. Luo and published by Springer. This book was released on 2015-07-30 with total page 316 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Vibro-impact Dynamics by : Albert C. J. Luo
Download or read book Vibro-impact Dynamics written by Albert C. J. Luo and published by John Wiley & Sons. This book was released on 2013-01-25 with total page 269 pages. Available in PDF, EPUB and Kindle. Book excerpt: Presents a systematic view of vibro-impact dynamics based on the nonlinear dynamics analysis Comprehensive understanding of any vibro-impact system is critically impeded by the lack of analytical tools viable for properly characterizing grazing bifurcation. The authors establish vibro-impact dynamics as a subset of the theory of discontinuous systems, thus enabling all vibro-impact systems to be explored and characterized for applications. Vibro-impact Dynamics presents an original theoretical way of analyzing the behavior of vibro-impact dynamics that can be extended to discontinuous dynamics. All topics are logically integrated to allow for vibro-impact dynamics, the central theme, to be presented. It provides a unified treatment on the topic with a sound theoretical base that is applicable to both continuous and discrete systems Vibro-impact Dynamics: Presents mapping dynamics to determine bifurcation and chaos in vibro-impact systems Offers two simple vibro-impact systems with comprehensive physical interpretation of complex motions Uses the theory for discontinuous dynamical systems on time-varying domains, to investigate the Fermi-oscillator Essential reading for graduate students, university professors, researchers and scientists in mechanical engineering.
