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An Introduction to Dynamical Systems and Chaos

  • Textbook
  • © 2024

Overview

Authors:
  1. G. C. Layek

    1. Department of Mathematics, The University of Burdwan, Burdwan, India

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  • Discusses continuous and discrete nonlinear systems by using a systematic, sequential and logical approach
  • Presents solved examples with physical explanations of oscillations, bifurcations, Lie symmetry analysis
  • Explores the concepts of multifractals and global spectrum for quantifying inhomogeneous chaotic attractors

Part of the book series: University Texts in the Mathematical Sciences (UTMS)

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Table of contents (13 chapters)

  1. Front Matter

    Pages i-xvii
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  2. Continuous Dynamical Systems

    • G. C. Layek
    Pages 1-39

  3. Linear Systems

    • G. C. Layek
    Pages 41-76

  4. Phase Plane Analysis

    • G. C. Layek
    Pages 77-120

  5. Stability Theory

    • G. C. Layek
    Pages 121-155

  6. Oscillations

    • G. C. Layek
    Pages 157-200

  7. Theory of Bifurcations

    • G. C. Layek
    Pages 201-266

  8. Hamiltonian Systems

    • G. C. Layek
    Pages 267-321

  9. Symmetry Analysis

    • G. C. Layek
    Pages 323-414

  10. Discrete Dynamical Systems

    • G. C. Layek
    Pages 415-446

  11. Some Maps

    • G. C. Layek
    Pages 447-496

  12. Conjugacy of Maps

    • G. C. Layek
    Pages 497-518

  13. Chaos

    • G. C. Layek
    Pages 519-618

  14. Fractals

    • G. C. Layek
    Pages 619-681

  15. Back Matter

    Pages 683-688
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Keywords

About this book

This book discusses continuous and discrete nonlinear systems in systematic and sequential approaches. The unique feature of the book is its mathematical theories on flow bifurcations, nonlinear oscillations, Lie symmetry analysis of nonlinear systems, chaos theory, routes to chaos and multistable coexisting attractors. The logically structured content and sequential orientation provide readers with a global overview of the topic. A systematic mathematical approach has been adopted, featuring a multitude of detailed worked-out examples alongside comprehensive exercises. The book is useful for courses in dynamical systems and chaos and nonlinear dynamics for advanced undergraduate, graduate and research students in mathematics, physics and engineering. 

The second edition of the book is thoroughly revised and includes several new topics: center manifold reduction, quasi-periodic oscillations, Bogdanov–Takens, periodbubbling and Neimark–Sacker bifurcations, and dynamics on circle. The organized structures in bi-parameter plane for transitional and chaotic regimes are new active research interest and explored thoroughly. The connections of complex chaotic attractors with fractals cascades are explored in many physical systems. Chaotic attractors may attain multiple scaling factors and show scale invariance property. Finally, the ideas of multifractals and global spectrum for quantifying inhomogeneous chaotic attractors are discussed.

Authors and Affiliations

  • Department of Mathematics, The University of Burdwan, Burdwan, India

    G. C. Layek

About the author

G. C. LAYEK is a Professor of the Department of Mathematics, The University of Burdwan, India. He received his Ph.D. degree from Indian Institute of Technology, Kharagpur and did his Post doctoral studies at Indian Statistical Institute, Kolkata. His areas of research are nonlinear dynamics, chaos theory, turbulence, boundary layer flows and thermal sciences. Professor Layek has published more than 100 research papers in international journals of repute. He taught more than two decades at the post-graduate level in the University of Burdwan. He made several international academic visits, such asLaboratoire de Me ́canique des Fluides de Lille (LMFL), Centrale Lille, France as ‘Professeur invitaé’, Saint Petersburg State University and Kazan State Technological University, Russia for collaborative research works. Layek and Pati’s model (Physics Letters A, 381: 3568-3575, 2017) got recognition for exploring bifurcations and Shil’nikov chaos in Rayleigh-Bénard convection of a Boussinesq fluid layer heated underneath taking non-Fourier heat-flux. The existence of non-Kolmogorov turbulence is established for free-shear turbulent flows, viz., turbulent wake, jet and thermal plume flows through Lie symmetry analysis on statistical turbulent model equations. He has made significant contributions for identification of organized structures in transitional routes and chaotic regimes of many physical phenomena.He now focuses research works on organized structures in chaos and turbulence.

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