Dynamical Systems: Theories and Applications, 1st Edition (Hardback) book cover

Dynamical Systems

Theories and Applications, 1st Edition

By Zeraoulia Elhadj

CRC Press

400 pages | 80 B/W Illus.

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Chaos is the idea that a system will produce very different long-term behaviors when the initial conditions are perturbed only slightly. Chaos is used for novel, time- or energy-critical interdisciplinary applications. Examples include high-performance circuits and devices, liquid mixing, chemical reactions, biological systems, crisis management, secure information processing, and critical decision-making in politics, economics, as well as military applications, etc. This book presents the latest investigations in the theory of chaotic systems and their dynamics. The book covers some theoretical aspects of the subject arising in the study of both discrete and continuous-time chaotic dynamical systems. This book presents the state-of-the-art of the more advanced studies of chaotic dynamical systems.

Table of Contents


Review of Chaotic Dynamics


Poincare map technique

Smale horseshoe

Symbolic dynamics

Strange attractors

Basins of attraction

Density, robustness and persistence of chaos

Entropies of chaotic attractors

Period 3 implies chaos

The Snap-back repeller and the Li-Chen-Marotto theorem

Shi'lnikov criterion for the existence of chaos


Human Immunodeficiency Virus and Urbanization Dynamics


Definition of Human Immunodeficiency Virus (HIV)

Modelling the Human Immunodeficiency Virus (HIV)

Dynamics of sexual transmission of the Human Immunodeficiency Virus

The effects of variable infectivity on the HIV dynamics

The CD4+ Lymphocyte dynamics in HIV infection

The viral dynamics of a highly pathogenic simian/human immunodeficiency virus

The effects of morphine on Simian Immunodeficiency Virus Dynamics

The dynamics of the HIV therapy system

Dynamics of urbanization


Chaotic Behaviors in Piecewise Linear Mappings


Chaos in one-dimensional piecewise smooth maps

Chaos in one-dimensional singular maps

Chaos in 2-D piecewise smooth maps


Robust Chaos in Neural Networks Models


Chaos in neural networks models

Robust chaos in discrete time neural networks

The importance of robust chaos in mathematics and some open problems


Estimating Lyapunouv Exponents of 2-D Discrete Mappings


Lyapunouv exponents of the discrete hyperchaotic double scroll map

Lyapunouv exponents for a class of 2-D piecewise linear mappings

Lyapunouv exponents of a family of 2-D discrete mappings with separate variables

Lyapunouv exponents of a discontinuous piecewise linear mapping of the plane governed by a simple switching law

Lyapunouv exponents of a modified map-based BVP model


Control, Synchronization and Chaotification of Dynamical Systems


Compound synchronization of different chaotic systems

Synchronization of 3-D continuous-time quadratic systems using a universal nonlinear control law

Co-existence of certain types of synchronization and its inverse

Synchronization of 4-D continuous-time quadratic systems using a universal nonlinear control law

Quasi-synchronization of systems with different dimension

Chaotification of 3-D linear continuous-time systems using the signum function feedback

Chaos control problem of a 3-D cancer model with structured uncertainties

Controlling homoclinic chaotic attractor

Robustification of 2-D piecewise smooth mappings

Chaotifying stable n-D linear maps via the controller of any bounded function




Boundedness of Some Forms of Quadratic Systems


Boundedness of certain forms of 3-D quadratic continuous-time systems

Bounded jerky dynamics

Bounded hyperjerky dynamics

Boundedness of the generalized 4-D hyperchaotic model containing Lorenz-Steno and Lorenz-Haken systems

Boundedness of 2-D Henon-like mapping

Examples of fully bounded chaotic attractors


Some Forms of Globally Asymptotically Stable Attractors


Direct Lyapunov stability for ordinary differential equations

Exponential stability of nonlinear time-varying

Lasalle's Invariance Principle

Direct Lyapunov-type stability for fractional-like systems

Construction of globally asymptotically stable n-D discrete mappings

Construction of superstable n-D mappings

Examples of globally superstable 1-D quadratic mappings

Construction of globally superstable 3-D quadratic mappings

Hyperbolicity of dynamical systems

Consequences of uniform hyperbolicity

Structural stability for 3-D quadratic mappings

Construction of globally asymptotically stable partial differential systems

Construction of globally stable system of delayed differential equations

Stabilization by the Jurdjevic-Quinn method


Transformation of Dynamical Systems to Hyperjerky Motions


Transformation of 3-D dynamical systems to jerk form

Transformation of 3-D dynamical systems to rational and cubic jerks forms

Transformation of 4-D dynamical systems to hyperjerk form

Examples of crackle and top dynamics




About the Author

Zeraoulia Elhadj, born on February 23, 1976, in Yabous, Khenchela, Algeria, received his B.S. degree in mathematics from the Institute of Mathematics (University of Batna, Batna, Algeria) in 1998 and a Ph.D. degree in mathematics from the University of Constantine, Constantine, Algeria, in 2006. He joined the Department of Mathematics, University of Tébessa, Tébessa, Algeria, in 2001 as a Research Associate, and in the same year, Elhadj became an Assistant Professor. Since 2001, he has been teaching undergraduate and graduate courses on applied mathematics. His primary research interests include bifurcations and chaos. He has authored or coauthored more than 150 journal and conference papers and 15 books. He is the Editor-in-Chief of the Annual Review of Chaos Theory, Bifurcations and Dynamical Systems Journal.

Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Arithmetic
MATHEMATICS / Differential Equations
SCIENCE / Mathematical Physics