This book treats nonlinear dynamics in both Hamiltonian and dissipative systems. The emphasis is on the mechanics for generating chaotic motion, methods of calculating the transitions from regular to chaotic motion, and the dynamical and statistical properties of the dynamics when it is chaotic. The new edition brings the subject matter in a rapidly expanding field up to date, and has greatly expanded the treatment of dissipative dynamics to include most important subjects.

A clear introduction to chaotic phenomena for undergraduate students in science, engineering, and mathematics.

This textbook is aimed at newcomers to nonlinear dynamics and chaos, especially students taking a first course in the subject. The presentation stresses analytical methods, concrete examples, and geometric intuition. The theory is developed systematically, starting with first-order differential equations and their bifurcations, followed by phase plane analysis, limit cycles and their bifurcations, and culminating with the Lorenz equations, chaos, iterated maps, period doubling, renormalization, fractals, and strange attractors.

This book treats stochastic motion in nonlinear oscillator systems. It describes a rapidly growing field of nonlinear mechanics with applications to a number of areas in science and engineering, including astronomy, plasma physics, statistical mechanics and hydrodynamics. The main em phasis is on intrinsic stochasticity in Hamiltonian systems, where the stochastic motion is generated by the dynamics itself and not by external noise. However, the effects of noise in modifying the intrinsic motion are also considered. A thorough introduction to chaotic motion in dissipative systems is given in the final chapter. Although the roots of the field are old, dating back to the last century when Poincare and others attempted to formulate a theory for nonlinear perturbations of planetary orbits, it was new mathematical results obtained in the 1960's, together with computational results obtained using high speed computers, that facilitated our new treatment of the subject. Since the new methods partly originated in mathematical advances, there have been two or three mathematical monographs exposing these developments. However, these monographs employ methods and language that are not readily accessible to scientists and engineers, and also do not give explicit tech niques for making practical calculations. In our treatment of the material, we emphasize physical insight rather than mathematical rigor. We present practical methods for describing the motion, for determining the transition from regular to stochastic behavior, and for characterizing the stochasticity. We rely heavily on numerical computations to illustrate the methods and to validate them.

Chaotic Dynamics and Fractals covers the proceedings of the 1985 Conference on Chaotic Dynamics, held at the Georgia Institute of Technology. This conference deals with the research area of chaos, dynamical systems, and fractal geometry. This text is organized into three parts encompassing 16 chapters. The first part describes the nature of chaos and fractals, the geometric tool for some strange attractors, and other complicated sets of data associated with chaotic systems. This part also considers the Henon-Hiles Hamiltonian with complex time, a Henon family of maps from C2 into itself, and the idea of turbulent maps in the course of presenting results on iteration of continuous maps from the unit interval to itself. The second part discusses complex analytic dynamics and associated fractal geometry, specifically the bursts into chaos, algorithms for obtaining geometrical and combinatorial information, and the parameter space for iterated cubic polynomials. This part also examines the differentiation of Julia sets with respects to a parameter in the associated rational map, permitting the formulation of Taylor series expansion for the sets. The third part highlights the applications of chaotic dynamics and fractals. This book will prove useful to mathematicians, physicists, and other scientists working in, or introducing themselves to, the field.

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.

This book is a collection of papers contributed by some of the greatest names in the areas of chaos and nonlinear dynamics. Each paper examines a research topic at the frontier of the area of dynamical systems. As well as reviewing recent results, each paper also discusses the future perspectives of each topic. The result is an invaluable snapshot of the state of the ?eld by some of the most important researchers in the area. The ?rst contribution in this book (the section entitled “How did you get into Chaos?”) is actually not a paper, but a collection of personal accounts by a number of participants of the conference held in Aberdeen in September 2007 to honour Celso Grebogi’s 60th birthday. At the instigation of James Yorke, many of the most well-known scientists in the area agreed to share their tales on how they got involved in chaos during a celebratory dinner in Celso’s honour during the conference. This was recorded in video, we felt that these accounts were a valuable historic document for the ?eld. So we decided to transcribe it and include it here as the ?rst section of the book.

The Solar System is a complex and fascinating dynamical system. This is the first textbook to describe comprehensively the dynamical features of the Solar System and to provide students with all the mathematical tools and physical models they need to understand how it works. It is a benchmark publication in the field of planetary dynamics and destined to become a classic. Clearly written and well illustrated, Solar System Dynamics shows how a basic knowledge of the two- and three-body problems and perturbation theory can be combined to understand features as diverse as the tidal heating of Jupiter's moon Io, the origin of the Kirkwood gaps in the asteroid belt, and the radial structure of Saturn's rings. Problems at the end of each chapter and a free Internet Mathematica® software package are provided. Solar System Dynamics provides an authoritative textbook for courses on planetary dynamics and celestial mechanics. It also equips students with the mathematical tools to tackle broader courses on dynamics, dynamical systems, applications of chaos theory and non-linear dynamics.

Over the past two decades scientists, mathematicians, and engineers have come to understand that a large variety of systems exhibit complicated evolution with time. This complicated behavior is known as chaos. In the new edition of this classic textbook Edward Ott has added much new material and has significantly increased the number of homework problems. The most important change is the addition of a completely new chapter on control and synchronization of chaos. Other changes include new material on riddled basins of attraction, phase locking of globally coupled oscillators, fractal aspects of fluid advection by Lagrangian chaotic flows, magnetic dynamos, and strange nonchaotic attractors. This new edition will be of interest to advanced undergraduates and graduate students in science, engineering, and mathematics taking courses in chaotic dynamics, as well as to researchers in the subject.

Distinguishing chaoticity from regularity in deterministic dynamical systems and specifying the subspace of the phase space in which instabilities are expected to occur is of utmost importance in as disparate areas as astronomy, particle physics and climate dynamics. To address these issues there exists a plethora of methods for chaos detection and predictability. The most commonly employed technique for investigating chaotic dynamics, i.e. the computation of Lyapunov exponents, however, may suffer a number of problems and drawbacks, for example when applied to noisy experimental data. In the last two decades, several novel methods have been developed for the fast and reliable determination of the regular or chaotic nature of orbits, aimed at overcoming the shortcomings of more traditional techniques. This set of lecture notes and tutorial reviews serves as an introduction to and overview of modern chaos detection and predictability techniques for graduate students and non-specialists. The book covers theoretical and computational aspects of traditional methods to calculate Lyapunov exponents, as well as of modern techniques like the Fast (FLI), the Orthogonal (OFLI) and the Relative (RLI) Lyapunov Indicators, the Mean Exponential Growth factor of Nearby Orbits (MEGNO), the Smaller (SALI) and the Generalized (GALI) Alignment Index and the ‘0-1’ test for chaos.