UID:
almahu_9949199626802882
Format:
online resource.
Edition:
1st ed. 1997.
ISBN:
9781468400328
Content:
Philosophy of the Text This text has been designed to be an introductory survey of the basic concepts and applied mathematical methods of nonlinear science. Students in engineer ing, physics, chemistry, mathematics, computing science, and biology should be able to successfully use this text. In an effort to provide the students with a cutting edge approach to one of the most dynamic, often subtle, complex, and still rapidly evolving, areas of modern research-nonlinear physics-we have made extensive use of the symbolic, numeric, and plotting capabilities of Maple V Release 4 applied to examples from these disciplines. No prior knowledge of Maple or computer programming is assumed, the reader being gently introduced to Maple as an auxiliary tool as the concepts of nonlinear science are developed. The diskette which accompanies the text gives a wide variety of illustrative nonlinear examples solved with Maple. An accompanying laboratory manual of experimental activities keyed to the text allows the student the option of "hands on" experience in exploring nonlinear phenomena in the REAL world. Although the experiments are easy to perform, they give rise to experimental and theoretical complexities which are not to be underestimated. The Level of the Text The essential prerequisites for the first eight chapters of this text would nor mally be one semester of ordinary differential equations and an intermediate course in classical mechanics.
Note:
1 Introduction -- 1.1 It's a Nonlinear World -- 1.2 Symbolic Computation -- 1.3 Nonlinear Activities -- 1.4 Structure of the Text -- 2 Some Nonlinear Systems -- 2.1 Nonlinear Mechanics -- 2.2 Competition Phenomena -- 2.3 Nonlinear Electrical Phenomena -- 2.4 Chemical and Other Oscillators -- 2.5 Pattern Formation -- 2.6 Solitons -- 2.7 Chaos and Maps -- 3 Topological Analysis -- 3.1 Introductory Remarks -- 3.2 Types of Simple Singular Points -- 3.3 Classifying Simple Singular Points -- 3.4 Examples of Phase Plane Analysis -- 3.5 Isoclines -- 4 Analytic Methods -- 4.1 Introductory Remarks -- 4.2 Some Exact Methods -- 4.3 Some Approximate Methods -- 4.4 The Krylov-Bogoliubov (KB) Method -- 4.5 Ritz and Galerkin Methods -- 5 The Numerical Approach -- 5.1 Finite-Difference Approximations -- 5.2 Euler and Modified Euler Methods -- 5.3 Rungé-Kutta (RK) Methods -- 5.4 Adaptive Step Size -- 5.5 Stiff Equations -- 5.6 Implicit and Semi-Implicit Schemes -- 6 Limit Cycles -- 6.1 Stability Aspects -- 6.2 Relaxation Oscillations -- 6.3 Bendixson's First Theorem: The Negative Criterion -- 6.4 The Poincaré-Bendixson Theorem -- 6.5 The Brusselator Model -- 7 Forced Oscillators -- 7.1 Duffing's Equation -- 7.2 The Jump Phenomenon and Hysteresis -- 7.3 Subharmonic and Other Periodic Oscillations -- 7.4 Power Spectrum -- 7.5 Chaotic Oscillations -- 7.6 Entrainment and Quasiperiodicity -- 7.7 The Rössler and Lorenz Systems -- 8 Nonlinear Maps -- 8.1 Introductory Remarks -- 8.2 The Logistic Map -- 8.3 Fixed Points and Stability -- 8.4 The Period-Doubling Cascade to Chaos -- 8.5 Period Doubling in the Real World -- 8.6 The Lyapunov Exponent -- 8.7 Stretching and Folding -- 8.8 The Circle Map -- 8.9 Chaos versus Noise -- 8.10 2-Dimensional Maps -- 9 Nonlinear PDE Phenomena -- 9.1 Introductory Remarks -- 9.2 Burger's Equation -- 9.3 Bäcklund Transformations -- 9.4 Solitary Waves -- 10 Numerical Simulation -- 10.1 Finite Difference Approximations -- 10.2 Explicit Methods -- 10.3 Von Neumann Stability Analysis -- 10.4 Implicit Methods -- 10.5 Method of Characteristics -- 10.6 Higher Dimensions -- 11 Inverse Scattering Method -- 11.1 Lax's Formulation -- 11.2 Application to KdV Equation -- 11.3 Multi-Soliton Solutions -- 11.4 General Input Shapes -- 11.5 The Zakharov-Shabat/AKNS Approach.
In:
Springer Nature eBook
Additional Edition:
Printed edition: ISBN 9781468400342
Additional Edition:
Printed edition: ISBN 9780817638382
Additional Edition:
Printed edition: ISBN 9781468400335
Language:
English
DOI:
10.1007/978-1-4684-0032-8
URL:
https://doi.org/10.1007/978-1-4684-0032-8
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