Periodic Motions / Edition 1

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A summary of the most important results in the existence and stability of periodic solutions for ordinary differential equations achieved in the twentieth century, along with relevant applications. It differs from standard classical texts on non-linear oscillations in that it also contains linear theory; theorems are proved with mathematical rigor; and, besides the classical applications such as Van der Pol's, Linard's and Duffing's equations, most applications come from biomathematics.
For graduate and Ph.D students in mathematics, physics, engineering, and biology, and as a standard reference for use by researchers in the field of dynamical systems and their applications.

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Product Details

  • ISBN-13: 9780387942049
  • Publisher: Springer New York
  • Publication date: 7/28/1994
  • Series: Applied Mathematical Sciences Series, #104
  • Edition description: 1994
  • Edition number: 1
  • Pages: 578
  • Product dimensions: 9.21 (w) x 6.14 (h) x 1.31 (d)

Table of Contents

1 Introduction 1
1.1 Existence, Uniqueness and Analytic Properties of Solutions 2
1.2 Linear Systems 10
1.3 Dynamical Systems 16
1.4 Stability 20
1.5 Liapunov's Direct Method 33
2 Periodic Solutions of Linear Systems 45
2.1 Linear Systems with Constant Coefficients 45
2.2 Homogeneous Linear Systems with Periodic Coefficients 52
2.3 Forced Linear Oscillations 60
2.4 Stability of Linear Systems 66
2.5 Hill's and Mathieu's Equations 71
3 Autonomous Systems in the Plane 83
3.1 The Poincare-Bendixson Theory 84
3.2 Lienard's Equation 97
3.3 Duffing's Equation 108
3.4 The Lotka-Volterra Predator-Prey Model and Generalizations 114
3.5 The Poincare Index and Non-existence of Cycles 122
3.6 Hilbert's Sixteenth Problem 141
4 Periodic Solutions of Periodic Systems 151
4.1 Existence of Periodic Solutions 152
4.2 Stability and Isolation of Periodic Solutions 161
4.3 Periodically Forced Lienard and Duffing Equations 171
4.4 Two Competing Species in a Periodically Changing Environment 183
4.5 Applications in Higher Dimensions 194
5 Autonomous Systems of Arbitrary Dimension 205
5.1 Orbital Stability 206
5.2 Poincare Map, Isolation and Isochronism 219
5.3 D-periodic Solutions of Cylindrical Systems 234
5.4 Existence of Periodic Solutions 250
5.5 Competitive and Cooperative Systems, Existence in Dimension Three 274
5.6 Invariant and Integral Manifolds of Periodic Solutions 290
6 Perturbations 301
6.1 Periodic Perturbations of Periodic Systems 302
6.2 Controllably Periodic Perturbations of Autonomous Systems 314
6.3 The Stability of Perturbed Periodic Solutions 341
6.4 Controllably Periodic Perturbations of Van der Pol's Equation 353
6.5 Averaging 361
6.6 Singular Perturbations and Relaxation Oscillations 371
6.7 Aperiodic Perturbations 381
7 Bifurcations 399
7.1 Structural Stability and Bifurcations 400
7.2 The Andronov-Hopf Bifurcation 411
7.3 A Predator-Prey Model with Memory 439
7.4 Zip Bifurcation in Competitive Systems 457
7.5 Functional Differential Equations 476
7.6 Through Periodic Motions to Chaos 491
App. A1 Matrices 503
App. A2 Topological Degree and Fixed Point Theorems 519
App. A3 Invariant Manifolds 531
References 545
Symbols 569
Index 571
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