Periodic Motions / Edition 1

Periodic Motions / Edition 1

by Miklos Farkas
     
 

ISBN-10: 0387942041

ISBN-13: 9780387942049

Pub. Date: 07/28/1994

Publisher: Springer New York

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

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Overview

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.

Product Details

ISBN-13:
9780387942049
Publisher:
Springer New York
Publication date:
07/28/1994
Series:
Applied Mathematical Sciences Series, #104
Edition description:
1994
Pages:
578
Product dimensions:
9.21(w) x 6.14(h) x 1.31(d)

Table of Contents

Preface
1Introduction1
1.1Existence, Uniqueness and Analytic Properties of Solutions2
1.2Linear Systems10
1.3Dynamical Systems16
1.4Stability20
1.5Liapunov's Direct Method33
2Periodic Solutions of Linear Systems45
2.1Linear Systems with Constant Coefficients45
2.2Homogeneous Linear Systems with Periodic Coefficients52
2.3Forced Linear Oscillations60
2.4Stability of Linear Systems66
2.5Hill's and Mathieu's Equations71
3Autonomous Systems in the Plane83
3.1The Poincare-Bendixson Theory84
3.2Lienard's Equation97
3.3Duffing's Equation108
3.4The Lotka-Volterra Predator-Prey Model and Generalizations114
3.5The Poincare Index and Non-existence of Cycles122
3.6Hilbert's Sixteenth Problem141
4Periodic Solutions of Periodic Systems151
4.1Existence of Periodic Solutions152
4.2Stability and Isolation of Periodic Solutions161
4.3Periodically Forced Lienard and Duffing Equations171
4.4Two Competing Species in a Periodically Changing Environment183
4.5Applications in Higher Dimensions194
5Autonomous Systems of Arbitrary Dimension205
5.1Orbital Stability206
5.2Poincare Map, Isolation and Isochronism219
5.3D-periodic Solutions of Cylindrical Systems234
5.4Existence of Periodic Solutions250
5.5Competitive and Cooperative Systems, Existence in Dimension Three274
5.6Invariant and Integral Manifolds of Periodic Solutions290
6Perturbations301
6.1Periodic Perturbations of Periodic Systems302
6.2Controllably Periodic Perturbations of Autonomous Systems314
6.3The Stability of Perturbed Periodic Solutions341
6.4Controllably Periodic Perturbations of Van der Pol's Equation353
6.5Averaging361
6.6Singular Perturbations and Relaxation Oscillations371
6.7Aperiodic Perturbations381
7Bifurcations399
7.1Structural Stability and Bifurcations400
7.2The Andronov-Hopf Bifurcation411
7.3A Predator-Prey Model with Memory439
7.4Zip Bifurcation in Competitive Systems457
7.5Functional Differential Equations476
7.6Through Periodic Motions to Chaos491
App. A1 Matrices503
App. A2 Topological Degree and Fixed Point Theorems519
App. A3 Invariant Manifolds531
References545
Symbols569
Index571

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