This book focuses on the development of Melnikov-type methods applied to high dimensional dynamical systems governed by ordinary differential equations. Although the classical Melnikov's technique has found various applications in predicting homoclinic intersections, it is devoted only to the analysis of three-dimensional systems (in the case of mechanics, they represent one-degree-of-freedom nonautonomous systems). This book extends the classical Melnikov's approach to the study of high dimensional dynamical systems, and uses simple models of dry friction to analytically predict the occurrence of both stick-slip and slip-slip chaotic orbits, research which is very rarely reported in the existing literature even on one-degree-of-freedom nonautonomous dynamics.
This pioneering attempt to predict the occurrence of deterministic chaos of nonlinear dynamical systems will attract many researchers including applied mathematicians, physicists, as well as practicing engineers. Analytical formulas are explicitly formulated step-by-step, even attracting potential readers without a rigorous mathematical background.
Author(s): Jan Awrejcewicz, Mariusz M. Holicke
Series: World Scientific Series on Nonlinear Science Series a
Publisher: World Scientific Publishing Company
Year: 2007
Language: English
Pages: 320
Tags: Математика;Нелинейная динамика;
Contents......Page 10
Preface......Page 6
1.1 Introduction......Page 12
1.2 Application of the Melnikov-type methods......Page 14
2.1 Introduction......Page 22
2.2 Geometric interpretation......Page 24
2.3 Melnikov's function......Page 31
3.1 Mathematical Model......Page 36
3.2 Homoclinic Chaos Criterion......Page 64
3.3 Numerical Simulations......Page 65
4.1 Stick-Slip Oscillator with Periodic Excitation......Page 68
4.2 Analysis of the Wandering Trajectories......Page 70
4.3 Comparison of Analytical and Numerical Results......Page 73
5.1 Mechanical Systems with Finite Number of Degrees-of- Freedom......Page 76
5.2 2-DOFs Mechanical Systems......Page 79
5.3 Reduction of the Melnikov-Gruendler Method for 1-DOF Systems......Page 89
6.1 Analytical Prediction of Chaos......Page 90
6.2 Numerical Results......Page 110
7.1 Analytical Prediction of Chaos......Page 114
7.2 Numerical Simulations......Page 191
7.3 Additional Numerical Example......Page 199
8.1 Physical and Mathematical Models......Page 204
8.2 Analytical Prediction of Homoclinic Intersections......Page 205
Bibliography......Page 296
Index......Page 302
