Bifurcation and Chaos in Coupled Oscillators

The following sections are included:

• Contents

This book consists of four chapters. In the first chapter, a two degree of freedom self-excited roll-slide oscillator with oscillations due to friction, is analyzed. First, the nonlinear nondimensional set of ordinary equations governing the dynamics of the system is established, and an analytical discussion is presented, leading to an investigation of the behavior of equilibria in the parameter space. The analytical approach is supported by numerical calculations and includes an analysis of birth and annihilation of equilibria. A possible physical interpretation of this behavior is also presented. Based on the prediction of the behavior of the system obtained from analytical considerations, further numerical calculations were performed for certain special parameters. By applying a numerical technique, transitions from a slidingstate to a sticking state can be studied. During numerical calculations attention is focused on the behavior of periodic orbits, which are observed with changes of the most important control parameter. Some scenarios from regular to irregular motion and vice versa are illustrated, and the physical interpretation of the behavior of the investigated system in each dynamical state is given. This chapter is based on results received from a collaboration with J. Delfs…

The following sections are included:

• Introduction

• Equations of Motion and Phase Flow

• Trivial Solutions - Analytical Investigations

• Discussion of the Analytical Results

• Stability of Equilibria - Numerical Investigation

• The Integration of the Equations of Motion

• Slip-stick transition

• Slip-slip transition

• Stick-slip transition

• Numerical Integrations

• Calculations of Periodic Orbits and Their Stability

• Evolution of Periodic Orbits

• Observations of Chaos

• Summary and Concluding Remarks

The following sections are included:

• Introduction

• The System and the Method

• The Influence of Coupling for Periodic and Chaotic Orbits

• Equilibria

• Periodic and Chaotic Orbits

• Conclusions

• Chaotic Dynamics in the Case of Two External Harmonic Excitations with Commensurable Frequencies 1:2, 1:3 and 2:3

• Results

• Conclusions

• The Influence of Friction on the Chaotic Dynamics

• Analysis

• Conclusions

• Quasiperiodicity, and Strange Non-Chaotic and Chaotic Attractors

• Analysis

• Conclusions

• Observation of Chaos in a Sinusoidally-Driven System

• Analysis

• Conclusions

The following sections are included:

• Introduction

• The System Analyzed

• Examples of Different Chaotic Dynamics

• Results of numerical calculations

• Concluding remarks

• Periodic, Quasiperiodic and Chaotic Orbits and Their Bifurcations

• Results of numerical calculations

• Conclusions

• A Particular Scenario Leading to Chaos

• Results

• Conclusions

The following sections are included:

• Cronjaeger’s Model of the Human Vocal Cords

• Equations of Motion

• Analysis of the Equations of Motion Based on an Analytical Approach [62]

• Introduction

• Equilibria

• Stability of equilibria

• The stability limit for one pair of complex conjugate eigenvalues

• The stability limit for two pairs of complex conjugate eigenvalues

• Analysis of the Regular Motion Based on the Numerical Investigation

• Introduction

• Constant solutions and their stability

• Influence of the air stream rate Q and the position of the cartilage X_o

• Influence of the Parameter E

• Influence of elastic coupling K_xy and the parameter K_D

• Numerical calculations of the periodic orbits

• Examples of periodic self-excited oscillations

• Subharmonics in the neighborhood of the bifurcation point

• Bifurcation diagram

• Irregular Motion

• Introduction

• From periodicity through quasiperiodicity to chaos

• Influence of parametric excitation on the behavior of a strange attractor

• Concluding remarks

The following sections are included:

• REFERENCES

• Subject Index