Bifurcation and Chaos in Engineering de Yushu Chen

Bifurcation and Chaos in Engineering

Yushu Chen

Andrew Y.T. Leung

For the many different deterministic non-linear dynamic systems (physical, mechanical, technical, chemical, ecological, economic, and civil and structural engineering), the discovery of irregular vibrations in addition to periodic and almost periodic vibrations is one of the most significant achievements of modern science. An in-depth study of the theory and application of non-linear science will certainly change one's perception of numerous non-linear phenomena and laws considerably, together with its great effects on many areas of application. As the important subject matter of non-linear science, bifurcation theory, singularity theory and chaos theory have developed rapidly in the past two or three decades. They are now advancing vigorously in their applications to mathematics, physics, mechanics and many technical areas worldwide, and they will be the main subjects of our concern. This book is concerned with applications of the methods of dynamic systems and subharmonic bifurcation theory in the study of non-linear dynamics in engineering. It has grown out of the class notes for graduate courses on bifurcation theory, chaos and application theory of non-linear dynamic systems, supplemented with our latest results of scientific research and materials from literature in this field. The bifurcation and chaotic vibration of deterministic non-linear dynamic systems are studied from the viewpoint of non-linear vibration.

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Cuprins

1 Dynamical Systems, Ordinary Differential Equations and Stability of Motion.- 1.1 Concepts of Dynamical Systems.- 1.2 Ordinary Differential Equations.- 1.3 Properties of Flow.- 1.4 Limit Point Sets.- 1.5 Liapunov Stability of Motion.- 1.6 Poincaré-Bendixson Theorem and its Applications.- 2 Calculation of Flows.- 2.1 Divergence of Flows 3.- 2.2 Linear Autonomous Systems and Linear Flows and the Calculation of Flows about the IVP.- 2.3 Hyperbolic Operator (or Generality).- 2.4 Non-linear Differential Equations and the Calculation of their Flows.- 2.5 Stable Manifold Theorem.- 3 Discrete Dynamical Systems.- 3.1 Discrete Dynamical Systems and Linear Maps.- 3.2 Non-linear Maps and the Stable Manifold Theorem.- 3.3 Classification of Generic Systems.- 3.4 Stability of Maps and Poincaré Mapping.- 3.5 Structural Stability Theorem.- 4 Liapunov—Schmidt Reduction.- 4.1 Basic Concepts of Bifurcation.- 4.2 Classification of Bifurcations of Planar Vector Fields.- 4.3 The Implicit Function Theorem.- 4.4 Liapunov—Schmidt Reduction.- 4.5 Methods of Singularity.- 4.6 Simple Bifurcations.- 4.7 Bifurcation Solution of the 1/2 Subharmonic Resonance Case of Non-linear Parametrically Excited Vibration Systems.- 4.8 Hopf Bifurcation Analyzed by Liapunov—Schmidt Reduction.- 5 Centre Manifold Theorem and Normal Form of Vector Fields.- 5.1 Centre Manifold Theorem.- 5.2 Saddle—Node Bifurcation.- 5.3 Normal Form of Vector Fields.- 6 Hopf Bifurcation.- 6.1 Hopf Bifurcation Theorem.- 6.2 Complex Normal Form of the Hopf Bifurcation.- 6.3 Normal Form of the Hopf Bifurcation in Real Numbers.- 6.4 Hopf Bifurcation with Parameters.- 6.5 Calculating Formula for the Hopf Bifurcation Solution.- 6.6 Stability of the Hopf Bifurcation Solution.- 6.7 Effective Method for Computing the Hopf BifurcationSolution Coefficients.- 6.8 Bifurcation Problem Involving Double Zero Eigenvalues.- 7 Application of the Averaging Method in Bifurcation Theory.- 7.1 Standard Equation.- 7.2 Averaging Method and Poincaré Maps.- 7.3 The Geometric Description of the Averaging Method.- 7.4 An Example of the Averaging Method — the Duffing Equation.- 7.5 The Averaging Method and Local Bifurcation.- 7.6 The Averaging Method, Hamiltonian Systems and Global Behaviour.- 8 Brief Introduction to Chaos.- 8.1 What is Chaos?.- 8.2 Some Examples of Chaos.- 8.3 A Brief Introduction to the Analytical Method of Chaotic Study.- 8.4 The Hamiltonian System.- 8.5 Some Statistical Characteristics.- 8.6 Conclusions.- 9 Construction of Chaotic Regions.- 9.1 Incremental Harmonic Balance Method (IHB Method).- 9.2 The Newtonian Algorithm.- 9.3 Number of Harmonic Terms.- 9.4 Stability Characteristics.- 9.5 Transition Sets in Physical Parametric Space.- 9.6 Example of the Duffing Equation with Multi—Harmonic Excitation.- 10 Computational Methods.- 10.1 Normal Form Theory.- 10.2 Symplectic Integration and Geometric Non-Linear Finite Element Method.- 10.3 Construction of the Invariant Torus.- 11 Non-linear Structural Dynamics.- 11.1 Bifurcations in Solid Mechanics.- 11.2 Non-Linear Dynamics of an Unbalanced Rotating Shaft.- 11.3 Galloping Vibration Analysis for an Elastic Structure.- 11.4 Other Applications of Bifurcation Theory.- References.