This work deals effectively and systematically with the main contents of modern analytical methods of nonlinear science, dynamic systems and the basic concepts of the theory and its applications in engineering. While many books consider systems involving only a few unknowns, a real engineering system involves thousands, and this work emphasises the reduction of the number of these unknowns while capturing the essential physical phenomena. Methods of Liapunov-Schmidt reduction, central manifold, normal form and averaging are discussed in detail. Computational methods for harmonic balance,…mehr
This work deals effectively and systematically with the main contents of modern analytical methods of nonlinear science, dynamic systems and the basic concepts of the theory and its applications in engineering. While many books consider systems involving only a few unknowns, a real engineering system involves thousands, and this work emphasises the reduction of the number of these unknowns while capturing the essential physical phenomena. Methods of Liapunov-Schmidt reduction, central manifold, normal form and averaging are discussed in detail. Computational methods for harmonic balance, normal form, symplectic integration and invariant torus are studied. Finally, applications to solid mechanics, rotating shaft, flutter and galloping are given. This book can be used as a textbook or reference guide to the subject by undergraduate and postgraduate students studying in areas such as mechanics, mathematics, physics and a wide range of related scientific disciplines. It will also be valuable as a reference book for teachers, researchers and engineering designers.
1. Dynamic Systems, Ordinary Differential Equations and Stability of Motion
2. Calculation of Flows
3. Discrete Dynamic Systems
4. Liapunov-Schmidt Reduction
5. Center Manifold Theorem and Normal Form
6. Hopf Bifurcation
7. Averaging Method in Bifurcation Theory
8. Introduction to Chaos
9. Construction of Chaotic Regions
10. Numerical Methods
11. Nonlinear Structural Dynamics
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.
1. Dynamic Systems, Ordinary Differential Equations and Stability of Motion
2. Calculation of Flows
3. Discrete Dynamic Systems
4. Liapunov-Schmidt Reduction
5. Center Manifold Theorem and Normal Form
6. Hopf Bifurcation
7. Averaging Method in Bifurcation Theory
8. Introduction to Chaos
9. Construction of Chaotic Regions
10. Numerical Methods
11. Nonlinear Structural Dynamics
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.
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