The Method of Normal Forms 2nd Edition by Ali Hasan Nayfeh ISBN 0471015247 9780471015244
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ISBN 10: 0471015247
ISBN 13: 9780471015244
Author: Ali Hasan Nayfeh
In this introductory treatment Ali Nayfeh presents different concepts from dynamical systems theory and nonlinear dynamics in a rigorous yet plan way. He systematically introduces models and techniques and states the relevant ranges of validity and applicability.
The reader is provided with a clear operational framework for consciously use rather than focused on the underlying mathematical apparatus. The exposition is largely by means of examples, dealt with up to their final outcome. For most of the examples, the results obtained with the method of normal forms are equivalent to those obtained with other perturbation methods, such as the method of multiple scales and the method of averaging.
The previous edition had a remarkable success by researchers from all over the world working in the area of nonlinear dynamics and their applications in engineering. Additions to this new edition concern major topics of current interest. In particular, the author added three new chapters dedicated to Maps, Bifurcations of Continuous Systems, and Retarded Systems. In particular the latter has become of major importance in several applications, both in mechanics and in different areas.
Accessible to engineers and applied scientist involved with nonlinear dynamics and their applications in a wide variety of fields. It is assumed that readers have a knowledge of basic calculus as well as the elementary properties of ordinary-differential equations.
The Method of Normal Forms 2nd Edition Table of contents:
Chapter 1: Introduction to Nonlinear Dynamics
- The Role of Nonlinearity in Physical Systems
- Basic Concepts: Phase Space, Trajectories, and Stability
- Linear vs. Nonlinear Systems
Chapter 2: The Method of Normal Forms
- Definitions and Concepts
- The Role of Normal Forms in Simplifying Nonlinear Systems
- Bifurcation Theory and Its Relation to Normal Forms
- Transformation Techniques for Simplifying Nonlinear Equations
Chapter 3: Canonical Forms and Their Application
- The Process of Obtaining Normal Forms
- One-Dimensional and Multi-Dimensional Systems
- Examples of Canonical Forms in Mechanical and Electrical Systems
Chapter 4: Bifurcations and Instabilities
- Definition of Bifurcation Points
- Stability Analysis and Phase Portraits
- Saddle-Node, Hopf, and Pitchfork Bifurcations
- Application to Real-World Systems
Chapter 5: Nonlinear Oscillations and Resonance
- Analysis of Nonlinear Oscillatory Systems
- The Method of Averaging
- Parametric and Forced Resonance
- Application to Mechanical and Electrical Oscillators
Chapter 6: Application of Normal Forms to Control Systems
- The Role of Normal Forms in Control Theory
- Nonlinear Control Systems and Stabilization
- Use of Normal Forms in Feedback Linearization
Chapter 7: Perturbation Methods and Asymptotic Expansions
- Techniques for Approximate Solutions of Nonlinear Systems
- Singular Perturbations and Multiple Scales
- The Role of Normal Forms in Asymptotic Analysis
Chapter 8: Advanced Topics in Normal Forms
- Higher-Order Normal Forms and Their Applications
- Symmetry and Group Theory in Normal Form Theory
- Numerical Methods for Normal Form Calculations
Chapter 9: Case Studies and Examples
- Application to Physical Systems: Mechanical, Electrical, and Fluid Systems
- Engineering Applications and Examples from Control Systems
- Real-World Examples of Bifurcations and Nonlinear Phenomena
Chapter 10: Conclusion and Future Directions
- Summary of Key Concepts and Methods
- Current Research and Emerging Trends in Normal Form Theory
- Future Applications and Challenges
Appendices
- A: Mathematical Tools and Notations
- B: List of Important Theorems and Proofs
- C: Selected References and Further Reading
Index
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