The Method of Normal Forms

In this introductory treatment the author Ali Nayfeh presents different concepts of dynamical systems theory and nonlinear dynamics. He systematically introduces models and techniques and states the relevant ranges of their validity and applicability. The text provides the reader with a clear operational framework instead of focusing on the underlying mathematical rigor. The exposition mainly features examples, which are shown through to their fianl outcome. For most of the examples, the results obtained with the method of normal forms are shown to be equivalent to to those obtained with other perturbation methods, such as the method of multiple scales and the method of averaging.
Additions to this new edition concern major topics of current interest. In particular, the author has added three new chapters dedicated to maps, bifurcations of continuous systems, and retarded systems. Especially the latter topic has gained major importance in several applications in mechanics and other areas.
From the contents:SDOF Autonomous SystemsSystems of First–Order EquationsMapsBifurcations of Continuous SystemsForced Oscillations of the Duffing OscillatorForced Oscillations of SDOF SystemsParametrically Excited SystemsMDOF Systems with Quadratic NonlinearitiesTDOF Systems with Cubic NonlinearitiesSystems with Quadratic and Cubis NonlinearitiesRetarded Systems

Spis treści:
Preface.
Introduction 1 
1 SDOF Autonomous Systems 7 
1.1 Introduction 7
1.2 Duffing Equation 9 
1.3 Rayleigh Equation 13 
1.4 Duffing–Rayleigh–van der Pol Equation 15 
1.5 An Oscillator with Quadratic and Cubic Nonlinearities 17 
1.6 A General System with Quadratic and Cubic Nonlinearities 22 
1.7 The van der Pol Oscillator 24 
1.8 Exercises 27 
2 Systems of Fir st–Order Equations 31 
2.1 Introduction 31 
2.2 A Two–Dimensional System with Diagonal Linear Part 34 
2.3 A Two–Dimensional System with a Nonsemisimple Linear Form 39 
2.4 An n–Dimensional System with Diagonal Linear Part 40 
2.5 A Two–Dimensional System with Purely Imaginary Eigenvalues 42 
2.6 A Two–Dimensional System with Zero Eigenvalues 48 
2.7 A Three–Dimensional System with Zero and Two Purely Imaginary Eigenvalues 48 
2.8 The Mathieu Equation 54 
2.9 Exercises 57 
3 Maps 61 
3.1 Linear Maps 61 
3.2 NonlinearMaps 66 
3.3 Center–Manifold Reduction 72 
3.4 Local Bifurcations 76 
3.5 Exercises 91 
4 Bifurcations of Continuous Systems 97 
4.1 Linear Systems 97 
4.2 Fixed Points of Nonlinear Systems 100
4.3 Center–Manifold Reduction 103 
4.4 Local Bifurcations of Fixed Points 107 
4.5 Normal Forms of Static Bifurcations 117 
4.6 Normal Form of Hopf Bifurcation 137 
4.7 Exercises 146 
5 Forced Oscillations of the Duffing Oscillator 161 
5.1 Primary Resonance 161 
5.2 Subharmonic Resonance of Order One–Third 164 
5.3 Superharmonic Resonance of Order Three 167 
5.4 An Alternate Approach 169 
5.5 Exercises 172 
6 Forced Oscillations of SDOF Systems 175 
6.1 Introduction 175 
6.2 Primary Resonance 176 
6.3 Subharmonic Resonance of Order One–Half 178 
6.4 Superharmonic Resonance of Order Two 180 
6.5 Subharmonic Resonance of Order One–Third 182 
7 Parametrically Excited Systems 187 
7.1 The Mathieu Equation 187&n bsp;
7.2 Multiple–Degree–of–Freedom Systems 191 
7.3 Linear Systems Having Repeated Frequencies 195 
7.4 Gyroscopic Systems 205
7.5 A Nonlinear Single–Degree–of–Freedom System 208 
7.6 Exercises 212 
8 MDOF Systems with Quadratic Nonlinearities 217 
8.1 Nongyroscopic Systems 217 
8.2 Gyroscopic Systems 225 
8.3 Two Linearly Coupled Oscillators 229 
8.4 Exercises 232 
9 TDOF Systems with Cubic Nonlinearities 235 
9.1 Nongyroscopic Systems 235 
9.2 Gyroscopic Systems 249 
10 Systems with Quadratic and Cubic Nonlinearities 257 
10.1 Introduction 257
10.2 The Case of No Internal Resonance 262 
10.3 The Case of Three–to–One Internal Resonance 263 
10.4 The Case of One–to–One Internal Resonance 264 
10.5 The Case of Two–to–One Internal Resonance 266 
10.6 Method of Multiple Scales 267 
10.7 Generalized Method of Averaging 276 
10.8 A Nonsemisimple One–to–One Internal Resonance 279 
10.9 Exercises 285 
11 Retarded Systems 287 
11.1 A Scalar Equation 287 
11.2 A Single–Degree–of–Freedom System 295 
11.3 A Three–Dimensional System 304 
11.4 Crane Control with Time–Delayed Feedback 311 
11.5 Exercises 313 
References 315 
Further Reading 319 
Index 325

Nota biograficzna:
Ali Hasan Nayfeh received his B.S. on engineering science and his M.S. and PhD in aeronautics and astronautics from Stanford University. He established and served as Dean of the College of Engineering, Yarmouk University, Jordan from 1980–1984. He is currently Universi ty Distinguished Professor of Engineering at Virginia Polytechnic Institute and State University. He is the Editor of Wiley Series in Nonlinear Science and editor in Chief of Nonlinear Dynamics and the Journal of Vibration and Control.
Prof. Nayfeh is a fellow of the American Physical Society (APS), the American Institute of Aeronautics and Astronautics (AIAA), the American Society of Mechanical Engineers (ASME), the Society of Design and Process Science, and the American Academy of Mechanics (AAM). He holds honrary doctorates from Marine Technical University, Russia, Technical University of Munich, Germany, and Politechnika Szczecinsksa, Poland.
Prof. Nayfeh received AIAA′s Pendray Aerospace Literature Award in 1995; ASME′s J. P. Den Hartog Award in 1997; the Frank J. Maher Award for Excellence in Engineering Education in 1997; ASME′s Lyapunov Award in 2005; the Virginia Academy of Science′s Life Achievement in Science Award in 2005; the Gold Medal of Honor from the Academy of Trans–Disciplinary Learning and Advanced Studies in 2007; and the Thomas K. Caughey Dynamics Award in 2008.

Okładka tylna:
In this introductory treatment the author Ali Nayfeh presents different concepts of dynamical systems theory and nonlinear dynamics. He systematically introduces models and techniques and states the relevant ranges of their validity and applicability. The text provides the reader with a clear operational framework instead of focusing on the underlying mathematical rigor. The exposition mainly features examples, which are shown through to their fianl outcome. For most of the examples, the results obtained with the method of normal forms are shown to be equivalent to to those obtained with other perturbation methods, such as the method of multiple scales and the method of averaging.
Additions to this new edition concern major topics of current interest. In particular, the author has added three new chapters dedic ated to maps, bifurcations of continuous systems, and retarded systems. Especially the latter topic has gained major importance in several applications in mechanics and other areas.
From the contents:SDOF Autonomous SystemsSystems of First–Order EquationsMapsBifurcations of Continuous SystemsForced Oscillations of the Duffing OscillatorForced Oscillations of SDOF SystemsParametrically Excited SystemsMDOF Systems with Quadratic NonlinearitiesTDOF Systems with Cubic NonlinearitiesSystems with Quadratic and Cubis NonlinearitiesRetarded Systems