Applied Nonlinear Dynamics Analytical, Computational, and Experimental Methods

by Nayfeh, Ali H.; Balachandran, Balakumar

• ISBN13: 9780471593485
• ISBN10: 0471593486
• Edition: 1st
• Format: Hardcover
• Copyright: 1995-02-17
• Publisher: Wiley-VCH

Summary

A unified and coherent treatment of analytical, computational and experimental techniques of nonlinear dynamics with numerous illustrative applications. Features a discourse on geometric concepts such as Poincar? maps. Discusses chaos, stability and bifurcation analysis for systems of differential and algebraic equations. Includes scores of examples to facilitate understanding.

Author Biography

Ali H. Nayfeh is a University Distinguished Professor of Engineering Science and Mechanics at the Virginia Polytechnic Institute and State University, Blacksburg, Virginia. Professor Nayfeh is the Editor-in-Chief of the journal Nonlinear Dynamics and the Journal of Vibration and Control. He is the author of Perturbation Methods (Wiley, 1973), Nonlinear Oscillations (coauthored with Dean T. Mook; Wiley,1979), Introduction to Perturbation techniques (Wiley, 1981), Problems in Perturbation (Wiley, 1985), and Method of Normal Forms (Wiley, 1993). Professor Nayfeh's areas of interest include nonlinear vibrations and dynamics, wave propagation, ship and submarine motions, structural dynamics, acoustics, aerodynamic/dynamic/structure/control interactions, flight mechanics, and transition from laminar to turbulent flows. Balakumar Balachandran is Assistant Professor of Mechanical Engineering at the University of Maryland, College Park, Maryland. His areas of interest include vibration and acoustics control, nonlinear dynamics, structural dynamics, and system identification.

Table of Contents

Preface	xiii
Introduction	1	(34)
Discrete-Time Systems	2	(4)
Continuous-Time Systems	6	(9)
Nonautonomous Systems	6	(5)
Autonomous Systems	11	(2)
Phase Portraits and Flows	13	(2)
Attracting Sets	15	(5)
Concepts of Stability	20	(9)
Lyapunov Stability	20	(3)
Asymptotic Stability	23	(2)
Poincare Stability	25	(2)
Lagrange Stability (Bounded Stability)	27	(1)
Stability Through Lyapunov Function	27	(2)
Attractors	29	(2)
Comments	31	(1)
Exercises	31	(4)
Equilibrium Solutions	35	(112)
Continuous-Time Systems	35	(26)
Linearization Near an Equilibrium Solution	36	(3)
Classification and Stability of Equilibrium Solutions	39	(8)
Eigenspaces and Invariant Manifolds	47	(11)
Analytical Construction of Stable and Unstable Manifolds	58	(3)
Fixed Points of Maps	61	(7)
Bifurcations of Continuous Systems	68	(53)
Local Bifurcations of Fixed Points	70	(11)
Normal Forms for Bifurcations	81	(2)
Bifurcation Diagrams and Sets	83	(13)
Center Manifold Reduction	96	(12)
The Lyapunov-Schmidt Method	108	(1)
The Method of Multiple Scales	108	(7)
Structural Stability	115	(1)
Stability of Bifurcations to Perturbations	116	(3)
Codimension of a Bifurcation	119	(2)
Global Bifurcations	121	(1)
Bifurcations of Maps	121	(7)
Exercises	128	(19)
Periodic Solutions	147	(84)
Periodic Solutions	147	(11)
Autonomous Systems	148	(8)
Nonautonomous Systems	156	(2)
Comments	158	(1)
Floquet Theory	158	(14)
Autonomous Systems	159	(10)
Nonautonomous Systems	169	(2)
Comments on the Monodromy Matrix	171	(1)
Manifolds of a Periodic Solution	172	(1)
Poincare Maps	172	(15)
Nonautonomous Systems	176	(5)
Autonomous Systems	181	(6)
Bifurcations	187	(21)
Symmetry--Breaking Bifurcation	189	(6)
Cyclic--Fold Bifurcation	195	(5)
Period--Doubling or Flip Bifurcation	200	(4)
Transcritical Bifurcation	204	(1)
Secondary Hopf or Neimark Bifurcation	205	(3)
Analytical Constructions	208	(11)
Method of Multiple Scales	209	(3)
Center Manifold Reduction	212	(5)
General Case	217	(2)
Exercises	219	(12)
Quasiperiodic Solutions	231	(46)
Poincare Maps	233	(9)
Winding Time and Rotation Number	238	(2)
Second-Order Poincare Map	240	(1)
Comments	241	(1)
Circle Map	242	(6)
Constructions	248	(6)
Method of Multiple Scales	249	(2)
Spectral Balance Method	251	(2)
Poincare Map Method	253	(1)
Stability	254	(1)
Synchronization	255	(14)
Exercises	269	(8)
Chaos	277	(146)
Maps	278	(10)
Continuous-Time Systems	288	(7)
Period-Doubling Scenario	295	(1)
Intermittency Mechanisms	296	(18)
Type I Intermittency	300	(5)
Type III Intermittency	305	(6)
Type II Intermittency	311	(3)
Quasiperiodic Routes	314	(20)
Ruelle-Takens Scenario	315	(2)
Torus Breakdown	317	(14)
Torus Doubling	331	(3)
Crises	334	(22)
Melnikov Theory	356	(34)
Homoclinic Tangles	356	(3)
Heteroclinic Tangles	359	(4)
Numerical Prediction of Manifold Intersections	363	(3)
Analytical Prediction of Manifold Intersections	366	(8)
Application of Melnikov's Method	374	(16)
Comments	390	(1)
Bifurcations of Homoclinic Orbits	390	(20)
Planar Systems	391	(6)
Orbits Homoclinic to a Saddle	397	(5)
Orbits Homoclinic to a Saddle Focus	402	(5)
Comments	407	(3)
Exercises	410	(13)
Numerical Methods	423	(38)
Continuation of Fixed Points	423	(13)
Sequential Continuation	425	(3)
Davidenko-Newton-Raphson Continuation	428	(1)
Arclength Continuation	428	(4)
Pseudo-Arclength Continuation	432	(3)
Comments	435	(1)
Simple Turning and Branch Points	436	(2)
Hopf Bifurcation Points	438	(3)
Homotopy Algorithms	441	(4)
Construction of Periodic Solutions	445	(10)
Finite-Difference Method	446	(3)
Shooting Method	449	(6)
Poincare Map Method	455	(1)
Continuation of Periodic Solutions	455	(6)
Sequential Continuation	456	(1)
Arclength Continuation	456	(2)
Pseudo-Arclength Continuation	458	(2)
Comments	460	(1)
Tools to Analyze Motions	461	(102)
Introduction	462	(3)
Time Histories	465	(7)
State Space	472	(6)
Pseudo-State Space	478	(24)
Choosing the Embedding Dimension	483	(12)
Choosing the Time Delay	495	(5)
Two or More Measured Signals	500	(2)
Fourier Spectra	502	(12)
Poincare Sections and Maps	514	(6)
Systems of Equations	514	(2)
Experiments	516	(3)
Higher-Order Poincare Sections	519	(1)
Comments	519	(1)
Autocorrelation Functions	520	(5)
Lyapunov Exponents	525	(13)
Concept of Lyapunov Exponents	525	(4)
Autonomous Systems	529	(2)
Maps	531	(3)
Reconstructed Space	534	(3)
Comments	537	(1)
Dimension Calculations	538	(12)
Capacity Dimension	538	(3)
Pointwise Dimension	541	(4)
Information Dimension	545	(2)
Correlation Dimension	547	(1)
Generalized Correlation Dimension	548	(1)
Lyapunov Dimension	549	(1)
Comments	549	(1)
Higher-Order Spectra	550	(7)
Exercises	557	(6)
Control	563	(26)
Control of Bifurcations	563	(8)
Static Feedback Control	564	(4)
Dynamic Feedback Control	568	(3)
Comments	571	(1)
Chaos Control	571	(13)
The OGY Scheme	572	(5)
Implementation of the OGY Scheme	577	(3)
Pole Placement Technique	580	(2)
Traditional Control Methods	582	(2)
Synchronization	584	(5)
Bibliography	589	(74)
Subject Index	663