Iutam Symposium on Nonlinear Stochastic Dynamics

by Iutam Symposium on Nonlinear Stochastic Dynamics; Namachchivaya, N. Sri; Lin, Y. K.

• ISBN13: 9781402014710
• ISBN10: 1402014716
• Format: Hardcover
• Copyright: 2003-12-01
• Publisher: Kluwer Academic Pub

Summary

Non-linear stochastic systems are at the center of many engineering disciplines and progress in theoretical research had led to a better understanding of non-linear phenomena. This book provides information on new fundamental results and their applications which are beginning to appear across the entire spectrum of mechanics. The outstanding points of these proceedings are Coherent compendium of the current state of modelling and analysis of non-linear stochastic systems from engineering, applied mathematics and physics point of view. Subject areas include: Multiscale phenomena, stability and bifurcations, control and estimation, computational methods and modelling. For the Engineering and Physics communities, this book will provide first-hand information on recent mathematical developments. The applied mathematics community will benefit from the modelling and information on various possible applications.

Table of Contents

Preface	xv
Part I Multi-scale Phenomena
Nonlinear Diffusion Approximation of Slow Motion	5	(14)
Ludwig Arnold
Introduction	5	(2)
Averaging	7	(1)
Linear Diffusion Approximation	8	(3)
Nonlinear Diffusion Approximation	11	(1)
Toy Examples	12	(2)
The Lorenz-Maas Atmosphere-Ocean Model	14	(5)
Appendix: Sketch of the Proof of Theorem 4.3	15	(4)
Experiments on Large Fluctuations	19	(10)
A. Bandrivskyy
S. Beri
D. G. Luchinsky
P. V. E. McClintock
Introduction	19	(1)
Basis of the Theory	20	(1)
Basis of the Experiments	21	(1)
Finite Noise Intensity	22	(3)
Fast Monte Carlo Technique	25	(1)
Conclusion	26	(3)
Stochastic Averaging	29	(14)
P. Bernard
Introduction	30	(1)
Averaging Method for ODE's	30	(3)
Averaging result	31	(1)
Example	31	(1)
Amplitude-angle variables	32	(1)
Example	33	(1)
Stochastic Averaging	33	(2)
Limit theorems	34	(1)
Averaging a Stochastic Nonlinear Oscillator	35	(1)
Application to Approximations of the Law of the Maximum of the Displacement of a Nonlinear Oscillator	35	(4)
Some results about one-dimensional diffusions	36	(1)
Definitions	36	(1)
Properties	36	(1)
Approximation of the probability distribution of the maximum of a one-dimensional diffusion	37	(1)
Approximating the law of the maximum of the stationary displacement of a stochastic nonlinear oscillator	38	(1)
Conclusion	39	(4)
Noise Sensitivity of Stochastic Resonance	43	(14)
Mark Freidlin
Introduction	43	(2)
Action Functional for the Process Xtδε	45	(3)
Wavefront Propagation Problem	48	(4)
Exit Problem and Stochastic Resonance	52	(5)
Stochastic Resonance	57	(14)
Peter Imkeller
Ilya Pavlyukevich
Background and Paradigm	58	(1)
Freidlin's Approach	59	(2)
Spectral Power Amplification	61	(2)
The Spectral Gap	63	(1)
Asymptotics of the SPA Coefficient	64	(2)
The `Effective Dynamics': Two-state Markov Chains	66	(5)
Metastability of Diffusion Processes	71	(12)
C. Schutte
W. Huisinga
S. Meyn
Introduction	71	(1)
V-uniform Ergodicity and Spectral Gaps	72	(4)
Diffusion semigroups	73	(2)
Non-probabilistic semigroups	75	(1)
Metastability and Exit Rates	76	(7)
Exit rates	76	(1)
The twisted process	77	(2)
Consequences for exit times	79	(4)
Stochastic Averaging Near Homoclinic Orbits via Singular Perturbations	83	(14)
Richard B. Sowers
Introduction	83	(2)
Problem Formulation	85	(12)
Part II Stability and Bifurcations
Stochastic Internal Resonance	97	(14)
S. T. Ariaratnam
N. M. Abdelrahman
Introduction	97	(1)
Conservative Systems	98	(6)
Double Fourier Expansion	101	(1)
Calculation of Lyapunov Exponent	102	(2)
Non-conservative Systems	104	(4)
Conclusions	108	(3)
Dynamics of Globally Coupled Noisy Excitable Elements	111	(14)
J. A. Acebron
A. R. Bulsara
W.-J. Rappel
Introduction	112	(1)
Model Equations	112	(4)
Bifurcation Analysis	116	(7)
Conclusion	123	(2)
Stochastic Duffing-van der Pol Oscillator	125	(12)
Peter H. Baxendale
Introduction	125	(3)
Behavior Near 0	128	(1)
Khasminskii-Carverhill Formula	128	(1)
The Duffing Case B = 0, β < 0	129	(1)
The Case A > 0, B > 0	130	(7)
Numerical calculations for λave	133	(4)
Slow Sweep Through a Period-Doubling Cascade	137	(10)
Huw G. Davies
Philotas Kyriakidis
Introduction	138	(1)
A Coupled-modes Model	139	(2)
A Period-doubling of a Map	141	(3)
Conclusion	144	(3)
Multi-Scale Analysis of Noise-Sensitivity Near a Bifurcation	147	(10)
R. Kuske
Introduction	147	(1)
Multi-scale analysis	148	(5)
Additive noise: β < 0	149	(1)
Multiplicative noise: β < 0	150	(3)
Supercritical Case: β > 0	153	(1)
Conclusion	154	(3)
Appendix: Details of the Multi-scale Analysis	154	(3)
Hopf Meets Hamilton Under Whitney's Umbrella	157	(10)
William F. Langford
Introduction	157	(1)
Classical Hopf Bifurcation Theorem	158	(1)
Hamiltonian-Hopf Bifurcation	159	(2)
Whitney's Umbrella	161	(2)
Near-Hamiltonian Hopf Bifurcation	163	(1)
Conclusion	164	(3)
Stochastic Stability of Two Coupled Oscillators in Resonance	167	(12)
N. Sri Namachchivaya
N. Ramakrishnan
H. J. Van Roessel
L. Vedula
Introduction	168	(1)
Problem Formulation	169	(1)
Moment Lyapunov Exponents	170	(3)
The Eigenvalue Problem	173	(1)
Numerical Results	174	(5)
Appendix: Drift and Diffusion Coefficients	176	(3)
On Stabilizing the Double Oscillator by Mean Zero Noise	179	(12)
Volker Wihstutz
Introduction	179	(1)
Stabilizing the Inverted Pendulum	180	(1)
Admissible Noise	181	(2)
A Stabilization Result	183	(1)
Proofs	184	(7)
Proof of the lemma	184	(3)
Proof of the lemma	187	(4)
Stability of a Two-Dimensional System	191	(14)
Wei-Chau Xie
Introduction	191	(2)
Formulation	193	(1)
Moment Lyapunov Exponents	194	(6)
Zeroth-order perturbation	195	(1)
First-order perturbation	196	(1)
Second-order perturbation	196	(1)
Determination of Λ2	197	(1)
Determination of λ2	198	(2)
Conclusion	200	(5)
Part III Control and Estimation
The Distributed Nonlinear Stochastic World of Networks	205	(12)
P. R. Kumar
Introduction	205	(1)
Some Questions of Perhaps Permanent Interest	206	(2)
A Single Queue	208	(1)
Queueing Networks	209	(2)
Computational Proofs of Stability for Queueing Networks	211	(2)
Determining the Capacity of a Wireless Network	213	(2)
Conclusion	215	(2)
Noise effects in Nonlinear System Identification	217	(10)
Patricia Davies
Rong Deng
Timothy Doughty
Anil K. Bajaj
Introduction	217	(1)
System Models	218	(2)
Asymptotic Bias Analysis	220	(1)
Results and Discussion	221	(3)
Conclusion	224	(3)
Structural Vibration Mitigation using Dissipative Smart Damping Devices	227	(10)
Erik A. Johnson
Baris Erkus
Introduction	227	(3)
EVP Representation of an LQR Problem	230	(1)
Definitions	230	(1)
Dissipativity constraint	231	(1)
A Numerical Example	231	(4)
9-story benchmark and EVP verification	233	(1)
The effects of the dissipativity constraint	233	(2)
Conclusion	235	(2)
Parameter Estimation for Stochastic Systems	237	(10)
Rafail Z. Khasminskii
Linear Systems	237	(2)
Nonlinear Systems	239	(3)
Examples	242	(2)
Conclusion	244	(3)
Control Against Large Deviation for Oscillatory Systems	247	(10)
Agnessa Kovaleva
Introduction	247	(3)
Main Equations and Performance Criteria	250	(2)
Controlled Synchronization in the Presence of Noise	252	(2)
Conclusion	254	(3)
Parameter Analysis	257	(12)
F. Ma
H. Zhang
A. Bockstedte
G. C. Foliente
P. Paevere
Introduction	258	(1)
Differential Hysteresis	259	(1)
Elimination of Redundant Parameter	260	(2)
Parameter Sensitivity Analysis	262	(4)
Local sensitivity analysis	262	(1)
Global sensitivity analysis	263	(3)
Conclusion	266	(3)
Stochastic Parameter Estimation of Non-Linear Systems	269	(10)
Marcello Vasta
Introduction	269	(1)
The Estimation Problem	270	(1)
The Spectral Method	271	(2)
Energy Based Stochastic Estimation Method	273	(2)
Identification to Spectral Representation Input Processes	275	(2)
Conclusion	277	(2)
Lyapunov Exponent and Stability of Controlled Systems	279	(12)
W. Q. Zhu
Z. L. Huang
Introduction	279	(1)
Stochastic Averaging	280	(1)
Ergodic Control	281	(1)
The Largest Lyapunov Exponent	282	(2)
Example	284	(3)
Conclusion	287	(4)
Part IV Modeling
From Earth's Ice Ages to Human Sensory Systems	291	(8)
Kurt Wiesenfeld
Fernan Jaramillo
Introduction	291	(1)
Stochastic Resonance in Physics	292	(3)
Brownian motion and hair bundles	295	(4)
Response Spectral Densities	299	(8)
G. Q. Cai
Introduction	299	(1)
Analysis	300	(2)
Example	302	(3)
Conclusion	305	(2)
Stochastic Lotka-Volterra Systems	307	(12)
Mikhail F. Dimentberg
Stochastic L-V Model in Population Dynamics and its Analysis	307	(2)
Intermittency in Population Sizes and its Analytical Description	309	(5)
Expected Time for Quasiextinction: First-Passage Problem	314	(5)
Dynamic Systems Driven By Poisson/Levy White Noise	319	(12)
Mircea Grigoriu
Gennady Samorodnitsky
Introduction	319	(1)
Why Non-Gaussian White Noise?	320	(1)
Linear Systems	321	(1)
Nonlinear Systems	322	(7)
Multiplicative noise	322	(4)
Additive noise	326	(3)
Conclusion	329	(2)
Stochastic Dynamics of Friction-Induced Vibration in Disc Brakes	331	(12)
S. Qiao
R. A. Ibrahim
Stochasticity of Friction	331	(1)
Analytical Modeling	332	(9)
Dynamics under Constant Friction Coefficient	338	(1)
Dynamics under Random Friction Coefficient	339	(2)
Conclusion	341	(2)
Multiplicative Random Impulse Process	343	(10)
Radoslaw Iwankiewicz
Introduction	343	(2)
Dynamical Systems With Multiplicative Random Impulse Noise: Statement of the Problem	345	(3)
Ito's Stochastic Differential Equation for Dynamical Systems with Multiplicative Random Impulse Noise	348	(3)
General procedure	348	(1)
Dynamical systems with linear term b(t, Y(t))	349	(1)
Dynamical systems with quadratic term b(t, Y (t))	350	(1)
Conclusion	351	(2)
Multiscale Impact Models: Multibody Dynamics and Wave Propagation	353	(10)
W. Schiehlen
R. Seifried
Introduction	353	(1)
Multibody Systems with Impact	354	(1)
Elastodynamic Contact Using Wave Propagation	355	(1)
Contact Simulation Using Finite Elements	356	(1)
Experimental Validation	356	(2)
Computation of the Coefficient of Restitution	358	(2)
Signal Analysis	360	(1)
Conclusion	361	(2)
Nonstationary Response of Nonlinear Systems	363	(12)
Andrew W. Smyth
Sami F. Masri
Overview of Nonstationary Excitation Compact Representation	364	(1)
Statistical Linearization and Nonstationary Excitation	364	(1)
Example Application	365	(5)
Wide-banded real world excitation data example	367	(1)
Narrow-banded excitation	368	(1)
Response estimation with a highly non-Gaussian excitation process	369	(1)
Conclusion	370	(5)
Hilbert Spectral Description and Simulation	375	(12)
Y. K. Wen
Ping Gu
Introduction	375	(2)
Hilbert-Huang Transform (HHT) and Empirical Mode Decomposition (EMD)	377	(2)
Hilbert Spectral Description and Representation of a Nonstationary process	379	(3)
Simulation of Nonstationary Process Based on Sample Observation	382	(2)
Conclusion	384	(3)
Part V Computational Methods
Random Vibrations of Riding Cars with Bilinear Damping	387	(14)
Walter V. Wedig
Introduction into the Problem	387	(3)
Asymptotic Solutions of the FP-Equation	390	(3)
Expansions by Orthogonal Polynomials	393	(1)
Riding Cars Excited by Colored Noise	394	(4)
Conclusion	398	(3)
Solutions of the First-Passage Problem by Importance Sampling	401	(14)
Christian Bucher
Michael Macke
Introduction	401	(1)
Importance Sampling Method	402	(3)
Linear Oscillator under Parametric Excitation	405	(2)
Oscillator with Nonlinear Damping	407	(2)
Hysteretic Oscillator under Non-Gaussian Excitation	409	(2)
Conclusion	411	(4)
Passive Fields and Particles	415	(10)
Bruno Eckhardt
Erwan Hascoet
Wolfgang Braun
Introduction	415	(1)
Relaxation in Chaotic Advection	416	(3)
Polymer Stretching in Chaotic Flows	419	(6)
Sample Point Uniformity Measures	425	(10)
Erik A. Johnson
Steven F. Wojtkiewicz
Lawrence A. Bergman
Introduction	426	(1)
Uniform Sample Distributions	427	(1)
Uniformity in stochastic dynamics	427	(1)
Uniformity in response surface methodologies	428	(1)
Quantitative Measure of Uniformity	428	(3)
Discrepancy: A measure of uniformity	428	(2)
Discrepancy sensitivity: Contribution to uniformity	430	(1)
Numerical Examples	431	(1)
Numerical example in uncertainty	431	(1)
Discrepancy sensitivity in response surfaces	432	(1)
Conclusion	432	(3)
Kinks in a Stochastic PDE	435	(10)
Grant Lythe
Salman Habib
Numerical solution	436	(2)
Counting kinks	438	(1)
The Steady State Density of Kinks	439	(6)
Path Integration as a Tool for Investigating Chaotic Behaviour	445	(10)
Arvid Naess
Christian Skaug
Introduction	445	(1)
The Nonlinear Oscillators	446	(1)
The Stochastic Differential Equation and its Solution	447	(2)
Numerical Examples	449	(3)
Conclusion	452	(3)
Time-Varying Cardiovascular Oscillations	455	(10)
V. N. Smelyanskiy
D. A. Timucin
D. G. Luchinsky
A. Stefanovska
A. Bandrivskyy
P. V. E. McClintock
Introduction	456	(1)
The Problem	457	(1)
Mode Decomposition	458	(1)
Basic Ideas of the Modelling	458	(5)
Inference	463	(1)
Conclusion	463	(2)
Index	465