Nonlinear Dynamics and Chaos: Where do we go from here?

by Hogan; J

• ISBN13: 9780750308625
• ISBN10: 0750308621
• Format: Hardcover
• Copyright: 2002-08-01
• Publisher: CRC Press

Summary

Nonlinear dynamics has been successful in explaining complicated phenomena in well-defined low-dimensional systems. Now it is time to focus on real-life problems that are high-dimensional or ill-defined, for example, due to delay, spatial extent, stochasticity, or the limited nature of available data. How can one understand the dynamics of such systems? Written by international experts, Nonlinear Dynamics and Chaos: Where Do We Go from Here? assesses what the future holds for dynamics and chaos. The chapters address one or more of the broad and interconnected main themes: neural and biological systems, spatially extended systems, and experimentation in the physical sciences. The contributors offer suggestions as to what they see as the way forward, often in the form of open questions for future research.

Table of Contents

Preface	ix
Bifurcation and degenerate decomposition in multiple time scale dynamical systems	1	(20)
John Guckenheimer
Definitions and background	4	(4)
Slow--fast decompositions	8	(2)
Degenerate decomposition and bifurcation	10	(3)
The forced Van der Pol equation	13	(8)
Acknowledgment	18	(1)
References	19	(2)
Many-body quantum mechanics	21	(34)
Robert S MacKay
Signs of nonlinearity in quantum mechanics	24	(3)
Discrete breathers	27	(6)
Experimental evidence for quantum discrete breathers	33	(2)
Towards a mathematical theory of quantum discrete breathers	35	(4)
Obstructions to the theory	39	(2)
A proposed solution	41	(6)
A tentative application to 4-methyl pyridine	47	(4)
Conclusion	51	(4)
Acknowledgments	52	(1)
References	52	(3)
Unfolding complexity: hereditary dynamical systems---new bifurcation schemes and high dimensional chaos	55	(18)
Uwe an der Heiden
Hereditary systems	55	(5)
Difference equations with continuous argument: idealized turbulence	60	(1)
First-order difference-differential equations: a singular perturbation problem with bifurcation gaps	61	(3)
Prime number dynamics of a retarded difference equation	64	(3)
Second-order non-smooth difference-differential equations	67	(3)
Outlook	70	(3)
Acknowledgment	70	(1)
References	70	(3)
Creating stability out of instability	73	(18)
Christopher K R T Jones
Nonlinear optical fibres	74	(10)
Can two unstable waves make a stable pulse?	84	(4)
Some predictions, suggestions and questions	88	(3)
References	89	(2)
Signal or noise? A nonlinear dynamics approach to spatio-temporal communication	91	(26)
Gregory D Van Wiggeren
Jordi Garcia-Ojalvo
Rajarshi Roy
Communication with dynamically fluctuating states of light polarization	94	(11)
Spatio-temporal communication with synchronized optical chaos	105	(7)
Conclusions	112	(5)
Acknowledgments	113	(1)
References	113	(4)
Outstanding problems in the theory of pattern formation	117	(50)
Edgar Knobloch
Pattern selection on lattices	128	(4)
Imperfection sensitivity	132	(6)
Coupled Ginzburg--Landau equations	138	(3)
The nearly-inviscid Faraday system	141	(8)
Nonlinear waves in extended systems with broken reflection symmetry	149	(9)
Summary and conclusions	158	(9)
Acknowledgments	159	(1)
References	159	(8)
Is chaos relevant to fluid mechanics?	167	(20)
Tom Mullin
Taylor--Couette flow	167	(3)
Preliminary observations	170	(3)
Symmetry considerations	173	(1)
Codimension-two bifurcations	174	(6)
Imperfect gluing bifurcations	180	(2)
Conclusion	182	(5)
Acknowledgments	183	(1)
References	184	(3)
Time-reversed acoustics and chaos	187	(22)
Mathias Fink
Time-reversal mirrors	188	(4)
Time-reversal experiments	192	(6)
Time reversal in chaotic cavities	198	(8)
Conclusion	206	(3)
Acknowledgments	206	(1)
References	206	(3)
Reduction methods applied to non-locally coupled oscillator systems	209	(20)
Yoshiki Kuramoto
When coupling non-locality becomes crucial	211	(2)
Rotating spiral waves without phase singularity	213	(5)
Simpler case: coexistence of coherent and incoherent domains	218	(6)
Concluding remarks	224	(5)
References	225	(4)
A prime number of prime questions about vortex dynamics in nonlinear media	229	(40)
Art Winfree
Stable organizing centres	230	(18)
Persistent organizing centres	248	(3)
Igniting resistance to curvature-induced contraction	251	(7)
Dynamic C: synchronization	258	(2)
Ball-park estimation of pertinent quantities	260	(2)
Passing in silence from 1994 to 2002	262	(7)
Acknowledgments	264	(1)
Addendum while in press	264	(1)
References	264	(5)
Spontaneous pattern formation in primary visual cortex	269	(52)
Paul C Bressloff
Jack D Cowan
The Turing mechanism and its role in cooperative cortical dynamics	274	(7)
A continuum model of V1 and its intrinsic circuitry	281	(4)
Orientation tuning and O(2) symmetry	285	(3)
Amplitude equation for interacting hypercolumns	288	(4)
Cortical pattern formation and E(2) symmetry	292	(15)
Spatial frequency tuning and SO(3) symmetry	307	(9)
Future directions	316	(5)
References	317	(4)
Models for pattern formation in development	321	(28)
Bard Ermentrout
Remus Osan
Competition	326	(4)
Spatial scales	330	(6)
Orientation maps and feature maps	336	(5)
Kohonen maps and abstract feature models	341	(4)
Conclusions	345	(4)
References	346	(3)
Spatio-temporal nonlinear dynamics: a new beginning	349	(8)
William L Ditto
Fibrillation of the heart	350	(3)
Neuro computing	353	(2)
Where do we go from here?	355	(2)
Acknowledgments	355	(1)
References	355	(2)
Author index	357