william@ronzoni.berkeley.edu (W. E. Grosso) (03/07/91)
I'm doing some research in dynamical systems for a paper and presentation. The question : does anybody out there know about intermittency in dynamical systems (from what I gather, it seems to be associated with numerical experimentation). Thanks, William Grosso (william@math.berkeley.edu)
scavo@cie.uoregon.edu (Tom Scavo) (03/08/91)
In article <1991Mar6.232931.2553@agate.berkeley.edu> william@ronzoni.berkeley.edu (W. E. Grosso) writes: >I'm doing some research in dynamical systems for a paper >and presentation. The question : does anybody out there >know about intermittency in dynamical systems >(from what I gather, it seems to be associated with >numerical experimentation). Consider E : R --> R with E(x) = \lambda exp(x) , for example. This map experiences a saddle-node bifurcation at \lambda = 1/e. Now, choose \lambda ever so slightly greater than 1/e and compute the orbit of 0 , say. This orbit will eventually "escape" to infinity, but only after many iterations; E^n(0) is said to be an _intermittent orbit_ for this value of \lambda. See section 6.8.1 in Guckenheimer & Holmes, _Nonlinear Oscil- lations, Dynamical Systems, and Bifurcations of Vector Fields_, Springer-Verlag, 1983. See also Devaney's chapter in _The Science of Fractal Images_, Springer-Verlag, 1988 (Peitgen & Saupe, editors) on pp.163-167. Tom Scavo scavo@cie.uoregon.edu
guy@physics (Guy Metcalfe) (03/08/91)
In article <1991Mar6.232931.2553@agate.berkeley.edu> william@ronzoni.berkeley.edu (W. E. Grosso) writes: >and presentation. The question : does anybody out there >know about intermittency in dynamical systems >(from what I gather, it seems to be associated with >numerical experimentation). There are indeed numerical experiments using 1- and 2-dimensional maps that exhibit intermittent behavior, but by no means is the phenomena so limited: it is observed in certain systems of differential equations, and in the lab. I am most familiar with examples from convection experiments, and, while these are the most well explored, there are other example systems. There are in fact 3 types of `classical' intermittency that are distinguished by how eigenvalues leave the unit circle in the complex plane. These were first discussed in \bibitem{Poman} Y. Pomeau and P. Manneville, Commun. Math. Phys. {\bf 74}, 189 (1980). and examples observed in convective systems by, among others, \bibitem{intermittency_folk} P. Berg\'e, M. Dubois, P. Manneville, and Y. Pomeau, J. Phys. (Paris) Lett. {\bf 41}, L341 (1980); M. Dubois, M. A. Rubio, and P. Berg\'e, Phys. Rev. Lett. {\bf 51}, 1446 (1983); H. Haucke, R. E. Ecke, Y. Maeno, and J. C. Wheatley, Phys. Rev. Lett. {\bf 53}, 2090 (1984). The recent book by Manneville ("Dissapative Structures and Weak Turbulence" Academic Press, 1990?) has good discussion and more references. Lately, other types of intermittency associated with chaotic dynamics have been explored. I'm thinking of the work by Grebogi, Ott, Yorke and coworkers at Maryland. Some references are \bibitem{goy} Celso Grebogi, Edward Ott, Filipe Romeiras and James A. Yorke, Phys. Rev. A {\bf 36}, 5365 (1987); Celso Grebogi, Edward Ott and James A. Yorke, Physica {\bf 7D}, 181 (1983); Phys. Rev. Lett. {\bf 48}, 1507 (1982). This intermittency occurs as (chaotic) attractors collide with manifolds of other objects in phase space, and then are modified/destroyed by the interaction. Read the references to find out what really is thought to happen:-). Some lab examples are \bibitem{experiment_eg} M. Iansiti, Qing Hu, R. M. Westervelt, and M. Tinkham, Phys. Rev. Lett. {\bf 55}, 746 (1985); Didier Dangoisse, Pierre Glorieux, and Daniel Hennequin, Phys. Rev. Lett. {\bf 57}, 2657 (1986); T. L. Carroll, L. M. Pecora, and F. J. Rachford, Phys. Rev. Lett. {\bf 59}, 2891 (1987); W. L. Ditto, S. Rauseo, R. Cawley, C. Grebogi, G.-H. Hsu, E. Kostelich, E. Ott, H. T. Savage, R. Segnan, M. L. Spano, and J. A. Yorke, Phys. Rev. Lett. {\bf 63}, 923 (1989). And as you might guess from the bibliography entries, I plan to add to the list of experimental examples shortly. Hope your talk goes well and this info is helpful. -- Guy Metcalfe Duke University Dept. of Physics guy@phy.duke.edu & Center for Nonlinear Studies guy@physics.phy.duke.edu Durham, N.C. 27706 guy%phy.duke.edu@cs.duke.edu