Hypermeander of spirals: Local bifurcations and statistical properties

Physica D: Nonlinear Phenomena

Volumn 156, Issue 3-4, 2001, Pages 364-382

  Ashwin, Peter ^a   Melbourne, Ian ^b   Nicol, Matthew ^c  

^a UNIVERSITY OF EXETER   (United Kingdom)

^b University of Houston   (United States)

^c UNIVERSITY OF SURREY   (United Kingdom)

Author keywords

Bifurcation; Center bundle; Deterministic Brownian motion; Hypermeander; Invariance principle; Spiral wave

Indexed keywords

EID: 0035882632     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0167-2789(01)00296-2     Document Type: Article

References (47)

1. Ashwin P., Melbourne I. Noncompact drift for relative equilibria and relative periodic orbits. Nonlinearity. 10:1997;595-616.
2. Ashwin P., Melbourne I., Nicol M. Drift bifurcations of relative equilibria and transitions of spiral waves. Nonlinearity. 12:1999;741-755.
3. P. Ashwin, I. Melbourne, M. Nicol, Drift for Euclidean extensions of dynamical systems, in: B. Fiedler, et al. (Eds.), Proceedings of the International Conference on Differential Equations, Equadiff'99, World Scientific, Singapore, 2000, pp. 145-150.
4. Barkley D. Euclidean symmetry and the dynamics of rotating spiral waves. Phys. Rev. Lett. 72:1994;164-167.
5. Barkley D., Kness M., Tuckerman S. Spiral wave dynamics in a simple model of excitable media: the transition from simple to compound rotation. Phys. Rev. A. 42:1990;2489-2492.
6. Biktashev V.N., Holden A., Nikolaev E.V. Spiral wave meander and symmetry of the plane. Int. J. Bifurc. Chaos. 6:1996;2433-2440.
7. Biktashev V.N., Holden A. Deterministic Brownian motion in the hypermeander of spiral waves. Physica D. 116:1998;342-354.
8. P. Billingsley, Convergence of Probability Measures, Wiley, New York, 1968.
9. R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer, Berlin, 1975.
10. P. Coullet, K. Emilsson, Chaotically induced defect diffusion, in: Instabilities and Nonequilibrium Structures, Kluwer Academic Publishers, Dordrecht, 1996, pp. 55-62.
11. Denker M., Philipp W. Approximation by Brownian motion for Gibbs measures and flows under a function. Ergod. Theory Dyn. Syst. 4:1984;541-552.
12. D. Dolgopyat, On mixing properties of compact group extensions of hyperbolic systems, Preprint.
13. Dolgopyat D. Prevalence of rapid mixing in hyperbolic flows. Ergod. Theory Dyn. Syst. 18:1998;1097-1114.
14. Dolgopyat D. Prevalence of rapid mixing. II. Topological prevalence. Ergod. Theory Dyn. Syst. 20:2000;1045-1059.
15. Fiedler B., Sandstede B., Scheel A., Wulff C. Bifurcation from relative equilibria to non-compact group actions: skew products, meanders, and drifts. Doc. Math. J. DMV. 1:1996;479-505.
16. Fiedler B., Turaev D.V. Normal forms, resonances, and meandering tip motions near relative equilibria of Euclidean group actions. Arch. Rat. Mech. Anal. 145:1998;129-159.
17. M.J. Field, I. Melbourne, A. Török, Decay of correlations, central limit theorems and approximation by Brownian motion for compact Lie group extensions, University of Houston Research Report UH/MD-268, submitted for publication.
18. Field M.J., Parry W. Stable ergodicity of skew extensions by compact Lie groups. Topology. 38:1999;167-187.
19. Golubitsky M., LeBlanc V.G., Melbourne I. Meandering of the spiral tip - an alternative approach. J. Nonlinear Sci. 7:1997;557-586.
20. Golubitsky M., LeBlanc V.G., Melbourne I. Hopf bifurcation from rotating waves and patterns in physical space. J. Nonlinear Sci. 10:2000;69-101.
21. J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York. Appl. Math. Sci. 42 (1990).
22. J. Guckenheimer, M.R. Myers, F.J. Wicklin, P.A. Worfolk, Dstool: A Dynamical Systems Toolkit with an Interactive Graphical Interface, User's Manual, Center for Applied Mathematics, Cornell University, Ithaca, NY, 1991.
23. Jahnke W., Skaggs W.E., Winfree A.T. Chemical vortex dynamics in the Belousov-Zhabotinskii reaction and in the two-variable Oregonator model. J. Chem. Phys. 93:1989;740-749.
24. Jahnke W., Winfree A.T. A survey of spiral-wave behaviour in the Oregonator model. Int. J. Bifurc. Chaos. 1:1991;445-466.
25. Karma A. Meandering transition in two-dimensional excitable media. Phys. Rev. Lett. 65:1990;2824-2827.
26. J.S.W. Lamb, I. Melbourne, C. Wulff, in preparation.
27. Li G., Ouyang Q., Petrov V., Swinney H.L. Transition from simple rotating chemical spirals to meandering and traveling spirals. Phys. Rev. Lett. 77:1996;2105-2108.
28. Lugosi E. Analysis of meandering in Zykov kinetics. Physica D. 40:1989;331-337.
29. MacKay R.S., Tresser C. Transition to topological chaos for circle maps. Physica D. 19:1986;206-237.
30. Mikhailov A.S., Zykov V.S. Kinematical theory of spiral waves in excitable media: comparison with numerical simulations. Physica D. 52:1991;379-397.
31. Nicol M., Melbourne I., Ashwin P. Euclidean extensions of dynamical systems. Nonlinearity. 14:2001;275-300.
32. Parry W., Pollicott M. Stability of mixing for toral extensions of hyperbolic systems. Proc. Steklov Inst. 216:1997;354-363.
33. W. Parry, M. Pollicott, Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics, Astérique, Vols. 187 and 188, Société Mathématique de France, Montrouge, 1990.
34. W. Philipp, W.F. Stout, Almost Sure Invariance Principles for Partial Sums of Weakly Dependent Random Variables, Memoirs of the American Mathematical Society, Vol. 161, American Mathematical Society, Providence, RI, 1975.
35. Plesser T., Müller S.C., Hess B. Spiral wave dynamics as a function of proton concentration in the ferroin-catalyzed Belousov-Zhabotinskii reaction. J. Phys. Chem. 94:1990;7501-7507.
36. Pollicott M. On the rate of mixing of Axiom A flows. Invent. Math. 81:1985;413-426.
37. Rössler O.E., Kahlert C. Winfree meandering in a two-dimensional two-variable excitable medium. Z. Naturforsch. 34:1979;565-570.
38. D. Ruelle, Thermodynamic Formalism, Encyclopedia of Mathematics and its Applications, Vol. 5, Addison-Wesley, Reading, MA, 1978.
39. Ruelle D. Flows which do not exponentially mix. C.R. Acad. Sci. Paris. 296:1983;191-194.
40. Sandstede B., Scheel A., Wulff C. Center-manifold reduction for spiral waves. C.R. Acad. Sci. Sér. I. 324:1997;153-158.
41. Sandstede B., Scheel A., Wulff C. Dynamics of spiral waves on unbounded domains using center-manifold reductions. J. Diff. Eqs. 141:1997;122-149.
42. Scheel A. Bifurcation to spiral waves in reaction-diffusion systems. SIAM J. Math. Anal. 29:1998;1399-1418.
43. Skinner G.S., Swinney H.L. Periodic to quasiperiodic transition of chemical spiral rotation. Physica D. 48:1991;1-16.
44. Winfree A.T. Varieties of spiral wave behaviour: an experimentalist's approach to the theory of excitable media. Chaos. 1:1991;303-334.
45. C. Wulff, Theory of meandering and drifting spiral waves in reaction-diffusion systems, Dissertation, FU Berlin, 1996.
46. Zykov V.S. Cycloidal circulation of spiral waves in an excitable medium. Biofizika. 31:1986;862-865.
47. I. Melbourne, A. Török, Central limit theorems and invariance principles for hyperbolic flows. University of Houston preprint (2001).