Two-state intermittency near a symmetric interaction of saddle-node and Hopf bifurcations: A case study from dynamo theory

Physica D: Nonlinear Phenomena

Volumn 194, Issue 1-2, 2004, Pages 30-48

  Ashwin, Peter ^a   Rucklidge, Alastair M ^b   Sturman, Rob ^a  

^a UNIVERSITY OF EXETER   (United Kingdom)

^b UNIVERSITY OF LEEDS   (United Kingdom)

Author keywords

Bifurcation with symmetry; Cycling chaos; Dynamo theory; Intermittency

Indexed keywords

CHAOS THEORY; COMPUTER SIMULATION; MARKOV PROCESSES; MATHEMATICAL MODELS; ORDINARY DIFFERENTIAL EQUATIONS; PROBABILITY;

DYNAMO MODELS; INVARIANT SUBSPACES; TWO STATE INTERMITTENCY; UNSTABLE DYNAMICS;

BIFURCATION (MATHEMATICS);

EID: 2942566386     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physd.2004.02.002     Document Type: Article

References (30)

1. V.S. Anishchenko, V.V. Astakhov, A.B. Neiman, T.E. Vadivasova, L. Schimansky-Geier, Nonlinear Dynamics of Chaotic and Stochastic Systems, Springer Series in Synergetics, Springer Verlag, Berlin, 2002.
2. Ashwin P., Buescu J., Stewart I., From attractor to chaotic saddle: a tale of transverse instability. Nonlinearity. 9:1996;703.
3. Ashwin P., Aston P.J., Blowout bifurcations of codimension two. Phys. Lett. A. 244:1998;261-270.
4. Ashwin P., Covas E., Tavakol R., Influence of noise on scalings for in-out intermittency. Phys. Rev. E. 64:2001;066204.
5. Ashwin P., Covas E., Tavakol R., Transverse instability for non-normal parameters. Nonlinearity. 12:1999;563-577.
6. Ashwin P., Field M., Rucklidge A.M., Sturman R., Phase resetting effects for robust cycles between chaotic sets. Chaos. 13:2003;973-981.
7. Ashwin P., Rucklidge A.M., Cycling chaos: its creation, persistence and loss of stability in a model of nonlinear magnetoconvection. Physica D. 122:1998;134-154.
8. Çenys A., Anagnostopoulos A.N., Bleris G.L., Distribution of laminar lengths for noisy on-off intermittency. Phys. Lett. A. 224:1997;346.
9. Covas E., Tavakol R., Ashwin P., Tworkowski A., Brooke J.M., In-out intermittency in PDE and ODE models of axisymmetric mean field dynamos. Chaos. 11:2001;404-409.
10. Ding M., Yang W., Stability of synchronous chaos and on-off intermittency in coupled map lattices. Phys. Rev. E. 56:1997;4009.
11. Fujisaka H., Yamada H., A new intermittency in coupled dynamical systems. Prog. Theor. Phys. 74:1985;918-921.
12. J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Applied Mathematical Sciences, vol. 42, Springer, New York, 1983.
13. A. Hasegawa, M. Komuro, T. Endo, A new type of intermittency from a ring of four coupled phase-locked loops, in: Proceedings of the ECCTD'97, European Circuit Society, Budapest, September 1997.
14. Yu. Ilyashenko, W. Li, Nonlocal Bifurcations Mathematical Surveys and Monographs, vol. 66, AMS, Providence, RI, 1999.
15. Kirk V., Breaking of symmetry in the saddle-node Hopf-bifurcation. Phys. Lett. A. 154:1991;243.
16. Knobloch E., Tobias S.M., Weiss N.O., Modulation and symmetry changes in stellar dynamos. Mon. Not. R. Astron. Soc. 297:1998;1123-1138.
17. B. Krauskopf, B.E. Oldeman, The saddle-node Hopf bifurcation with global reinjection, Preprint, University of Bristol, 2003.
18. Krauskopf B., Wieczorek S., Accumulating regions of winding periodic orbits in optically driven lasers. Physica D. 173:2002;97-113.
19. Martel C., Knobloch E., Vega J.M., Dynamics of counterpropagating waves in parametrically forced systems. Physica D. 137:2000;94.
20. Lai Y.-C., Distinct small-distance scaling behavior of on-off intermittency in chaotic dynamical systems. Phys. Rev. E. 54:1996;321.
21. Lai Y.-C., Grebogi C., Intermingled basins and two-state on-off intermittency. Phys. Rev. E. 52:1995;3313. (rapid communications).
22. I. Melbourne, M.R.E. Proctor, A.M. Rucklidge, A heteroclinic model of geodynamo reversals and excursions, in: P. Chossat, D. Armbruster, I. Oprea (Eds.), Dynamo and Dynamics, A Mathematical Challenge, Kluwer Academic Publishers, Dordrecht, 2001, pp. 363-370.
23. Ott E., Sommerer J., Blowout bifurcations: the occurrence of riddled basins and on-off intermittency. Phys. Lett. A. 188:1994;39.
24. Pikovsky A.S., On the interaction of strange attractors. Z. Phys. B. 55:1984;149.
25. A. Pikovsky, M. Rosenblum, J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge Nonlinear Science Series, vol. 12, Cambridge University Press, Cambridge, 2001.
26. Platt N., Hammel S.M., Heagy J.F., Effects of additive noise on on-off intermittency. Phys. Rev. Lett. 72:1994;3498.
27. Platt N., Spiegel E.A., Tresser C., On-off intermittency: a mechanism for bursting. Phys. Rev. Lett. 70:1993;279.
28. W. Press, S. Teukolsky, W. Vetterling, B. Flannery, Numerical Recipes in C, Cambridge University Press, Cambridge, 1988.
29. Tobias S.M., Weiss N.O., Kirk V., Chaotically modulated stellar dynamos. Mon. Not. R. Astron. Soc. 273:1995;1150-1166.
30. Woltering M., Marcus M., Riddled basins of coupled elastic arches. Phys. Lett. A. 260:1999;453-461.