Coexistence of infinitely many attractors in a simple flow

Physica D: Nonlinear Phenomena

Volumn 109, Issue 3-4, 1997, Pages 201-241

  Chawanya, Tsuyoshi ^a,b  

^a TOHOKU UNIVERSITY   (Japan)

^b OSAKA UNIVERSITY   (Japan)

Author keywords

Fractal basin boundary; Heteroclinic orbit; Infinitely many attractors; Population dynamics; Replicator equation

Indexed keywords

EID: 0000190211     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0167-2789(97)00067-5     Document Type: Article

References (31)

1. R.M. May and W.J. Leonard, Nonlinear aspects of competition between three species, SIAM J. Appl. Math. 29 (1975) 243-252.
2. J. Guckenheimer and P. Holmes, Structurally stable heteroclinic cycles, Math. Proc. Comb. Phil. Soc. 103 (1988) 189-192.
3. I. Melbourne, An example of a non-asymptotically stable attractor, Nonlinearity 4 (1991) 835-844.
4. V. Kirk and M. Silber, A competition between heteroclinic cycles, Nonlinearity 7 (1994) 1367-1384.
5. T. Chawanya, A new type of irregular motion in a class of game dynamics systems, Prog. Theoret. Phys. 94 (1995) 163-179.
6. S. Sasa and T. Chawanya, Weakly chaotic population dynamics in random ecological networks, preprint.
7. L.J. Diaz, Persistence of cycles and nonhyperbolic dynamics at heteroclinic bifurcations, Nonlinearity 8 (1995) 693-713.
8. T. Chawanya, Infinitely many attractors in game dynamics system, Prog. Theoret. Phys. 95 (1996) 679-684.
9. S. Newhouse, Diffeomorphisms with infinitely many sinks, Topology 13 (1974) 9-18.
10. J. Palis and F. Takens, Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations, Cambridge Studies in Advanced Mathematics, Vol. 35 (Cambridge University Press, Cambridge, 1993).
11. P.D. Taylor and L.B. Jonker, Evolutionarily stable strategies and game dynamics, Math. Biosci. 40 (1978) 145-156.
12. J. Hofbauer and K. Sigmund, The Theory of Evolution and Dynamical Systems (Cambridge University Press, Cambridge, 1988).
13. I.M. Bomze, Lotka-Volterra equation and replicator dynamics: New issues in classification, Biol. Cybern. 72 (1995) 447-453.
14. E.C. Zeeman, Population dynamics from game theory, in: Global Theory of Dynamical Systems, Springer Lecture Notes in Mathematics 819 (1980).
15. J. Hofbauer, On the occurrence of limit cycles in the Volterra-Lotka equation, Nonlinear Analysis 5 (1981) 1003-1007.
16. M.E. Gilpin, Spiral chaos in a predator-prey model, American Naturalist 113 (1979) 306-308.
17. A. Arneodo, P. Coullet and C. Tresser, Occurrence of strange attractors in three-dimensional Volterra equations, Phys. Lett. A 79 (1980) 259-263.
18. W.H. Press, B.P. Flannery, S.A. Teukolsky and W.T. Vetterling, Numerical Recipes in C (Cambridge University Press, Cambridge, 1988).
19. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Applied Mathematics Sciences, Vol. 42 (Springer, New York, 1983).
20. P. Ashwin and J.W. Swift, The dynamics of n weakly coupled identical oscillators, J. Nonlinear Sci. 2 (1992) 69-108.
21. K. Okuda, New aspects of collective dynamics in globally coupled phase oscillators, Thesis, Kyoto University (1994).
22. D. Hansel, G. Mato and C. Meunier, Clustering and slow switching in globally coupled phase oscillators, Phys. Rev. E 48 (1994) 3470-3477.
23. E. Knobloch and M. Silber, Oscillatory convection in a rotating layer, Physica D 63 (1993) 213-232.
24. A. Gaunersdorfer, Time averages for heteroclinic attractors, SIAM J. Appl. Math. 52 (1992) 1476-1489.
25. S. Grossmann and H. Fujisaka, Diffusion in discrete nonlinear dynamical systems, Phys. Rev. A 26 (1982) 1779-1782.
26. J. Milnor, On the Concept of Attractor Commun. Math. Phys. 99 (1985) 177-195.
27. A.S. Pikovsky and P. Grassberger, Symmetry breaking bifurcation for couple chaotic attractors, J. Phys. A 24 (1991) 4587-4597.
28. J.C. Alexander, J.A. Yorke, Y. Zhiping and I. Kan, Riddled basins, Int. J. Bifur. and Chaos 2 (1992) 795-813.
29. E. Ott, J.C. Alexander, I. Kan, J.C. Sommerer and J.A. Yorke, The transition to chaotic attractors with riddled basins, Physica D 76 (1994) 384-410.
30. M. Dellnitz, M. Field, M. Golubitsky, J. Ma and A. Hohmann, Cycling chaos, Int. J. Bifur. and Chaos 5 (1995) 1243-1247.
31. Possible coexistence of infinitely many attractors in 4 dimensional flow has been reported by S.V. Gonchenko, L.P. Shil'nikov and D.V. Turaev, Dynamical phenomena in systems with structurally unstable Poincaré homoclinic orbits, Chaos 6 (1996) 15-31.