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Wada basin boundaries and basin cells. (English) Zbl 0886.58072
Summary: In dynamical systems examples are common in which two or more attractors coexist, and in such cases the basin boundary is nonempty. We consider a two-dimensional diffeomorphism \(F\) (that is, \(F\) is an invertible map and both \(F\) and its inverse are differentiable with continuous derivatives), which has at least three basins. Fractal basin boundaries contain infinitely many periodic points. Generally, only finitely many of these periodic points are “outermost” on the basin boundary, that is, “accessible” from a basin. For many systems, all accessible points lie on stable manifolds of periodic points. A point \(x\) on the basin boundary is a Wada point if every open neighborhood of \(x\) has a nonempty intersection with at least three different basins. We call the boundary of a basin a Wada basin boundary if all its points are Wada points. Our main goal is to have definitions and hypotheses for Wada basin boundaries that can be verified by computer. The basic notion “basin cell” will play a fundamental role in our results for numerical verifications. Assuming each accessible point on the boundary of a basin \(B\) is on the stable manifold of some periodic orbit, we show that \(\partial B\) is a Wada basin boundary if the unstable manifold of each of its accessible periodic orbits intersects at least three basins. In addition, we find conditions for basins \(B_{1}, B_{2},\dots, B_{N}\) \((N\geq{}3)\) to satisfy \(\partial B_{1}=\partial B_{2}=\cdots =\partial B_{N}\). Our results provide numerically verifiable conditions guaranteeing that the boundary of a basin is a Wada basin boundary. Our examples make use of an existing numerical procedure for finding the accessible periodic points on the basin boundary and another procedure for plotting stable and unstable manifolds to verify the existence of Wada basin boundaries.

MSC:
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37B99 Topological dynamics
Software:
Dynamics
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[1] Grebogi, C.; Ott, E.; Yorke, J.A., Chaos, strange attractors, and fractal basin boundaries in nonlinear dynamics, Science, 238, 632-638, (1987) · Zbl 1226.37015
[2] Grebogi, C.; Kostelich, E.; Ott, E.; Yorke, J.A., Multi-dimensioned intertwined basin boundaries: basin structure of the kicked double rotor, Physica D, 25, 347-360, (1987)
[3] Gwinn, E.G.; Westervelt, R.M., Fractal basin boundaries and intermittency in the driven damped pendulum, Phys. rev. A, 33, 4143-4155, (1986)
[4] Mira, C., Frontiére floue séparant LES domaines d’attraction de deux attracteurs, exemples, C.R. acad. sci. Paris, 288A, 591-594, (1979) · Zbl 0397.34063
[5] McDonald, S.W.; Grebogi, C.; Ott, E.; Yorke, J.A., Fractal basin boundaries, Physica D, 17, 125-153, (1985) · Zbl 0588.58033
[6] Moon, F.C.; Li, G.-X., Fractal basin boundaries and homoclinic orbits for periodic motions in a two-well potential, Phys. rev. lett., 55, 1439-1442, (1985)
[7] Nusse, H.E.; Yorke, J.A., The equality of fractal dimension and uncertainty dimension for certain dynamical systems, Commun. math. phys., 150, 1-21, (1992) · Zbl 0770.58030
[8] Takesue, S.; Kaneko, K., Fractal basin structure, Prog. theor. phys., 71, 35-49, (1984)
[9] Yamaguchi, Y.; Mishima, N., Fractal basin boundary of a two-dimensional cubic map, Phys. lett. A, 109, 196-200, (1985)
[10] Alligood, K.T.; Sauer, T., Rotation numbers of periodic orbits in the Hénon map, Commun. math. phys., 120, 105-119, (1988) · Zbl 0729.58032
[11] Alligood, K.T.; Yorke, J.A., Accessible saddles on fractal basin boundaries, Ergod. theor. dynam. syst., 12, 377-400, (1992) · Zbl 0767.58023
[12] Grebogi, C.; Ott, E.; Yorke, J.A., Basin boundary metamorphoses: changes in accessible boundary orbits, Physica D, 24, 243-262, (1987) · Zbl 0613.58018
[13] Hammel, S.M.; Jones, C.K.R.T., Jumping stable manifolds for dissipative maps of the plane, Physica D, 35, 87-106, (1989) · Zbl 0686.58008
[14] Nusse, H.E.; Yorke, J.A., A numerical procedure for finding accessible trajectories on basin boundaries, Nonlinearity, 4, 1183-1212, (1991) · Zbl 0739.58038
[15] Kennedy, J.; Yorke, J.A., Basins of Wada, Physica D, 51, 213-225, (1991) · Zbl 0746.58054
[16] Nusse, H.E.; Yorke, J.A., A procedure for finding numerical trajectories on chaotic saddles, Physica D, 36, 137-156, (1989) · Zbl 0728.58027
[17] You, Z.; Kostelich, E.; Yorke, J.A., Calculating stable and unstable manifolds, Int. J. bifurc. chaos, 1, 605-624, (1991) · Zbl 0874.58053
[18] Nusse, H.E.; Yorke, J.A., Dynamics: numerical explorations, () · Zbl 0805.58007
[19] Newhouse, S.E., Lectures on dynamical systems, () · Zbl 0561.58025
[20] H.E. Nusse and J.A. Yorke, The structure of basins of attraction and their trapping regions, Ergod. Theor. Dynam. Syst., to be published. · Zbl 0893.58039
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