×

zbMATH — the first resource for mathematics

Fractal basin boundaries. (English) Zbl 0588.58033
Basin boundaries for dynamical systems can be either smooth or fractal. This paper investigates fractal basin boundaries. One practical consequence of such boundaries is that they can lead to great difficulty in predicting to which attractor a system eventually goes. The structure of fractal basin boundaries can be classified as being either locally connected or locally disconnected. Examples and discussion of both types of structures are given, and it appears that fractal basin boundaries should be common in typical dynamical systems. Lyapunov numbers and the dimension for the measure generated by inverse orbits are also discussed.

MSC:
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Ott, E., Rev. mod. phys., 53, 655, (1981), and references therein
[2] Grebogi, C.; McDonald, S.W.; Ott, E.; Yorke, J.A., Phys. lett., 99A, 415, (1983)
[3] Grebogi, C.; Ott, E.; Yorke, J.A., Phys. rev. lett., Physica, 7D, 181, (1983)
[4] Grebogi, C.; Ott, E.; Yorke, J.A., Phys. rev. lett., 50, 935, (1983)
[5] Cartwright, M.L.; Littlewood, J.E., Ann. math., J. London math. soc., 20, 180, (1945)
[6] Levinson, N., Ann. math., 50, 127, (1949)
[7] Levi, M., Mem. amer. math. soc., 32, 244, (1981)
[8] Kaplan, J.L.; Yorke, J.A., Comm. math. phys., 67, 93, (1979)
[9] Lorenz, E.N., J. atmos. sci., 20, 130, (1963)
[10] Li, T.-Y.; Yorke, J.A., Am. math. mon., 82, 985, (1975)
[11] Julia, G.; Fatou, P.; Mandelbrot, B.B., J. math. pure appl., Bull. soc. math. France, Ann. N.Y. acad. sci., 357, 249, (1980)
[12] Farmer, J.D.; Ott, E.; Yorke, J.A., Physica, 7D, 153, (1983)
[13] Pelikan, S., A dynamical meaning of fractal dimension, U. minn. IMA preprint #73, (1984) · Zbl 0532.58013
[14] Kaplan, J.L.; Mallet-Paret, J.; Yorke, J.A., Ergod, th.&dyn. sys., 4, 261, (1984)
[15] Smale, S., Bull. amer. math. soc., 73, 747, (1967)
[16] Flaherty, J.E.; Hoppensteadt, F.C., Stud. appl. math., 58, 5, (1978)
[17] Guckenheimer, J.; Holmes, P., Nonlinear oscillations, dynamical systems, and bifurcation of vector fields, () · Zbl 0515.34001
[18] Smale, S.; Williams, R.F., J. math. biol., 3, 1, (1976)
[19] Pianigiani, G.; Yorke, J.A., Transactions amer. math. soc., 252, 351, (1979)
[20] Yorke, J.A.; Yorke, E.D., J. stat. phys., 21, 263, (1979)
[21] Bergé, P.; Dubois, M., Phys. lett., 93A, 365, (1983)
[22] Kaplan, J.L.; Yorke, J.A., Functional differential equations and approximation of fixed points, (), 228
[23] McDonald, S.; Grebogi, C.; Ott, E.; Yorke, J.A., Phys. lett., 107A, 51, (1985)
[24] Manning, A., Ann. math., 119, 425, (1984)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.