Abstract
The form and structure of an attractor are characterized by the types of unstable periodic points (saddle points) that it contains. Changing the value of a bifurcation parameter can cause various saddle points to enter and leave attractors. Chaotic bifurcations result from the collision of an attractor with such points and their resultant inclusion into the attractor.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Cvitanović, P. (1989) Universality in Chaos, 2nd edn. Hilger, Bristol
Devaney, R.L. (1989) An Introduction to Chaotic Dynamical Systems. Benjamin/Cummings, Redwood City, CA
Feigenbaum, M.J. (1979) The universal metric properties of nonlinear transformations. J. Stat. Phys. 21, 669–706
Grassberger, P., Badii, R., Politi, A. (1988) Scaling laws for invariant measures on hyperbolic and non-hyperbolic attractors. J. Stat. Phys. 51, 135–178
Grebogi, C., Ott, E., Yorke, J.A. (1983) Crises, sudden changes in chaotic attractors, and transient chaos. Physica D 7, 181–200
Greene, J.M. (1979) A method for determining a stochastic transition. J. Math. Phys. 20, 1183–1201
Hata, H., Horita, T., Mori, H., Morita, T., Tomita, K. (1988) Characterization of local structures of chaotic attractors in terms of coarse-grained local expansion rate. Prog. Theor. Phys. 80, 809–826
Horita, T., Hata, H., Mori H., Morita, T., Tomita, K., Kuroki, S., Okamoto, H. (1988) Local structures of chaotic attractors and q-phase transitions at attractor-merging crises in the sine-circle maps. Prog. Theor. Phys. 80, 793–808
Hénon, M. (1976) A two-dimensional mapping with a strange attractor. Common. Math. Phys. 50, 69–77
Horita, T., Hata, H., Mori, H. (1990) Cascade of attractor-merging crises to the critical golden torus and universal expansion-rate spectra. Prog. Theor. Phys. 84, 558–562
Jensen, M.H., Kadanoff, L.P., Libchaber, A., Procaccia, I., Stavans, J. (1985) Global universality at the onset of chaos: results of a forced Rayleigh- Benard experiment. Phys. Rev. Lett. 55, 2798–2801
MacKay, R.S., Tresser, C. (1986) Transition to topological chaos for circle maps. Physica D 19, 206–237
Šarkovskii, A.N. (1964) Coexistence of cycles of a continuous map of a line into itself. Ukr. Math. Z. 16, 61–71
Tominaga, H., Mori, H. (1991) Crossover to the f(α) spectra between critical and chaotic regime, and universal critical scaling laws for two-dimensional fractality. Prog. Theor. Phys. 86, 355–369
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Mori, H., Kuramoto, Y. (1998). Bifurcation Phenomena of Dissipative Dynamical Systems. In: Dissipative Structures and Chaos. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-80376-5_10
Download citation
DOI: https://doi.org/10.1007/978-3-642-80376-5_10
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-80378-9
Online ISBN: 978-3-642-80376-5
eBook Packages: Springer Book Archive