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Bifurcation Phenomena of Dissipative Dynamical Systems

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Dissipative Structures and Chaos
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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.

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References

  • Cvitanović, P. (1989) Universality in Chaos, 2nd edn. Hilger, Bristol

    MATH  Google Scholar 

  • Devaney, R.L. (1989) An Introduction to Chaotic Dynamical Systems. Benjamin/Cummings, Redwood City, CA

    Google Scholar 

  • Feigenbaum, M.J. (1979) The universal metric properties of nonlinear transformations. J. Stat. Phys. 21, 669–706

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Grassberger, P., Badii, R., Politi, A. (1988) Scaling laws for invariant measures on hyperbolic and non-hyperbolic attractors. J. Stat. Phys. 51, 135–178

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Grebogi, C., Ott, E., Yorke, J.A. (1983) Crises, sudden changes in chaotic attractors, and transient chaos. Physica D 7, 181–200

    Article  MathSciNet  ADS  Google Scholar 

  • Greene, J.M. (1979) A method for determining a stochastic transition. J. Math. Phys. 20, 1183–1201

    Article  ADS  Google Scholar 

  • 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

    Article  MathSciNet  ADS  Google Scholar 

  • 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

    Article  MathSciNet  ADS  Google Scholar 

  • Hénon, M. (1976) A two-dimensional mapping with a strange attractor. Common. Math. Phys. 50, 69–77

    Article  ADS  MATH  Google Scholar 

  • 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

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • 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

    Article  ADS  Google Scholar 

  • MacKay, R.S., Tresser, C. (1986) Transition to topological chaos for circle maps. Physica D 19, 206–237

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Å arkovskii, A.N. (1964) Coexistence of cycles of a continuous map of a line into itself. Ukr. Math. Z. 16, 61–71

    MathSciNet  Google Scholar 

  • 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

    Article  MathSciNet  ADS  Google Scholar 

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© 1998 Springer-Verlag Berlin Heidelberg

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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

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  • 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

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