Abstract
A special type of turbulence, called boiling type turbulence, can be obtained by coupling a set of identical single-threshold relaxation oscillators in such a manner that the slow variables are locally cross-inhibitory. If the cross-inhibition is overcritical, leading to ‘morphogenesis’ between the slow variables, a sudden slackening occurs wherever the the slowly moving-up ‘skyline’ hits the threshold (with the consequence of a rearrangement of the skyline); and so forth. A simple equation is considered on a ring in a cellular approximation. One cell is not chaotic; two are capable of chaos; three apparently produce higher chaos of the first order; and so forth. In the 3-cellular case, geometrical arguments backed by simulation suggest the presence of a cross-section which is folded over (between one passage and the next) in several independent directions. By implication, the formation of singular sets of Alexandrov type can be expected.
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References
Alexandrov, P.S. (1948): Introduction to Set Theory and the Theory of Real Functions (in German), VEB-Verlag der Wissenschaften, Berlin, 1956, p. 224. (First Russian ed.: Ogie, Moscow-Leningrad, 1948.)
Babloyantz, A. (1980): These proceedings.
Fife, P. (1980): These proceedings.
Guckenheimer, J. (1979): On a codimension two bifurcation, Preprint.
Haken, H. (1980): These proceedings.
Kuramoto, Y. (1979): Diffusion-induced chaos in reaction systems, Progr. Theor. Phys. Suppl. 64, 346.
Kuramoto, Y. (1980): These proceedings.
Kuramoto, Y. and T. Yamada (1976): Turbulent state in chemical reactions, Progr. Theor. Phys. 56, 679–683.
Lorenz, E.N. (1963): Deterministic nonperiodic flow, J. Atmos. Sci. 20, 130–141.
Othmer, H. (1980): These proceedings.
Plykin, R.V. (1974): Math. Sbornik 94, 243–246.
Rössler, O.E. (1976): Chemical turbulence: chaos in a simple reaction-diffusion system, Z. Naturforsch. 31a, 1168–1172.
Rössler, O.E. (1979): An equation for hyperchaos, Phys. Lett. 71A, 155–157.
Ruelle, D. and F. Takens (1971): On the nature of turbulence, Commun. Math. Phys. 20, 167–192.
Smale, S. (1967): Differentiate dynamical systems, Bull. Amer. Math. Soc. 73, 747–817.
Turing, A.M. (1952): The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. London B 237, 37–72.
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Rössler, O.E. (1980). Chaos and Turbulence. In: Haken, H. (eds) Dynamics of Synergetic Systems. Springer Series in Synergetics, vol 6. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-67592-8_12
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DOI: https://doi.org/10.1007/978-3-642-67592-8_12
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