Abstract
From a large class of diffeomorphisms in the plane, which are known to produce chaotic dynamics, we explicitly construct their continuous suspension on a three dimensional cylinder. This suspension is smooth (C 1) and can be characterized by the choice of two smooth functions on the unit interval, which have to fulfill certain boundary conditions. For the case of entire Cremona transformations, we are able to construct the corresponding autonomous differential equations of the flow explicitly. Thus it is possible to relate properties of discrete maps to those of ordinary differential equations in a quantitative manner. Furthermore, our construction makes it possible to study the exact solutions of chaotic differential-equations directly.
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References
Collet, P., Eckmann, J.-P.: Iterated maps on the interval as dynamical systems. Boston: Birkhäuser 1980
Eckmann, J.-P.: Roads to turbulence in dissipative dynamical systems. Rev. Mod. Phys.53, 643 (1981);
Ruelle, D., Takens, F.: On the nature of turbulence. Commun. Math. Phys.20, 167 (1971);
Pomeau, Y., Manneville, P.: Intermittent transition to turbulence in dissipative dynamical systems. Commun. Math. Phys.77, 789 (1980)
Feigenbaum, M.: Quantitative universality for a class of nonlinear transformations. J. Stat. Phys.19, 25 (1978)
Grossmann, S.: Private communication
See, e.g., Haken, H.: Advanced synergetics. Berlin, Heidelberg, New York: Springer 1983
Gonzalez, O.L., Piro, O.: Chaos in a nonlinear driven oscillator with exact solution. Phys. Rev. Lett.50, 870 (1983)
Farmer, J.D., Crutchfield, J.P., Froehling, H., Packard, N., Shaw, R.: Power spectra and mixing properties of strange attractors. Proc. of the 1979 Conf. on Nonlin. Dyn. Helleman, R.H.G. (ed.). Ann. N.Y. Acad. Sci.375, 353 (1980)
Smale, S.: Bull. Am. Math. Soc.73, 747 (1967)
Hénon, M.: A two-dimensional mapping with a strange attractor. Commun. Math. Phys.50, 69 (1976)
Bunz, H.: Private communication
Crutchfield, J.P., Farmer, J.D., Packard, N., Shaw, R.: Mixing properties of chaotic attractors. 16 mm film
Rössler, O.E.: An equation for continuous chaos. Phys. Lett.57A, 397 (1976)
Mayer-Kress, G.: Zur Persistenz von Chaos und Ordnung in nichtlinearen dynamischen Systemen. Ph. D. thesis, Universität Stuttgart 1984
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Communicated by O. E. Lanford
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Mayer-Kress, G., Haken, H. An explicit construction of a class of suspensions and autonomous differential equations for diffeomorphisms in the plane. Commun.Math. Phys. 111, 63–74 (1987). https://doi.org/10.1007/BF01239015
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DOI: https://doi.org/10.1007/BF01239015