Measures of Complexity and Chaos pp 245-248 | Cite as

# Phase Transitions Induced by Deterministic Delayed Forces

Chapter

## Abstract

The frontier between noise and deterministic chaos was recently shown with ∂V/∂x = x was shown to behave as a linear Langevin equation with Gaussian noise, in the limit A → ∞. In deterministic systems the statistics of the feedback term cannot be stated a priori, it follows from internal properties of the equation. Let us point out that the Gaussian behavior of x results from the analytical form choosen for the feedback f(x) in the sense that the periodic character of f(x) is responsible for the short memory effects in f[ax(t)]. On the contrary, in the case of Mackey-Glass equation

^{1}to be free, in the sense that the very simple deterministic retarded equation$$\frac{{dx}}{{dt}} + \frac{{\partial v}}{{\partial x}}\left( {x\left( t \right)} \right) = \sin \left[ {Ax\left( {t - d} \right)} \right]$$

(1)

^{2}, the feedback will never get short memory as the parameter A increases, because the corresponding feedback f(x) = x / 1+x^{c}has only one maximum.## Keywords

Deterministic Chaos Short Memory Feedback Term Gaussian Behavior Noise Induce Transition
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## References

- 1.a) B. Dorizzi, B. Grammaticos, M. Le Berre, Y. Pommeau, E. Ressayre, A. Tallet, Phys. Rev. A 35, 328, 1987 ;Google Scholar
- b) M. Le Berre, Y. Pommeau, E. Ressayre, A. Tallet, H.M. Gibbs, D.L. Kaplan, M.J. Rose, in “Far from equilibrium phase transitions”, Proc. Sitges, Barcelona 1988, ed. by L. Garrido, Springer Verlag, n°319, Berlin 1988 ;Google Scholar
- c) The Gaussian character of Eq.(1) was mentioned by K. Ikeda, O. Akimoto, in Coherence and Quantum Optics V, Rochester, 1983, ed. by L. Mandel and E. Wolf (Plenum N.Y., 1984 ).Google Scholar
- 2.M.C. Mackey, L. Glass, Science 197, 287 1977. See also J.D. Farmer, Physica 4D, 366, 1982.Google Scholar
- 3.E. Hairer, S.P. Norsett, G. Wanner, “Solving ordinary differential equations”, Springer Verlag, 1987.zbMATHGoogle Scholar
- 4.M. Le Berre, E. Ressayre, A. Tallet, Proc. of IQEC Conference in Rochester, June 1989.Google Scholar
- 5.See for example J. Masoliver, B.J. West, H. Lindenberg, Phys. Rev. A 35, 3086, 1987 and references herein.Google Scholar
- 6.W. Horsthemke, R. Lefever “Noise induced transitions : Theory and applications in Physics, Chemistry and Biology”, Springer Verlag, 1984. See also R.R. Grigolini, L.A. Lugiato, R. Mannella and P.V.E. McClintock, M. Meni, M. Pernigo, Phys. Rev. A38, 1966, 1988 and references herein.Google Scholar

## Copyright information

© Plenum Press, New York 1989