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
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## References

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## Copyright information

© Plenum Press, New York 1989