A scaling law: How an attractor's volume depends on noise level |
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| Authors: | Edward Ott Ellen D Yorke James A Yorke |
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| Institution: | Laboratory for Plasma and Fusion Energy Studies, Department of Electrical Engineering and Department of Physics and Astronomy, University of Maryland, College Park, MD 20742, USA;Department of Physics, University of Maryland, Baltimore Country, MD 21228, USA;Institute for Physical Science and Technology, and Department of Mathematics, University of Maryland, College Park, MD. 20742, USA |
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| Abstract: | We investigate the meaning of the dimension of a strange attractor for systems with noise. More specifically, we investigate the effect of adding noise of magnitude ε to a deterministic system with D degrees of freedom. If the attractor has dimension d and d < D, then its volume is zero. The addition of noise may be an important physical probe for experimental situations, useful for determining how much of the observed phenomena in a system is due to noise already present. When the noise is added, the attractor Aε has positive volume. We conjecture that the generalized volume of Aε is proportional to εD ? d for ε near 0 and show this relationship is valid in several cases. For chaotic attractors there are a variety of ways of defining d and the generalized volume definition must be chosen accordingly. |
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