A Scaling Law: How an Attractor's Volume Depends on Noise Level

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 A_ε has positive volume. We conjecture that the generalized volume of A_ε is proportional to ε_{D − 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.