Abstract
We present a simple example of the infinite-dimensional map which has a scrambled set. It is the "logistic map with diffusion" which is a natural extension of the well known one-dimensional logistic map and derived by a semi-implicit discretization in t of a semilinear parabolic equation obtained by adding the diffusion term to the logistic equation. This map is neither expansive nor invertible even locally and hence outside of existing theories of scrambled and hyperbolic invariant sets. We will establish a sufficient condition for it to become a contractive perturbation of the logistic map and construct a scrambled set with several modifications of the method developed in Refs. 7 and 8.
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