Characterization of nonuniform chaos in area-preserving nonlinear maps through a continuous archetype

Numerical investigations conducted over a wealth of nonlinear area-preserving smooth maps (e.g. the Standard Map) showed that these systems possess physically relevant features that are not captured by any continuous archetype of two-dimensional conservative dynamics. Among these properties are the dispersive behavior of stretch factor statistics, the multifractal character of the measure associated with invariant foliations, the sign-alternating property, accounting for the nestedly bent structure of invariant foliations, and the strict inequality between the topological entropy, htop, and the Lyapunov exponent, Λ. We refer to systems possessing all of these properties as nonuniformly chaotic. In this article, we present a globally continuous, piecewise-smooth area-preserving transformation, the toral homeomorphism H, as an archetype of nonuniformly chaotic behavior. The relatively simple structure of point set dynamics and the closed-form knowledge of the pointwise expanding and contracting invariant directions associated with H, permits to derive either analytically, or with arbitrary numerical precision, the standard chaotic properties as well as the dynamics of the physically relevant properties that define nonuniform chaos. Potentialities and limitations of the model proposed in representing geometric and statistical properties of physically relevant smooth systems are discussed in detail.