Abstract
We propose a piecewise linear, area-preserving, continuous map of the two-torus as a prototype of nonlinear two-dimensional mixing transformations that preserve a smooth measure (e.g., the Lebesgue measure). The model lends itself to a closed-form analysis of both statistical and geometric properties. We show that the proposed model shares typical features that characterize chaotic dynamics associated with area-preserving nonlinear maps, namely, strict inequality between the line-stretching exponent and the Lyapunov exponent, the dispersive behavior of stretch-factor statistics, the singular spatial distribution of expanding and contracting fibers, and the sign-alternating property of cocycle dynamics. The closed-form knowledge of statistical and geometric properties (in particular of the invariant contracting and dilating directions) makes the proposed model a useful tool for investigating the relationship between stretching and folding in bounded chaotic systems, with potential applications in the fields of chaotic advection, fast dynamo, and quantum chaos theory.
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Cerbelli, S., Giona, M. A Continuous Archetype of Nonuniform Chaos in Area-Preserving Dynamical Systems. J Nonlinear Sci 15, 387–421 (2005). https://doi.org/10.1007/s00332-004-0673-2
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DOI: https://doi.org/10.1007/s00332-004-0673-2