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.

A continuous archetype of nonuniform chaos in area-preserving dynamical systems / Cerbelli, Stefano; Giona, Massimiliano. - In: JOURNAL OF NONLINEAR SCIENCE. - ISSN 0938-8974. - 15:6(2005), pp. 387-421. [10.1007/s00332-004-0673-2]

A continuous archetype of nonuniform chaos in area-preserving dynamical systems

CERBELLI, Stefano;GIONA, Massimiliano
2005

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.
2005
01 Pubblicazione su rivista::01a Articolo in rivista
A continuous archetype of nonuniform chaos in area-preserving dynamical systems / Cerbelli, Stefano; Giona, Massimiliano. - In: JOURNAL OF NONLINEAR SCIENCE. - ISSN 0938-8974. - 15:6(2005), pp. 387-421. [10.1007/s00332-004-0673-2]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/236418
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