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Nondeterministic chaos, and the two-fold singularity in piecewise smooth flows


Reference:

Colombo, A. and Jeffrey, M. R., 2011. Nondeterministic chaos, and the two-fold singularity in piecewise smooth flows. SIAM Journal on Applied Dynamical Systems, 10 (2), pp. 423-451.

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    Official URL:

    http://dx.doi.org/10.1137/100801846

    Abstract

    A vector field is piece wise smooth if its value jumps across a hyper surface, and a two-fold singularity is a point where the flow is tangent to the hypersurface from both sides. Two-folds are generic in piecewise smooth systems of three or more dimensions. We derive the local dynamics of all possible two-folds in three dimensions, including nonlinear effects around certain bifurcations, finding that they admit a flow exhibiting chaotic but nondeterministic dynamics. In cases where the flow passes through the two-fold, upon reaching the singularity it is unique in neither forward nor backward time, meaning the causal link between inward and outward dynamics is severed. In one scenario this occurs recurrently. The resulting flow makes repeated, but nonperiodic, excursions from the singularity, whose path and amplitude is not determined by previous excursions. We show that this behavior is robust and has many of the properties associated with chaos. Local geometry reveals that the chaotic behavior can be eliminated by varying a single parameter: the angular jump of the vector field across the two-fold.

    Details

    Item Type Articles
    CreatorsColombo, A.and Jeffrey, M. R.
    DOI10.1137/100801846
    DepartmentsFaculty of Science > Mathematical Sciences
    Publisher Statement2011SIADS_2fold2.pdf: ©SIAM
    RefereedYes
    StatusPublished
    ID Code26959

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