Dimension prints and the avoidance of sets for flow solutions of non-autonomous ordinary differential equations

Robinson, James and Sharples, Nicholas ORCID: https://orcid.org/0000-0003-1722-5647 (2013) Dimension prints and the avoidance of sets for flow solutions of non-autonomous ordinary differential equations. Journal of Differential Equations, 254 . pp. 4144-4167. [Article]

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Abstract

We provide a criterion for a generalised
ow solution of a non-autonomous
ordinary differential equation to avoid a subset of the phase space. This
improves on that established by Aizenman for the autonomous case, where
avoidance is guaranteed if the underlying vector field is sufficiently regular
and the subset has sufficiently small box-counting dimension. We define the
r-codimension print of a subset $S\subset \R^{n}\times [0,T]$, which is a subset of $(0,\infty]^{2}$ that encodes the dimension of S in a way that distinguishes spatial and temporal detail. We prove that the subset S is avoided by a generalised flow solution with underlying vector field in $L^{p}(0, T; L^{q}(R^{n}))$ if the Holder conjugates (q^{*}; p^{*}) are in the r-codimension print of S.

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