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Article Dans Une Revue Journal of Dynamics and Differential Equations Année : 2024

Specific properties of the ODE's flow in dimension two versus dimension three

Résumé

This paper deals with the asymptotics of the ODE's flow induced by a regular vector field $b$ on the $d$-dimensional torus $\R^d/\Z^d$. First, we start by revisiting the Franks-Misiurewicz theorem which claims that the Herman rotation set of any two-dimensional continuous flow is a closed line segment of $\R^2$. Various general examples illustrate this result, among which a complete study of the Stepanoff flow associated with a vector field $b=a\,\zeta$, where $\zeta$ is a constant vector in $\R^2$. Furthermore, several extensions of the Franks-Misiurewicz theorem are obtained in the two-dimensional ODE's context. On the one hand, we provide some interesting stability properties in the case where the Herman rotation set has a commensurable direction. On the other hand, we present new results highlighting the exceptional character of the opposite case, {\em i.e.} when the Herman rotation set is a closed line segment with $0_{\R^2}$ at one end and with an irrational slope, if it is not reduced to a single point. Besides this, given a pair $(\mu,\nu)$ of invariant probability measures for the flow, we establish new Fourier relations between the determinant $\det\,(\widehat{\mu b}(j),\widehat{\nu b}(k))$ and the determinant $\det\,(j,k)$ for any pair $(j,k)$ of non null integer vectors, which can be regarded as an extension of the Franks-Misiurewicz theorem. Next, in contrast with dimension two, any three-dimensional closed convex polyhedron with rational vertices is shown to be the rotation set associated with a suitable vector field~$b$. Finally, in the case of an invariant measure $\mu$ with a regular density and a non null mass $\mu(b)$ with respect to~$b$, we show that the homogenization of the two-dimensional transport equation with the oscillating velocity $b(x/\ep)$ as $\ep$ tends to $0$, leads us to a nonlocal limit transport equation, but with the effective constant velocity~$\mu(b)$.
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Dates et versions

hal-03412162 , version 1 (02-11-2021)

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Marc Briane, Loïc Hervé. Specific properties of the ODE's flow in dimension two versus dimension three. Journal of Dynamics and Differential Equations, 2024, 36 (1), pp.421-461. ⟨10.1007/s10884-022-10154-1⟩. ⟨hal-03412162⟩
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