Specific Properties of the ODE’s Flow in Dimension Two Versus Dimension Three

• Published in: Journal of dynamics and differential equations Vol. 36; no. 1; pp. 421 - 461
• Main Authors: Briane, Marc, Hervé, Loïc
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.03.2024; Springer Nature B.V; Springer Verlag
• Subjects: Analysis of PDEs; Apexes; Applications of Mathematics; Continuous flow; Determinants; Fields (mathematics); Invariants; Mathematics; Mathematics and Statistics; Ordinary Differential Equations; Partial Differential Equations; Rotation; Segments; Theorems; Toruses; Transport equations; Two dimensional flow; Vectors (mathematics); Transport equation; 37C40; Rotation set; 34E10; Invariant measure; 37C10; ODE’s flow; Homogenization; Asymptotics; Fourier coefficients; 42B05; transport equation Mathematics Subject Classification: 34E10; homogenization; asymptotics; ODE's flow; invariant measure; rotation set
• ISSN: 1040-7294; 1572-9222
• DOI: 10.1007/s10884-022-10154-1

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 ζ , where ζ 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, 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 ( μ , ν ) of invariant probability measures for the flow, we establish new Fourier relations between the determinant det ( μ b ^ ( j ) , ν 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 μ with a regular density and a non null mass μ ( b ) with respect to b , we show that the homogenization of the two-dimensional transport equation with the oscillating velocity b ( x / ε ) as ε tends to 0, leads us to a nonlocal limit transport equation, but with the effective constant velocity  μ ( b ) .