Dynamical systems with multivalued integrals on a torus

Abstract

Abstract-Properties of the solutions to differential equations on the torus with a complete set of multivalued first integrals are considered, including the existence of an invariant measure, the averaging principle, and the infiniteness of the number of zeros for integrals of zero-mean functions along trajectories. The behavior of systems with closed trajectories of large period is studied. It is shown that a generic system acquires a limit mixing property as the periods tend to infinity.

On a conjecture of De Giorgi related to homogenization

“…Then there is a Lipschitz continuous function X 0 ptq such that (12) |X ε ptq´X 0 ptq| ď CpT qε, t P r0, T s where T ą 0 is the length of the time interval t P r0, T s, CpT q is a positive constant depending only on T and G ‹ . Furthermore, if G ‹ pt, ηq does not depend on t and is periodic in η, then X 0 ptq " p 0`β t for some p 0 , β P R.…”

“…It needs to be mentioned that Kolmogorov's proof is not constructive i.e., he did not write explicit form of such transformation. In [19] Tassa found a simple argument that renders the explicit form of f. Such coordinate transformation exists for d ě 3 under various assumptions [1], [12].…”

“…For d ě 3 Kolmogorov's theorem has been generalized by Kozlov which we state below without proof, see [12]. Proposition 1.…”

The dynamics of systems with servoconstraints. II

“…Such transformations must necessarily preserve the volume element in R (i.e., their Jacobians are equal to 1). The conditions for the reducibility of a dynamical system on an n-dimensional torus with a full set of (n − 1) multivalued integrals to the form (4.8) are found, up to time rescaling, in [18] (see also references therein).…”

The dynamics of systems with servoconstraints. II

“…Такие преобразования с необходимостью должны сохранять элемент объема в R (то есть их якобианы равны единице). В [18] указаны условия приводимости динамической системы на n-мерном торе с полным набором многозначных интегралов (в количестве n − 1) к виду (4.8) с точностью до замены времени. Там же указаны ссылки на другие работы по этой теме.…”