In the present paper we investigate two-parametric family of nonautonomous ordinary differential equations on the two-torus that model the Josephson effect from superconductivity. We study its rotation number as a function of parameters and its Arnold tongues (also called phase locking domains): the level sets of the rotation number that have non-empty interior. The Arnold tongues of the equation under consideration have many non-typical properties: the phase locking happens only for integer values of the rotation number [5, 11]; the boundaries of the tongues are given by analytic curves [4, 7], the tongues have zero * Laboratoire J.-V.Poncelet (UMI 2615 du
The famous conjecture of V.Ya.Ivrii (1978) says that in every billiard with infinitely-smooth boundary in a Euclidean space the set of periodic orbits has measure zero. In the present paper we study the complex algebraic version of Ivrii's conjecture for quadrilateral orbits in two dimensions, with reflections from complex algebraic curves. We present the complete classification of 4-reflective algebraic counterexamples: billiards formed by four complex algebraic curves in the projective plane that have open set of quadrilateral orbits. As a corollary, we provide classification of the so-called real algebraic pseudo-billiards with open set of quadrilateral orbits: billiards formed by four real algebraic curves; the reflections allow to change the side with respect to the reflecting tangent line.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.