On second-order differential equations on the circle with the first degree of structural instability

Abstract

We consider autonomous differential equations of the second order the right-hand sides of which are polynomials of degree n with respect to the first derivative with periodic continuous coefficients. In addition, it is assumed that the leading coefficient and the free term do not vanish. Such equation define on the cylindrical phase space a dynamical system without singular points and closed trajectories homotopic to zero. Structurally stable are equations for which the topological structure of the phase portrait of the corresponding dynamical system does not change under small perturbations in the class of such equations. An equation is structurally stable if and only if all of its closed trajectories are hyperbolic. Structurally stable equations form an open and everywhere dense set in the space of the equations under consideration. The paper investigates equations of the first degree of structural instability – structurally unstable equations for which the topological structure of the phase portrait does not change when passing to a sufficiently close structurally unstable equation. The set of equations of the first degree of structural instability is an embedded smooth submanifold of codimension one in the space of all equations under consideration; it is open and everywhere dense in the set of structurally unstable equations and consists of equations that have a single nonhyperbolic closed trajectory – a double cycle.