Star-shaped periodic solutions of a system of delay differential equations: A computer-assisted proof

Abstract

In this paper, we consider a planar system of nonlinear delay differential equations and prove the existence of the star-shaped periodic solutions conjectured by Hara (Nonlinear Analysis: Theory, Methods \& Applications, 30:4555--4563 (1997)) for seven parameter values. These solutions cannot be obtained by a direct application of the standard Fourier spectral method, since the corresponding linearized operator is degenerate. From numerical observations of the Fourier coefficients, we derive an ansatz that writes a star-shaped periodic solution as a periodic profile in a rotating frame. Restricting the profile to real Fourier coefficients, we show that the resonance between the delay and the period of the profile fixes the rotation frequency exactly. We then validate a zero of the resulting real system with the radii polynomial approach and interval arithmetic, with bounds that cover all Fourier modes. For each parameter value, we also determine the fundamental period and the winding number of the solution.