Center problem for trigonometric Liénard systems (Q2013158)

The second-order differential equation \(\ddot \theta = g(\theta) + f(\theta) \dot \theta\), where \(f\) and \(g\) are trigonometric polynomials with real coefficients, is equivalent to the first order system \[ \dot \theta = y,\quad \dot y = g(\theta) + f(\theta) y.\eqno{(S)} \] Supposing \(f(0) = g(0) = 0\) and \(g'(0) < 0\) the origin is a focus or a center for (S). The authors prove that the origin is a center if and only if either (a) \(f = \alpha g\) for some real constant \(\alpha\) or (b) there exists a real trigonometric polynomial \(p\) and two real polynomials \(f_1\) and \(g_1\) such that (i) \(p'(0) < 0\), (ii) \(g_1(p(0)) p''(0) < 0\), and (iii) \(f(\theta) = f_1(p(\theta)) p'(\theta)\) and \(g(\theta) = g_1(p(\theta)) p'(\theta)\). The result and techniques used to prove it hark back to the work of L.~A.~Cherkas, C.~J.~Christopher, N.~G.~Lloyd, and J.~M.~Pearson on the characterization of centers in real analytic and polynomial Liénard equations.