Constructive proof of the existence of periodic solutions to the Duffing equation
The authors consider the periodic boundary value problem \[ {c} x''(t) + Cx'(t) +g(t,x(t)) = e(t), \;t \in \mathbb{R}, \quad x(0) -x(2\pi) = x'(0) - x'(2\pi) = 0, \] where \(g\) is continuous, of class \(C^{1}\) with respect to \(x\) and \(2\pi\)-periodic with respect to \(t\), \(C\in \mathbb{R}\) and \(e\) is continuous and \(2\pi\)-periodic. Under a well-known additional assumption on \(g'_{x}(t,x)\), related to a suitable interaction of this function with the eigenvalues of the corresponding linear homogeneous problem, the authors provide a constructive proof of the existence of a unique \(2\pi\)-periodic solution to the previous problem, computing such solution. This result is illustrated with two graphical concrete examples.
scientific article; zbMATH DE number 1159107