Sunflower model: time-dependent coefficients and topology of the periodic solutions set
The authors investigate \(T\)-periodic solutions of parametrized delay differential equations of the form \[ \ddot y(t) = a(t)\dot y(t) + \lambda b(t) \phi(y(t),y(t-r)) , \quad \lambda \geq 0, \tag{1} \] where \(\phi\) is a continuous function, \(r>0\) is a finite time delay, and \(a\) and \(b\) are assumed to be continuous, \(T\)-periodic with \(a\) of nonzero average and \(b(t) \neq 0\) for all \(t\). These equations generalize in some sense the sunflower equation, i.e., \[ \ddot y(t) = -\frac{\alpha}{r} \dot y(t) - \frac{\beta}{r} \sin(y(t-r)), \tag{2} \] in which \(\alpha\) and \(\beta\) are experimental parameters. By a combination of topological methods, such as degree theory and fixed point index theory, along with a classical Liénard-plane analysis, the authors are able to obtain information on the structure of the set of \(T\)-periodic solutions of (1) as \(\lambda \geq 0\) varies. Roughly speaking, the main result of the paper gives a topological condition on the function \(p\mapsto\phi(p,p)\) which implies that there exists a connected set of ``nontrivial'' \(T\)-periodic solutions of (1) whose closure is not compact and intersects the set of constant solutions of (1) for \(\lambda =0\). The paper closes with an application to a variant of the original sunflower equation (2), namely, to an equation with time-periodic coefficients.
scientific article; zbMATH DE number 6506036
scientific article