Existence of periodic solutions for certain differential equations. (English) Zbl 0061.19002

Lefschetz, S.

Proc. Natl. Acad. Sci. USA 29, 29-32 (1943).

From the text: In the present note we prove the existence of periodic solutions for the equation \[ \frac{d^x}{dt^2}+ g'(x)\frac{dx}{dt} + f(x) = e(t) \tag{1} \] where \(e(t)\) has the period \(T\), and \(g\), \(f\) are restricted as stated below. As we shall see, the proof is essentially elementary. The type of equation under discussion generalizes the equation for the response of an electrical series circuit with resistance \(R\), capacity \(C\) (both constant) and an inductor with current-flux saturation curve \(i=h(Q)\). Here (1) is the differential equation of the flux with \(f=h/C\), \(g(x) = Rh\). The function \(h(x)\) may be satisfactorily represented by an odd polynomial such that \(xh(x)>0\). This particular case suffices to indicate the importance of the periodic solutions of (1).

MSC:

34C25

Periodic solutions to ordinary differential equations

Keywords:

ordinary differential equations