Steady-state solutions of Minorsky’s quasi-linear equation

Abstract

A quasi-linear Mathieu-type equation is investigated by means of the averaging method in the neighborhood of the main resonance. All possible types of phase portraits are found, and steady-state solutions are identified. The periodic solutions of the primary equation that correspond to the steady-state solutions of the averaged equation are determined. Analytical expressions for probabilities of the dissipation-induced capture into feasible steady-state solutions are obtained.

Appendix: averaging

Introduce in (9) the following notation:

$$\begin{aligned} G_2= & {} (P\sin t+Q\cos t)^2, \end{aligned}$$

(A.1)

$$\begin{aligned} G_4= & {} (P\sin t+Q\cos t)^4. \end{aligned}$$

(A.2)

To calculate ${\overline{H}}$, calculating thereby the right-hand sides of (12), one should find $\overline{G_2}$, $\overline{G_4}$, $\overline{G_2\cos 2t}$, $\overline{G_4\cos 2t}$. One can readily obtain:

$$\begin{aligned} \overline{G_2}= & {} (P^2+Q^2)/2, \end{aligned}$$

(A.3)

$$\begin{aligned} \overline{G_4}= & {} 3(P^2+Q^2)^2/8, \end{aligned}$$

(A.4)

$$\begin{aligned} \overline{G_2\cos 2t}= & {} (Q^2-P^2)/4, \end{aligned}$$

(A.5)

$$\begin{aligned} \overline{G_4\cos 2t}= & {} (Q^4-P^4)/4. \end{aligned}$$

(A.6)