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Periodic solutions of homogeneous equations. (English) Zbl 0747.34030
The author investigates stability properties of special solutions of differential equations (1) \(\dot x=f(t,x)\), where \(f:\mathbb{R} \times \mathbb{R}^ n \to \mathbb{R}^ n\) is assumed to be homogeneous of degree 1 in \(x\) with respect to positive factors, \(f(t,\alpha x)=\alpha f(t,x)\) for \(\alpha >0\). As an auxiliary construction, using a homogeneous mapping \(\varphi :\mathbb{R}^ n \to \mathbb{R}^ +\), he defines a sort of “sphere” \(S_ \varphi=\{x \in \mathbb{R}^ n:\varphi (x)=1\}\). The projections of solutions of (1) onto \(S_ \varphi\), \(y(t)=x(t)/\varphi(x(t))\), satisfy (2) \(\dot y=f(t,y)-\varphi'(y)f(t,y)y\). The author obtains stability results for constant solutions of (2) (which correspond to exponential solutions of (1)) and for periodic solutions of (2) (corresponding to spiral solutions of (1)). In addition, he presents a generalization of the concept of a Lyapunov function to homogeneous systems which works on the projected system (2) but is independent of the choice of the projection \(\varphi\), and a Bendixson-Dulac criterion for the projected system. Certain applications (to a Kermack-McKendrick desease transmission model and to Fisher’s equation) are also given.
Reviewer: W.Müller (Berlin)

34D20 Stability of solutions to ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
92D30 Epidemiology
Full Text: DOI
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