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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)

##### MSC:
 34D20 Stability of solutions to ordinary differential equations 34C25 Periodic solutions to ordinary differential equations 92D30 Epidemiology
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##### References:
 [1] Bonckaert, P; Dumortier, F; van Strien, S, Singularities of vector fields on $$R$$^3 determined by their first nonvanishing jet, Ergodic theory dynamical systems, 9, 281-308, (1989) · Zbl 0661.58005 [2] Busenberg, S.N; Jaderberg, L.K, Decay characteristics for differential equations without linear terms, J. differential equations, 18, 87-102, (1975) · Zbl 0297.34006 [3] Busenberg, S; van den Driessche, P, Analysis of a disease transmission model in a population with varying size, J. math. biol., 28, 257-270, (1990) · Zbl 0725.92021 [4] Busenberg, S; Hadeler, K.P, Demography and epidemics, preprint, emory university, 1989, Math. biosci., 101, 63-74, (1990) · Zbl 0751.92012 [5] Busenberg, S; Cooke, K; Thieme, H, Investigation of the transmission and persistence of HIV/AIDS in a heterogeneous population, SIAM J. appl. math., (1989), to appear [6] Camacho, M.I.T, Geometric properties of homogeneous vector fields of degree two in $$R$$^3, Trans. amer. math. soc., 268, 79-101, (1981) · Zbl 0484.58020 [7] Coleman, C, Equivalence of planar dynamical and differential systems, J. differential equations, 1, 222-233, (1965) · Zbl 0135.30704 [8] Coleman, C, Growth and decay estimates near non-elementary stationary points, Canad. J. math., 22, 1156-1167, (1970) · Zbl 0231.34038 [9] Dietz, K; Hadeler, K.P, Epidemiological models for sexually transmitted diseases, J. math. biol., 26, 1-25, (1988) · Zbl 0643.92015 [10] Dumortier, F; Roussarie, R, Smooth linearization of germs of $$R$$^2-actions and holomorphic vector fields, Ann. inst. Fourier (Grenoble), 30, 31-64, (1980) · Zbl 0418.58015 [11] Golubitsky, M; Stewart, I; Schaeffer, D.G, () [12] Hadeler, K.P, On copositive matrices, Linear algebra appl., 49, 79-89, (1983) · Zbl 0506.15016 [13] Hadeler, K.P; Waldstätter, R; Wörz, A, Models for pair formation in two-sex populations, J. math. biol., 26, 635-649, (1988) · Zbl 0714.92018 [14] Hadeler, K.P, Homogeneous delay equations and models for pair formation, preprint, Georgia institute of technology, J. math. biol., (1990), to appear [15] Hahn, W, Stability of motion, () · Zbl 0189.38503 [16] Hahn, W, Über differentialgleichungen erster ordnung mit homogenen rechten seiten, Z. angew. math. mech., 46, 357-361, (1966) · Zbl 0173.10701 [17] hofbauer, J; Sigmund, K, The theory of evolution and dynamical systems, (1988), Cambridge Univ. Press London
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