Periodic solutions of homogeneous equations.

*(English)*Zbl 0747.34030The 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 |

##### Keywords:

stability properties; differential equations; homogeneous; sphere; exponential solutions; periodic solutions; spiral solutions; Lyapunov function; projected system; Bendixson-Dulac criterion; Kermack-McKendrick desease transmission model; Fisher’s equation
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\textit{K. P. Hadeler}, J. Differ. Equations 95, No. 1, 183--202 (1992; Zbl 0747.34030)

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##### References:

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