Periodic solutions for $dot x=Ax+g(x,,t)+varepsilon p(t)$

Peter J. Ponzo
1971 Canadian mathematical bulletin  
We wish to establish the existence of a periodic solution to (1) x = Ax+g(x, t) + ep(t), ( # = d/dt) where x, g and p are «-vectors, A is an n x n constant matrix, and € is a small scalar parameter. We assume that g and p are locally Lipschitz in x and continuous and T-periodic in t, and that the origin is a point of asymptotically stable equilibrium, when € = 0. Although the result below is not new ([1], [2]), the proof is simple and of some interest and provides an explicit bound on c which
more » ... ll guarantee the existence of a T-periodic solution. It also gives a bound on the norm of the periodic solution. In what follows, ||. || denotes the Euclidean norm. THEOREM. If (i) p(t + T)=p(t) and \\p(t)\\ < 1 for all t, (ii) g(x, t-\-T)=g(x, t) and \g(x, t)\\ =o(\\x\\) uniformly in t, (iii) the eigenvalues of A have negative real part, then (1) possesses a T-periodic solution for e sufficiently small. Proof. We wish to select the constant c>0 such that the surface V(x)=x T Bx-c 2 confines interior trajectories, where B is the unique, real, symmetric, positive definite matrix which satisfies A T B+BA= -I (/, the unit matrix). Assuming the existence of such a constant, we may apply Brouwer's fixed point theorem to the region V(x)0 and A be the smallest and largest eigenvalues of B, respectively. Then A||x|| 2 <x r^< A||x|| 2 so that V(x) = c 2 lies in (2) c/VA < ||x|| < c/Vx In order that the surface V=c 2 confine interior trajectories, it is sufficient to have dV/dt<0 everywhere on the surface. We have dV/dt = -x T x+2(g T +€ P T )Bx. 575 Downloaded from https://www.cambridge.org/core. 12 Dec 2020 at 20:04:58, subject to the Cambridge Core terms of use.
doi:10.4153/cmb-1971-105-4 fatcat:ep3sac2utjbenpilsvnnwvhv3y