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Massera’s convergence theorem for periodic nonlinear differential equations. (English) Zbl 0603.34033
The paper provides a new method for proving the existence of at least one stable periodic solution and the absence of chaotic motions for vector differential equations (*) $$dx/dt=f(t,x)$$ which are $$\sigma$$-periodic in t, locally Lip in x and satisfy the following hypothesis: there exist positive scalars $$\lambda$$, $$\epsilon$$ and a real symmetric non-singular $$n\times n$$ matrix P, with signature n-2, such that $(x_ 1-x_ 2)^ TP[f(t,x_ 1)-f(t,x_ 2)+\lambda (x_ 1-x_ 2)]\leq -\epsilon | x_ 1-x_ 2|^ 2,$ for $$t\in {\mathbb{R}}$$ and $$x_ 1$$, $$x_ 2\in open$$ $$S\subset {\mathbb{R}}^ n$$. It is proved that if a solution x(t) of (*) lies in a compact $$S_ 0\subset S$$ for $$t_ 0\leq t<\infty$$ then x(t) converges to a $$\sigma$$-periodic solution u(t) as $$t\to +\infty$$. If, in addition, there exists a bounded open D, with $$\bar D\subset S$$, whose boundary $$\partial D$$ is crossed inwards by all solutions of (*) which meet it then D contains at least one $$\sigma$$-periodic solution u(t) which is Lyapunov stable. This u(t) is also asymptotically stable if f(t,x) is analytic in $${\mathbb{R}}\times \bar D$$. These results extend a theorem concerning periodic scalar equations by J. L. Massera [Duke Math. J. 17, 457-475 (1950; Zbl 0038.250)].

##### MSC:
 34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations 34D20 Stability of solutions to ordinary differential equations
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##### References:
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