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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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