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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:
  Brockett, R.W, Finite dimensional linear systems, (1970), Wiley New York · Zbl 0216.27401  Cartwright, M.L, Almost periodic flows and solutions of differential equations, (), 355-380 · Zbl 0155.41901  Cronin, J, A criterion for asymptotic stability, J. math. anal. appl., 74, 247-269, (1980) · Zbl 0448.34055  Cronin, J, Differential equations: introduction and qualitative theory, (1980), Dekker New York · Zbl 0429.34002  Demidovich, B.P, On the dissipativity of certain non-linear systems of differential equations I, Vestnik Moscow. univ. ser. I mat. mek., 6, 19-27, (1961), [Russian] · Zbl 0121.31304  Kolesov, Ju.S; Krasnoselskii, M.A, Ljapunov stability and equations with concave operators, Soviet math. dokl., 3, 1192-1195, (1962) · Zbl 0125.32604  Kolesov, Ju.S, Schauder’s principle and the stability of periodic solutions, Soviet math. dokl., 10, 1290-1293, (1969) · Zbl 0196.16202  Lloyd, N.G, Degree theory, (1978), Cambridge Univ. Press London · Zbl 0367.47001  Massera, J.L, The existence of periodic solutions of systems of differential equations, Duke math. J., 17, 457-475, (1950) · Zbl 0038.25002  Pliss, V.A, Nonlocal problems of the theory of oscillations, (1966), Academic Press New York · Zbl 0151.12104  Reissig, R; Sansone, G; Conti, R, Non-linear differential equations of higher order, (1974), Noordhoff Leyden · Zbl 0275.34001  Rouche, N; Mawhin, J, Ordinary differential equations, stability and periodic solutions, (1980), Pitman London · Zbl 0433.34001  Sell, G.R, Periodic solutions and asymptotic stability, J. differential equations, 2, 143-157, (1966) · Zbl 0136.08501  Sell, G.R, Topological dynamics and ordinary differential equations, (1971), Van Nostrand-Reinhold London · Zbl 0212.29202  Smith, R.A, Absolute stability of certain differential equations, J. London math. soc., 7, 203-210, (1973) · Zbl 0269.34043  Smith, R.A, Poincaré index theorem concerning periodic orbits of differential equations, (), 341-362 · Zbl 0509.34046  Smith, R.A, Certain differential equations have only isolated periodic orbits, Ann. mat. pura appl., 137, 217-244, (1984) · Zbl 0561.34028  Yoshizawa, T, Stable sets and periodic solutions in a perturbed system, Contrib. differential equations, 2, 407-420, (1963) · Zbl 0131.08605
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