## 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

Zbl 0038.250
Full Text:

### References:

 [1] Brockett, R. W., Finite Dimensional Linear Systems (1970), Wiley: Wiley New York · Zbl 0216.27401 [2] Cartwright, M. L., Almost periodic flows and solutions of differential equations, (Proc. London Math. Soc., 17 (1967)), 355-380 · Zbl 0155.41901 [3] Cronin, J., A criterion for asymptotic stability, J. Math. Anal. Appl., 74, 247-269 (1980) · Zbl 0448.34055 [4] Cronin, J., Differential Equations: Introduction and Qualitative Theory (1980), Dekker: Dekker New York · Zbl 0429.34002 [5] 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 [6] Kolesov, Ju. S.; Krasnoselskii, M. A., Ljapunov stability and equations with concave operators, Soviet Math. Dokl., 3, 1192-1195 (1962) · Zbl 0125.32604 [7] Kolesov, Ju. S., Schauder’s principle and the stability of periodic solutions, Soviet Math. Dokl., 10, 1290-1293 (1969) · Zbl 0196.16202 [8] Lloyd, N. G., Degree Theory (1978), Cambridge Univ. Press: Cambridge Univ. Press London · Zbl 0367.47001 [9] Massera, J. L., The existence of periodic solutions of systems of differential equations, Duke Math. J., 17, 457-475 (1950) · Zbl 0038.25002 [10] Pliss, V. A., Nonlocal Problems of the Theory of Oscillations (1966), Academic Press: Academic Press New York · Zbl 0151.12104 [11] Reissig, R.; Sansone, G.; Conti, R., Non-linear Differential Equations of Higher Order (1974), Noordhoff: Noordhoff Leyden · Zbl 0275.34001 [12] Rouche, N.; Mawhin, J., Ordinary Differential Equations, Stability and Periodic Solutions (1980), Pitman: Pitman London · Zbl 0433.34001 [13] Sell, G. R., Periodic solutions and asymptotic stability, J. Differential Equations, 2, 143-157 (1966) · Zbl 0136.08501 [14] Sell, G. R., Topological Dynamics and Ordinary Differential Equations (1971), Van Nostrand-Reinhold: Van Nostrand-Reinhold London · Zbl 0212.29202 [15] Smith, R. A., Absolute stability of certain differential equations, J. London Math. Soc., 7, 203-210 (1973) · Zbl 0269.34043 [16] Smith, R. A., Poincaré index theorem concerning periodic orbits of differential equations, (Proc. London Math. Soc., 48 (1984)), 341-362 · Zbl 0509.34046 [17] Smith, R. A., Certain differential equations have only isolated periodic orbits, Ann. Mat. Pura Appl., 137, 217-244 (1984) · Zbl 0561.34028 [18] 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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