Stabilité des solutions périodiques ou anti-périodiques de certains systèmes différentiels

Daad Abou Saleh

Abstract: We consider the non linear differential system in $\R^2$
$$ \left\{\begin{array}{l}
u'(t)+ku(t)\Bigl(u(t)^2+v(t)^2\Bigr)-\lambda u(t)=h_1(t),
v'(t)+kv(t)\Bigl(u(t)^2+v(t)^2\Bigr)-\lambda v(t)=h_2(t).
\end{array}\right.$$
In order to study the stability of anti-periodic solutions of this system, a result of Liapounov in the stability theory of autonomous nonlinear ODE is enlarged to diffenrentials systems of the form $u'=F(t,u)$ in the Hilbert space framework with finite dimension. On the other hand, a result of R. Bellman in the instability theory of autonomous nonlinear ODE is enlarged to diffenrentials systems where $L$, the linearized operator for $F$, is a nonautonomous periodic operator.

Keywords: periodic solutions; stability.

Classification (MSC2000): 34A34, 34D20, 34D23.

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