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Sign changing Lyapunov functions and perturbation of invariant tori of dynamical systems. (English) Zbl 0698.34044
Dynamical systems and ergodic theory, 28th Sem. St. Banach Int. Math. Cent., Warsaw/Pol. 1986, Banach Cent. Publ. 23, 385-389 (1989).
[For the entire collection see Zbl 0686.00015.]
The author introduces the notion of the Green function \(G_ 0(\tau,\phi)\) for the differential system \({\dot \phi}=a(\phi)\), \(\dot x=A(\phi)x\), where \(a(\phi)\) is a vector-valued function and \(A(\phi)\) is a matrix-valued function \(2\pi\)-periodic in the coordinates of \(\phi\). The vector-valued function \(x=\int^{\infty}_{-\infty}G_ 0(\tau,\phi)f(\phi_{\tau}(\phi))d\tau\) is an invariant torus of the system \({\dot \phi}=a(\phi)\), \(\dot x=A(\phi)x+f(\phi)\), where f is also \(2\pi\)-periodic. Sufficient conditions for the existence of the Green function are given.
Reviewer: L.Hatvani
MSC:
34D20 Stability of solutions to ordinary differential equations
37-XX Dynamical systems and ergodic theory
34B27 Green’s functions for ordinary differential equations
Keywords:
Green function