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Persistence of invariant tori in systems of coupled oscillators. II: Degenerate problems. (English) Zbl 0760.34043

The authors continue their study of persistence of invariant tori to the class of differential equations \((*)\;dx_ i/dt=f_ i(x_ i)+\delta g_ i(x_ 0,x_ 1,\dots,x_ N,\delta)\), \(i=1,\dots,N\), \(dx_ 0/dt=\varepsilon\delta(Bx_ 0+\sum^ N_{i=1}C_ ix_ i)\) describing indirectly coupled oscillators. Here \(x_ 0\in R^{N_ 0}\), \(x_ i\in R^{n_ i}\), the positive parameter \(\varepsilon\) and \(\delta\) measure the relative capacity of the coupling medium and the coupling strength, respectively. It is assumed that each subunit \(dx_ i/dt=f_ i(x_ i)\) has a nonconstant periodic solution \(x_ i=\pi_ i(t)\) with least period \(T_ i>0\) and 1 as a simple multiplier, \(f_ i\) and \(g_ i\) are sufficiently smooth, all eigenvalues of \(B\) have negative real parts, \(\varepsilon=\varepsilon_ 0\delta^{-p}\) with \(p>0\), \(\varepsilon>0\), \(\delta\geq 0\).
For \(\delta=0\), \((*)\) has a \(N_ 0\)-parameter family of \(N\)-dimensional invariant tori. The authors address the problem which, if any, of these tori persist for \(\delta>0\). In a preceding paper [Differ. Integral Equ. 4, 333-368 (1991)] the authors considered the case \(p\geq 1\) (the unperturbed torus is hyperbolic), in the paper under consideration they are concerned with the case \(0<p<1\), that is, the coupling vanishes faster than the capacitance, and the invariant manifold of the unperturbed problem is not normally hyperbolic. To this end they transform \((*)\) into some normal form, prove the persistence of invariant tori of \((*)\) under some conditions by means of a fixed point argument (contraction principle), investigate their smoothness and study the asymptotic behaviour of solutions near these invariant tori.

MSC:

34C30 Manifolds of solutions of ODE (MSC2000)
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34C45 Invariant manifolds for ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
37-XX Dynamical systems and ergodic theory
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