Behavior of solutions of linear quasiperiodic systems near resonances
Denote (1) \(\dot x=\Lambda x\), \({\dot \phi}=\omega\) where \(x\in C^ n\) or \(x\in R^ n\), \(\phi \in T^{\ell}\), \(T^{\ell}\) is the \(\ell\)- dimensional torus, \(\omega\) is a frequency vector with rationally independent components, \(\Lambda\) is a fixed matrix in Jordan form fulfilling some additional assumptions. Denote by \(\theta^ r(\Lambda,\omega)\) the set of \(C^ r\) systems (2) \(\dot y=B(\phi)y\), \({\dot \phi}=\omega\) reducible to (1) by means of a \(C^{r+1}\)-smooth mapping \(y=H(\phi)x\). System (2) generates a dynamical system \([X^ t_ B]\) on \(T^{\ell}\times P^{n-1}\) where \(P^{n-1}\) is the corresponding projective space. Let \(X_ B(\phi,t)\) be the fundamental matrix of (2). If \(\phi \in T^{\ell}\), \([x]\in P^{n-1}\) then we define \([X^ t_ B](\phi,x)=(\phi +\omega t,[X_ B(\phi,t)x]),\) \(t\in R\). Denote by \(M^ r(\Lambda,\omega)\subset \overline{\theta^ r}(\Lambda,\omega)\) the subset of linear systems for which the corresponding projective system is recurrent. The bar denotes closure in \(C^ r\)-topology. Let (1) satisfy: there exists \(\nu >0\) such that \((3)\quad | \lambda_ a-\lambda_ b-i(k,\omega)| <c| k|^{-2r-\nu}\) for \(a,b=1,...,n\) has a solution \(k\in Z^{\ell}\) for any \(c>0\). Theorem. Assume (1) satisfies (3). Then \(M^ r(\Lambda,\omega)\) is an everywhere dense \(G_{\delta}\)-subset in \(\overline{\theta^ r}(\Lambda,\omega)\). The same result is proved for proximal systems. The results answer the question about the structure of perturbed systems if the condition of incommensurability is cancelled.
scientific article; zbMATH DE number 4148598
scientific article