Realization of tangent perturbations in discrete and continuous time conservative systems
Hassan Najafi Alishah João Lopes Dias
Discrete & Continuous Dynamical Systems - A 2014, 34(12): 5359-5374 doi: 10.3934/dcds.2014.34.5359
We prove that any perturbation of the symplectic part of the derivative of a Poisson diffeomorphism can be realized as the derivative of a $C^1$-close Poisson diffeomorphism. We also show that a similar property holds for the Poincaré map of a Hamiltonian on a Poisson manifold. These results are the conservative counterparts of the Franks lemma, a perturbation tool used in several contexts most notably in the theory of smooth dynamical systems.
keywords: Poisson Hamiltonian flowbox coordinates. Franks lemma symplectic and Hamiltonian $C^1$ perturbations
Brjuno condition and renormalization for Poincaré flows
João Lopes Dias
Discrete & Continuous Dynamical Systems - A 2006, 15(2): 641-656 doi: 10.3934/dcds.2006.15.641
In this paper we give a new proof of the local analytic linearization of flows on T2 with a Brjuno rotation number, using renormalization techniques.
keywords: analytic conjugacy to linear flow. Poincaré flows continued fractions Brjuno numbers small divisors Renormalization of vector fields
Renormalization of diophantine skew flows, with applications to the reducibility problem
Hans Koch João Lopes Dias
Discrete & Continuous Dynamical Systems - A 2008, 21(2): 477-500 doi: 10.3934/dcds.2008.21.477
We introduce a renormalization group framework for the study of quasiperiodic skew flows on Lie groups of real or complex $n\times n$ matrices, for arbitrary Diophantine frequency vectors in $R^{d}$ and dimensions $d,n$. In cases where the Lie algebra component of the vector field is small, it is shown that there exists an analytic manifold of reducible skew systems, for each Diophantine frequency vector. More general near-linear flows are mapped to this case by increasing the dimension of the torus. This strategy is applied for the group of unimodular $2\times 2$ matrices, where the stable manifold is identified with the set of skew systems having a fixed fibered rotation number. Our results apply to vector fields of class Cγ, with $\gamma$ depending on the number of independent frequencies, and on the Diophantine exponent.
keywords: reducibility of skew flows. Key words and phrases: Renormalization of flows

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