We present Kolmogorov–Arnold–Moser (or KAM) Theory regarding typicality of quasi-periodic invariant tori, partly from a historical and partly from a pedagogical point of view At the same time we aim at a unified approach of the theory in various dynamical settings: the ‘classical’ Hamiltonian setting of Lagrangean tori, the Hamiltonian lower dimensional isotropic tori, the dissipative case of quasi-periodic attractors, etc Also we sketch the theory of quasi-periodic bifurcations, where resonances cause Cantorization and fraying of the bifurcation sets known from the cases of equilibrium points and periodic orbits Here the concept of Whitney differentiability plays a central role, which locally organizes the nowhere dense union of persistent quasi-periodic invariant tori, of positive measure At the level of torus bundles this Cantorization is observed as well, where the geometry of the torus bundles turns out to be persistent In the meantime we briefly deal with the natural affine structure of quasi-periodic tori, with uniqueness of most of the KAM tori, and with the mechanisms of the destruction of resonant unperturbed tori Other parts of the theory, such as the Hamiltonian higher

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KAM Theory : Quasi-periodicity in Dynamical Systems

A sharpened version of Moser's 'modifying terms' KAM theorem is
 derived, and it is shown how this theorem can be used to
 investigate the persistence of invariant tori in general
 situations, including those where some of the Floquet exponents
 of the invariant torus may vanish. The result is 'structural'
 and can be applied to dissipative, Hamiltonian, and symmetric
 vector fields; moreover, we give variants of the result for real
 analytic, Gevrey regular ultradifferentiable
 and finitely differentiable vector fields. In the first two
 cases, the conjugacy constructed in the theorem is shown to be
 Gevrey smooth in the sense of Whitney on the set of parameters
 that satisfy a "Diophantine'' non-resonance condition.

/pdf/a-parametrised-version-of-moser-s-modifying-terms-theorem-jbs6c6v6ts.pdf

A parametrised version of Moser's modifying terms theorem

Under a small perturbation of a completely integrable Hamiltonian system, invariant tori with Diophantine frequencies of motion are not destroyed but only slightly deformed, provided that the Hessian (with respect to the action variables) of the unperturbed Hamiltonian vanishes nowhere (the Kolmogorov nondegeneracy). The motion on every perturbed torus is quasiperiodic with the same frequencies. In this sense the frequencies of invariant tori of the unperturbed system are preserved. Recently, it has been found that the Kolmogorov nondegeneracy condition can be weakened so as to guarantee the preservation of only some subset of frequencies. Such partial preservation of frequencies can also be defined for lower dimensional invariant tori, whose dimension is less than the number of degrees of freedom. We consider a more general problem of partial preservation not only of the frequencies of invariant tori but also of their Floquet exponents (the eigenvalues of the coefficient matrix of the variational equation along the torus). The results are formulated for Hamiltonian, reversible, and dissipative systems (with a complete proof for the reversible case).

Partial preservation of frequencies and floquet exponents in KAM theory

Linearization of Gevrey flows on Td with a Brjuno type arithmetical condition

In this paper we prove the persistence of invariant tori for analytic perturbation of constant vector field under weaker non-degeneracy condition. In the proof we introduce a parameter q and make the steps of KAM iteration infinitely small in the speed of function \({q^{n} \epsilon}\), \({0 < q < 1}\), rather than super exponential function.

/pdf/on-invariant-tori-of-vector-field-under-weaker-non-3lm7wxcp12.pdf

On invariant tori of vector field under weaker non-degeneracy condition

In this paper we prove the persistence of elliptic lower
dimensional invariant tori for nearly integrable Gevrey-smooth
Hamiltonian systems under Russmann's non-degeneracy condition by
an improved KAM iteration, and the persisting invariant tori are
Gevrey smooth with respect to parameters in the sense of Whitney,
with a Gevrey index depending on the Gevrey class of Hamiltonian
systems and on the exponent in the Diophantine condition. Moreover
the Gevrey index should be optimal for the Diophantine condition
in the proof of our theorem.

Dongfeng Zhang

Papers

On elliptic lower dimensional tori for Gevrey-smooth Hamiltonian systems under Rüssmann's non-degeneracy condition