Renormalization of continuous-time dynamical systems with KAM applications

In this dissertation, we construct a sequence of renormalization group transformations on a space of analytic vector fields. We apply these transformations to study the persistence of quasiperiodic motion (invariant tori) with sufficiently incommensurate frequency vectors w in near-integrable systems. The renormalization transformations preserve geometrical “classes” of the vector fields, such as Hamiltonian, divergence-free, time-reversible, and symmetric with respect to an involution. Two different approaches have been developed. One approach makes use of a recent multidimensional generalization of the continued fraction algorithm and applies to Diophantine frequency vectors w. The other approach applies to the larger set of Brjuno frequency vectors. We prove the existence of an integrable limit set of the renormalization and show that there exists a finite-codimension stable manifold W for the sequence of renormalization maps, associated to this set. We show that every vector field on W has an analytic elliptic invariant torus on which the flow is conjugate to a rotation with a Diophantine or, more generally, Brjuno frequency vector w. Consequently, every family of vector fields that intersects W has a member which has an analytic invariant torus with frequency vector w. We show that the number of parameters of a family can be reduced if a non-degeneracy condition is satisfied. In certain classes of vector fields, e.g. Hamiltonian vector fields, the number of parameters can be reduced to zero, and analogous statements are true for individual vector fields. In the special case of two degree of freedom Hamiltonian vector fields we also construct a sequence of renormalization group transformations with an attracting integrable limit set, directly on a space of Hamiltonian functions. As an application of the scheme we give a proof of KAM theorem for Hamiltonians satisfying a nondegeneracy condition. On a numerical level, the scheme can be applied to obtain the critical function of one-parameter families of two-degree of freedom Hamiltonian systems.