Extension and Unification of Singular Perturbation Methods for ODEs Based on the Renormalization Group Method

The renormalization group (RG) method is one of the singular perturbation methods which is used in search for asymptotic behavior of solutions of differential equations. In this article, time-independent vector fields and time (almost) periodic vector fields are considered. Theorems on error estimates for approximate solutions, existence of approximate invariant manifolds and their stability, inheritance of symmetries from those for the original equation to those for the RG equation, are proved. Further it is proved that the RG method unifies traditional singular perturbation methods, such as the averaging method, the multiple time scale method, the (hyper-) normal forms theory, the center manifold reduction, the geometric singular perturbation method and the phase reduction. A necessary and sufficient condition for the convergence of the infinite order RG equation is also investigated.Comment: publised as SIAM j. on Appl. Dyn.Syst., Vol.8, 1066-1115 (2009

Stability of an [N/2]-dimensional invariant torus in the Kuramoto model at small coupling

When the natural frequencies are allocated symmetrically in the Kuramoto model there exists an invariant torus of dimension [N/2]+1 (N is the population size). A global phase shift invariance allows to reduce the model to $N-1$ dimensions using the phase differences, and doing so the invariant torus becomes [N/2]-dimensional. By means of perturbative calculations based on the renormalization group technique, we show that this torus is asymptotically stable at small coupling if N is odd. If N is even the torus can be stable or unstable depending on the natural frequencies, and both possibilities persist in the small coupling limit.Comment: to appear in Physica

Periodic orbits and chaos in fast-slow systems with Bogdanov-Takens type fold points

The existence of stable periodic orbits and chaotic invariant sets of singularly perturbed problems of fast-slow type having Bogdanov-Takens bifurcation points in its fast subsystem is proved by means of the geometric singular perturbation method and the blow-up method. In particular, the blow-up method is effectively used for analyzing the flow near the Bogdanov-Takens type fold point in order to show that a slow manifold near the fold point is extended along the Boutroux's tritronqu\'{e}e solution of the first Painlev\'{e} equation in the blow-up space

Geometry of integrable non-Hamiltonian systems

This is an expanded version of the lecture notes for a minicourse that I gave at a summer school called "Advanced Course on Geometry and Dynamics of Integrable Systems" at CRM Barcelona, 9--14/September/2013. In this text we study the following aspects of integrable non-Hamiltonian systems: local and semi-local normal forms and associated torus actions for integrable systems, and the geometry of integrable systems of type $(n,0)$. Most of the results presented in this text are very recent, and some theorems in this text are even original in the sense that they have not been written down explicitly elsewhere.Comment: 54 pages, 12 figures. arXiv admin note: substantial text overlap with arXiv:1203.276

Renormalization Group Method

Renormalization Group (RG) method is a general method whose aim is to globally approximate solutions to differential equations involving a small parameter. In this thesis, we will give an algorithm for the RG method to generate the RG equation needed in the process of finding an approximate solution for ODEs. In chapter 1, we have some introduction to perturbation theory and introducing some traditional methods in perturbation theory. In chapter 2 we compare the results of RG and other conventional methods using numerical or explicit methods. Thereafter, in chapter 3, we rigorously compare the approximate solution obtained using the RG method and the true solution using two classes of system of ordinary differential equations. In chapter 4, we present a simplified RG method and apply it to the second order RG. In chapter 5 we briefly explain the first order Normal Form (NF) theory and then its relation to the RG method. Also a similar geometric interpretation for the RG equation and NF's outcome has been provided. In the Appendix, we have added definitions and proofs used in this thesis. The RG method is much more straightforward than other traditional methods and does not require prior information about the solutions. One begins with a naive perturbative expansion which already contains all the necessary information that we need to construct a solution. Using RG, there is no need to asymptotically match the solutions in the overlapping regions, which is a key point in some other methods. In addition, the RG method is applicable to most of perturbed differential equations and will produce a closed form solution which is, most of the times, as accurate as or even more accurate than the solutions obtained by other conventional methods