Normal forms of C∞ vector fields based on the renormalization group

2021 ◽  
Vol 62 (6) ◽  
pp. 062703
Author(s):  
Hayato Chiba
Keyword(s):  
Renormalization Group ◽  
Vector Fields ◽  
Normal Forms

scholarly journals On the Computation of Degenerate Hopf Bifurcations forn-Dimensional Multiparameter Vector Fields

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Michail P. Markakis ◽  
Panagiotis S. Douris
Keyword(s):  
Vector Fields ◽  
Center Manifold ◽  
Normal Forms ◽  
Computer Assisted ◽  
Hopf Bifurcations ◽  
Symbolic Form ◽  
Parametric System ◽  
Analytical Expressions

The restriction of ann-dimensional nonlinear parametric system on the center manifold is treated via a new proper symbolic form and analytical expressions of the involved quantities are obtained as functions of the parameters by lengthy algebraic manipulations combined with computer assisted calculations. Normal forms regarding degenerate Hopf bifurcations up to codimension 3, as well as the corresponding Lyapunov coefficients and bifurcation portraits, can be easily computed for any system under consideration.

Normal Forms for Codimension One Planar Piecewise Smooth Vector Fields

2014 ◽  
Vol 24 (07) ◽  
pp. 1450090 ◽  
Author(s):  
Tiago de Carvalho ◽  
Durval José Tonon
Keyword(s):  
Vector Field ◽  
Normal Form ◽  
Vector Fields ◽  
Normal Forms ◽  
Piecewise Smooth ◽  
Topological Equivalence ◽  
Smooth Vector ◽  
Codimension One ◽  
Piecewise Smooth Vector Fields ◽  
Piecewise Smooth Vector Field

In this paper, we are dealing with piecewise smooth vector fields in a 2D-manifold. In such a scenario, the main goal of this paper is to exhibit the homeomorphism that gives the topological equivalence between a codimension one piecewise smooth vector field and the respective C0-normal form.

scholarly journals Stability and boundedness in AdS/CFT with double trace deformations

2019 ◽  
Vol 34 (18) ◽  
pp. 1950138 ◽  
Author(s):  
Steven Casper ◽  
William Cottrell ◽  
Akikazu Hashimoto ◽  
Andrew Loveridge ◽  
Duncan Pettengill
Keyword(s):  
Renormalization Group ◽  
Boundary Conditions ◽  
Vector Fields ◽  
Scalar Fields ◽  
Legendre Transform ◽  
Renormalization Group Flow ◽  
Physical Content ◽  
Effective Masses ◽  
Group Flow

Scalar fields on the bulk side of AdS/CFT correspondence can be assigned unconventional boundary conditions related to the conventional one by Legendre transform. One can further perform double trace deformations which relate the two boundary conditions via renormalization group flow. Thinking of these operators as S and T transformations, respectively, we explore the SL(2, R) family of models which naively emerges from repeatedly applying these operations. Depending on the parameters, the effective masses vary and can render the theory unstable. However, unlike in the SL(2, Z) structure previously seen in the context of vector fields in AdS4, some of the features arising from this exercise, such as the vacuum susceptibility, turns out to be scheme dependent. We explain how scheme independent physical content can be extracted in spite of some degree of scheme dependence in certain quantities.

scholarly journals Normal forms for perturbations of systems possessing a Diophantine invariant torus

2017 ◽  
Vol 39 (8) ◽  
pp. 2176-2222 ◽  
Author(s):  
JESSICA ELISA MASSETTI
Keyword(s):  
Vector Fields ◽  
Normal Forms ◽  
Invariant Tori ◽  
Inverse Function ◽  
Hamiltonian Mechanics ◽  
Inverse Function Theorem ◽  
Unified Framework ◽  
Normal Form Theorem ◽  
Real Analytic ◽  
Twist Theorem

We give a new proof of Moser’s 1967 normal-form theorem for real analytic perturbations of vector fields possessing a reducible Diophantine invariant quasi-periodic torus. The proposed approach, based on an inverse function theorem in analytic class, is flexible and can be adapted to several contexts. This allows us to prove in a unified framework the persistence, up to finitely many parameters, of Diophantine quasi-periodic normally hyperbolic reducible invariant tori for vector fields originating from dissipative generalizations of Hamiltonian mechanics. As a byproduct, generalizations of Herman’s twist theorem and Rüssmann’s translated curve theorem are proved.

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