scholarly journals Partial Preservation of Frequencies and Floquet Exponents of Invariant Tori in the Reversible KAM Context 2

2017 ◽  
Vol 63 (3) ◽  
pp. 516-541
Author(s):  
M B Sevryuk
Keyword(s):  
Fixed Point ◽  
Invariant Torus ◽  
Variational Equation ◽  
Invariant Tori ◽  
Kam Theory ◽  
Coefficient Matrix ◽  
Nondegeneracy Conditions

We consider the persistence of smooth families of invariant tori in the reversible context 2 of KAM theory under various weak nondegeneracy conditions via Herman’s method. The reversible KAM context 2 refers to the situation where the dimension of the fixed point manifold of the reversing involution is less than half the codimension of the invariant torus in question. The nondegeneracy conditions we employ ensure the preservation of any prescribed subsets of the frequencies of the unperturbed tori and of their Floquet exponents (the eigenvalues of the coefficient matrix of the variational equation along the torus).

scholarly journals THE QUASI-PERIODIC REVERSIBLE HOPF BIFURCATION

2007 ◽  
Vol 17 (08) ◽  
pp. 2605-2623 ◽  
Author(s):  
HENK W. BROER ◽  
M. CRISTINA CIOCCI ◽  
HEINZ HANßMANN
Keyword(s):  
Hopf Bifurcation ◽  
Bifurcation Point ◽  
Invariant Tori ◽  
Kam Theory ◽  
Cantor Sets ◽  
Reversible Systems ◽  
Periodic Dynamics

We consider the perturbed quasi-periodic dynamics of a family of reversible systems with normally 1:1 resonant invariant tori. We focus on the generic quasi-periodic reversible Hopf bifurcation and address the persistence problem for integrable quasi-periodic tori near the bifurcation point. Using KAM theory, we describe how the resulting invariant tori of maximal and lower dimensions are parameterized by Cantor sets.

scholarly journals Analytic Lagrangian tori for the planetary many-body problem

2009 ◽  
Vol 29 (3) ◽  
pp. 849-873 ◽  
Author(s):  
LUIGI CHIERCHIA ◽  
FABIO PUSATERI
Keyword(s):  
Body Problem ◽  
Invariant Tori ◽  
Kam Theory ◽  
Many Body ◽  
Complete Proof ◽  
Small Denominators ◽  
C Dynamics ◽  
Lagrangian Tori ◽  
Real Analytic ◽  
Nearly Integrable Hamiltonian Systems

AbstractIn 2004, Féjoz [Démonstration du ‘théoréme d’Arnold’ sur la stabilité du système planétaire (d’après M. Herman). Ergod. Th. & Dynam. Sys.24(5) (2004), 1521–1582], completing investigations of Herman’s [Démonstration d’un théoréme de V.I. Arnold. Séminaire de Systémes Dynamiques et manuscripts, 1998], gave a complete proof of ‘Arnold’s Theorem’ [V. I. Arnol’d. Small denominators and problems of stability of motion in classical and celestial mechanics. Uspekhi Mat. Nauk. 18(6(114)) (1963), 91–192] on the planetary many-body problem, establishing, in particular, the existence of a positive measure set of smooth (C∞) Lagrangian invariant tori for the planetary many-body problem. Here, using Rüßmann’s 2001 KAM theory [H. Rüßmann. Invariant tori in non-degenerate nearly integrable Hamiltonian systems. R. & C. Dynamics2(6) (2001), 119–203], we prove the above result in the real-analytic class.

scholarly journals Existence of Positive Solutions for a Class of Nonlinear Algebraic Systems

2016 ◽  
Vol 2016 ◽  
pp. 1-7
Author(s):  
Yongqiang Du ◽  
Guang Zhang ◽  
Wenying Feng
Keyword(s):  
Fixed Point ◽  
Positive Solutions ◽  
Nonlinear Term ◽  
Zero Point ◽  
Coefficient Matrix ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Algebraic Systems ◽  
Infinite Point ◽  
Nonlinear Algebraic Systems ◽  
Existence Of Positive Solutions

Based on Guo-Krasnoselskii’s fixed point theorem, the existence of positive solutions for a class of nonlinear algebraic systems of the formx=GFxis studied firstly, whereGis a positiven×nsquare matrix,x=col⁡(x1,x2,…,xn), andF(x)=col⁡(f(x1),f(x2),…,f(xn)), where,F(x)is not required to be satisfied sublinear or superlinear at zero point and infinite point. In addition, a new cone is constructed inRn. Secondly, the obtained results can be extended to some more general nonlinear algebraic systems, where the coefficient matrixGand the nonlinear term are depended on the variablex. Corresponding examples are given to illustrate these results.

A NEWTON METHOD FOR LOCATING INVARIANT TORI OF MAPS

2006 ◽  
Vol 16 (05) ◽  
pp. 1491-1503 ◽  
Author(s):  
HARRY DANKOWICZ ◽  
GUNJAN THAKUR
Keyword(s):  
Dynamical Systems ◽  
Invariant Torus ◽  
Local Stability ◽  
Discrete System ◽  
Invariant Tori ◽  
A Priori ◽  
Spatial Location ◽  
Fixed Time ◽  
Periodic Trajectory ◽  
Shift Function

This paper presents an iterative method for locating invariant tori of maps. Specifically, through the introduction of a shift function on the torus corresponding to the projection of the dynamics on the torus onto toral parameters, a discrete system of equations may be formulated whose solution approximates the spatial location of the torus. As invariant tori of continuous flows may be considered invariant under the application of the flow for a fixed time, or may correspond to invariant tori (of one less dimension) of suitably introduced Poincaré maps, the methodology applies to discrete and continuous dynamical systems alike. Moreover, the insensitivity of the method to the local stability characteristics of the invariant torus as well as to the precise nature of the flow on the torus imply, for example, that the method can be employed for the continuation of unstable invariant tori on which the dynamics are attracted to a periodic trajectory as long as the torus is sufficiently smooth. The proposed methodology as well as reduced formulations that result from a priori knowledge about the invariant torus are illustrated through some sample dynamical systems.

On Invariant Tori with Prescribed Frequency in Hamiltonian Systems

2016 ◽  
Vol 16 (4) ◽  
Author(s):  
Dongfeng Zhang ◽  
Junxiang Xu ◽  
Hao Wu
Keyword(s):  
Hamiltonian Systems ◽  
Small Perturbation ◽  
Topological Degree ◽  
Invariant Torus ◽  
Invariant Tori ◽  
Rational Approximations ◽  
Integrable Hamiltonian Systems ◽  
Resonant Condition ◽  
Degeneracy Condition ◽  
Nearly Integrable Hamiltonian Systems

AbstractIn this paper we are mainly concerned with the persistence of invariant tori with prescribed frequency for analytic nearly integrable Hamiltonian systems under the Brjuno–Rüssmann non-resonant condition, when the Kolmogorov non-degeneracy condition is violated. As it is well known, the frequency of the persisting invariant tori may undergo some drifts, when the Kolmogorov non-degeneracy condition is violated. By the method of introducing external parameters and rational approximations, we prove that if the Brouwer topological degree of the frequency mapping is nonzero at some Brjuno–Rüssmann frequency, then the invariant torus with this frequency persists under small perturbation.

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