scholarly journals The Existence of Invariant Tori and Quasiperiodic Solutions of the Nosé–Hoover Oscillator

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Yanmin Niu ◽  
Xiong Li
Keyword(s):  
Invariant Tori ◽  
Equivalent Form ◽  
Real Parameter ◽  
Dimensional System ◽  
Reversible System ◽  
Positive Real ◽  
3 Dimensional ◽  
Quasiperiodic Solutions ◽  
Positive Real Parameter ◽  
Twist Theorem

In this paper, we consider an equivalent form of the Nosé–Hoover oscillator, x ′ = y , y ′ = − x − y z ,   and   z ′ = y 2 − a , where a is a positive real parameter. Under a series of transformations, it is transformed into a 2-dimensional reversible system about action-angle variables. By applying a version of twist theorem established by Liu and Song in 2004 for reversible mappings, we find infinitely many invariant tori whenever a is sufficiently small, which eventually turns out that the solutions starting on the invariant tori are quasiperiodic. The discussion about quasiperiodic solutions of such 3-dimensional system is relatively new.

scholarly journals EXISTENCE OF COHERENT SYSTEMS

2008 ◽  
Vol 19 (04) ◽  
pp. 449-454 ◽  
Author(s):  
MONTSERRAT TEIXIDOR I. BIGAS
Keyword(s):  
Vector Bundle ◽  
Vector Space ◽  
Moduli Spaces ◽  
Stability Condition ◽  
Real Parameter ◽  
Coherent System ◽  
Coherent Systems ◽  
Positive Real ◽  
Mild Conditions ◽  
Positive Real Parameter

A coherent system of type (r, d, k) on a curve consists of a vector bundle of rank r and degree d together with a vector space of dimension k of the sections of this vector bundle. There is a stability condition depending on a positive real parameter α that allows to construct moduli spaces for these objects. This paper shows non-emptiness of these moduli spaces when k > r for any α under some mild conditions on the degree and genus.

scholarly journals Existence of Three Solutions for a Nonlinear Fractional Boundary Value Problem via a Critical Points Theorem

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Chuanzhi Bai
Keyword(s):  
Boundary Value Problem ◽  
Critical Points ◽  
Boundary Value ◽  
Real Parameter ◽  
Fractional Boundary Value Problem ◽  
Positive Real ◽  
Three Solutions ◽  
Positive Real Parameter

This paper is concerned with the existence of three solutions to a nonlinear fractional boundary value problem as follows:(d/dt)((1/2)0Dtα-1(0CDtαu(t))-(1/2)tDTα-1(tCDTαu(t)))+λa(t)f(u(t))=0, a.e.  t∈[0,T],u(0)=u(T)=0,whereα∈(1/2,1], andλis a positive real parameter. The approach is based on a critical-points theorem established by G. Bonanno.

EXISTENCE OF COHERENT SYSTEMS II

2008 ◽  
Vol 19 (10) ◽  
pp. 1269-1283 ◽  
Author(s):  
MONTSERRAT TEIXIDOR I BIGAS
Keyword(s):  
Vector Bundle ◽  
Vector Space ◽  
Moduli Spaces ◽  
Stability Condition ◽  
Real Parameter ◽  
Coherent System ◽  
Coherent Systems ◽  
Positive Real ◽  
Mild Conditions ◽  
Positive Real Parameter

A coherent system of type (r, d, k) on a curve consists of a vector bundle of rank r and degree d together with a vector space of dimension k of the sections of this vector bundle. There is a stability condition depending on a positive real parameter α that allows to construct moduli spaces for these objects. This paper shows non-emptiness of these moduli spaces when k > r for any α under some mild conditions on the degree and genus.

scholarly journals On the Periodicity of a Difference Equation with Maximum

2008 ◽  
Vol 2008 ◽  
pp. 1-11 ◽  
Author(s):  
Ali Gelisken ◽  
Cengiz Cinar ◽  
Ibrahim Yalcinkaya
Keyword(s):  
Difference Equation ◽  
Open Problem ◽  
Initial Conditions ◽  
Rational Numbers ◽  
Real Parameter ◽  
Positive Real ◽  
Positive Real Parameter

We investigate the periodic nature of solutions of the max difference equationxn+1=max⁡{xn,A}/(xnxn−1),n=0,1,…, whereAis a positive real parameter, and the initial conditionsx−1=Ar−1andx0=Ar0such thatr−1andr0are positive rational numbers. The results in this paper answer the Open Problem 6.2 posed by Grove and Ladas (2005).

scholarly journals Chaotic Behaviour and Bifurcation in Real Dynamics of Two-Parameter Family of Functions including Logarithmic Map

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Mohammad Sajid
Keyword(s):  
Fixed Points ◽  
Research Work ◽  
Period Doubling ◽  
Periodic Points ◽  
Real Parameter ◽  
Chaotic Behaviour ◽  
Positive Real ◽  
The Real ◽  
Positive Real Parameter ◽  
Two Parameters

The focus of this research work is to obtain the chaotic behaviour and bifurcation in the real dynamics of a newly proposed family of functions fλ,ax=x+1−λxlnax;x>0, depending on two parameters in one dimension, where assume that λ is a continuous positive real parameter and a is a discrete positive real parameter. This proposed family of functions is different from the existing families of functions in previous works which exhibits chaotic behaviour. Further, the dynamical properties of this family are analyzed theoretically and numerically as well as graphically. The real fixed points of functions fλ,ax are theoretically simulated, and the real periodic points are numerically computed. The stability of these fixed points and periodic points is discussed. By varying parameter values, the plots of bifurcation diagrams for the real dynamics of fλ,ax are shown. The existence of chaos in the dynamics of fλ,ax is explored by looking period-doubling in the bifurcation diagram, and chaos is to be quantified by determining positive Lyapunov exponents.

(t, ℓ)-STABILITY AND COHERENT SYSTEMS

2019 ◽  
Vol 62 (3) ◽  
pp. 661-672
Author(s):  
L. BRAMBILA-PAZ ◽  
O. MATA-GUTIÉRREZ
Keyword(s):  
Moduli Spaces ◽  
Finite Family ◽  
Real Parameter ◽  
Projective Curve ◽  
Coherent Systems ◽  
Positive Real ◽  
Positive Real Parameter ◽  
Complex Projective ◽  
Complex Projective Curve

AbstractLet X be a non-singular irreducible complex projective curve of genus g ≥ 2. The concept of stability of coherent systems over X depends on a positive real parameter α, given then a (finite) family of moduli spaces of coherent systems. We use (t, ℓ)-stability to prove the existence of coherent systems over X that are α-stable for all allowed α > 0.

A 60GHz-Band 3-Dimensional System-in-Package Transmitter Module with Integrated Antenna

2012 ◽  
Vol E95.C (7) ◽  
pp. 1141-1146 ◽  
Author(s):  
Noriharu SUEMATSU ◽  
Satoshi YOSHIDA ◽  
Shoichi TANIFUJI ◽  
Suguru KAMEDA ◽  
Tadashi TAKAGI ◽  
...  
Keyword(s):  
Band 3 ◽  
Dimensional System ◽  
3 Dimensional ◽  
Integrated Antenna

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.

scholarly journals Identification of focus and center in a 3-dimensional system

2014 ◽  
Vol 19 (2) ◽  
pp. 485-522 ◽  
Author(s):  
Lingling Liu ◽  
◽  
Bo Gao ◽  
Dongmei Xiao ◽  
Weinian Zhang ◽  
...  
Keyword(s):  
Dimensional System ◽  
3 Dimensional

scholarly journals The Existence of Periodic Orbits and Invariant Tori for Some 3-Dimensional Quadratic Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-12
Author(s):  
Yanan Jiang ◽  
Maoan Han ◽  
Dongmei Xiao
Keyword(s):  
Normal Form ◽  
Limit Cycles ◽  
Averaging Method ◽  
Integral Manifold ◽  
Invariant Tori ◽  
Normal Form Theory ◽  
Quadratic Systems ◽  
3 Dimensional ◽  
Volterra Systems ◽  
Form Theory

We use the normal form theory, averaging method, and integral manifold theorem to study the existence of limit cycles in Lotka-Volterra systems and the existence of invariant tori in quadratic systems inℝ3.

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