scholarly journals On the Effective Reducibility of a Class of Quasi-Periodic Linear Hamiltonian Systems Close to Constant Coefficients

2018 ◽  
Vol 2018 ◽  
pp. 1-7 ◽  
Author(s):  
Nina Xue ◽  
Wencai Zhao
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Linear Hamiltonian System ◽  
Constant Matrix ◽  
Change Of Variables ◽  
Periodic Linear ◽  
Multiple Eigenvalues ◽  
Linear Hamiltonian Systems ◽  
Constant Coefficients ◽  
Exponentially Small

In this paper, we consider the effective reducibility of the quasi-periodic linear Hamiltonian system x˙=A+εQt,εx, ε∈0,ε0, where A is a constant matrix with possible multiple eigenvalues and Q(t,ε) is analytic quasi-periodic with respect to t. Under nonresonant conditions, it is proved that this system can be reduced to y˙=A⁎ε+εR⁎t,εy, ε∈0,ε⁎, where R⁎ is exponentially small in ε, and the change of variables that perform such a reduction is also quasi-periodic with the same basic frequencies as Q.

scholarly journals On the Reducibility of Quasiperiodic Linear Hamiltonian Systems and Its Applications in Schrödinger Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Nina Xue ◽  
Wencai Zhao
Keyword(s):  
Hamiltonian System ◽  
Schrödinger Equation ◽  
Small Parameter ◽  
Hamiltonian Systems ◽  
Schrodinger Equation ◽  
Linear Hamiltonian System ◽  
Constant Matrix ◽  
Change Of Variables ◽  
Multiple Eigenvalues ◽  
Linear Hamiltonian Systems

In this paper, we consider the reducibility of the quasiperiodic linear Hamiltonian system ẋ=A+εQt, where A is a constant matrix with possible multiple eigenvalues, Qt is analytic quasiperiodic with respect to t, and ε is a small parameter. Under some nonresonant conditions, it is proved that, for most sufficiently small ε, the Hamiltonian system can be reduced to a constant coefficient Hamiltonian system by means of a quasiperiodic symplectic change of variables with the same basic frequencies as Qt. Applications to the Schrödinger equation are also given.

scholarly journals New Interval Oscillation Criteria for Certain Linear Hamiltonian Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Jing Shao ◽  
Fanwei Meng
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Linear Matrix ◽  
Riccati Transformation ◽  
Integral Means ◽  
Oscillation Criteria ◽  
General Integral ◽  
Generalized Riccati Transformation ◽  
Linear Hamiltonian Systems ◽  
Interval Oscillation

Using a generalized Riccati transformation and the general integral means technique, some new interval oscillation criteria for the linear matrix Hamiltonian systemU'=(A(t)-λ(t)I)U+B(t)V,V'=C(t)U+(μ(t)I-A*(t))V,t≥t0are obtained. These results generalize and improve the oscillation criteria due to Zheng (2008). An example is given to dwell upon the importance of our results.

scholarly journals Bifurcation Diagrams and Global Phase Portraits for Some Hamiltonian Systems with Rational Potentials

2018 ◽  
Vol 28 (13) ◽  
pp. 1850168
Author(s):  
Ting Chen ◽  
Jaume Llibre
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Normal Forms ◽  
Dynamical Behavior ◽  
Phase Portraits ◽  
Bifurcation Diagrams ◽  
Linear Change ◽  
Rational Potential ◽  
Change Of Variables ◽  
Poincaré Disk

In this paper, we study the global dynamical behavior of the Hamiltonian system [Formula: see text], [Formula: see text] with the rational potential Hamiltonian [Formula: see text], where [Formula: see text] and [Formula: see text] are polynomials of degree 1 or 2. First we get the normal forms for these rational Hamiltonian systems by some linear change of variables. Then we classify all the global phase portraits of these systems in the Poincaré disk and provide their bifurcation diagrams.

scholarly journals Some Oscillation Results for Linear Hamiltonian Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-13
Author(s):  
Nan Wang ◽  
Fanwei Meng
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Riccati Technique ◽  
Averaging Technique ◽  
Oscillation Criteria ◽  
Integral Averaging Technique ◽  
Linear Hamiltonian Systems ◽  
Interval Oscillation ◽  
Generalized Matrix ◽  
Positive Functionals

The purpose of this paper is to develop a generalized matrix Riccati technique for the selfadjoint matrix Hamiltonian systemU′=A(t)U+B(t)V,V′=C(t)U−A∗(t)V. By using the standard integral averaging technique and positive functionals, new oscillation and interval oscillation criteria are established for the system. These criteria extend and improve some results that have been required before. An interesting example is included to illustrate the importance of our results.

Parameterization of the M(λ) function for a Hamiltonian system of limit circle type

1983 ◽  
Vol 93 (3-4) ◽  
pp. 349-360 ◽  
Author(s):  
D. B. Hinton ◽  
J. K. Shaw
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Deficiency Index ◽  
Limit Circle ◽  
Weyl Matrix ◽  
Linear Hamiltonian Systems ◽  
Limit Circle Case ◽  
Special Case ◽  
Circle Case

SynopsisThe authors continue their study of Titchmarch-Weyl matrix M(λ) functions for linear Hamiltonian systems. A representation for the M(λ) function is obtained in the case where the system is limit circle, or maximum deficiency index, type. The representation reduces, in a special case, to a parameterization for scalar m-coefficients due to C. T. Fulton. A proof that matrix M(λ) functions are meromorphic in the limit circle case is given.

scholarly journals Principal solutions of positive linear Hamiltonian systems

1976 ◽  
Vol 22 (4) ◽  
pp. 411-420 ◽  
Author(s):  
Don Hinton
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Deficiency Index ◽  
Linear Hamiltonian Systems ◽  
Scalar Operator ◽  
Index Problem

AbstractThe Hamiltonian system Y′ = BY + CZ, Z′ = – AY – B*Z is considered where the coefficients are continuous on I = [a, ∞, C = C* ≧ 0, and A = A* ≦ 0. A solution (Y, Z) satisfying Y*Z = Z*Y is defined to be principal (coprincipal) provided that (i) Y−1 exists on I (Z−1 exists on I) and (ii) as t→∞ ( as t → ∞). Three conditions are given which are separtely equivalent to the condition that a solution is principal iff it is coprincipal. For a self-adjoint scalar operator L of order 2n, this problem is related to the deficiency index problem and to a problem of Anderson and Lazer (1970) which concerns the number of lnearly independent solutions of L (y) =0 satisfying y(k) ∈ (a, ∞) (k = 0, …, n).

Stability of Equilibrium Solutions in Planar Hamiltonian Difference Systems

2015 ◽  
Vol 67 (6) ◽  
pp. 1270-1289
Author(s):  
Cristian Carcamo ◽  
Claudio Vidal
Keyword(s):  
Hamiltonian Systems ◽  
Linear Hamiltonian System ◽  
Equilibrium Solutions ◽  
Order Difference ◽  
First Order ◽  
Difference Systems ◽  
Linear Hamiltonian Systems ◽  
The Stability ◽  
Cubic Polynomials ◽  
Lyapunov Sense

AbstractIn this paper, we study the stability in the Lyapunov sense of the equilibrium solutions of discrete or difference Hamiltonian systems in the plane. First, we perform a detailed study of linear Hamiltonian systems as a function of the parameters. In particular we analyze the regular and the degenerate cases. Next, we give a detailed study of the normal form associated with the linear Hamiltonian system. At the same time we obtain the conditions under which we can get stability (in linear approximation) of the equilibrium solution, classifying all the possible phase diagrams as a function of the parameters. After that, we study the stability of the equilibrium solutions of the first order difference system in the plane associated with mechanical Hamiltonian systems and Hamiltonian systems defined by cubic polynomials. Finally, we point out important diòerences with the continuous case.

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