On the reducibility of two-dimensional linear quasi-periodic systems with small parameter

2014 ◽  
Vol 35 (7) ◽  
pp. 2334-2352 ◽  
Author(s):  
JUNXIANG XU ◽  
XUEZHU LU
Keyword(s):  
Differential Equations ◽  
Small Parameter ◽  
Lebesgue Measure ◽  
Periodic System ◽  
Coefficient Matrix ◽  
Periodic Systems ◽  
Two Dimensional ◽  
Constant Matrix ◽  
Small Parameters ◽  
Real Analytic

In this paper we consider a linear real analytic quasi-periodic system of two differential equations, whose coefficient matrix analytically depends on a small parameter and closes to constant. Under some non-resonance conditions about the basic frequencies and the eigenvalues of the constant matrix and without any non-degeneracy assumption of the small parameter, we prove that the system is reducible for most of the sufficiently small parameters in the sense of the Lebesgue measure.

Reducibility of a class of nonlinear quasi-periodic systems with Liouvillean basic frequencies

2020 ◽  
pp. 1-38
Author(s):  
DONGFENG ZHANG ◽  
JUNXIANG XU
Keyword(s):  
Periodic Solution ◽  
Small Parameter ◽  
Small Perturbation ◽  
Periodic System ◽  
Order Term ◽  
High Order ◽  
Periodic Systems ◽  
Elliptic Type ◽  
Constant Matrix ◽  
Image Position

In this paper we consider the following nonlinear quasi-periodic system: $$\begin{eqnarray}{\dot{x}}=(A+\unicode[STIX]{x1D716}P(t,\unicode[STIX]{x1D716}))x+\unicode[STIX]{x1D716}g(t,\unicode[STIX]{x1D716})+h(x,t,\unicode[STIX]{x1D716}),\quad x\in \mathbb{R}^{d},\end{eqnarray}$$ where $A$ is a $d\times d$ constant matrix of elliptic type,  $\unicode[STIX]{x1D716}g(t,\unicode[STIX]{x1D716})$ is a small perturbation with $\unicode[STIX]{x1D716}$ as a small parameter, $h(x,t,\unicode[STIX]{x1D716})=O(x^{2})$ as $x\rightarrow 0$ , and $P,g$ and $h$ are all analytic quasi-periodic in $t$ with basic frequencies $\unicode[STIX]{x1D714}=(1,\unicode[STIX]{x1D6FC})$ , where $\unicode[STIX]{x1D6FC}$ is irrational. It is proved that for most sufficiently small $\unicode[STIX]{x1D716}$ , the system is reducible to the following form: $$\begin{eqnarray}{\dot{x}}=(A+B_{\ast }(t))x+h_{\ast }(x,t,\unicode[STIX]{x1D716}),\quad x\in \mathbb{R}^{d},\end{eqnarray}$$ where $h_{\ast }(x,t,\unicode[STIX]{x1D716})=O(x^{2})~(x\rightarrow 0)$ is a high-order term. Therefore, the system has a quasi-periodic solution with basic frequencies $\unicode[STIX]{x1D714}=(1,\unicode[STIX]{x1D6FC})$ , such that it goes to zero when $\unicode[STIX]{x1D716}$ does.

scholarly journals Second-order differential equations with random perturbations and small parameters

2017 ◽  
Vol 147 (4) ◽  
pp. 763-779
Author(s):  
M. Kamenskii ◽  
S. Pergamenchtchikov ◽  
M. Quincampoix
Keyword(s):  
Differential Equations ◽  
Small Parameter ◽  
Random Perturbation ◽  
Stochastic Averaging ◽  
Second Order ◽  
Random Perturbations ◽  
Existence And Uniqueness Theorem ◽  
Second Order Differential Equations ◽  
Averaging Theorem ◽  
Small Parameters

We consider boundary-value problems for differential equations of second order containing a Brownian motion (random perturbation) and a small parameter and prove a special existence and uniqueness theorem for random solutions. We study the asymptotic behaviour of these solutions as the small parameter goes to zero and show the stochastic averaging theorem for such equations. We find the explicit limits for the solutions as the small parameter goes to zero.

A correction to ‘Analyticity and metric transitivity on the torus’

1999 ◽  
Vol 19 (1) ◽  
pp. 259-261
Author(s):  
SOL SCHWARTZMAN
Keyword(s):  
Differential Equations ◽  
Two Dimensional ◽  
Dimensional Torus ◽  
Topological Transitivity ◽  
Topological Disc ◽  
Real Analytic

In [2], flows on the standard two-dimensional torus given by the differential equations \begin{equation*} \frac{dx}{dt}=a-Fy(x,y),\quad \frac{dv}{dt}=b+Fx(x,y) \end{equation*} were considered. It was assumed that $F(x,y)$ was real analytic and of period one in both $x$ and $y$. A key step in proving the results in [2] was to show that one could conclude topological transitivity for the flow provided one assumed: \begin{enumerate} \item[(a)] $a/b$ is irrational; \item[(b)] there does not exist a topological disc on the torus that is invariant under the flow. \end{enumerate}

scholarly journals Examples of conservative diffeomorphisms of the two-dimensional torus with coexistence of elliptic and stochastic behaviour

1982 ◽  
Vol 2 (3-4) ◽  
pp. 439-463 ◽  
Author(s):  
Feliks Przytycki
Keyword(s):  
Fixed Point ◽  
Lebesgue Measure ◽  
Bifurcation Parameter ◽  
Two Dimensional ◽  
Open Area ◽  
Dimensional Torus ◽  
Characteristic Exponents ◽  
Almost Everywhere ◽  
Lyapunov Characteristic Exponents ◽  
Real Analytic

AbstractWe find very simple examples of C∞-arcs of diffeomorphisms of the two-dimensional torus, preserving the Lebesgue measure and having the following properties: (1) the beginning of an arc is inside the set of Anosov diffeomorphisms; (2) after the bifurcation parameter every diffeomorphism has an elliptic fixed point with the first Birkhoff invariant non-zero (the KAM situation) and an invariant open area with almost everywhere non-zero Lyapunov characteristic exponents, moreover where the diffeomorphism has Bernoulli property; (3) the arc is real-analytic except on two circles (for each value of parameter) which are inside the Bernoulli property area.

Partially Invariant Solutions for Two-dimensional Ideal MHD Equations

1990 ◽  
Vol 45 (11-12) ◽  
pp. 1219-1229 ◽  
Author(s):  
D.-A. Becker ◽  
E. W. Richter
Keyword(s):  
Differential Equations ◽  
Linear Equations ◽  
Mhd Equations ◽  
Two Dimensional ◽  
Invariant Solutions ◽  
First Order ◽  
Partially Invariant Solutions ◽  
Quasilinear Systems ◽  
Ideal Mhd ◽  
Non Linear Equations

AbstractA generalization of the usual method of similarity analysis of differential equations, the method of partially invariant solutions, was introduced by Ovsiannikov. The degree of non-invariance of these solutions is characterized by the defect of invariance d. We develop an algorithm leading to partially invariant solutions of quasilinear systems of first-order partial differential equations. We apply the algorithm to the non-linear equations of the two-dimensional non-stationary ideal MHD with a magnetic field perpendicular to the plane of motion.

scholarly journals Dynamics with Cattaneo–Christov heat and mass flux theory of bioconvection Oldroyd-B nanofluid

2020 ◽  
Vol 12 (8) ◽  
pp. 168781402093046 ◽  
Author(s):  
Noor Saeed Khan ◽  
Qayyum Shah ◽  
Arif Sohail
Keyword(s):  
Mass Transfer ◽  
Porous Media ◽  
Differential Equations ◽  
Mass Flux ◽  
Homotopy Analysis Method ◽  
Two Dimensional ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Transfer Flow ◽  
Insight Into

Entropy generation in bioconvection two-dimensional steady incompressible non-Newtonian Oldroyd-B nanofluid with Cattaneo–Christov heat and mass flux theory is investigated. The Darcy–Forchheimer law is used to study heat and mass transfer flow and microorganisms motion in porous media. Using appropriate similarity variables, the partial differential equations are transformed into ordinary differential equations which are then solved by homotopy analysis method. For an insight into the problem, the effects of various parameters on different profiles are shown in different graphs.

A Difference Method for Plane Problems in Magnetoelastodynamics

1972 ◽  
Vol 39 (3) ◽  
pp. 689-695 ◽  
Author(s):  
W. W. Recker
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Method ◽  
Difference Method ◽  
Necessary Condition ◽  
Two Dimensional ◽  
Symmetric Hyperbolic System ◽  
Von Neumann ◽  
First Order ◽  
Independent Variables

The two-dimensional equations of magnetoelastodynamics are considered as a symmetric hyperbolic system of linear first-order partial-differential equations in three independent variables. The characteristic properties of the system are determined and a numerical method for obtaining the solution to mixed initial and boundary-value problems in plane magnetoelastodynamics is presented. Results on the von Neumann necessary condition are presented. Application of the method to a problem which has a known solution provides further numerical evidence of the convergence and stability of the method.

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