scholarly journals Meromorphic integrability of the Hamiltonian systems with homogeneous potentials of degree -4

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jaume Llibre ◽  
Yuzhou Tian
Keyword(s):  
Hamiltonian Systems ◽  
Homogeneous Polynomial ◽  
First Integral ◽  
Liouville Integrability ◽  
Linear Change ◽  
Change Of Variables ◽  
Polynomial First Integral ◽  
Homogeneous Potentials

<p style='text-indent:20px;'>We characterize the meromorphic Liouville integrability of the Hamiltonian systems with Hamiltonian <inline-formula><tex-math id="M2">\begin{document}$ H = \left(p_1^2+p_2^2\right)/2+1/P(q_1, q_2) $\end{document}</tex-math></inline-formula>, being <inline-formula><tex-math id="M3">\begin{document}$ P(q_1, q_2) $\end{document}</tex-math></inline-formula> a homogeneous polynomial of degree <inline-formula><tex-math id="M4">\begin{document}$ 4 $\end{document}</tex-math></inline-formula> of one of the following forms <inline-formula><tex-math id="M5">\begin{document}$ \pm q_1^4 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ 4q_1^3q_2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ \pm 6q_1^2q_2^2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ \pm \left(q_1^2+q_2^2\right)^2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M9">\begin{document}$ \pm q_2^2\left(6q_1^2-q_2^2\right) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M10">\begin{document}$ \pm q_2^2\left(6q_1^2+q_2^2\right) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M11">\begin{document}$ q_1^4+6\mu q_1^2q_2^2-q_2^4 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M12">\begin{document}$ -q_1^4+6\mu q_1^2q_2^2+q_2^4 $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M13">\begin{document}$ \mu&gt;-1/3 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M14">\begin{document}$ \mu\neq 1/3 $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M15">\begin{document}$ q_1^4+6\mu q_1^2q_2^2+q_2^4 $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M16">\begin{document}$ \mu \neq \pm 1/3 $\end{document}</tex-math></inline-formula>. We note that any homogeneous polynomial of degree <inline-formula><tex-math id="M17">\begin{document}$ 4 $\end{document}</tex-math></inline-formula> after a linear change of variables and a rescaling can be written as one of the previous polynomials. We remark that for the polynomial <inline-formula><tex-math id="M18">\begin{document}$ q_1^4+6\mu q_1^2q_2^2+q_2^4 $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M19">\begin{document}$ \mu\in\left\{-5/3, -2/3\right\} $\end{document}</tex-math></inline-formula> we only can prove that it has no a polynomial first integral.</p>

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 On the integrability of Hamiltonian systems with d degrees of freedom and homogenous polynomial potential of degree n

2018 ◽  
Vol 20 (08) ◽  
pp. 1750045 ◽  
Author(s):  
Jaume Llibre ◽  
Clàudia Valls
Keyword(s):  
Hamiltonian Systems ◽  
Degrees Of Freedom ◽  
First Integral ◽  
Polynomial Potential ◽  
Polynomial First Integral ◽  
Darboux First Integral

We consider Hamiltonian systems with [Formula: see text] degrees of freedom and a Hamiltonian of the form [Formula: see text] where [Formula: see text] is a homogenous polynomial of degree [Formula: see text]. We prove that such Hamiltonian systems with [Formula: see text] odd or [Formula: see text], have a Darboux first integral if and only if they have a polynomial first integral.

scholarly journals Application of the Gaussian anamorphosis to assimilation in a 3-D coupled physical-ecosystem model of the North Atlantic with the EnKF: a twin experiment

2009 ◽  
Vol 6 (1) ◽  
pp. 617-652 ◽  
Author(s):  
E. Simon ◽  
L. Bertino
Keyword(s):  
North Atlantic ◽  
Coupled Model ◽  
Surface Concentration ◽  
Ecosystem Model ◽  
Concentration Data ◽  
Linear Change ◽  
The North ◽  
Change Of Variables ◽  
Non Linear ◽  
Non Gaussian

Abstract. We consider the application of the Ensemble Kalman Filter (EnKF) to a coupled ocean ecosystem model (HYCOM-NORWECOM). Such models, especially the ecosystem models, are characterized by strongly non-linear interactions active in ocean blooms and present important limitations for the use of data assimilation methods based on linear statistical analysis. Besides the non-linearity of the model, one is confronted with physical/biological limitations, the analysis state having to be consistent with the model, especially with the constraints of positiveness of some variables. Furthermore the non-Gaussian distributions of the biogeochemical variables break an important assumption of the linear analysis, leading to a loss of optimality of the filter. We present an extension of the EnKF dealing with these limitations by introducing a non-linear change of variables (anamorphosis function) in order to execute the analysis step in a Gaussian space. We present also the initial results of the application of this non-Gaussian extension of the EnKF to the assimilation of simulated chlorophyll surface concentration data in a North Atlantic configuration of the HYCOM NORWECOM coupled model.

scholarly journals Fokas method for linear boundary value problems involving mixed spatial derivatives

2020 ◽  
Vol 476 (2239) ◽  
pp. 20200076
Author(s):  
A. Batal ◽  
A. S. Fokas ◽  
T. Özsarı
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Initial Boundary ◽  
Parabolic Pdes ◽  
Linear Change ◽  
Transform Method ◽  
Change Of Variables ◽  
Mixed Derivatives ◽  
Fokas Method ◽  
Initial Boundary Value

We obtain solution representation formulae for some linear initial boundary value problems posed on the half space that involve mixed spatial derivative terms via the unified transform method (UTM), also known as the Fokas method. We first implement the method on the second-order parabolic PDEs; in this case one can alternatively eliminate the mixed derivatives by a linear change of variables. Then, we employ the method to biharmonic problems, where it is not possible to eliminate the cross term via a linear change of variables. A basic ingredient of the UTM is the use of certain invariant maps. It is shown here that these maps are well defined provided that certain analyticity issues are appropriately addressed.

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