ON THE DEGENERATE CENTER PROBLEM

2011 ◽  
Vol 21 (05) ◽  
pp. 1383-1392 ◽  
Author(s):  
JAUME GINÉ
Keyword(s):  
First Integral ◽  
New Method ◽  
Center Problem ◽  
Linear Type ◽  
Analytic First Integral

In this work, it is proved that any degenerate center is limit of a [Formula: see text] linear type center and when the degenerate center has an analytic first integral then it is limit of an analytic linear type center. A new method to detect integrability developed in [Giné & Santallusia, 2011] is applied to the degenerate center problem. Moreover, a review of the most important recent contributions to the degenerate center problem is given.

Analytic nilpotent centers with analytic first integral

2010 ◽  
Vol 72 (9-10) ◽  
pp. 3732-3738 ◽  
Author(s):  
Isaac A. García ◽  
Jaume Giné
Keyword(s):  
First Integral ◽  
Analytic First Integral

scholarly journals An Algorithm for the Numerical Integration of Perturbed and Damped Second-Order ODE Systems

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2028
Author(s):  
Fernando García-Alonso ◽  
José Antonio Reyes ◽  
Mónica Cortés-Molina
Keyword(s):  
Numerical Integration ◽  
Structural Dynamics ◽  
Truncation Error ◽  
First Integral ◽  
Second Order ◽  
New Method ◽  
Series Method ◽  
Initial Value ◽  
Satellite Problem ◽  
Damped Systems

A new method of numerical integration for a perturbed and damped systems of linear second-order differential equations is presented. This new method, under certain conditions, integrates, without truncation error, the IVPs (initial value problems) of the type: x″(t)+Ax′(t)+Cx(t)=εF(x(t),t), x(0)=x0, x′(0)=x0′, t∈[a,b]=I, which appear in structural dynamics, astrodynamics, and other fields of physics and engineering. In this article, a succession of real functions is constructed with values in the algebra of m×m matrices. Their properties are studied and we express the solution of the proposed IVP through a serial expansion of the same, whose coefficients are calculated by means of recurrences involving the perturbation function. This expression of the solution is used for the construction of the new numerical method. Three problems are solved by means of the new series method; we contrast the results obtained with the exact solution of the problem and with its first integral. In the first problem, a quasi-periodic orbit is integrated; in the second, a problem of structural dynamics associated with an earthquake is studied; in the third, an equatorial satellite problem when the perturbation comes from zonal harmonics J2 is solved. The good behavior of the series method is shown by comparing the results obtained against other integrators.

Jets of differential equations having integrable extensions

1998 ◽  
Vol 18 (6) ◽  
pp. 1527-1544
Author(s):  
MASSIMO VILLARINI
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
First Integral ◽  
Jet Space ◽  
Analytic First Integral

We characterize the set of $n$-jets admitting an extension which is a germ of a differential equation with an analytic first integral, and compute its codimension in the $n$-jet space. Some applications in the case of the centre-focus problem are given.

A new method for solving the two-center problem with realistic potentials

1977 ◽  
Vol 286 (3) ◽  
pp. 512-522 ◽  
Author(s):  
F.A. Gareev ◽  
M.Ch. Gizzatkulov ◽  
J. Revai
Keyword(s):  
New Method ◽  
Center Problem ◽  
Realistic Potentials

scholarly journals Analytic integrability of quadratic–linear polynomial differential systems

2009 ◽  
Vol 31 (1) ◽  
pp. 245-258 ◽  
Author(s):  
JAUME LLIBRE ◽  
CLÀUDIA VALLS
Keyword(s):  
Singular Point ◽  
Explicit Expression ◽  
First Integral ◽  
First Integrals ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Analytic Integrability ◽  
Linear Polynomial ◽  
Analytic First Integral

AbstractFor the quadratic–linear polynomial differential systems with a finite singular point, we classify the ones which have a global analytic first integral, and provide the explicit expression of their first integrals.

A note on analytic integrability of planar vector fields

2012 ◽  
Vol 23 (5) ◽  
pp. 555-562 ◽  
Author(s):  
A. ALGABA ◽  
C. GARCÍA ◽  
M. REYES
Keyword(s):  
Vector Field ◽  
Integrable Systems ◽  
Analytic Functions ◽  
Vector Fields ◽  
First Integral ◽  
Analytic Integrability ◽  
Planar Vector Fields ◽  
Planar Vector ◽  
Analytic First Integral ◽  
Image Position

We give a new characterisation of integrability of a planar vector field at the origin. This allows us to prove that the analytic systemswhereh,K, Ψ and ξ are analytic functions defined in the neighbourhood ofOwithK(O) ≠ 0 or Ψ(O) ≠ 0 andn≥ 1, have a local analytic first integral at the origin. We show new families of analytically integrable systems that are held in the above class. In particular, this class includes all the nilpotent and generalised nilpotent integrable centres that we know.

Selective Sampling and Orientation of Non-Homogeneous Tissues

1968 ◽  
Vol 26 ◽  
pp. 98-99
Author(s):  
C. C. Clawson ◽  
L. W. Anderson ◽  
R. A. Good
Keyword(s):  
Electron Microscopy ◽  
Electron Microscope ◽  
Light Microscopy ◽  
Random Sampling ◽  
New Method ◽  
Microscope Examination ◽  
Small Areas ◽  
Selective Sampling ◽  
Large Numbers ◽  
Specific Orientation

Investigations which require electron microscope examination of a few specific areas of non-homogeneous tissues make random sampling of small blocks an inefficient and unrewarding procedure. Therefore, several investigators have devised methods which allow obtaining sample blocks for electron microscopy from region of tissue previously identified by light microscopy of present here techniques which make possible: 1) sampling tissue for electron microscopy from selected areas previously identified by light microscopy of relatively large pieces of tissue; 2) dehydration and embedding large numbers of individually identified blocks while keeping each one separate; 3) a new method of maintaining specific orientation of blocks during embedding; 4) special light microscopic staining or fluorescent procedures and electron microscopy on immediately adjacent small areas of tissue.

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