Abstract
In this paper we study the center problem for polynomial differential systems and we prove that any center of an analytic differential system is analytically reducible. x˙=y,y˙=P0(x)+P1(x)y+P2(x)y2, We also study the centers for the Cherkas polynomial differential systems where Pi(x) are polynomials of degree n, P0(0)=0 and P′0(0)<0. Computing the focal values we find the center conditions for such systems for degree 3, and using modular arithmetics for degree 4. Finally we do a conjecture about the center conditions for Cherkas polynomial differential systems of degree n.
Highlights
IntroductionWhere f (x), g(x) and h(x) are polynomials
The well-known polynomial Liénard equation x + f (x)x + g(x) = 0, (1.1)where f (x) and g(x) are polynomials, which we can be rewritten as the differential system in the plane x = y, y = −g(x) − y f (x), (1.2)can be generalized into what we call the polynomial Cherkas polynomial differential equation x + h(x)x2 + f (x)x + g(x) = 0, (1.3)where f (x), g(x) and h(x) are polynomials
Where f (x) and g(x) are polynomials, which we can be rewritten as the differential system in the plane x = y, y = −g(x) − y f (x), (1.2)
Summary
Where f (x), g(x) and h(x) are polynomials This Cherkas equation can be transformed into the differential system in the plane x = y, y = −g(x) − y f (x) − y2h(x). The Liénard systems (1.2) with a center are time-reversible (see below the definition) through an analytic invertible transformation followed by a rescalling of time. This type of symmetry after transformation is called generalized symmetry. Cherkas gave necessary and sufficient conditions for the existence of a center at the origin of equation (1.5). These results permit to check if a particular system of the form (1.2), (1.4), (1.6), and (1.8) has a center at the origin. In practice from the conditions obtained it is not easy to get the explicit form of the families with center even for systems of small degree
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