Algebraic limit cycles in polynomial systems of differential equations

2007 ◽  
Vol 40 (47) ◽  
pp. 14207-14222 ◽  
Author(s):  
Jaume Llibre ◽  
Yulin Zhao
Keyword(s):  
Differential Equations ◽  
Limit Cycles ◽  
Polynomial Systems ◽  
Systems Of Differential Equations ◽  
Algebraic Limit Cycles ◽  
Algebraic Limit

scholarly journals Algebraic Limit Cycles Bifurcating from Algebraic Ovals of Quadratic Centers

2018 ◽  
Vol 28 (12) ◽  
pp. 1850145 ◽  
Author(s):  
Jaume Llibre ◽  
Yun Tian
Keyword(s):  
Limit Cycles ◽  
Algebraic Curves ◽  
Polynomial Systems ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Period Annulus ◽  
Invariant Algebraic Curves ◽  
Relevant Role ◽  
Algebraic Limit Cycles ◽  
Algebraic Limit

In the integrability of polynomial differential systems it is well known that the invariant algebraic curves play a relevant role. Here we will see that they can also play an important role with respect to limit cycles.In this paper, we study quadratic polynomial systems with an algebraic periodic orbit of degree [Formula: see text] surrounding a center. We show that there exists only one family of such systems satisfying that an algebraic limit cycle of degree [Formula: see text] can bifurcate from the period annulus of the mentioned center under quadratic perturbations.

Planar Polynomial Systems with Non-Algebraic Limit Cycles

2009 ◽  
Author(s):  
Khalil I. T. Al-Dosary ◽  
George Maroulis ◽  
Theodore E. Simos
Keyword(s):  
Limit Cycles ◽  
Polynomial Systems ◽  
Algebraic Limit Cycles ◽  
Algebraic Limit

NON-ALGEBRAIC LIMIT CYCLES FOR PARAMETRIZED PLANAR POLYNOMIAL SYSTEMS

2007 ◽  
Vol 18 (02) ◽  
pp. 179-189 ◽  
Author(s):  
KHALIL I. T. AL-DOSARY
Keyword(s):  
Limit Cycles ◽  
Polynomial Systems ◽  
Homogeneous Polynomials ◽  
The Family ◽  
Planar Systems ◽  
Algebraic Limit Cycles ◽  
Algebraic Limit

In this paper, we determine conditions for planar systems of the form [Formula: see text] where a, b and c are real constants, to possess non-algebraic limit cycles. This is done as an application of a former theorem gives description of the existence of the non-algebraic limit cycles of the family of systems: [Formula: see text] where Pn(x,y), Qn(x,y) and Rm(x,y) are homogeneous polynomials of degrees n, n and m respectively with n < m and n is odd, m is even. The tool for proving these results is based on a method developed in [7].

scholarly journals The Cubic Polynomial Differential Systems with two Circles as Algebraic Limit Cycles

2018 ◽  
Vol 18 (1) ◽  
pp. 183-193 ◽  
Author(s):  
Jaume Giné ◽  
Jaume Llibre ◽  
Claudia Valls
Keyword(s):  
Limit Cycles ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Cubic Polynomial ◽  
Algebraic Limit Cycles ◽  
Algebraic Limit

AbstractIn this paper we characterize all cubic polynomial differential systems in the plane having two circles as invariant algebraic limit cycles.

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