Invariant Algebraic Curves and Hyperelliptic Limit Cycles of Liénard Systems

2021 ◽  
Vol 20 (2) ◽  
Author(s):  
Xinjie Qian ◽  
Yang Shen ◽  
Jiazhong Yang
Keyword(s):  
Limit Cycles ◽  
Algebraic Curves ◽  
Liénard Systems ◽  
Invariant Algebraic Curves

Classification of Invariant Algebraic Curves and Nonexistence of Algebraic Limit Cycles in Quadratic Systems from Family (I) of the Chinese Classification

2020 ◽  
Vol 30 (04) ◽  
pp. 2050056 ◽  
Author(s):  
Maria V. Demina ◽  
Claudia Valls
Keyword(s):  
Differential System ◽  
Limit Cycles ◽  
Algebraic Curves ◽  
Complete Classification ◽  
Quadratic Systems ◽  
Correct Proof ◽  
Invariant Algebraic Curves ◽  
Algebraic Limit Cycles ◽  
Algebraic Limit

We give the complete classification of irreducible invariant algebraic curves in quadratic systems from family [Formula: see text] of the Chinese classification, that is, of differential system [Formula: see text] with [Formula: see text]. In addition, we provide a complete and correct proof of the nonexistence of algebraic limit cycles for these equations.

INTEGRABILITY AND BIFURCATIONS OF LIMIT CYCLES IN A CUBIC KOLMOGOROV SYSTEM

2013 ◽  
Vol 23 (04) ◽  
pp. 1350061 ◽  
Author(s):  
FENG LI
Keyword(s):  
Critical Point ◽  
Computer Algebra ◽  
Limit Cycles ◽  
Computer Algebra System ◽  
Algebraic Curves ◽  
Necessary Conditions ◽  
Invariant Algebraic Curves ◽  
Kolmogorov System

We investigate the planar cubic Kolmogorov systems with three invariant algebraic curves which have a equilibrium at (1,1). With the help of computer algebra system MATHEMATICA, we prove that five limit cycles can be bifurcated from a critical point in the first quadrant. Moreover, the necessary conditions of center are obtained, by technical transformation, and its sufficiencies are proved.

Coexistence of limit cycles and invariant algebraic curves for a Kukles system

2004 ◽  
Vol 59 (5) ◽  
pp. 673-693 ◽  
Author(s):  
J CHAVARRIGA ◽  
E SAEZ ◽  
I SZANTO ◽  
M GRAU
Keyword(s):  
Limit Cycles ◽  
Algebraic Curves ◽  
Invariant Algebraic Curves ◽  
Kukles System

scholarly journals Limit Cycles and Invariant Curves in a Class of Switching Systems with Degree Four

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Xinli Li ◽  
Huijie Yang ◽  
Binghong Wang
Keyword(s):  
Limit Cycles ◽  
Large Amplitude ◽  
Small Amplitude ◽  
Algebraic Curves ◽  
Switching Systems ◽  
Invariant Curves ◽  
Invariant Algebraic Curves

In this paper, a class of switching systems which have an invariant conic x2+cy2=1,c∈R, is investigated. Half attracting invariant conic x2+cy2=1,c∈R, is found in switching systems. The coexistence of small-amplitude limit cycles, large amplitude limit cycles, and invariant algebraic curves under perturbations of the coefficients of the systems is proved.

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.

Coexistence of limit cycles and invariant algebraic curves for a Kukles system

2004 ◽  
Vol 59 (5) ◽  
pp. 673-693 ◽  
Author(s):  
J. Chavarriga ◽  
E. Sáez ◽  
I. Szántó ◽  
M. Grau
Keyword(s):  
Limit Cycles ◽  
Algebraic Curves ◽  
Invariant Algebraic Curves ◽  
Kukles System

Invariant algebraic curves and conditions for a centre

1994 ◽  
Vol 124 (6) ◽  
pp. 1209-1229 ◽  
Author(s):  
C. J. Christopher
Keyword(s):  
Limit Cycles ◽  
Algebraic Curves ◽  
Two Dimensional ◽  
Invariant Algebraic Curves ◽  
Kukles System ◽  
Degenerate Cases

Conditions for the existence of a centre in two-dimensional systems are considered along the lines of Darboux. We show how these methods can be used in the search for maximal numbers of bifurcating limit cycles. We also extend the method to include more degenerate cases such as are encountered in less generic systems. These lead to new classes of integrals. In particular, the Kukles system is considered, and new centre conditions for this system are obtained.

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