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.

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 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.

On nearly-commensurable periods in the restricted problem of three bodies, with calculations of the long-period variations in the interior 2:1 case

1966 ◽  
Vol 25 ◽  
pp. 197-222 ◽  
Author(s):  
P. J. Message
Keyword(s):  
Periodic Solution ◽  
Periodic Solutions ◽  
Large Amplitude ◽  
Restricted Problem ◽  
Small Amplitude ◽  
Minor Planets ◽  
Long Period ◽  
Numerical Integrations ◽  
Period Variations ◽  
The Mean

An analytical discussion of that case of motion in the restricted problem, in which the mean motions of the infinitesimal, and smaller-massed, bodies about the larger one are nearly in the ratio of two small integers displays the existence of a series of periodic solutions which, for commensurabilities of the typep+ 1:p, includes solutions of Poincaré'sdeuxième sortewhen the commensurability is very close, and of thepremière sortewhen it is less close. A linear treatment of the long-period variations of the elements, valid for motions in which the elements remain close to a particular periodic solution of this type, shows the continuity of near-commensurable motion with other motion, and some of the properties of long-period librations of small amplitude.To extend the investigation to other types of motion near commensurability, numerical integrations of the equations for the long-period variations of the elements were carried out for the 2:1 interior case (of which the planet 108 “Hecuba” is an example) to survey those motions in which the eccentricity takes values less than 0·1. An investigation of the effect of the large amplitude perturbations near commensurability on a distribution of minor planets, which is originally uniform over mean motion, shows a “draining off” effect from the vicinity of exact commensurability of a magnitude large enough to account for the observed gap in the distribution at the 2:1 commensurability.

A Cubic System with Eight Small-Amplitude Limit Cycles

1991 ◽  
Vol 47 (2) ◽  
pp. 163-171 ◽  
Author(s):  
E. M. JAMES ◽  
N. G. LLOYD
Keyword(s):  
Limit Cycles ◽  
Small Amplitude ◽  
Cubic System
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