Rational limit cycles on generalized Bernouilli polynomial equations

2021 ◽  
Vol 62 (3) ◽  
pp. 033501
Author(s):  
Clàudia Valls
Keyword(s):  
Limit Cycles ◽  
Polynomial Equations

scholarly journals Rational Limit Cycles on Abel Polynomial Equations

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 885 ◽  
Author(s):  
Claudia Valls
Keyword(s):  
Limit Cycles ◽  
Upper Bounds ◽  
Polynomial Equations ◽  
Abel Polynomial ◽  
Real Polynomials ◽  
Abel Equations

In this paper we deal with Abel equations of the form d y / d x = A 1 ( x ) y + A 2 ( x ) y 2 + A 3 ( x ) y 3 , where A 1 ( x ) , A 2 ( x ) and A 3 ( x ) are real polynomials and A 3 ≢ 0 . We prove that these Abel equations can have at most two rational (non-polynomial) limit cycles when A 1 ≢ 0 and three rational (non-polynomial) limit cycles when A 1 ≡ 0 . Moreover, we show that these upper bounds are sharp. We show that the general Abel equations can always be reduced to this one.

TWELVE LIMIT CYCLES IN A CUBIC CASE OF THE 16th HILBERT PROBLEM

2005 ◽  
Vol 15 (07) ◽  
pp. 2191-2205 ◽  
Author(s):  
P. YU ◽  
M. HAN
Keyword(s):  
Limit Cycles ◽  
Multiple Scales ◽  
Hilbert Problem ◽  
Perturbation Technique ◽  
Local Limit ◽  
Polynomial Equations ◽  
Computationally Efficient ◽  
Degree Polynomial ◽  
Planar System ◽  
16Th Hilbert Problem

In this paper, we prove the existence of twelve small (local) limit cycles in a planar system with third-degree polynomial functions. The best result so far in literature for a cubic order planar system is eleven limit cycles. The system considered in this paper has a saddle point at the origin and two focus points which are symmetric about the origin. This system was studied by the authors and shown to exhibit ten small limit cycles: five around each of the focus points. It will be proved in this paper that the system can have twelve small limit cycles. The major tasks involved in the proof are to compute the focus values and solve coupled enormous large polynomial equations. A computationally efficient perturbation technique based on multiple scales is employed to calculate the focus values. Moreover, the focus values are perturbed to show that the system can exactly have twelve small limit cycles.

scholarly journals Lac Operon Boolean Models: Dynamical Robustness and Alternative Improvements

Mathematics ◽  
2021 ◽  
Vol 9 (6) ◽  
pp. 600 ◽  
Author(s):  
Marco Montalva-Medel ◽  
Thomas Ledger ◽  
Gonzalo A. Ruz ◽  
Eric Goles
Keyword(s):  
Escherichia Coli ◽  
Limit Cycles ◽  
The Other ◽  
Lac Operon ◽  
Boolean Models ◽  
Bistable Behavior

In Veliz-Cuba and Stigler 2011, Boolean models were proposed for the lac operon in Escherichia coli capable of reproducing the operon being OFF, ON and bistable for three (low, medium and high) and two (low and high) parameters, representing the concentration ranges of lactose and glucose, respectively. Of these 6 possible combinations of parameters, 5 produce results that match with the biological experiments of Ozbudak et al., 2004. In the remaining one, the models predict the operon being OFF while biological experiments show a bistable behavior. In this paper, we first explore the robustness of two such models in the sense of how much its attractors change against any deterministic update schedule. We prove mathematically that, in cases where there is no bistability, all the dynamics in both models lack limit cycles while, when bistability appears, one model presents 30% of its dynamics with limit cycles while the other only 23%. Secondly, we propose two alternative improvements consisting of biologically supported modifications; one in which both models match with Ozbudak et al., 2004 in all 6 combinations of parameters and, the other one, where we increase the number of parameters to 9, matching in all these cases with the biological experiments of Ozbudak et al., 2004.

Sign in / Sign up

Export Citation Format

Share Document