Bifurcation in a quartic polynomial system arising in biology

1990 ◽  
pp. 356-368 ◽  
Author(s):  
Franz Rothe ◽  
Douglas S. Shafer
Keyword(s):  
Polynomial System ◽  
Quartic Polynomial

BIFURCATION OF LIMIT CYCLES AND ISOCHRONOUS CENTERS FOR A QUARTIC SYSTEM

2013 ◽  
Vol 23 (10) ◽  
pp. 1350172 ◽  
Author(s):  
WENTAO HUANG ◽  
AIYONG CHEN ◽  
QIUJIN XU
Keyword(s):  
Limit Cycles ◽  
Necessary Conditions ◽  
Polynomial System ◽  
Isochronous Center ◽  
Eighth Order ◽  
Bifurcation Of Limit Cycles ◽  
Quartic Polynomial ◽  
Fine Focus ◽  
Sufficient And Necessary Conditions ◽  
Isochronous Centers

For a quartic polynomial system we investigate bifurcations of limit cycles and obtain conditions for the origin to be a center. Computing the singular point values we find also the conditions for the origin to be the eighth order fine focus. It is proven that the system can have eight small amplitude limit cycles in a neighborhood of the origin. To the best of our knowledge, this is the first example of a quartic system with eight limit cycles bifurcated from a fine focus. We also give the sufficient and necessary conditions for the origin to be an isochronous center.

scholarly journals Iterative stability analysis for general polynomial control systems

2021 ◽  
Author(s):  
Bo Xiao ◽  
Hak-Keung Lam ◽  
Zhixiong Zhong
Keyword(s):  
Stability Analysis ◽  
Lyapunov Function ◽  
Control Systems ◽  
Polynomial System ◽  
Stability Conditions ◽  
General Polynomial ◽  
Main Challenge ◽  
Detailed Simulation ◽  
Feasible Solutions ◽  
The Stability

AbstractThe main challenge of the stability analysis for general polynomial control systems is that non-convex terms exist in the stability conditions, which hinders solving the stability conditions numerically. Most approaches in the literature impose constraints on the Lyapunov function candidates or the non-convex related terms to circumvent this problem. Motivated by this difficulty, in this paper, we confront the non-convex problem directly and present an iterative stability analysis to address the long-standing problem in general polynomial control systems. Different from the existing methods, no constraints are imposed on the polynomial Lyapunov function candidates. Therefore, the limitations on the Lyapunov function candidate and non-convex terms are eliminated from the proposed analysis, which makes the proposed method more general than the state-of-the-art. In the proposed approach, the stability for the general polynomial model is analyzed and the original non-convex stability conditions are developed. To solve the non-convex stability conditions through the sum-of-squares programming, the iterative stability analysis is presented. The feasible solutions are verified by the original non-convex stability conditions to guarantee the asymptotic stability of the general polynomial system. The detailed simulation example is provided to verify the effectiveness of the proposed approach. The simulation results show that the proposed approach is more capable to find feasible solutions for the general polynomial control systems when compared with the existing ones.

THE NUMBER OF ROOTS OF A POLYNOMIAL SYSTEM

2020 ◽  
pp. 1-10
Author(s):  
NGUYEN CONG MINH ◽  
LUU BA THANG ◽  
TRAN NAM TRUNG
Keyword(s):  
Polynomial Ring ◽  
Polynomial System ◽  
Combinatorial Nullstellensatz ◽  
Roots Of A Polynomial

Abstract Let I be a zero-dimensional ideal in the polynomial ring $K[x_1,\ldots ,x_n]$ over a field K. We give a bound for the number of roots of I in $K^n$ counted with combinatorial multiplicity. As a consequence, we give a proof of Alon’s combinatorial Nullstellensatz.

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