scholarly journals A globally asymptotically stable polynomial vector field with rational coefficients and no local polynomial Lyapunov function

2018 ◽  
Vol 121 ◽  
pp. 50-53 ◽  
Author(s):  
Amir Ali Ahmadi ◽  
Bachir El Khadir
Keyword(s):  
Vector Field ◽  
Lyapunov Function ◽  
Asymptotically Stable ◽  
Polynomial Vector ◽  
Globally Asymptotically Stable ◽  
Polynomial Vector Field ◽  
Stable Polynomial ◽  
Local Polynomial

scholarly journals Global Dynamics of an SIQR Model with Vaccination and Elimination Hybrid Strategies

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 328 ◽  
Author(s):  
Yanli Ma ◽  
Jia-Bao Liu ◽  
Haixia Li
Keyword(s):  
Lyapunov Function ◽  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Endemic Equilibrium ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Disease Free ◽  
The Basic Reproduction Number

In this paper, an SIQR (Susceptible, Infected, Quarantined, Recovered) epidemic model with vaccination, elimination, and quarantine hybrid strategies is proposed, and the dynamics of this model are analyzed by both theoretical and numerical means. Firstly, the basic reproduction number R 0 , which determines whether the disease is extinct or not, is derived. Secondly, by LaSalles invariance principle, it is proved that the disease-free equilibrium is globally asymptotically stable when R 0 < 1 , and the disease dies out. By Routh-Hurwitz criterion theory, we also prove that the disease-free equilibrium is unstable and the unique endemic equilibrium is locally asymptotically stable when R 0 > 1 . Thirdly, by constructing a suitable Lyapunov function, we obtain that the unique endemic equilibrium is globally asymptotically stable and the disease persists at this endemic equilibrium if it initially exists when R 0 > 1 . Finally, some numerical simulations are presented to illustrate the analysis results.

Application of Gaussian moment method to a gene autoregulation model of rational vector field

2016 ◽  
Vol 30 (20) ◽  
pp. 1650264 ◽  
Author(s):  
Yan-Mei Kang ◽  
Xi Chen
Keyword(s):  
Vector Field ◽  
Moment Method ◽  
Stochastic Systems ◽  
Polynomial Vector ◽  
Polynomial Vector Field ◽  
Model Driven ◽  
Rational Polynomial ◽  
Resonant Effect ◽  
Biochemical Systems ◽  
Gaussian Moment

We take a lambda expression autoregulation model driven by multiplicative and additive noises as example to extend the Gaussian moment method from nonlinear stochastic systems of polynomial vector field to noisy biochemical systems of rational polynomial vector field. As a direct application of the extended method, we also disclose the phenomenon of stochastic resonance. It is found that the transcription rate can inhibit the stochastic resonant effect, but the degradation rate may enhance the phenomenon. These observations should be helpful in understanding the functional role of noise in gene autoregulation.

scholarly journals Zero Counting and Invariant Sets of Differential Equations

2017 ◽  
Vol 2019 (13) ◽  
pp. 4119-4158
Author(s):  
Gal Binyamini
Keyword(s):  
Vector Field ◽  
Differential Equations ◽  
Upper Bound ◽  
Linear Differential Equations ◽  
Invariant Sets ◽  
Polynomial Vector ◽  
Algebraic Hypersurface ◽  
Polynomial Differential Equations ◽  
Polynomial Vector Field ◽  
Linear Differential

Abstract Consider a polynomial vector field $\xi$ in ${\mathbb C}^n$ with algebraic coefficients, and $K$ a compact piece of a trajectory. Let $N(K,d)$ denote the maximal number of isolated intersections between $K$ and an algebraic hypersurface of degree $d$. We introduce a condition on $\xi$ called constructible orbits and show that under this condition $N(K,d)$ grows polynomially with $d$. We establish the constructible orbits condition for linear differential equations over ${\mathbb C}(t)$, for planar polynomial differential equations and for some differential equations related to the automorphic $j$-function. As an application of the main result, we prove a polylogarithmic upper bound for the number of rational points of a given height in planar projections of $K$ following works of Bombieri–Pila and Masser.

scholarly journals Computing the topological degree of polynomial maps

1997 ◽  
Vol 56 (1) ◽  
pp. 87-94
Author(s):  
Takis Sakkalis ◽  
Zenon Ligatsikas
Keyword(s):  
Vector Field ◽  
Topological Degree ◽  
Recursive Algorithm ◽  
Main Idea ◽  
Dimensional Problem ◽  
Polynomial Vector ◽  
Polynomial Vector Field ◽  
Polynomial Maps

Let C be a cube in Rn+1 and let F = (f1, …, fn+1) be a polynomial vector field. In this note we propose a recursive algorithm for the computation of the degree of F on C. The main idea of the algorithm is that the degree of F is equal to the algebraic sum of the degrees of the map (f1, f2, …, fi−1, fi, fi+1, …, fn+1) over all sides of C, thereby reducing an (n + 1)–dimensional problem to an n–dimensional one.

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