Bifurcations of limit cycles in a -equivariant planar polynomial vector field of degree 7

2010 ◽  
Vol 216 (1) ◽  
pp. 35-50 ◽  
Author(s):  
Hongxian Zhou ◽  
Yanmin Zhao
Keyword(s):  
Vector Field ◽  
Limit Cycles ◽  
Polynomial Vector ◽  
Polynomial Vector Field

ON THE CONTROL OF PARAMETERS OF DISTRIBUTIONS OF LIMIT CYCLES FOR A Z2-EQUIVARIANT PERTURBED PLANAR HAMILTONIAN POLYNOMIAL VECTOR FIELD

2005 ◽  
Vol 15 (01) ◽  
pp. 137-155 ◽  
Author(s):  
JIBIN LI ◽  
HONGXIAN ZHOU
Keyword(s):  
Vector Field ◽  
Limit Cycles ◽  
Control Parameter ◽  
Compound Eyes ◽  
Polynomial Vector ◽  
Polynomial Vector Field ◽  
Control Of Parameters ◽  
The Given

In this paper, a Z2-equivariant perturbed planar Hamiltonian polynomial vector field of degree 5 is used to show how to control parameter conditions such that this system has more limit cycles and various configurations of these limit cycles. By using the method of detection functions, we prove that under different determined parameter conditions, the given system has at least 20–23 limit cycles having different configurations of the compound eyes. We give a corollary to point out that the assertion is incorrect that this system exists as 29 limit cycles in a reference.

Limit Cycles Close to Infinity of Certain Non-Linear Differential Equations

1990 ◽  
Vol 33 (1) ◽  
pp. 55-59
Author(s):  
Víctor Guíñez ◽  
Eduardo Sáez ◽  
Iván Szántó
Keyword(s):  
Vector Field ◽  
Differential Equations ◽  
Limit Cycles ◽  
Linear Differential Equations ◽  
Polynomial Vector ◽  
Polynomial Vector Field ◽  
Non Linear ◽  
Linear Differential

AbstractThrough successive radial perturbations of a certain planar Hamiltonian polynomial vector field of degree 2K + 1, we obtain a least K limit cycles containing (2K + 1)2 singularities.

BIFURCATION OF LIMIT CYCLES IN Z10-EQUIVARIANT VECTOR FIELDS OF DEGREE 9

2006 ◽  
Vol 16 (08) ◽  
pp. 2309-2324 ◽  
Author(s):  
SHARON WANG ◽  
PEI YU ◽  
JIBIN LI
Keyword(s):  
Vector Field ◽  
Hamiltonian Systems ◽  
Dynamic Systems ◽  
Limit Cycles ◽  
Bifurcation Theory ◽  
Vector Fields ◽  
Polynomial Vector ◽  
Bifurcation Of Limit Cycles ◽  
Polynomial Vector Field ◽  
Closed Orbits

In this paper, we consider the weakened Hilbert's 16th problem for symmetric planar perturbed polynomial Hamiltonian systems. In particular, a perturbed Hamiltonian polynomial vector field of degree 9 is studied, and an example of Z10-equivariant planar perturbed Hamiltonian systems is constructed. With maximal number of closed orbits, it gives rise to different configurations of limit cycles. By applying the bifurcation theory of planar dynamic systems and the method of detection functions, with the aid of numerical simulations, we show that a polynomial vector field of degree 9 with Z10 symmetry can have at least 80 limit cycles, i.e. H(9) ≥ 92 - 1.

BIFURCATIONS OF LIMIT CYCLES IN A Z2-EQUIVARIANT PLANAR POLYNOMIAL VECTOR FIELD OF DEGREE 7

2006 ◽  
Vol 16 (04) ◽  
pp. 925-943 ◽  
Author(s):  
JIBIN LI ◽  
MINGJI ZHANG ◽  
SHUMIN LI
Keyword(s):  
Vector Field ◽  
Dynamical Systems ◽  
Limit Cycles ◽  
Bifurcation Theory ◽  
Vector Fields ◽  
Compound Eyes ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Planar Dynamical Systems ◽  
Equivariant Vector Field

By using the bifurcation theory of planar dynamical systems and the method of detection functions, the bifurcations of limit cycles in a Z2-equivariant planar perturbed Hamiltonian polynomial vector fields of degree 7 are studied. An example of a special Z2-equivariant vector field having 50 limit cycles with a configuration of compound eyes are given.

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.

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