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Jihua Yang

Publications and source records attributed to Jihua Yang.

11 recordsLinked to original sources
arXiv2026

Zeros of complete elliptic integrals and its application to Melnikov functions

In this paper, we first discuss the linear independence of the complete elliptic integrals of the first, second and third kinds $K(k)$, $E(k)$ and $\Pi(\mu(k),k)$, and then obtain an upper bound for the number of zeros of a function of the form \begin{eqnarray*} p(k)K(k)+q(k)E(k)+r(k)\Pi(\mu(k),k),\ k\in(-1,1), \end{eqnarray*} where $p(k)$, $q(k)$ and $r(k)$ are real polynomials, $\mu(k)$ is a real polynomial or rational function. Finally, we apply it to a Hamiltonian triangle with three invariant straight lines under small real polynomials piecewise smooth perturbation.

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arXiv2025

The cyclicity of period annulus of cubic isochronous Hamiltonian systems

Cima, Ma\~{n}osas and Villadelprat (J. Differ. Equations, 157, 373--413, 1999) proved that a cubic Hamiltonian system possesses an isochronous center at the origin if and only if its Hamiltonian function can be expressed as \begin{eqnarray*}H_1(x,y)=k_1^2x^2+(k_2y+k_3x+k_4x^2)^2, \end{eqnarray*} where $k_1,k_2,k_3,k_4\in\mathbb{R}$, $k_1k_2\neq0$. This paper is devoted to investigating the weak Hilbert's 16th problem for the dynamical system associated with the above Hamiltonian function. We show that the maximum number of limit cycles is $n-1$. Furthermore, this number is reached. That is, we solve the weak Hilbert's 16th problem restricted to cubic Hamiltonian systems with an isochronous center at the origin.

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arXiv2025

Bifurcation of limit cycles from a cubic reversible isochrone

This paper is devoted to study the limit cycle problem of a cubic reversible system with an isochronous center, when it is perturbed inside a class of polynomials. An upper bound of the number of limit cycles is obtained using the Abelian integral. The algebraic structure of the Abelian integral is acquired thanks to some iterative formulas, which differs in many aspects from other methods. Some numerical simulations verify the existence of limit cycles.

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arXiv2025

Limit cycles appearing from the perturbation of a cubic isochronous center

For a polynomial differential system $$\dot{x}=-y+\sum\limits_{i+j=3}α_{i,j}x^iy^j,\quad \dot{y}=x+\sum\limits_{i+j=3}β_{i,j}x^iy^j,$$ Pleshkan (Differ. Equations, 1969) proved that the origin is an isochronous center of this system iff it can be brought to one of $S^*_1$, $S^*_2$, $S^*_3$ or $S^*_4$. The bifurcation of limit cycles for these four types of isochronous differential systems have not yet been studied, except for $S^*_1$. This paper is devoted to study the limit cycle problem of $S^*_2$ when we perturb it with an arbitrary polynomial vector field. An upper bound of the number of limit cycles is obtained using the Abelian integral.

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arXiv2022

Bifurcation of limit cycles by perturbing piecewise smooth integrable differential systems with four zones

This paper deals with the problem of limit cycle bifurcations for piecewise smooth integrable differential systems with four zones. When the unperturbed system has a family of periodic orbits, the first order Melnikov function is derived which can be used to study the number of limit cycles bifurcated from the periodic orbits. As an application, using the first order Melnikov function and Picard-Fuchs equation, we obtain an upper bound of the number of bifurcated limit cycles of a concrete piecewise smooth differential system.

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arXiv2019

Limit cycles appearing from the perturbation of differential systems with multiple switching curves

This paper deals with the problem of limit cycle bifurcations for a piecewise near-Hamilton system with four regions separated by algebraic curves $y=\pm x^2$. By analyzing the obtained first order Melnikov function, we give an upper bound of the number of limit cycles which bifurcate from the period annulus around the origin under $n$-th degree polynomial perturbations. In the case $n=1$, we obtain that at least 4 (resp. 3) limit cycles can bifurcate from the period annulus if the switching curves are $y=\pm x^2$ (resp. $y=x^2$ or $y=-x^2$). The results also show that the number of switching curves affects the number of limit cycles.

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arXiv2018

Bifurcation of limit cycles from a quadratic global center with two switching lines

In this paper, we generalize the Picard-Fuchs equation method to study the bifurcation of limit cycles of perturbed differential systems with two switching lines. We obtain the detailed expression of the corresponding first order Melnikov function which can be used to get the upper bound of the number of limit cycles for the perturbed system by using Picard-Fuchs equation. It is worth noting that we greatly simplify the computations and this method can be applied to study the number of limit cycles of other differential systems with two switching lines. Our results also show that the number of switching lines has essentially impact on the number of limit cycles bifurcating from a period annulus.

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arXiv2018

Limit cycles appearing from perturbations of cubic piecewise smooth center with double invariant real straight lines

This paper investigates the exact number of limit cycles given by the averaging theory of first order for the piecewise smooth integrable non-Hamiltonian system \begin{eqnarray*} (\dot{x},\ \dot{y})=\begin{cases} (-y(x+a)^2+\varepsilon f^+(x,y),\ x(x+a)^2+\varepsilon g^+(x,y)),\ \ x\geq0,\\ (-y(x+b)^2+\varepsilon f^-(x,y),\ x(x+b)^2+\varepsilon g^-(x,y)),\ ~ \, x<0,\\ \end{cases}\end{eqnarray*} where $ab\neq 0$, $0<|\varepsilon|\ll 1$, and $f^\pm(x,y)$ and $g^\pm(x,y)$ are polynomials of degree $n$. It is proved that the exact number of limit cycles emerging from the period annulus surrounding the origin is linear depending on $n$ and it is at least twice the associated estimation of smooth systems.

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arXiv2018

On the number of limit cycles for generic Lotka-Volterra system and Bogdanov-Takens system under perturbations of piecewise smooth polynomials

In this paper, we consider the bifurcation of limit cycles for generic L-V system ($\dot{x}=y+x^2-y^2\pm\frac{4}{\sqrt{3}}xy,~\dot{y}=-x+2xy$) and B-T system ($\dot{x}=y,~\dot{y}=-x+x^2$) under perturbations of piecewise smooth polynomials with degree $n$. Here the switching line is $y=0$. By using Picard-Fuchs equations, we bound the number of zeros of first order Melnikov function which controls the number of limit cycles bifurcating from the center. It is proved that the upper bounds of the number of limit cycles for generic L-V system and B-T system are respectively $36n-65~(n\geq4),~37,57,93~(n=1,2,3)$ and $12n+6$.

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arXiv2017

On the number of zeros of Abelian integrals for discontinuous quadratic differential systems

Applying the Picard-Fuchs equation to the discontinuous differential system, we obtain the upper bounds of the number of zeros for Abelian integrals of four kinds of quadratic differential systems when it is perturbed inside all discontinuous polynomials with degree $n$. Furthermore, by using the {\it Chebyshev criterion}, we obtain the sharp upper bounds on each period annulus for $n=2$.

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