arXiv subjects

Haihua Liang

Publications and source records attributed to Haihua Liang.

8 recordsLinked to original sources

arXiv2026

Bifurcation of limit cycles in a class of piecewise smooth generalized Abel equations with two asymmetric zones

This paper studies the number of limit cycles, known as the Smale-Pugh problem, for the generalized Abel equation \begin{align*} \frac{dx}{dθ}=A(θ)x^p+B(θ)x^q, \end{align*} where $A$ and $B$ are are piecewise trigonometrical polynomials of degree $ m $ with two zones $0\leqθ<θ_1$ and $θ_1\leqθ\leq2π$. By means of the first and second order analysis using the Melnikov theory and applying the new Chebyshev criterion that established by \cite{HLZ2023}, we estimate the maximum number of positive and negative limit cycles that such equations can have, and reveal how this maximum number, denoted by $H_{θ_1}(m)$, is affected by the location of the separation line $θ=θ_1$. For the equation of classical Abel type, our result not only includes the estimates provided in the recent paper (Huang et al., SIAM J. Appl. Dyn. Syst., 2020), i.e., $H_{2π}(m)\geq 4m-2$ for $θ_1=2π$, but also shows that the equation in the discontinuous case can possess more than two times as many limit cycles as in the continuous case. More accurately, $H_π(m)\geq 8m+2$ and $H_{θ_1}(m)\geq 14m-6$ for $θ_1\in (0,π)\cup (π,2π)$.

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arXiv2024

U-SEANNet: A Simple, Efficient and Applied U-Shaped Network for Diagnosis of Nasal Diseases on Nasal Endoscopic Images

Numerous studies have affirmed that deep learning models can facilitate early diagnosis of lesions in endoscopic images. However, the lack of available datasets stymies advancements in research on nasal endoscopy, and existing models fail to strike a good trade-off between model diagnosis performance, model complexity and parameters size, rendering them unsuitable for real-world application. To bridge these gaps, we created the first large-scale nasal endoscopy dataset, named 7-NasalEID, comprising 11,352 images that contain six common nasal diseases and normal samples. Subsequently, we proposed U-SEANNet, an innovative U-shaped architecture, underpinned by depth-wise separable convolution. Moreover, to enhance its capacity for detecting nuanced discrepancies in input images, U-SEANNet employs the Global-Local Channel Feature Fusion module, enabling it to utilize salient channel features from both global and local contexts. To demonstrate U-SEANNet's potential, we benchmarked U-SEANNet against seventeen modern architectures through five-fold cross-validation. The experimental results show that U-SEANNet achieves a commendable accuracy of 93.58%. Notably, U-SEANNet's parameters size and GFLOPs are only 0.78M and 0.21, respectively. Our findings suggest U-SEANNet is the state-of-the-art model for nasal diseases diagnosis in endoscopic images.

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arXiv2023

SFUSNet: A Spatial-Frequency domain-based Multi-branch Network for diagnosis of Cervical Lymph Node Lesions in Ultrasound Images

Booming deep learning has substantially improved the diagnosis for diverse lesions in ultrasound images, but a conspicuous research gap concerning cervical lymph node lesions still remains. The objective of this work is to diagnose cervical lymph node lesions in ultrasound images by leveraging a deep learning model. To this end, we first collected 3392 cervical ultrasound images containing normal lymph nodes, benign lymph node lesions, malignant primary lymph node lesions, and malignant metastatic lymph node lesions. Given that ultrasound images are generated by the reflection and scattering of sound waves across varied bodily tissues, we proposed the Conv-FFT Block. It integrates convolutional operations with the fast Fourier transform to more astutely model the images. Building upon this foundation, we designed a novel architecture, named SFUSNet. SFUSNet not only discerns variances in ultrasound images from the spatial domain but also adeptly captures micro-structural alterations across various lesions in the frequency domain. To ascertain the potential of SFUSNet, we benchmarked it against 12 popular architectures through five-fold cross-validation. The results show that SFUSNet is the state-of-the-art model and can achieve 92.89% accuracy. Moreover, its average precision, average sensitivity and average specificity for four types of lesions achieve 90.46%, 89.95% and 97.49%, respectively.

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arXiv2020

A new Chebyshev criterion and its application to planar differential systems

This paper establishes a new Chebyshev criterion for some family of integrals. By virtue of this criterion we obtain several new Chebyshev families. With the help of these new families we can answer the conjecture posed by Gasull et al in 2015. %[J. Differential Equations, 258 (2015), 3286--3303]. Their applications to other two planar differential systems also show that our approach is simpler and in a unified way to handle many kinds of planar differential systems for estimating the number of limit cycles bifurcating from period annulus.

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arXiv2018

Planar Semi-quasi Homogeneous Polynomial differential systems with a given degree

This paper study the planar semi-quasi homogeneous polynomial differential systems (short for PSQHPDS), which can be regard as a generalization of semi-homogeneous and of quasi-homogeneous systems. By using the algebraic skills, several important properties of PSQHPDS are derived and are employed to establish an algorithm for obtaining all the explicit expressions of PSQHPDS with a given degree. Afterward, we apply this algorithm to research the center problem of quadratic and cubic PSQHPDS. It is proved that the quadratic one hasn't center, and, that the cubic one has center if and only if it can be written as $\dot{x}=x^2-y^3, \dot{y}=x$ after a linear transformation of coordinate and a rescaling of time.

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arXiv2017

Limit cycles of planar system defined by the sum of two quasi-homogeneous vecter fields

In this paper we consider the limit cycles of the planar system $$\frac{d}{dt}(x,y)=\mathbf X_n+\mathbf X_m, $$ where $\mathbf X_n$ and $\mathbf X_m$ are quasi-homogeneous vector fields of degree $n$ and $m$ respectively. We prove that under a new hypothesis, the maximal number of limit cycles of the system is $1$. We also show that our result can be applied to some systems when the previous results are invalid. The proof is based on the investigations for the Abel equation and the generalized-polar equation associated with the system, respectively. Usually these two kinds of equations need to be dealt with separately, and for both equations, an efficient approach to estimate the number of periodic solutions is constructing suitable auxiliary functions. In the present paper we develop a formula on the divergence, which allows us to construct an auxiliary function of one equation with the auxiliary function of the other equation, and vice versa.

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arXiv2016

A geometric criterion for equation $\dot{x}=\sum^m_{i=0}a_i(t)x^i$ having at most $m$ isolated periodic solutions

This paper is devoted to the investigation of generalized Abel equation $\dot{x}=S(x,t)=\sum^m_{i=0}a_i(t)x^i$, where $a_i\in\mathrm C^{\infty}([0,1])$. A solution $x(t)$ is called a {\em periodic solution} if $x(0)=x(1)$. In order to estimate the number of isolated periodic solutions of the equation, we propose a hypothesis (H) which is only concerned with $S(x,t)$ on $m$ straight lines: There exist $m$ real numbers $λ_1<\cdots<λ_m$ such that either $(-1)^i\cdot S(λ_i,t)\geq0$ for $i=1,\cdots,m$, or $(-1)^i\cdot S(λ_i,t)\leq0$ for $i=1,\cdots,m$. By means of Lagrange interpolation formula, we proves that the equation has at most $m$ isolated periodic solutions (counted with multiplicities) if hypothesis (H) holds, and the upper bound is sharp. Furthermore, this conclusion is also obtained under some weaker geometric hypotheses. Applying our main result for the trigonometrical generalized Abel equation with coefficients of degree one, we give a criterion to obtain the upper bound for the number of isolated periodic solutions. This criterion is "almost equivalent" to hypothesis (H) and can be much more effectively checked.

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arXiv2014

Abelian integrals and limit cycles for a class of cubic polynomial vector fields of Lotka-Volterra type with a rational first integral of degree 2

In this paper, we study the number of limit cycles which bifurcate from the periodic orbits of cubic polynomial vector fields of Lotka-Volterra type having a rational first integral of degree 2, under polynomial perturbations of degree $n$. The analysis is carried out by estimating the number of zeros of the corresponding Abelian integrals. Moreover, using \emph{Chebyshev criterion}, we show that the sharp upper bound for the number of zeros of the Abelian integrals defined on each period annulus is 3 for $n=3$. The simultaneous bifurcation and distribution of limit cycles for the system with two period annuli under cubic polynomial perturbations are considered. All configurations $(u,v)$ with $0\leq u, v\leq 3, u+v\leq5$ are realizable.

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