arXiv subjects

Hebai Chen

Publications and source records attributed to Hebai Chen.

13 recordsLinked to original sources

arXiv2026

New criteria for upper bounds on the number of limit cycles in Liénard systems

We establish criteria for at most one or at most two limit cycles of generalized Liénard systems in a single potential well. Two independent uniqueness tests use modified quotients of the damping and restoring forces; both allow a degenerate restoring force, and one extends to continuous potential profiles with corners. Opposite convexity of the two branches of the damping quotient in the potential coordinate gives an upper bound of two. Further criteria apply to weak damping with an analytic normalized profile on a bounded potential interval under an explicit smallness condition, to small profile perturbations vanishing near the equilibrium on a bounded range of half-widths, and to parameter families satisfying curvature, monotonicity and boundary conditions. Applications give another proof of the classical quartic bound. For Selkov systems, we obtain a two-cycle bound in a specified negative-parameter region and prove existence and uniqueness in a positive-parameter region. We also prove Conjecture~2.1 in [J. Math. Anal. Appl., 2025]. Examples attain the two-cycle bound.

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arXiv2026

Some new criteria for the uniqueness of limit cycles in Rayleigh--Lienard systems

In this paper, new uniqueness criteria for Rayleigh--Liénard systems with damping of positive odd degree in the velocity are established. Under suitable conditions, any limit cycle is unique and hyperbolic, with its stability determined by the damping signs. Neither symmetry nor monotonicity of the damping and restoring functions is required. Changes of variables and time yield further criteria for generalized Kukles systems. Applications to three polynomial families yield uniqueness and nonexistence results. In particular, we resolve an open uniqueness problem for a hybrid van der Pol--Rayleigh oscillator left open by Chen et al. (Phys. D, 2021). We also give an alternative uniqueness proof for a system with homogeneous components and obtain uniqueness and nonexistence results for a modified hybrid van der Pol--Rayleigh oscillator.

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arXiv2026

Limit cycles of slow-fast quintic Lienard systems

For every fixed real quintic polynomial $F$, we prove that the slow--fast Liénard system $\dot{x}=y-F(x),\quad \dot{y}=-\varepsilon x$ admits at most two limit cycles whenever $0<\varepsilon<\varepsilon_0(F)$ for a suitable $\varepsilon_0(F)>0$. Using scalar comparison near the equilibrium and a logarithmic fold-passage correction, we exclude \rev{canard singular limits}. Every sequence of limit cycles with $\varepsilon\to0^+$ has a subsequence converging in the Hausdorff metric to a common slow--fast path, whose regular slow arcs are either all normally attracting or all normally repelling. We also establish uniform Floquet separation for fixed $F$. The global counting argument uses shared regular arcs, nesting, and stability alternation. These results give a global small-parameter bound and describe the singular-limit structure, complementing the local cyclicity analysis of Li and Lu(J. Differential Equations, 2014).

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arXiv2026

Cyclicity of lips and center-focus problem at infinity

In this work, we discover an intriguing Liénard system which is integrable near infinity, integrable near zero and having Ilyashenko-Kotova lips in between. Moreover, we give a necessary and sufficient condition of polynomial Liénard system with center at infinity. In a different direction we improve a lower bound on Hilbert number for Liénard systems. The improvement is due to development of Brudnyi method of calculation of cyclicity of centers near infinity and near zero.

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arXiv2026

Bounds on the maximum number of limit cycles of piecewise linear Lienard systems II. The discontinuous case

This paper concerns the planar Liénard system $\dot x=y-F(x),\ \dot y=-x$, where $F(x)$ is a piecewise linear function with exactly $m$ jump points and no fold points. Tonnelier [SIAM J. Appl. Math. 63 (2002)] conjectured that the maximum number of limit cycles of the system is $2m$. The conjecture was confirmed for $m=1$ in [J. Lond. Math. Soc. 113 (2026)], and a lower bound of $2m$ for arbitrary $m$ was established by Chen et al. [arXiv:2608.19542]. In this paper, we construct systems with at least $4m-2$ hyperbolic crossing limit cycles for every positive integer $m$, thereby disproving Tonnelier's conjecture for $m\geq2$. We also establish the uniform upper bound $2^{224(m+1)^2}$ for the total number of crossing, grazing, sliding, and composite limit cycles.

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arXiv2026

Bounds on the maximum number of limit cycles of piecewise linear Lienard systems I. The continuous case

This paper concerns the planar Liénard system $\dot x=y-F(x),\ \dot y=-x$, where $F(x)$ is a continuous piecewise linear function with exactly $n$ fold points. Tonnelier [SIAM J. Appl. Math. 63 (2002)] conjectured that the maximum number of limit cycles of the system is $n$ when $F$ has exactly $n$ fold points. The conjecture was confirmed for $n=1$ and $n=2$ in [J. Nonlinear Sci. 25 (2015)] and [J. Lond. Math. Soc. 113 (2026)], respectively, whereas the case $n\ge3$ remained open. In this paper, we show that the maximal number of limit cycles is at least $|3n-4|$ for $n\in\mathbb N^+$, thereby disproving Tonnelier's conjecture for $n\ge3$. The proof reveals a unified multiscale perturbation mechanism underlying the creation and coexistence of multiple limit cycles. Moreover, we prove that the number of limit cycles admits the finite upper bound $2^{28(n+1)^2}$.

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arXiv2026

The number of limit cycles of piecewise linear Liénard systems

For the planar Liénard differential system $\dot{x}=F(x)-y$, $\dot{y}=x$, where $F(x)$ is a piecewise linear function, Tonnelier (SIAM J. Appl. Math., 2002) conjectured that the maximum number of limit cycles of the system is $n$ when $F(x)$ has $n$ fold points and no jump points, and $2n$ when $F(x)$ has $n$ jump points and no fold points. This conjecture was confirmed by Llibre et al. (J. Nonlinear Sci., 2015) (resp. Chen et al. (J. London Math. Soc., 2026a)) when $F(x)$ has one fold point and no jump points (resp. two fold points). More recently, Chen et al. (J. London Math. Soc., 2026b) proved that the conjecture is correct when $F(x)$ has no fold points and one jump point. All other cases remain open. Here we verify that the lower bound for the maximum number of limit cycles of the system can be $n$ when $F(x)$ has only $n$ fold points, and $2n$ when $F(x)$ has only $n$ jump points, thereby confirming the lower bound part of Tonnelier's conjecture. Moreover, when $F(x)$ has $m$ jump points and $n-m$ fold points, $0\le m\le n$, we also show that the system can have $n+m=(n-m)+2m$ limit cycles. In addition, a complete classification of the {dynamics} near infinity for this class of systems is provided.

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arXiv2025

The discontinuous limit case of an archetypal oscillator with constant excitation and van der Pol damping: A single equilibrium

This paper investigates the global dynamics of the discontinuous limit case of an archetypal oscillator with constant excitation that exhibits a single equilibrium. For parameter regions in which this oscillator possesses two or three equilibria, the global bifurcation diagram and the corresponding phase portraits on the Poincare disc have been presented in [Phys. D, 438 (2022) 133362]. The present work completes the global structure of the discontinuous limit case of an archetypal oscillator with constant excitation. Although the dynamical phenomena are less rich compared to systems with more than one equilibrium, the presence of a single equilibrium gives rise to additional limit cycles surrounding it, thereby enriching the overall dynamics and making the analysis substantially more intricate than in the previously studied cases.

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arXiv2024

A quintic Z2-equivariant Liénard system arising from the complex Ginzburg-Landau equation: (II)

We continue to study a quintic Z2-equivariant Liénard system $\dot x=y,\dot y=-(a_0x+a_1x^3+a_2x^5)-(b_0+b_1x^2)y$ with $a_2b_1\ne 0$, arising from the complex Ginzburg-Landau equation. Global dynamics of the system have been studied in [{\it SIAM J. Math. Anal.}, {\bf 55}(2023) 5993-6038] when the sum of the indices of all equilibria is $-1$, i.e., $a_2<0$. The aim of this paper is to study the global dynamics of this quintic Liénard system when the sum of the indices of all equilibria is $1$, i.e., $a_2>0$.

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arXiv2023

The number of limit cycles of Josephson equation

In this paper, the existence and number of non-contractible limit cycles of the Josephson equation $β\frac{d^{2}Φ}{dt^{2}}+(1+γ\cos Φ)\frac{dΦ}{dt}+\sin Φ=α$ are studied, where $ϕ\in \mathbb S^{1}$ and $(α,β,γ)\in \mathbb R^{3}$. Concretely, by using some appropriate transformations, we prove that such type of limit cycles are changed to limit cycles of some Abel equation. By developing the methods on limit cycles of Abel equation, we prove that there are at most two non-contractible limit cycles, and the upper bound is sharp. At last, combining with the results of the paper (Chen and Tang, J. Differential Equations, 2020), we show that the sum of the number of contractible and non-contractible limit cycles of the Josephson equation is also at most two, and give the possible configurations of limit cycles when two limit cycles appear.

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arXiv2022

A sufficient and necessary condition of generalized polynomial Liénard systems with global centers

The aim of this paper is to give a sufficient and necessary condition of the generalized polynomial Liénard system with a global center (including linear typer and nilpotent type). Recently, Llibre and Valls [J. Differential Equations, 330 (2022), 66-80] gave a sufficient and necessary condition of the generalized polynomial Liénard system with a linear type global center. It is easy to see that our sufficient and necessary condition is more easy by comparison. In particular, we provide the explicit expressions of all the generalized polynomial Liénard differential systems of degree 5 having a global center at the origin and the explicit expression of a generalized polynomial Liénard differential system of indefinite degree having a global center at the origin.

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arXiv2022

New criterions on nonexistence of periodic orbits of planar dynamical systems and their applications

Characterizing existence or not of periodic orbit is a classical problem and it has both theoretical importance and many real applications. Here, several new criterions on nonexistence of periodic orbits of the planar dynamical system $\dot x=y,~\dot y=-g(x)-f(x,y)y$ are obtained in this paper, and by examples showing that these criterions are applicable, but the known ones are invalid to them. Based on these criterions, we further characterize the local topological structures of its equilibrium, which also show that one of the classical results by A.F. Andreev [Amer. Math. Soc. Transl. 8 (1958), 183--207] on local topological classification of the degenerate equilibrium is incomplete. Finally, as another application of these results, we classify the global phase portraits of a planar differential system, which comes from the third question in the list of the 33 questions posed by A. Gasull and also from a mechanical oscillator under suitable restriction to its parameters.

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arXiv2019

Crossing limit cycles of nonsmooth Liénard systems and applications

Continuing the investigation for the number of crossing limit cycles of nonsmooth Liénard systems in [Nonlinearity 21(2008), 2121-2142] for the case of a unique equilibrium, in this paper we consider the case of any number of equilibria. We give results about the existence and uniqueness of crossing limit cycles, which hold not only for a unique equilibrium but also for multiple equilibria. Moreover, we find a sufficient condition for the nonexistence of crossing limit cycles. Finally, applying our results we prove the uniqueness of crossing limit cycles for planar piecewise linear systems with a line of discontinuity and without sliding sets.

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