M. Sabatini
We prove a uniqueness result for limit cycles of a class of second order ODE's. As a special case, we prove limit cycle's uniqueness for an ODE studied in \cite{ETBA}.
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In this note, we present an extension to second order nonlinear ordinary differential equations (ODEs) of the Nagumo-like uniqueness criterion for first order ODEs established in [A. Constantin, On Nagumo's theorem, Proc. Japan Acad. 86(A) (2010), pp. 41--44].
We prove a uniqueness result for limit cycles of the second order ODE $\ddot x + \dot x ϕ(x,\dot x) + g(x) = 0$. Under mild additional conditions, we show that such a limit cycle attracts every non-constant solution. As a special case, we prove limit cycle's uniqueness for an ODE studied in \cite{ETA} as a model of pedestrians' walk. This paper is an extension to equations with a non-linear $g(x)$ of the results presented in \cite{S}.
We study local, analytic solutions for a class of initial value problems for singular ODEs. We prove existence and uniqueness of such solutions under a certain non-resonance condition. Our proof translates the singular initial value problem to an equilibrium problem of a regular ODE. Then, we apply classical invariant manifold theory. We demonstrate that the class of ODEs under consideration captures models which describe the shape of axially symmetric surfaces which are closed on one side. Our main result guarantees smoothness at the tip of the surface.
Using ODE-methods and trapping regions derived by Mattingly and Sinai we give a new proof of the existence and uniqueness of solutions to Navier-Stokes equations with periodic boundary conditions on the plane.
In this paper we give sufficient conditions to ensure uniqueness of limit cycles for a class of planar vector fields. We also exhibit a class of examples with exactly one limit cycle.
We give an account of the results about limit cycle's uniqueness for Liénard equations, from Levinson-Smith's one to the most recent ones. We present a new uniqueness theorem in the line of Sansone-Massera's geometrical approach.
Periodic Center Manifolds for Nonhyperbolic Limit Cycles in ODEs
针对非双曲极限环附近中心流形存在性此前无严格证明的问题，采用Lyapunov-Perron方法给出其存在性的初等证明，并提供了具有不同光滑性与唯一性的周期中心流形的解析向量场实例。
Uniqueness of Limit Cycles for Abel Equations and Planar Systems
该研究针对$2\pi$周期系数的阿贝尔方程，提出半周期凸性方法证明其正极限环的唯一性与双曲性，并将结果应用于两类平面多项式系统，解答了Gasull公开问题的部分内容。
Continuation of connecting orbits in 3D-ODEs: (I) Point-to-cycle connections
提出使用投影边界条件进行三维自治常微分方程中点到周期轨道连接轨道的数值延拓新方法，通过伴随变分方程的特征函数避免单值矩阵计算，并在AUTO中实现。
Biologists have long observed periodic-like oxygen consumption oscillations in yeast populations under certain conditions and several unsatisfactory explanations for this phenomenon have been proposed. These "autonomous oscillations" have often appeared with periods that are nearly integer divisors of the calculated doubling time of the culture. We hypothesize that these oscillations could be caused by a weak form of cell cycle synchronization that we call clustering. We develop some novel ordinary differential equation models of the cell cycle. For these models, and for random and stochastic …
We show that there exist generic slow-fast systems with only one (time-scaling) parameter on the two-torus, which have canard cycles for arbitrary small values of this parameter. This is in drastic contrast with the planar case, where canards usually occur in two-parametric families. Here we treat systems with a convex slow curve. In this case there is a set of parameter values accumulating to zero for which the system has exactly one attracting and one repelling canard cycle. The basin of the attracting cycle is almost the whole torus.
Theoretical results regarding two-dimensional ordinary-differential equations (ODEs) with second-degree polynomial right-hand sides are summarized, with an emphasis on limit cycles, limit cycle bifurcations and multistability. The results are then used for construction of two reaction systems, which are at the deterministic level described by two-dimensional third-degree kinetic ODEs. The first system displays a homoclinic bifurcation, and a coexistence of a stable critical point and a stable limit cycle in the phase plane. The second system displays a multiple limit cycle bifurcation, and a c…
It is well known that the sum of points of the period five cycle of the quadratic polynomial $f_{c}(x)=x^{2}+c$ is generally not one-valued. In this paper we will show that the sum of cycle points of the curves of period five is at most three-valued on a new coordinate plane, and that this result is essentially the best possible. The method of our proof relies on a implementing Gröbner-bases and especially extension theory from the theory of polynomial algebra.
A proof of the uniqueness of the limit cycle of a quasi-homogeneous system
本文通过分析无穷远处异宿分界线的唯一性，证明了拟齐次系统极限环的唯一性，从而对 A. Gasull 于 2021 年提出的低维动力系统开放问题 3 给出了肯定回答。
Recent asymptotic results by the authors provided detailed information on the shape of solitary high-energy travelling waves in FPU atomic chains. In this note we use and extend the methods to understand the linearisation of the travelling wave equation. We show that there are not any other zero eigenvalues than those created by the translation symmetry and this implies a local uniqueness result. The key argument in our asymptotic analysis is to replace the linear advance-delay-differential equation for the eigenfunctions by an approximate ODE.
This paper applies a recent result determining periodic orbits on the basis of first integrals, for Liénard systems. By solving a first order ODE with singularities, a crucial result is proved to locate intervals of single and isolated maximum amplitudes periodic orbits (limit cycles). With this result an upper bound for the number of limit cycles is provided. Some examples are presented along with conclusions and future work
The dynamics of a chemical reaction network (CRN) is often modelled under the assumption of mass action kinetics by a system of ordinary differential equations (ODEs) with polynomial right-hand sides that describe the time evolution of concentrations of chemical species involved. Given an arbitrarily large integer $K \in {\mathbb N}$, we show that there exists a CRN such that its ODE model has at least $K$ stable limit cycles. Such a CRN can be constructed with reactions of at most second order provided that the number of chemical species grows linearly with $K$. Bounds on the minimal number o…
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 wit…
Uniqueness and stability of limit cycles in planar piecewise linear differential systems without sliding region
本文研究无滑动区域的平面分段线性微分系统，利用 Poincaré 半映射的新积分特征，证明极限环数量的最优一致上界为一，并给出其双曲性、稳定性及存在条件。
We consider the dynamics of point particles which are confined to a bounded, possibly nonconvex domain $Ω$. Collisions with the boundary are described as purely elastic collisions. This turns the description of the particle dynamics into a coupled system of second order ODEs with discontinuous right-hand side. The main contribution of this paper is to develop a precise solution concept for this particle system, and to prove existence of solutions. In this proof we construct a solution by passing to the limit in an auxiliary problem based on the Yosida approximation. In addition to existence of…
The phase--portrait of the second order differential equation: $$\ddot x+\sum_{l=0}^nf_l(x) \dot x^l=0 ,$$ is studied. Some results concerning existence, non--existence and uniqueness of limit cycles are presented. Among these, a generalization of the classical Massera uniqueness result is proved.
This Letter outlines 20 geometric mechanisms by which limit cycles are created locally in two-dimensional piecewise-smooth systems of ODEs. These include boundary equilibrium bifurcations of hybrid systems, Filippov systems, and continuous systems, and limit cycles created from folds and by the addition of hysteresis or time-delay. Scaling laws for the amplitude and period of the limit cycles are compared to (classical) Hopf bifurcations.
ODE and PDE based modeling of biological transportation networks
本文研究了基于图的离散（ODE基于）模型中解的全局存在的问题，提出了一种适应性模型，使其可以得到宏观（PDE基于）系统的正式连续极限，并证明了宏观PDE模型弱解的全局存在性，最后通过离散模型的数值模拟展示了向稳态收敛、稳态的非唯一性以及其对初始数据和模型参数的依赖性。
We present algorithms and their implementation to compute limit cycles and their isochrons for state-dependent delay equations (SDDE's) which are perturbed from a planar differential equation with a limit cycle. Note that the space of solutions of an SDDE is infinite dimensional. We compute a two parameter family of solutions of the SDDE which converge to the solutions of the ODE as the perturbation goes to zero in a neighborhood of the limit cycle. The method we use formulates functional equations among periodic functions (or functions converging exponentially to periodic). The functional equ…
Exactness, Cohomology, and Uniqueness in First-Order Differential Equations
本文通过 de Rham 上同调视角研究一阶 ODE 可解性与底层域拓扑的关系，证明第一 de Rham 上同调群的平凡性是全局可积性与解唯一性的根本要求，并揭示拓扑障碍导致的非唯一性或多值行为，为理解一阶 ODE 全局行为提供统一框架。
Stability analysis and long-time convergence of a partial differential equation model of two-phase ageing
本文研究了两相衰老模型的稳定性及长时间收敛性，通过年龄结构偏微分方程描述野生种群的两个衰老阶段，证明了弱解的存在性、正性和唯一性，并探讨了简化后的ODE和PDE-ODE模型的稳态稳定性。
We analyze the complex dynamics dynamics of a family of $\mathbb{Z}_{12}-$equivariant planar systems, by using their reduction to an Abel equation. We derive conditions in the parameter space that allow uniqueness and hyperbolicity of a limit cycle surrounding either $1,~13$ or $25$ equilibria.
Numerical periodic normalization for codim 2 bifurcations of limit cycles with center manifold of dimension higher than 3
针对中心流形维数为4或5的三种最复杂极限环余维2分岔，推导了周期规范形系数的显式计算格式，并应用于激光模型、种群生物学模型和机械振动模型，通过Lyapunov指数验证了规范形分析结果。
Planar chemical reaction systems with algebraic and non-algebraic limit cycles
研究平面化学反应系统的极限环数量，针对三类反应系统给出修正希尔伯特数的下界，并构造具有代数与非代数极限环的化学系统，包括四个稳定代数极限环的实例。
We analyze the dynamics of a class of $\mathbb{Z}_{2n}$-equivariant differential equations on the plane, depending on 4 real parameters. This study is the generalisation to $\mathbb{Z}_{2n}$ of previous works with $\mathbb{Z}_4$ and $\mathbb{Z}_6$ symmetry. We reduce the problem of finding limit cycles to an Abel equation, and provide criteria for proving in some cases uniqueness and hyperbolicity of the limit cycle that surrounds either 1, $2n+1$ or $4n+1$ equilibria, the origin being always one of these points.
We analyze the dynamics of a 4-parameter family of planar ordinary differential equations, given by a polynomial of degree 5 that is equivariant under a symmetry of order 6. We obtain the number of limit cycles as a function of the parameters, and provide criteria for proving in some cases uniqueness and hyperbolicity of the limit cycle surrounding either 1, 7 or 13 critical points, the origin being always one of these points. The method used is the reduction of the problem to an Abel equation.
Numerical Periodic Normalization for Codim 2 Bifurcations of Limit Cycles
针对一般自治ODE中极限环的余维2分岔，推导了周期规范型及其系数的计算公式，并通过数值边界值算法实现，用于区分不同分岔场景。
Cycles in Impulsive Goodwin's Oscillators of Arbitrary Order
本文证明任意阶脉冲古德温振荡器在标准假设下存在正1-周期解，并给出m≤10时唯一性的证明及m>10时唯一性成立的温和条件。
On the Existence and Uniqueness of Poincaré Maps for Systems with Impulse Effects
本文研究了具有脉冲效应的动力系统在可微流形上演化时Poincaré映射的存在性和唯一性条件，并将其应用于具有多个域的系统。
We study the effect of additive Brownian noise on an ODE system that has a stable hyperbolic limit cycle, for initial data that are attracted to the limit cycle. The analysis is performed in the limit of small noise - that is, we modulate the noise by a factor $\varepsilon \searrow 0$ - and on a long time horizon. We prove explicit estimates on the proximity of the noisy trajectory and the limit cycle up to times $\exp\left(c \varepsilon^{-2}\right)$, $c>0$, and we show both that on the time scale $\varepsilon^{-2}$ the "'dephasing" (i.e., the difference between noiseless and noisy system meas…
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