Claudio A. Buzzi, Tiago de Carvalho, Marco A. Teixeira
This paper presents results concerning bifurcations of 2D piecewise-smooth dynamical systems governed by vector fields. Generic three-parameter families of a class of Non-Smooth Vector Fields are studied and the bifurcation diagrams are exhibited. Our main results describe the unfolding of the so called Resonant $Fold-Saddle$ singularity.
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Existence and bifurcation of homoclinic orbits in planar piecewise linear systems
研究了具有间断边界的两个区域的平面分段线性齐次系统同宿轨道的存在性和分支。此外，在不作连续性假设的情况下，还讨论了平面分段线性齐次系统周期轨道的存在性和原点的稳定性。
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.
Local bifurcations in a class of piecewise-smooth Filippov systems with a nonregular switching curve via a nonlinear double regularization process
本文通过非线性双正则化过程研究一类具有非规则切换曲线的分段光滑Filippov系统中局部分岔的保持与变化
Resonance near Border-Collision Bifurcations in Piecewise-Smooth, Continuous Maps
由分段光滑连续映射的边界碰撞分叉形成的锁模区域（共振舌）通常表现出独特的香肠状几何形状，其夹点称为“收缩点”。在本文中，我们扩展了分段线性情况的展开[{ \em Nonlinearity}，22（5）：1123-1144，2009]，以显示收缩点如何被非线性破坏。对于$N$维映射，我们得到了这个收缩点分支的余维3开折。我们表明，破坏的收缩点一般发生的鞍结分叉，平滑的香肠的一个边界，在其他边界留下一个扭结的曲线的创建。
Shrinking Point Bifurcations of Resonance Tongues for Piecewise-Smooth, Continuous Maps
共振舌是参数空间的锁模区域，其中出现稳定的周期解;它们通常出现在例如Neimark-Sacker分叉附近。对于分段光滑的连续映射，这些舌头在双参数分叉图中通常具有独特的透镜链（或香肠）形状。我们给出了一类“旋转”周期解的符号描述，显示透镜链结构的一般$N$维映射。然后，我们展开余维2，收缩点分叉，其中舌头的宽度为零。一些余维一分叉曲线从收缩点发出，我们确定那些形成舌头的边界。
Bifurcation of limit cycles in piecewise holomorphic systems separated by two external circles
本文研究由两个外圆分隔的三区域分片全纯系统扰动线性中心时极限环的分支数量，并证明在特定构型下四次系统至少存在十六个极限环。
In this paper, the general perturbation problem of piecewise smooth integrable differential systems with two switching planes is considered. Firstly, when the unperturbed system has a family of periodic orbits, the first order Melnikov vector function is derived which can be used to study the number of periodic orbits bifurcated from the period annuli. Then, by using the obtained Melnikov vector function, we get an upper bound of the number of periodic orbits of a concrete $n$-dimensional piecewise smooth differential system.
In this paper, we perturb the global center of the planar polynomial vector fields $\mathcal{X}(x,y)=(-y(x^2+a^2),x(x^2+a^2))$ ($a\neq0$) inside cubic piecewise smooth polynomials with switching line $y=0$. By using average function of first order, we prove that the sharper bound of the number of limit cycles bifurcating from the period annulus is 6.
Canard explosion and relaxation oscillation in planar, piecewise-smooth, continuous systems
光滑系统中的经典鸭式爆炸结果要求向量场至少为$C^3$，因为鸭式循环是由霍普夫分叉产生的。非光滑平面系统中鸭翼的研究是最近的，迄今为止仅限于分段线性或分段光滑范德波尔系统，其中临界流形的极值来自非光滑性。在这两种情况下，鸭式（或鸭式）爆炸是通过一个非光滑的分叉创建的缓慢零倾通过临界流形的一个角落。此外，这些系统有可能出现超爆炸分叉，其中鸭式爆炸被跳过。本文将结果扩展到更一般的分段光滑系统，寻找通过光滑或非光滑分叉创建周期轨道的条件。在分叉是非光滑的情况下，建立了确定分叉是超爆炸还是鸭翼的条件。
We solve the center-focus problem in a class of piecewise quadratic polynomial differential systems with an invariant straight line. The separation curve is also a straight line which is not invariant. We provide families having at the origin a weak-foci of maximal order. In the continuous class, the cyclicity problem is also solved, being $3$ such maximal number. Moreover, for the discontinuous class but without sliding segment, we prove the existence of $7$ limit cycles of small amplitude.
Motivated by models that arise in controlled ship maneuvering, we analyze Hopf bifurcations in systems with piecewise smooth nonlinear part. In particular, we derive explicit formulas for the generalization of the first Lyapunov coefficient to this setting. This generically determines the direction of branching (super- versus sub-criticality), but in general this differs from any fixed smoothing of the vector field. We focus on nonsmooth nonlinearities of the form $u_i|u_j|$, but our results are formulated in broader generality for systems in any dimension with piecewise smooth nonlinear part.…
Piecewise linear recurrent neural networks (PLRNNs) form the basis of many successful machine learning applications for time series prediction and dynamical systems identification, but rigorous mathematical analysis of their dynamics and properties is lagging behind. Here we contribute to this topic by investigating the existence of n-cycles $(n\geq 3)$ and border-collision bifurcations in a class of n-dimensional piecewise linear continuous maps which have the general form of a PLRNN. This is particularly important as for one-dimensional maps the existence of 3-cycles implies chaos. It is sho…
For many physical systems the transition from a stationary solution to sustained small amplitude oscillations corresponds to a Hopf bifurcation. For systems involving impacts, thresholds, switches, or other abrupt events, however, this transition can be achieved in fundamentally different ways. This paper reviews 20 such `Hopf-like' bifurcations for two-dimensional ODE systems with state-dependent switching rules. The bifurcations include boundary equilibrium bifurcations, the collision or change of stability of equilibria or folds on switching manifolds, and limit cycle creation via hysteresi…
Existence of Closed Orbit in Piecewise Continuous Matsumoto-Chua System
本文研究了分段连续Matsumoto-Chua系统闭轨存在的条件。我们感兴趣的是在一个伪鞍结分支的第一个返回的应用程序。在这个分支中，闭轨被转移到稳定的不动点。
Noninvertibility and Bifurcation Phenomena in a Four-Partitions Piecewise Linear Map
研究由四个分区定义的二维分段线性映射的全局动力学与分岔结构，分析不动点稳定性，揭示周期2动力学由翻转分岔引发，更高周期周期通过折叠边界碰撞分岔出现，还研究了退化参数区域，证明吸收不变区间存在及确定混沌动力学参数区。
Regularizations of two-fold bifurcations in planar piecewise smooth systems using blow up
利用Blowup方法研究了平面分段光滑（PWS）动力系统中余维1二重奇点的正则化问题。我们重点研究了PWS系统中可能出现的奇异鸭翼、伪均衡和极限环。利用Sotomayor和Teixeira \cite{Sotomayor96}的正则化方法，我们严格地证明了奇异鸭翼是如何持续存在的，以及在正则化系统中伪平衡点的分支与平衡点的分支之间的关系.我们还表明，PWS极限环连接到正则化的Hopf分支。此外，我们展示了如何正则化可以创建另一种类型的极限环，似乎不存在于原来的PWS系统。对于这两种类型的极限环，我们表明，产生周期轨道的Hopf分支的临界性强烈依赖于精确形式的正则化。最后，我们分析的极限环的正则化的周期轨道的局部唯一的家庭和连接，如果可能的话，到极限环的PWS系统。我们用数值模拟说明了我们的分析，并展示了正则化系统如何发生鸭式爆炸现象。
Nonsmooth bifurcations in families of one-dimensional piecewise-linear quasiperiodically forced maps
本文研究了一类一维分段线性准周期强迫映射族的非光滑双曲点，证明了存在非光滑倍周期双曲点及奇怪非混沌吸引子。
The phenomenon of slow passage through a Hopf bifurcation is ubiquitous in multiple-timescale dynamical systems, where a slowly-varying quantity replacing a static parameter induces the solutions of the resulting slow-fast system to feel the effect of a Hopf bifurcation with a delay. This phenomenon is well understood in the context of smooth slow-fast dynamical systems. In the present work, we study for the first time this phenomenon in piecewise linear (PWL) slow-fast systems. This special class of systems is indeed known to reproduce all features of their smooth counterpart while being more…
Canard-like phenomena in piecewise-smooth Van der Pol systems
我们证明了一个非线性的、分段光滑的、平面动力系统可以表现出鸭翼现象。非线性分段光滑系统中的鸭式解和爆炸在定性上比分段线性系更类似于光滑系统中的现象，因为非线性允许鸭式从小循环过渡到鸭式“有头”。“鸭翼是在缓慢零倾线与分裂流形重合时产生的分叉。然而，在某些条件下，这种分叉会导致一种称为超级爆炸的现象，即从全局吸引周期轨道到弛豫振荡的瞬时过渡。此外，我们证明了分叉-无论是导致鸭式或超爆炸-可以是亚临界的。
Limit Cycles in Piecewise Quadratic Kolmogorov Systems
本文研究分段二次Kolmogorov系统，解决中心问题并分析穿越型极限环分岔，发现分段情形至少存在六个极限环，且最大弱焦点阶数不决定小振幅极限环的最大数量。
Bifurcations of limit cycles in piecewise smooth Hamiltonian system with boundary perturbation
本文研究边界扰动下分段光滑哈密顿系统，推导了Melnikov函数表达式，并证明极限环数量等于扰动函数正零点个数。
Classifications and bifurcations of tangent points and their loops of planar piecewise-smooth systems
本文针对平面分段光滑系统中切点及其连接环，提出按切触阶数和构型进行分类，并通过构造显式开折研究一般切触阶数下的分岔，得到了切触阶数与分岔切点、切轨道及环数目之间的关系。
Grazing-sliding bifurcations creating infinitely many attractors
当分段光滑常微分方程组的参数变化时，当周期轨道与系统不连续的曲面碰撞时，系统会发生分岔。在一定条件下，这是一个掠滑分岔。近掠滑分岔结构稳定的动力学捕获分段线性连续映射。最近，它表明，这类映射可以有无穷多个渐近稳定的简单型周期解。在这里，这个结果被用来表明，在一个掠滑分岔渐近稳定的周期轨道可以分叉成无穷多个渐近稳定的周期轨道。对于一个抽象的常微分方程系统的周期轨道继续数值揭示随后的分叉处，他们被摧毁。
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.
Global dynamics and unfolding of planar piecewise smooth quadratic quasi-homogeneous differential systems
本文研究了平面分段光滑二次拟齐次非齐次多项式微分系统的全局动力学和分支。给出了分段光滑二次拟齐次系统中心存在的充分和必要条件。而且，如果中心存在，则它是全局非等距的，这在光滑二次拟齐次系统中是不可能出现的。然后研究了分段光滑二次拟齐次非齐次系统的全局结构。最后研究了分段光滑二次拟齐次中心的极限环分支，并应用Melnikov方法对分段光滑近哈密顿系统给出了从中心的周期轨道分支出的极限环的最大个数.
Bifurcations of grazing loops of arbitrary tangent multiplicity in piecewise-smooth systems
研究分段光滑系统中 grazing 循环在任意切乘数下的分支特性，提出新的函数扰动方法和局部化技术，建立切乘数与极限环、滑动环数量的关系。
In this paper, we study the number of limit cycles that can bifurcate from a periodic annulus in discontinuous planar piecewise linear Hamiltonian differential system with three zones separated by two parallel straight lines, such that the linear differential systems that define the piecewise one have a center and two saddles. That is, the linear differential system in the region between the two parallel lines (i.e. the central subsystem) has a center and the others subsystems have saddles. We prove that if the central subsystem has a real or a boundary center, then we have at least six limit …
Unfolding homoclinic connections formed by corner intersections in piecewise-smooth maps
分段光滑映射的不变集的稳定流形和非稳定流形都是分段光滑的。因此，当分段光滑映射的参数变化时，当不变集的稳定流形与其非稳定流形的不可微点相交时，不变集可以形成同宿联络（反之亦然）。这是一个余维一分叉，类似于光滑映射的同宿切线，这里称为同宿角。本文给出了平面分段光滑连续映射鞍不动点的通有同宿角点的开折。结果表明，边界碰撞分岔序列仅限于一个同宿角，附近的所有周期解都是不稳定的。
On 3-parameter families of piecewise smooth vector fields in the plane
研究了分片光滑平面动力系统在典型奇点附近的局部分歧问题。研究了一类非$-$光滑向量场的三参数族，并给出了三维分歧图。我们的主要结果通过3个参数的变化描述了所谓的折叠尖点奇异性的展开。
One-dimensional Piecewise Smooth Rational Degree Maps
针对混合流与分段流擦边分岔中出现的一维连续映射，研究其一侧x^{q/p}奇异、另一侧线性的结构，并对包括倍周期分岔和鲁棒混沌在内的全部情形作出分类。
Bifurcation analysis of a smoothed model of a forced impacting beam and comparison with an experiment
本文从第一性原理出发，建立了具有局部质量和对称止动块的垂直悬臂梁的单自由度分段线性模型。通过一个开关函数（非线性同伦）来平滑所得到的分段线性动力系统。对于所选择的平滑函数，它示出的平滑可以诱导分叉在某些参数制度。这些诱导的分叉消失时，充分的切换和日益本地化的影响变得更加困难。通过驱动频率和/或强迫振幅的周期轨道的一个和两个参数的延续的碰撞振子响应的分岔结构进行了研究。计算结果与实验测量值吻合较好。
From two-dimensional continuous maps to one-dimensional discontinuous maps: a novel reduction explaining complex bifurcation structures in piecewise-linear families of maps
本文提出了一种新的降维方法，通过将二维连续分段线性映射近似为一维不连续映射，揭示了复杂分岔结构的形成机制。
Limit cycles of a class of piecewise smooth Liénard systems
研究了两个多参数Liénard型分段多项式系统的极限环分支问题。基于Melnikov函数理论，我们得到了这两个系统的最大极限环个数.
The bifurcation structure within robust chaos for two-dimensional piecewise-linear maps
本文研究二维分段线性映射中鲁棒混沌吸引子的分岔结构，通过数值算法识别连通分量，发现反向与不可逆情形下存在倍分量分岔及新型异宿分岔。
The limiting slow dynamics of slow-fast, piecewise-linear, continuous systems of ODEs occurs on critical manifolds that are piecewise-linear. At points of non-differentiability, such manifolds are not normally hyperbolic and so the fundamental results of geometric singular perturbation theory do not apply. In this paper it is shown that if the critical manifold is globally stable then the system is forward invariant in a neighbourhood of the critical manifold. It follows that in this neighbourhood the dynamics is given by a regular perturbation of the dynamics on the critical manifold and so d…
Slow passage through a transcritical bifurcation in piecewise linear differential systems: canard explosion and enhanced delay
本文研究穿越跨临界分岔的慢通道现象，通过构建分段线性极小模型量化最大延迟，识别出三种延迟行为，并构建对称系统揭示无界canard爆炸与增强延迟机制。
Generic behavior of a piecewise smooth vector field with non-smooth switching surface
本文讨论了当开关集是代数簇时，$2D$分片光滑向量场典型奇异点的一般局部分类问题。主要的重点是获得分类结果有关的结构稳定性和一般余维1分叉。
The collision of a fixed point with a switching manifold (or border) in a piecewise-smooth map can create many different types of invariant sets. This paper explores two techniques that, combined, establish a chaotic attractor is created in a border-collision bifurcation in $\mathbb{R}^d$ $(d \ge 1)$. First, asymptotic stability of the fixed point at the bifurcation is characterised and shown to imply a local attractor is created. Second, a lower bound on the maximal Lyapunov exponent is obtained from the determinants of the one-sided Jacobian matrices associated with the fixed point. Special …
Unfolding a Codimension-Two, Discontinuous, Andronov-Hopf Bifurcation
我们提出了一个展开的余维2的情况下，同时发生的不连续分支和Andronov-Hopf分支的分段光滑，连续系统的自治常微分方程在平面上。我们发现的Hopf循环经历了一个放牧分叉，可能是非常短的轨道的鞍结分叉。我们推导出的标度律的分叉曲线，从余维二分叉。
The instantaneous local transition of a stable equilibrium to a chaotic attractor in piecewise-smooth systems of differential equations
分段光滑连续微分方程系统的吸引子在参数变化的情况下与切换流形碰撞时，可以从稳定平衡点分叉到更复杂的不变集.这里提供的数值证据表明，这个不变集可以是混沌的。过渡发生局部（在一个点的邻域）和瞬间（对于一个单一的临界参数值）。这一现象说明了正常形式的边界平衡分岔在三维中使用的参数值适应的分段线性模型的混沌电路。一个次要参数的变化揭示了一个倍周期级联到混沌与窗口的周期性。的动态是很好地近似的一维单峰映射，解释了这种分叉结构。通过研究非线性项的影响，研究了吸引子的鲁棒性。
Analysis of Stable Periodic Orbits in the 1-D Linear Piecewise-smooth Discontinuous Map
本文考虑一维线性分段光滑不连续映射。众所周知，对于特定的参数区域，这类映射存在稳定的周期轨道。我们还知道，在这种情况下，相应的分岔现象称为“加周期分岔”，具有特殊的结构。在过去的几年中，许多作者分析了这种结构使用边界碰撞分岔（BCB）曲线和不同的参数区域的特征。在本文中，我们分析了一个特定的参数空间使用不同的方法。我们表明，这种方法使我们能够提出一些有趣的问题，例如：轨道的周期和轨道的基数之间是否存在任何关系？我们证明了这样的关系存在，并确切地说，在一个周期n，其中，是欧拉数的轨道存在<$（n）。我们提出了一种算法，它给出了所有这些不同的轨道的不动点的位置，并给出了它们的存在范围相对于参数。此外，我们表明，使用一种新的算法的计算，找到它们的存在范围可以减少。
Chaos in Piecewise Smooth Vector Fields on Two Dimensional Torus and Sphere
本文研究了二维环面和球面上分片光滑向量场的整体动力学。我们提供的条件下，这些家庭具有周期性和密集的轨迹，我们描述了一些全球性的分歧。我们还研究了它的极小集，并刻画了定义在环面和球面上的分片光滑向量场的混沌行为。
The influence of localised randomness on regular grazing bifurcations with applications to impacting dynamics
本文研究了与碰撞振动动力学相关的分段光滑常微分方程系统的随机扰动，其中碰撞事件构成了随机性的主要来源。这种系统的特征在于存在切换流形，切换流形将相空间划分为系统光滑的区域。碰撞的起始被掠食分叉捕获，在该分叉处，描述没有碰撞的运动的周期性轨道与切换歧管形成切向相交。正则掠食分岔附近的振荡动力学是由包含平方根奇异性的分段光滑映射描述的，称为Nordmark映射。我们考虑三种情况下，有色噪声只影响影响动力学，并推导出三个二维随机Nordmark地图的噪声出现在不同的非线性或乘法的方式，这取决于噪声的来源。因此，随机动力学不同的三个噪声源之间，是根本不同的Nordmark地图与加性噪声。这种关键的依赖性的噪声的性质说明了一个原型的一个自由度的影响振荡器。
A Piecewise Smooth $λ$-Lemma
本文为分段光滑向量场与映射扩展了 $λ$ 引理，通过研究穿越轨道上时间-T-映射的正则性，证明其既是同胚也是分段光滑微分同胚。
Seasonal Forcing Dominated Dynamics of a piecewise smooth Ghil-Zaliapin-Thompson ENSO model
本文提出了一个分段光滑的 GZT 模型以解析处理 ENSO 动力学，显式构造了周期一轨道解并确定了其存在的参数区域，通过 Floquet 乘子分析了稳定性与分岔，发现大延迟下发生环面分岔而小延迟下发生折叠分岔，解析结果与数值延拓高度吻合。
We consider a network of identical piecewise smooth systems that synchronizes on the manifold given by a periodic orbit of a single agent. We explicitly characterize the fundamental matrix solution of the network along the synchronous solution and extend the Master Stability Function tool to the present case of non-smooth dynamics.
In this paper some piecewise smooth perturbations of a three-dimensional differential system are considered. The existence of invariant manifolds filled by periodic orbits is obtained after suitable small perturbations of the original differential system. These manifolds emerge from a continuum of cylinders of $\mathbb{R}^3$ which does exist for the piecewise smooth differential systems after a rotation of some planar algebraic polynomial curves. The main tool used in order to obtain the results is the averaging theory for piecewise smooth differential systems.
Limit cycles bifurcating from a period annulus in continuous piecewise differential systems with three zones
研究了一类具有三个区域的平面连续分段线性向量场。利用Poincaré映射和证明光滑微分系统极限环存在性的技巧，证明了这类系统至少存在两个周期环扰动引起的极限环.此外，我们还通过两个两参数三带分段线性向量场族的例子描述了这类向量场的极限环分支。
Simultaneous bifurcation of limit cycles for Piecewise Holomorphic systems
本文研究分段全纯扰动系统的极限环分岔，给出周期环域的类Abel积分表达式，利用共形共轭刻画分岔极限环，并首次证明分段全纯系统存在非嵌套极限环。
Renormalization for piecewise smooth homeomorphisms on the circle
本文研究了作用在圆上分段光滑同胚上的重整化算子，其实质是研究亏格为1的广义区间交换映射的Rauzy-Veech重整化.特别是，我们证明了零平均非线性和满足一定的光滑性和组合假设的重正化的映射收敛到一组分段仿射区间交换映射。
In this paper we study some aspects of thermodynamic formalism, more specifically topological pressure and, as a consequence, topological entropy for piecewise smooth vector fields, using topological conjugation with shift maps and the Perron- Frobenius Operator. Some relationships between entropy and Hausdorff dimensions are also investigated.
The planar visible fold is a simple singularity in piecewise smooth systems. In this paper, we consider singularly perturbed systems that limit to this piecewise smooth bifurcation as the singular perturbation parameter $ε\rightarrow 0$. Alternatively, these singularly perturbed systems can be thought of as regularizations of their piecewise counterparts. The main contribution of the paper is to demonstrate the use of consecutive blowup transformations in this setting, allowing us to obtain detailed information about a transition map near the fold under very general assumptions. We apply this …
On the Hausdorff dimension of invariant measures of piecewise smooth circle homeomorphisms
本文证明具有无理旋转数和零平均非线性的充分正则分段光滑圆同胚的唯一不变测度通常具有零Hausdorff维数，并借助广义区间交换变换的组合旋转数概念编码该一般性条件。
An asymptotically stable cusp-fold singularity in 3D piecewise smooth vector fields
本文研究了由两个区域组成的$R^3$上分片光滑向量场的一个典型奇异性。在我们的研究对象，尖点-折叠奇点，我们考虑同时发生的尖点奇点的一个向量场和折叠奇点的另一个。我们展示了一个规范形式，提出了在分段光滑向量场搜索的最重要的属性之一：渐近稳定性。
Piecewise Smooth Dynamical Systems Regularized by Convolution
本文提出通过卷积正则化分段光滑动力系统的方法，通过吹胀操作将问题转化为光滑向量场的研究，揭示了分段常数向量场中可能存在的非预期动力学现象。
We show the existence of Shilnikov-type dynamics and bifurcation behaviour in general discrete three-dimensional piecewise smooth maps and give analytical results for the occurence of such dynamical behaviour. Our main example in fact shows a `two-sided' Shilnikov dynamics, i.e. simultaneous looping and homoclinic intersection of the one-dimensional eigenmanfolds of fixed points on both sides of the border. We also present two complementary methods to analyse the return time of an orbit to the border: one based on recursion and another based on complex interpolation.
Boundary Hopf bifurcations in three-dimensional Filippov systems
本文研究三维Filippov系统中边界Hopf分岔的特性，通过分析滑动运动导致的维度丧失，推导出两个参数的显式公式，并通过数值分析探讨了可能的混沌吸引子。
Normal forms of piecewise-smooth systems with a monodromic singular point
针对含 FF、FP 或 PP 平衡点的分段光滑系统，本文利用分段解析同胚给出任意阶显式规范形，并据此提出 Lyapunov 常数计算方法以解决中心问题与任意阶 Hopf 分岔。
The Conley theory has a tool to guarantee the existence of periodic trajectories in isolating neighborhoods of semi-dynamical systems. We prove that the positive trajectories generated by a piecewise-smooth vector field $Z=(X, Y)$ defined in a closed manifold of three dimensions without the scape region produces a semi-dynamical system. Thus, we have built a semiflow that allows us to apply the classical Conley theory. Furthermore, we use it to guarantee the existence of periodic orbits in this class of piecewise-smooth vector fields.
Crossing limit cycles in piecewise smooth Kolmogorov systems: an application to Palomba's model
本文研究了一类分段光滑Kolmogorov系统中交叉极限环的数量，确定了小振幅交叉极限环下界，并证明Palomba经济学模型中至少存在一个交叉极限环。
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