Jacobo Pejsachowicz, Robert Skiba
We prove the existence of an unbounded connected branch of nontrivial homoclinic trajectories of a family of discrete nonautonomous asymptotically hyperbolic systems parametrized by a circle under assumptions involving the topological properties of the asymptotic stable bundles.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We revisit a K-theoretical invariant that was invented by the first author some years ago for studying multiparameter bifurcation of branches of critical points of functionals. Our main aim is to apply this invariant to investigate bifurcation of homoclinic solutions of families of Hamiltonian systems which are parametrised by tori.
Collision of equilibria with a splitting manifold has been locally studied, but might also be a contributing factor to global bifurcations. In particular a boundary collision can be coincident with collision of a virtual equilibrium with a periodic orbit, giving an analogue to a homoclinic bifurcation. This type of bifurcation is demonstrated in a nonsmooth climate application. Here we describe the nonsmooth bifurcation structure, as well as the smooth bifurcation structure for which the nonsmooth homoclinic bifurcation is a limiting case.
Global bifurcation of homoclinic solutions
本文结合Fredholm算子分岔理论与动力系统工具，利用奇偶性、全局Evans函数和适当拓扑，对Carathéodory微分方程有界整体解的全局分岔分支形状给出分类判据。
We study main bifurcations of multidimensional diffeomorphisms having a non-transversal homoclinic orbit to a saddle-node fixed point. On a parameter plane we build a bifurcation diagram for single-round periodic orbits lying entirely in a small neighbourhood of the homoclinic orbit. Also a relation of our results to well-known codimension one bifurcations of a saddle fixed point with a quadratic homoclinic tangency and a saddle-node fixed point with a transversal homoclinic orbit is discussed.
We prove that any diffeomorphism of a compact manifold can be approximated in topology C1 by another diffeomorphism exhibiting a homoclinic bifurcation (a homoclinic tangency or a heterodimensional cycle) or by one which is essentially hyperbolic (it has a finite number of transitive hyperbolic attractors with open and dense basin of attraction).
We briefly describe our works in collaboration with Jean-Christophe Yoccoz, a great mathematician and friend, with special emphasis on those related to Homoclinic Bifurcations and Fractal Geometry. We also tell some related personal stories.
We prove that any diffeomorphism of a compact manifold can be C^1-approximated by a diffeomorphism which exhibits a homoclinic bifurcation (a homoclinic tangency or a heterodimensional cycle) or by a diffeomorphism which is partially hyperbolic (its chain-recurrent set splits into partially hyperbolic pieces whose centre bundles have dimensions less or equal to two). We also study in a more systematic way the central models introduced in arXiv:math/0605387.
In this article we construct the parameter region where the existence of a homoclinic orbit to a zero equilibrium state of saddle type in the Lorenz-like system will be analytically proved in the case of a nonnegative saddle value. Then, for a qualitative description of the different types of homoclinic bifurcations, a numerical analysis of the detected parameter region is carried out to discover several new interesting bifurcation scenarios.
We study bifurcations of cubic homoclinic tangencies in two-dimensional symplectic maps. We distinguish two types of cubic homoclinic tangencies, and each type gives different first return maps derived to diverse conservative cubic Hénon maps with quite different bifurcation diagrams. In this way, we establish the structure of bifurcations of periodic orbits in two parameter general unfoldings generalizing to the conservative case the results previously obtained for the dissipative case. We also consider the problem of 1:4 resonance for the conservative cubic Hénon maps.
Following Part~I, we consider a class of reversible systems and study bifurcations of homoclinic orbits to hyperbolic saddle equilibria. Here we concentrate on the case in which homoclinic orbits are symmetric, so that only one control parameter is enough to treat their bifurcations, as in Hamiltonian systems. First, we modify and extend arguments of Part~I to show in a form applicable to general systems discussed there that if such bifurcations occur in four-dimensional systems, then variational equations around the homoclinic orbits are integrable in the meaning of differential Galois theory…
The existence and bifurcation of homoclinic orbits in planar piecewise linear homogeneous systems with two regions separated by a discontinuity boundary are investigated in this paper. In addition, existence of periodic orbits and stability of the origin are also discussed without the assumption of continuity for the planar piecewise linear homogeneous systems.
We provide global bifurcation results for a class of nonlinear hamiltonian systems
We study bifurcations of non-orientable area-preserving maps with quadratic homoclinic tangencies. We study the case when the maps are given on non-orientable two-dimensional surfaces. We consider one and two parameter general unfoldings and establish results related to the emergence of elliptic periodic orbits.
It was established in 2006 that bifurcations of three-dimensional diffeomorphisms with a homoclinic tangency to a saddle-focus fixed point with the Jacobian equal to 1 can lead to Lorenz-like strange attractors. In the present paper we prove an analogous result for three-dimensional diffeomorphisms with a homoclinic tangency to a saddle fixed point with the Jacobian equal to 1, provided the quadratic homoclinic tangency under consideration is non-simple.
We study the dynamics of an Atomic Force Microscope (AFM) model, under the Lennard-Jones force with non-linear damping, and harmonic forcing. We establish the bifurcation diagrams for equilibria in a conservative system. Particularly, we present conditions that guarantee the local existence of saddle-node bifurcations. By using the Melnikov method, the region in the space parameters where the persistence of homoclinic orbits is determined in a non-conservative system.
We show that heterodimensional cycles can be born at the bifurcations of a pair of homoclinic loops to a saddle-focus equilibrium for flows in dimension 4 and higher. In addition to the classical heterodimensional connection between two periodic orbits, we found two new types of heterodimensional connections: one is a heteroclinic between a periodic orbit of index 2 and a homoclinic loop, and the other connects a periodic orbit of index 3 to the saddle-focus equilibrium.
We study bifurcations of area-preserving maps, both orientable (symplectic) and non-orientable, with quadratic homoclinic tangencies. We consider one and two parameter general unfoldings and establish results related to the appearance of elliptic periodic orbits. In particular, we find conditions for such maps to have infinitely many generic (KAM-stable) elliptic periodic orbits of all successive periods starting at some number.
We study the relationship between homoclinic orbits associated to repellors, usually called snap-back repellors, and expanding sets of smooth endomorphisms. Critical homoclinic orbits constitutes an interesting bifurcation that is locally contained in the boundary of the set of maps having homoclinic orbits. This and other possible routes to the creation of homoclinic orbits are considered in low dimensions.
In a smooth dynamical system, a homoclinic connection is a closed orbit returning to a saddle equilibrium. Under perturbation, homoclinics are associated with bifurcations of periodic orbits, and with chaos in higher dimensions. Homoclinic connections in nonsmooth systems are complicated by their interaction with discontinuities in their vector fields. A connection may involve a regular saddle outside a discontinuity set, or a pseudo-saddle on a discontinuity set, with segments of the connection allowed to cross or slide along the discontinuity. Even the simplest case, that of connection to a …
In this paper, we deal with limit cycle bifurcations near a double homoclinic loop with a nilpotent saddle of order 2 by studying expansions of the first order Melnikov functions near the loop and coefficients in these expansions. More precisely, we prove that the perturbed system can have 11, 13, 14 or 16 limit cycles in a neighborhood of the loop under certain conditions. Finally, we give an example to illustrate the effectiveness of our main results.
We study generic two- and three-parameter unfoldings of a pair of orbits of quadratic homoclinic tangency in strongly dissipative systems. We prove that the corresponding stability windows for periodic orbits have various universal forms: the so-called "shrimps" (cross-road areas), as well as spring and saddle areas, and (in three-parameter unfoldings) "pregnant shrimps" (specific types of transitions between the shrimps and spring or saddle areas).
We study a homoclinic flip bifurcation of case~\textbf{C}, where a homoclinic orbit to a saddle equilibrium with real eigenvalues changes from being orientable to nonorientable. This bifurcation is of codimension two, and it is the lowest codimension for a homoclinic bifurcation of a real saddle to generate chaotic behavior in the form of (suspended) Smale horseshoes and strange attractors. We present a detailed numerical case study of how global stable and unstable manifolds of the saddle equilibrium and of bifurcating periodic orbits interact close to such bifurcation. This is a step forward…
When a real saddle equilibrium in a three-dimensional vector field undergoes a homoclinic bifurcation, the associated two-dimensional invariant manifold of the equilibrium closes on itself in an orientable or non-orientable way. We are interested in the interaction between global invariant manifolds of saddle equilibria and saddle periodic orbits for a vector field close to a codimension-two homoclinic flip bifurcation, that is, the point of transition between having an orientable or non-orientable two-dimensional surface. Here, we focus on homoclinic flip bifurcations of case $\textbf{B}$, wh…
Bifurcation structure of interval maps with orbits homoclinic to a saddle-focus
研究 $\mathbb{Z}_2$-对称系统中与 (2, 1)-型鞍焦相关的区间映射同宿分岔，通过符号编码高效定位分岔集，揭示了由 Belyakov 分岔引导的分岔结构，促进了对低维混沌系统的理解。
We show that nontrivial homoclinic trajectories of a family of discrete, nonautonomous, asymptotically hyperbolic systems parametrized by a circle bifurcate from a stationary solution if the asymptotic stable bundles Es(+{\infty}) and Es(-{\infty}) of the linearization at the stationary branch are twisted in different ways.
We study bifurcations of homoclinic orbits to hyperbolic saddle equilibria in a class of four-dimensional systems which may be Hamiltonian or not. Only one parameter is enough to treat these types of bifurcations in Hamiltonian systems but two parameters are needed in general systems. We apply a version of Melnikov's method due to Gruendler to obtain saddle-node and pitchfork types of bifurcation results for homoclinic orbits. Furthermore we prove that if these bifurcations occur, then the variational equations around the homoclinic orbits are integrable in the meaning of differential Galois t…
Heterodimensional cycles derived from homoclinic tangencies via Hopf bifurcations
本研究分析满足$|λγ|=1$的三维$C^r$微分同胚的二次焦鞍同宿切，通过三参数开折证明其上存在Hopf分岔及分岔周期轨道的同宿点，得出原映射可被具余指标一异维环的微分同胚$C^r$逼近的结论。
We study several families of planar quadratic diffeomorphisms near a Bogdanov-Takens bifurcation. For each family, the associate bifurcation diagram can be deduced from the interpolating flow. However, a zone of chaos confined between two lines of homoclinic bifurcation that are exponentially close to one-another is observed. The goal of this paper is to test numerically an accurate asymptotic expansion for the width of this chaotic zone for different families.
Degenerate homoclinic bifurcations in complex dimension 2
研究二维复空间中同宿分岔，通过采用Takens论证证明展开非持久同宿相切的族含二次相切，结合相关成果得到复亨农映射分岔轨迹中鲁棒同宿相切丰富，还研究了由持久相切族引起的分岔，为复纽豪斯现象提供新途径。
In this paper, a dc-dc boost converter with sliding mode control and washout filter is analysed. This device is modelled as a three-dimensional Filippov system, characterized by the existence of sliding movement and restricted to the switching manifold. The operating point of the boost converter is a pseudo-equilibrium, and it, undergoes a subcritical Hopf bifurcation. Such a bifurcation occurs in the sliding vector field and creates, in this field, an unstable limit cycle. The limit cycle is confined to the switching manifold and disappears when it touches the visible-invisible two-fold point…
Continuation and collapse of homoclinic tangles
本文从数值延拓和分岔理论角度研究参数化系统中同宿缠结的演化，通过分岔方程刻画同宿切触附近的最大不变集，并利用图论解释Hénon族中多峰轨道的分岔结构。
An inverse problem framework for constructing reaction systems with prescribed properties is presented. Kinetic transformations are defined and analysed as a part of the framework, allowing an arbitrary polynomial ordinary differential equation to be mapped to the one that can be represented as a reaction network. The framework is used for construction of specific two- and three-dimensional bistable reaction systems undergoing a supercritical homoclinic bifurcation, and the topology of their phase spaces is discussed.
We consider the bifurcation diagram in a suitable parameter plane of a quadratic vector field in $\mathbb{R}^3$ that features a homoclinic flip bifurcation of the most complicated type. This codimension-two bifurcation is characterized by a change of orientability of associated two-dimensional manifolds and generates infinite families of secondary bifurcations. We show that curves of secondary $n$-homoclinic bifurcations accumulate on a curve of a heteroclinic bifurcation involving infinity. We present an adaptation of the technique known as Lin's method that enables us to compute such connect…
The Koper model is a three-dimensional vector field that was developed to study complex electrochemical oscillations arising in a diffusion process. Koper and Gaspard described paradoxical dynamics in the model: they discovered complicated, chaotic behavior consistent with a homoclinic orbit of Shil'nikov type, but were unable to locate the orbit itself. The Koper model has since served as a prototype to study the emergence of mixed-mode oscillations (MMOs) in slow-fast systems, but only in this paper is the existence of these elusive homoclinic orbits established. They are found first in a la…
A saddle to saddle-focus homoclinic transition when the stable leading eigenspace is 3-dimensional (called the 3DL bifurcation) is analyzed. Here a pair of complex eigenvalues and a real eigenvalue exchange their position relative to the imaginary axis, giving rise to a 3-dimensional stable leading eigenspace. This transition is different from the standard Belyakov bifurcation, where a double real eigenvalue splits either into a pair of complex-conjugate eigenvalues or two distinct real eigenvalues. In the wild case, we obtain sets of codimension 1 and 2 bifurcation curves and points that asym…
Homoclinic orbits in a porous pellet
研究多孔颗粒中同宿轨道的生成机制及影响因素
In this paper, we study a two-parameter family of two-dimensional diffeomorphisms such that it has a cubic homoclinic tangency unfolding generically which is associated with a dissipative saddle point. Our first theorem presents an open set in the parameter-plane such that, for any parameter value in the open set, there exists a one-parameter subfamily through this value exhibiting cubically related persistent contact-making and contact-breaking quadratic tangencies. Moreover, the second theorem shows that any such two-parameter family satisfies Wang-Young's conditions which guarantee that it …
We study dynamics and bifurcations of 2-dimensional reversible maps having a symmetric saddle fixed point with an asymmetric pair of nontransversal homoclinic orbits (a symmetric nontransversal homoclinic figure-8). We consider one-parameter families of reversible maps unfolding the initial homoclinic tangency and prove the existence of infinitely many sequences (cascades) of bifurcations related to the birth of asymptotically stable, unstable and elliptic periodic orbits.
This paper provides for the first time correct third-order homoclinic predictors in $n$-dimensional ODEs near a generic Bogdanov-Takens bifurcation point. To achieve this, higher-order time approximations to the nonlinear time transformation in the Lindstedt-Poincaré method are essential. Moreover, a correct transform between approximations to solutions in the normal form and approximations to solutions on the parameter-dependent center manifold is needed. A detailed comparison is done between applying different normal forms (smooth and orbital), different phase conditions, and different pertu…
Using the technique of Poincaré return maps, we disclose an intricate order of the subsequent homoclinics near the primary homoclinic bifurcation of the Shilnikov saddle-focus in systems with reflection symmetry. We also reveal the admissible shapes of the corresponding bifurcation curves in a parameter plane of such systems. The scalability ratio of geometry and organization is proven to be universal for such homoclinic bifurcations of higher orders. Two applications with similar dynamics due to the Shilnikov saddle-foci, a smooth adaptation of the Chua circuit and a 3D normal form, are used …
Stable intersections of Cantor sets and positive density of persistent tangencies for homoclinic bifurcations of automorphisms of $\mathbb{C}^2$
本文研究 $\mathbb{C}^2$ 自同构族的通有同宿切触，证明了若稳定与不稳定流形局部相交 Cantor 集的线性化版本稳定相交，则持久切触参数集在分岔点处具有正密度。
The well-studied Hindmarsh-Rose model of neural action potential is revisited from the point of view of global bifurcation analysis. This slow-fast system of three paremeterised differential equations is arguably the simplest reduction of Hodgkin-Huxley models capable of exhibiting all qualitatively important distinct kinds of spiking and bursting behaviour. First, keeping the singular perturbation parameter fixed, a comprehensive two-parameter bifurcation diagram is computed by brute force. Of particular concern is the parameter regime where lobe-shaped regions of irregular bursting undergo a…
We consider robust relative homoclinic trajectories (RHTs) for equivariant vector fields. We give some conditions on the group and representation that imply existence of equivariant vector fields with such trajectories. Using these result we show very simply that abelian groups cannot exhibit relative homoclinic trajectories. Examining a set of group theoretic conditions that imply existence of RHTs, we construct some new examples of robust relative homoclinic trajectories. We also classify RHTs of the dihedral and low order symmetric groups by means of their symmetries.
Oscillators - dynamical systems with stable periodic orbits - arise in many systems of physical, technological, and biological interest. The standard phase reduction, a model reduction technique based on isochrons, can be unsuitable for oscillators which have a small-magnitude negative nontrivial Floquet exponent. This necessitates the use of the augmented phase reduction, a recently devised two-dimensional reduction technique based on isochrons and isostables. In this article, we calculate analytical expressions for the augmented phase reduction for two dynamically different planar systems: p…
We numerically investigate the branching of temporally localized, two-pulse periodic traveling waves from one-pulse periodic traveling waves with non-oscillating tails in delay differential equations (DDEs) with large delay. Solutions of this type are commonly referred to as temporal dissipative solitons (TDSs) in applications, and we adopt this term here. We show by means of a prototypical example that -- analogous to traveling pulses in reaction-diffusion partial differential equations (PDEs) -- the branching of two-pulse TDSs from one-pulse TDSs with non-oscillating tails is organized by co…
Hopf-Bautin and homoclinic bifurcations in a Saltzman-Maasch model with cubic feedback
研究具有三次反馈的Saltzman-Maasch模型，通过降维和分岔分析识别Hopf和Bautin分岔，利用Melnikov方法构造同宿轨道，为数值观测提供严格解析基础。
The FitzHugh-Nagumo equation has been investigated with a wide array of different methods in the last three decades. Recently a version of the equations with an applied current was analyzed by Champneys, Kirk, Knobloch, Oldeman and Sneyd using numerical continuation methods. They obtained a complicated bifurcation diagram in parameter space featuring a C-shaped curve of homoclinic bifurcations and a U-shaped curve of Hopf bifurcations. We use techniques from multiple time-scale dynamics to understand the structures of this bifurcation diagram based on geometric singular perturbation analysis o…
The attractors of a dynamical system govern its typical long-term behaviour. The presence of many attractors is significant as it means the behaviour is heavily dependent on the initial conditions. To understand how large numbers of attractors can coexist, in this thesis we study the occurrence of infinitely many stable single-round periodic solutions associated with homoclinic connections in two-dimensional maps. We show this phenomenon has a relatively high codimension requiring a homoclinic tangency and `global resonance', as has been described previously in the area-preserving setting. How…
Hidden and unstable periodic orbits as a result of homoclinic bifurcations in the Suarez-Schopf delayed oscillator and the irregularity of ENSO
该研究通过惯性流形理论分析Suarez-Schopf延迟振子，揭示线性稳定区内的隐藏与自激周期轨道及同宿分岔，关联ENSO不规则性，并发现耦合传输线中的异步周期态。
Homoclinic Orbits of the FitzHugh-Nagumo Equation: Bifurcations in the Full System
本文从快慢动力系统角度研究FitzHugh-Nagumo方程的波前解，通过数值方法揭示参数空间中同宿分岔曲线的急转弯现象，并利用几何模型预测多脉冲同宿轨道和周期轨道。
we consider a system with homoclinic orbit, We decompose the corresponding variational equation on the space of solutions and provide sufficient conditions for the permanency of homoclinic in the space of $C^1$ vector fields. We also provide new sufficient conditions for the persistence and multiple bifurcations of the bounded solutions nearby. our results can be verified numerically and do not meet the limitations of classic methods (like Melnikon integrals and Poincare map)
In this paper we give the bifurcation diagram of the family of cubic vector fields $\dot z=z^3+ ε_1z+ε_0$ for $z\in \mathbb{C}\mathbb{P}^1$, depending on the values of $ε_1,ε_0\in\mathbb{C}$. The bifurcation diagram is in $\mathbb{R}^4$, but its conic structure allows describing it for parameter values in $\mathbb{S}^3$. There are two open simply connected regions of structurally stable vector fields separated by surfaces corresponding to bifurcations of homoclinic connections between two separatrices of the pole at infinity. These branch from the codimension 2 curve of double singular points.…
Consider a family of planar systems depending on two parameters $(n,b)$ and having at most one limit cycle. Assume that the limit cycle disappears at some homoclinic (or heteroclinic) connection when $Φ(n,b)=0.$ We present a method that allows to obtain a sequence of explicit algebraic lower and upper bounds for the bifurcation set ${Φ(n,b)=0}.$ The method is applied to two quadratic families, one of them is the well-known Bogdanov-Takens system. One of the results that we obtain for this system is the bifurcation curve for small values of $n$, given by $b=\frac5 7 n^{1/2}+{72/2401}n- {30024/4…
Homoclinic and Heteroclinic Trajectories of Differential Equations with Piecewise Constant Arguments of Generalized Type
本文研究广义型分段常数变元拟线性系统的渐近行为，利用离散时间映射和Banach不动点定理严格证明同宿与异宿解的存在性，并讨论解的双曲集合。
In this paper, we will perform the parameter-dependent center manifold reduction near the generic and transcritical codimension two Bogdanov-Takens bifurcation in classical delay differential equations (DDEs). Using a generalization of the Lindstedt-Poincaré method to approximate the homoclinic solution allows us to initialize the continuation of the homoclinic bifurcation curves emanating from these points. The normal form transformation is derived in the functional analytic perturbation framework for dual semigroups (sun-star calculus) using a normalization technique based on the Fredholm al…
We study homoclinic orbits of the Swift-Hohenberg equation near a Hamiltonian-Hopf bifurcation. It is well known that in this case the normal form of the equation is integrable at all orders. Therefore the difference between the stable and unstable manifolds is exponentially small and the study requires a method capable to detect phenomena beyond all algebraic orders provided by the normal form theory. We propose an asymptotic expansion for an homoclinic invariant which quantitatively describes the transversality of the invariant manifolds. We perform high-precision numerical experiments to su…
In this paper we study the existence of heteroclinic cycles in generic unfoldings of nilpotent singularities. Namely we prove that any nilpotent singularity of codimension four in $\mathbb{R}^4$ unfolds generically a bifurcation hypersurface of bifocal homoclinic orbits, that is, homoclinic orbits to equilibrium points with two pairs of complex eigenvalues. We also prove that any nilpotent singularity of codimension three in $\mathbb{R}^3$ unfolds generically a bifurcation curve of heteroclinic cycles between two saddle-focus equilibrium points with different stability indexes. Under generic a…
We develop a $K$-theoretic approach to multiparameter bifurcation theory of homoclinic solutions of discrete non-autonomous dynamical systems from a branch of stationary solutions. As a byproduct we obtain a family index theorem for asymptotically hyperbolic linear dynamical systems which is of independent interest. In the special case of a single parameter, our bifurcation theorem weakens the assumptions in previous work by Pejsachowicz and the first author.
We study bifurcation behavior in periodic perturbations of two-dimensional symmetric systems exhibiting codimension-two bifurcations with a double eigenvalue when the frequencies of the perturbation terms are small. We transform the periodically perturbed system to a simpler one which is a periodic perturbation of the normal form for codimension-two bifurcations with a double zero eigenvalue and symmetry, and apply the subharmonic and homoclinic Melnikov methods to analyze bifurcations occurring in the system. In particular, we show that there exist transverse homoclinic or heteroclinic orbits…
A case study of multiple wave solutions in a reaction-diffusion system using invariant manifolds and global bifurcations
本研究通过局部稳定性、数值分支和不变流形计算，系统分析了捕食者-猎物反应扩散系统中的行波解，揭示了同宿、异宿连接及混沌现象，并展示了波解的多重性。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。