Global bifurcation of homoclinic solutions of hamiltonian systems

Bifurcations of cubic homoclinic tangencies in two-dimensional symplectic maps

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.

Analytic and algebraic conditions for bifurcations of homoclinic orbits II: Reversible systems

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…

Existence and bifurcation of homoclinic orbits in planar piecewise linear systems

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.

On bifurcations of area-preserving and non-orientable maps with quadratic homoclinic tangencies

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.

Birth of discrete Lorenz attractors at the bifurcations of 3D maps with homoclinic tangencies to saddle points

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.

Saddle-Node Bifurcation and Homoclinic Persistence in AFM with Periodic Forcing

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.

Homoclinic Bifurcations that Give Rise to Heterodimensional Cycles near A Saddle-Focus Equilibrium

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.

On dynamics and bifurcations of area-preserving maps with homoclinic tangencies

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.

Low dimensional bifurcations of snap-back repellors

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.

Homoclinic boundary-saddle bifurcations in nonsmooth vector fields

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 …

Limit cycle bifurcations near a double homoclinic loop with a nilpotent saddle of order 2

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.

Universal bifurcation patterns in the unfolding of a pair of homoclinic tangencies

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).

Cascades of Global Bifurcations and Chaos near a Homoclinic Flip Bifurcation: A Case Study

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…

Saddle Invariant Objects and their Global Manifolds in a Neighborhood of a Homoclinic Flip Bifurcation of Case B

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…

具有同宿于鞍焦轨道的区间映射的分岔结构

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。

A K-theoretical Invariant and Bifurcation for Homoclinics of Hamiltonian Systems

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.

Global bifurcation of homoclinic trajectories of discrete dynamical systems

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.

Analytic and algebraic conditions for bifurcations of homoclinic orbits I: Saddle equilibria

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…

通过Hopf分岔源自同宿切的异维环

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。

Width of homoclinic zone for quadratic maps

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.

二维复空间中的退化同宿分岔

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。

Hopf and Homoclinic Loop Bifurcations on a DC-DC Boost Converter under a SMC Strategy

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…

同宿缠结的延拓与坍缩

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。

Chemical Reaction Systems with a Homoclinic Bifurcation: an Inverse Problem

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.

Computing connecting orbits to infinity associated with a homoclinic flip bifurcation

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…

Shilnikov Homoclinic Bifurcation of Mixed-Mode Oscillations

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…

Homoclinic saddle to saddle-focus transitions in 4D systems

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…

多孔颗粒中的同宿轨道

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

Persistent antimonotonic bifurcations and strange attractors for cubic homoclinic tangencies

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 …

Mixed dynamics of 2-dimensional reversible maps with a symmetric couple of quadratic homoclinic tangencies

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.

Interplay between normal forms and center manifold reduction for homoclinic predictors near Bogdanov-Takens bifurcation

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…

Ordered intricacy of Shilnikov saddle-focus homoclinics in symmetric systems

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 …

$\mathbb{C}^2$ 自同构同宿分岔的 Cantor 集稳定相交与持久切触的正密度

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。

同宿解的全局分岔

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。

Codimension-two homoclinic bifurcations underlying spike adding in the Hindmarsh-Rose burster

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…

Augmented Phase Reduction for Periodic Orbits Near a Homoclinic Bifurcation and for Relaxation Oscillators

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…

Pulse-adding of Temporal Dissipative Solitons: Resonant Homoclinic Points and the Orbit Flip of Case B with Delay

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…

具有三次反馈的Saltzman-Maasch模型中的Hopf-Bautin分岔和同宿分岔

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。

Homoclinic Orbits of the FitzHugh-Nagumo Equation: The Singular-Limit

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…

Globally Resonant Homoclinic Tangencies

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…

Suarez-Schopf延迟振子中同宿分岔导致的隐藏与不稳定周期轨道及ENSO的不规则性

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。

On homoclinic orbits to center manifolds of elliptic-hyperbolic equilibria in Hamiltonian systems

We consider a Hamiltonian system which has an elliptic-hyperbolic equilibrium with a homoclinic loop. We identify the set of orbits which are homoclinic to the center manifold of the equilibrium via a Lyapunov- Schmidt reduction procedure. This leads to the study of a singularity which inherits certain structure from the Hamiltonian nature of the system. Under non-degeneracy assumptions, we classify the possible Morse indices of this singularity, permitting a local description of the set of homoclinic orbits. We also consider the case of time-reversible Hamiltonian systems.

FitzHugh-Nagumo方程的同宿轨道：完整系统中的分岔

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。

A Note on Homoclinic Orbits for Second Order Hamiltonian Systems

In this paper, we study the existence for the homoclinic orbits for the second order Hamiltonian systems. Under suitable conditions on the potential $V$, we apply the direct method of variations and the Fourier analysis to prove the existence of homoclinc orbits.

Permanency and bifurcations of bounded solutions near homocilincs with symetric eigenvalues

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)

Infinitely many homoclinic solutions for a class of subquadratic second-order Hamiltonian systems

In this paper, we mainly consider the existence of infinitely many homoclinic solutions for a class of subquadratic second-order Hamiltonian systems $\ddot{u}-L(t)u+W_u(t,u)=0$, where $L(t)$ is not necessarily positive definite and the growth rate of potential function $W$ can be in $(1,3/2)$. Using the variant fountain theorem, we obtain the existence of infinitely many homoclinic solutions for the second-order Hamiltonian systems.

The bifurcation diagram of cubic polynomial vector fields on $\mathbb{C}\mathbb{P}^1$

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.…

Creation of Homoclinic Tangencies in Hamiltonians by the Suspension of Poincaré Sections

In this note we show that for any Hamiltonian defined on a symplectic 4-manifold M and any point p in M, there exists a C2-close Hamiltonian whose regular energy surface through p is either Anosov or it contains a homoclinic tangency. Our result is based on a general construction of Hamiltonian suspensions for given symplectomorphisms on Poincaré sections already known to yield similar properties.

Some results on homoclinic and heteroclinic connections in planar systems

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 orbits for a class of nonperiodic Hamiltonian systems with some twisted conditions

In this paper, by the Masolv index theory, we will study the existence and multiplicity of homoclinic orbits for a class of asymptotically linear nonperiodic Hamiltonian systems with some twisted conditions on the Hamiltonian functions

Infinitely many homoclinic orbits for a class of superquadratic Hamiltonian systems

In this paper, we prove the existence of infinitely many homoclinic orbits for the first order Hamiltonian systems $J\dot{x}-M(t)x+ R'(t,x)=0$, by the minimax methods in critical point theory, when $R(t,y)$ satisfies the superquadratic condition ${{R(t,x)}\over{\|x\r|^{2}}}\longrightarrow \pm\infty$ as $\|x\|\longrightarrow\infty$, uniformly in $t$, and need not satisfy the global Ambrosetti-Rabinowitz condition

Bifurcation analysis of Bogdanov-Takens bifurcations in delay differential equations

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…

Homoclinic connections with many loops near a $0^2 iw$ resonant fixed point for Hamiltonian systems

In this paper we study the dynamics near the equilibrium point of a family of Hamiltonian systems in the neighborhood of a $0^2 iw$ resonance. The existence of a family of periodic orbits surrounding the equilibrium is well-known and we show here the existence of homoclinic connections with several loops for every periodic orbit close to the origin, except the origin itself. To prove this result, we first show a Hamiltonian normal form theorem inspired by the Elphick-Tirapegui-Brachet-Coullet-Iooss normal form. We then use a local Hamiltonian normalization relying on a result of Moser. We obta…

Splitting of separatrices for the Hamiltonian-Hopf bifurcation with the Swift-Hohenberg equation as an example

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…