G. Tigan, D. Opris
A 3D nonlinear chaotic system, called the T system, is analyzed in this paper. Horseshoe chaos is investigated via the heteroclinic Shilnikov method constructing a heteroclinic connections between the saddle equilibrium points of the system. Partially numerical computations are carried out to support the analytical results.
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We show that the stable and unstable sets of non-uniformly hyperbolic horseshoes arising in some heteroclinic bifurcations of surface diffeomorphisms have the value conjectured in a previous work by the second and third authors of the present paper. Our results apply to first heteroclinic bifurcations associated to horseshoes with Hausdorff dimension $<22/21$ of conservative surface diffeomorphisms.
The purpose of this paper is to advance the knowledge of the dynamics arising from the creation and subsequent bifurcation of Poincaré heteroclinic cycles. The problem is central to dynamics: it has to be addressed if, for instance, one aims at describing the typical orbit behaviour of a typical system, thus providing a global scenario for the ensemble of dynamical systems - see the Introduction and [P1, P2]. Here, we shall consider smooth, i.e. $C^\infty$, one-parameter families of dissipative, meaning non-conservative, surface diffeomorphisms. An hetereoclinic cycle may appear when the param…
We present a comprehensive mechanism for the emergence of rotational horseshoes and strange attractors in a class of two-parameter families of periodically-perturbed differential equations defining a flow on a three-dimensional manifold. When both parameters are zero, its flow exhibits an attracting heteroclinic network associated to two periodic solutions. After slightly increasing both parameters, while keeping a two-dimensional connection unaltered, we focus our attention in the case where the two-dimensional invariant manifolds of the periodic solutions do not intersect. We prove a wide ra…
Three-dimensional horseshoes near an unfolding of a Hopf-Hopf singularity
本文研究了Hopf-Hopf奇点分支附近$\mathbb{R}^4$中向量场的异宿环，通过首次返回映射证明了三维双曲马蹄及无穷多鞍型不变集的存在，并应用于Gaspard型分支。
A generalized attracting horseshoe is introduced as a new paradigm for describing chaotic strange attractors (of arbitrary finite rank) for smooth and piecewise smooth maps f from Q to Q, where Q is a homeomorph of the unit interval in real m-space for any integer m > 1. The main theorems for generalized attracting horseshoes are shown to apply to Henon and Lozi maps, thereby leading to rather simple new chaotic strange attractor existence proofs that apply to a range of parameter values that includes those of earlier proofs.
A new geometric criterion is derived for the existence of chaos in continuous-time autonomous systems in three-dimensional Euclidean spaces, where a type of Smale horseshoe in a subshift of finite type exists, but the intersection of stable and unstable manifolds of two points on a hyperbolic periodic orbit does not imply the existence of a Smale horseshoe of the same type on cross-sections of these two points. This criterion is based on the existence of a hyperbolic periodic orbit, differing from the classical equilibrium-based Shilnikov criterion and the condition of transversal homoclinic o…
Systems of $N$ identical globally coupled phase oscillators can demonstrate a multitude of complex behaviours. Such systems can have chaotic dynamics for $N>4$ when a coupling function is biharmonic. The case $N = 4$ does not possess chaotic attractors when the coupling is biharmonic, but has them when the coupling includes nonpairwise interactions of phases. Previous studies showed that some of chaotic attractors in this system are organized by heteroclinic networks. In present paper we discuss which heteroclinic cycles are forbidden and which are supported by this particular system. We also …
Key features of biological activity can often be captured by transitions between a finite number of semi-stable states that correspond to behaviors or decisions. We present here a broad class of dynamical systems that are ideal for modeling such activity. The models we propose are chaotic heteroclinic networks with nontrivial intersections of stable and unstable manifolds. Due to the sensitive dependence on initial conditions, transitions between states are seemingly random. Dwell times, exit distributions, and other transition statistics can be built into the model through geometric design an…
Variation of entropy in the Duffing system with the amplitude of the external force
研究Duffing系统中外部力幅值对熵变的影响，发现当力幅值超过临界值时，拓扑鞋带退化为伪鞋带，混沌不变集仍存在，且拓扑熵下界随幅值增加而减小。
Paperfolding Structures as Templates for Horseshoes in Multidimensional Hénon Maps
本文提出折纸结构作为几何模板，用于分析和分类三维及四维Hénon映射中马蹄铁的几何与拓扑特征，为高维混沌提供新见解。
Arches of chaos, heteroclinic connections of first-order MMRs and the chaotic transport of small bodies in the Sun-Jupiter system
研究太阳-木星平面限制三体问题中主要平均运动共振的不稳定周期轨道稳定和不稳定流形之间的异宿连接，计算一阶内共振2:1、3:2和外共振2:3的流形，并通过FLI图展示混沌鞍点结构，解释小行星轨道平面的混沌拱门现象。
Essential dynamics in chaotic attractors
本文证明生成充分复杂异宿结的 $S^3$ 光滑向量场必然产生无穷多在光滑扰动下持续存在的周期轨道，并据此关联 Template 且基于 Thurston-Nielsen 分类定理得出三维流存在混沌动力学的拓扑判据，成功应用于研究 Rössler 和 Lorenz 吸引子。
Heteroclinic cycles and chaos in a system of four identical phase oscillators with global biharmonic coupling
针对四个全局双谐波耦合相同相位振子这一最小 Kuramoto 型系统，本文利用对称性与异宿环情景提出搜索方法，并首次给出混沌存在的数值实例。
We consider the nonlinear cubic Wave, the Hartree and the nonlinear cubic Beam equations on $T^2$ and we prove the existence of different types of solutions which exchange energy between Fourier modes in certain time scales. This exchange can be considered \emph{chaotic-like} since either the choice of activated modes or the time spent in each transfer can be chosen randomly. The key point of the construction of those orbits is the existence of heteroclinic connections between invariant objects and the construction of symbolic dynamics (a Smale horseshoe) for the Birkhoff Normal Form truncatio…
A continuous map $f$ from a compact interval $I$ into itself is densely (resp. generically) chaotic if the set of points $(x,y)$ such that $\limsup_{n\to+\infty}|f^n(x)-f^n(y)|>0$ and $\liminf_{n\to+\infty} |f^n(x)-f^n(y)|=0$ is dense (resp. residual) in $I\times I$. We prove that if the interval map $f$ is densely but not generically chaotic then there is a descending sequence of invariant intervals, each of which containing a horseshoe for $f^2$. It implies that every densely chaotic interval map is of type at most $6$ for Sharkovsky's order (that is, there exists a periodic point of period …
Chaotic dynamics in refraction galactic billiards
本文研究天体力学中一类折射台球系统，通过逐步几何假设、符号动力学、解析不可积性及异宿连接，证明其在高内能下具有拓扑混沌。
Strange Attractors in Fractional Differential Equations: A Topological Approach to Chaos and Stability
本文采用拓扑方法分析含 Caputo 导数的分数阶微分方程，提出分数阶 Conley 指数与 Lyapunov 指数计算方法，以识别混沌吸引子并揭示其分形、谱性质及稳定性。
Knots and Chaos in the Rössler System
本文针对Rössler系统，在满足异宿条件的特定参数值下给出混沌动力学存在的充分条件，进而证明无穷多条周期轨迹的存在并研究其分岔。
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