Isolas, cusps and global bifurcations in an electronic oscillator

The aim of the present work is to describe the bifurcation behaviour of a class of asymmetric periodic orbits, in an electronic oscillator. The first time we detected them they were organized in a closed branch: that is, their bifurcation diagram showed an eight-shaped isola, with a nice structure of secondary branches emerging from period-doubling bifurcations. In a two-parameter bifurcation set, the isola structure persists. We find the regions of its existence, and describe its destruction in an isola centre with a cusp of periodic orbits. Finally, the introduction of a third parameter allows us to find the relation of our orbits to symmetric periodic orbits (via a symmetry-breaking bifurcation) and to homoclinic connections of the non-trivial equilibria. The isolas are successively created by collision of two adjacent limbs of the wiggly bifurcation curve. The Shil?nikov homoclinic and heteroclinic connections, related to the symmetric and asymmetric periodic orbits, emerge from T-points and end at Shil?nikov-Hopf singularities     

Global first harmonic bifurcation diagram for odd piecewise linear control systems

The describing function method (used normally as a first approximation to study the existence and the stability of periodic orbits) is used here to analyze the dependence of periodic orbits on the parameters of autonomous systems. The method can be applied to a large class of nonlinear systems but, for simplicity, attention here is paid to a class of single-input, single-output control systems with a piecewise linear character- istic function. A first approach to the bifurcation diagram associated with the periodic orbits of such systems, called a first harmonic bifurcation diagram' is obtained for two and three dimensions. This diagram in two dimensions describes all the qualitative behaviour of such systems. Although this is not the case in three dimensions, the information contained in the corresponding first harmonic bifurcation diagram is of great value. It shows much of the complexity of the periodic structure that can be found in such three-dimensional systems; in fact, part of the first harmonic bifurcation diagram coincides with the actual bifurcation diagram     

Symmetric and non-symmetric periodic orbits for the digital filter map

We exhibit instances of non-symmetric periodic orbits for the digital filter map, resolving a question posed in the literature as to whether such orbits can exist. This piecewise irrational rotation, depending on a parameter a = 2cos θ, is an isometry of [?1, 1) × [?1, 1) and reflections in the two diagonals are time-reversing symmetries for the map. Symmetric orbits are plentiful and have been much investigated. Each periodic orbit is paired with a symbolic string, from the alphabet {?, 0, +}, arising under iteration of the map because of the presence of a line of discontinuity. We prove the existence of an infinite family of non-symmetric orbits where the period N starts at 29 and increases in steps of 5; they correspond to the strings (+00)⁵(+?)² 0 ^N?19. We describe several computer algorithms to find non-symmetric periodic orbits and their symbolic strings and list non-symmetric strings both for a = 0.5, and for N ≤ 100 across the parameter range. Our evidence suggests that non-symmetric orbits, though not plentiful, are characteristic of the dynamics of the map for all parameter values.     

Equilibrium points and periodic orbits of higher order autonomous generalized Birkhoff system

Equilibrium points and periodic orbits of a higher order autonomous generalized Birkhoff system are studied by using qualitative methods of ordinary differential equations and the Liapunov center theorem. First, the equilibrium points and their properties are obtained from the equations of equilibrium points. Then, the characteristic roots of the Fréchet derivative C are obtained, and the type of equilibrium points of the system is verified. Finally, the existence theorem of periodic orbits is given by the Liapunov center theorem.     

一类离心调速器的Hopf分岔及其混沌控制

研究了受外部扰动的离心调速器系统的复杂动力学行为。通过系统运动的拉格朗日方程和牛顿第二定律,建立了离心调速器系统的动力学方程,应用Lyapunov直接方法分析了该系统平衡点的稳定性。利用相图分析了系统超混沌吸引子的特性,通过Poincaré截面和Lyapunov指数研究了系统的超混沌行为,通过仿真系统的分岔图和相图分析了系统通向混沌的道路,并且验证了该系统的分岔图与Lyapunov指数谱是完全吻合的。通过对系统施加非线性反馈控制器,并选取合适的反馈系数,可以获得各种不同的所需的稳定周期轨道。对受外部扰动的离心调速器系统施于此控制,计算机数值模拟结果表明,这种控制方法简便有效,控制范围广。     

Homoclinic intersections and Mel'nikov method for perturbed sine-Gordon equation

We describe and characterize rigorously the homoclinic structure of the perturbed sine-Gordon equation under periodic boundary conditions. The existence of invariant manifolds for a perturbed sine-Gordon equation is established. The Mel'nikov method, together with geometric analysis are used to assess the persistence of the homoclinic orbits under bounded and time-periodic perturbations.     

Transitive sets for quasi-periodically forced monotone maps

There has been considerable interest in recent years in quasi-periodically forced systems, partly due to the fact that these commonly exhibit strange non-chaotic attractors. Relatively little is known rigorously about such systems. In this paper we concentrate on investigating the structure of the simplest possible invariant sets for a particular class of quasi-periodically forced maps, namely those that are monotone in each fibre. Due to the quasi-periodic nature of the forcing, periodic orbits cannot occur, and their role is played by various types of invariant graph. Any compact invariant set is bounded by two invariant graphs, which are respectively upper and lower semi-continuous. If the set is minimal, these two graphs intersect on a residual set, on which both are continuous. Any transitive set Ω contains either one or two minimal sets, which must be the closure of one or the other of the boundaries of Ω. If Ω contains only one minimal set, then again its upper and lower boundaries intersect on a residual set. This case contains the original example of Grebogi et al. and its generalizations by Keller and by Glendinning. If Ω contains two minimal sets, then its upper and lower boundaries cannot intersect, though as far as we are aware, there is no known example where the existence of such a transitive set has been proven rigorously.     

Stabilization of periodic orbits near a subcritical Hopf bifurcation in delay-coupled networks

We study networks of delay-coupled oscillators with the aim to extend time-delayed feedback control to networks. We show that unstable periodic orbits of a network can be stabilized by a noninvasive, delayed coupling. We state criteria for stabilizing the orbits by delay-coupling in networks and apply these to the case where the local dynamics is close to a subcritical Hopf bifurcation, which is representative of systems with torsion-free unstable periodic orbits. Using the multiple scale method and the master stability function approach, the network system is reduced to the normal form, and the characteristic equations for Floquet exponents are derived in an analytical form, which reveals the coupling parameters for successful stabilization. Finally, we illustrate the results by numerical simulations of the Lorenz system close to a subcritical Hopf bifurcation. The unstable periodic orbits in this system have no torsion, and hence cannot be stabilized by the conventional time delayed-feedback technique.     

Periodic dynamics of coupled cell networks I: rigid patterns of synchrony and phase relations

It has recently been proved by Golubitsky and coworkers that in any network of coupled dynamical systems, the possible 'rigid' patterns of synchrony of hyperbolic equilibria are determined by purely combinatorial properties of the network, known as 'balanced equivalence relations'. A pattern is 'rigid' if it persists under small 'admissible' perturbations of the differential equation — ones that respect the network structure. We discuss a natural generalisation of these ideas to time-periodic states, and motivate two basic conjectures, the Rigid Synchrony Conjecture and the Rigid Phase Conjecture. These conjectures state that for rigid hyperbolic time-periodic patterns, cells with synchronous dynamics must have synchronous input cells, and cells with phase-related dynamics must have input cells that have the same phase relations. We provide evidence supporting the two conjectures, by proving them for a special class of periodic orbits, which we call 'tame', under strong assumptions on the network architecture and the symmetries of the periodic state. The discussion takes place in the formal setting of coupled cell networks. We prove that rigid patterns of synchrony are balanced, together with the analogous result for rigid patterns of phase relations. The assumption on the network architecture simplifies the geometry of admissible vector fields, while tameness rules out patterns with non-trivial local or multilocal symmetry. The main idea is to perturb an admissible vector field in a way that retains sufficient control over the associated perturbed periodic orbit. We present two techniques for constructing these perturbations, both using a general theorem on groupoid-symmetrisation of vector fields, which has independent interest. In particular we introduce a method of 'patching' that makes local changes to an admissible vector field. Having established these results for all-to-all coupled networks and tame periodic orbits we prove more general versions that require these assumptions only on a suitable quotient network. These conditions are weaker and encompass a larger class of networks and periodic orbits. We give an example to show that rigidity cannot be relaxed to hyperbolicity. We also prove, without any technical assumptions, that rigidly synchronous or phase-related cells must be input-isomorphic, a necessary precondition for the two conjectures to hold.     

Topological characterization of deterministic chaos: enforcing orientation preservation

The determinism principle, which states that dynamical state completely determines future time evolution, is a keystone of nonlinear dynamics and chaos theory. Since it precludes that two state space trajectories intersect, it is a core ingredient of a topological analysis of chaos based on a knot-theoretic characterization of unstable periodic orbits embedded in a strange attractor. However, knot theory can be applied only to three-dimensional systems. Still, determinism applies in any dimension. We propose an alternative framework in which this principle is enforced by constructing an orientation-preserving dynamics on triangulated surfaces and find that in three dimensions our approach numerically predicts the correct topological entropies for periodic orbits of the horseshoe map.     

Dynamics of surface diffeomorphisms relative to homoclinic and heteroclinic orbits

We show how to obtain information about the dynamics of a two-dimensional discrete-time system from its homoclinic and heteroclinic orbits. The results obtained are based on the theory of 'trellises', which comprise finite-length subsets of the stable and unstable manifolds of a collection of saddle periodic orbits. For any collection of homoclinic or heteroclinic orbits, we show how to associate a canonical 'trellis type' which describes the orbits. Given a trellis type, we show how to compute a 'graph representative' which gives a combinatorial invariant of the trellis type. The orbits of the graph give the dynamics forced by the homoclinic/heteroclinic orbits in the sense that every orbit of the graph representative is 'globally shadowed' by some orbit of the system, and periodic, homoclinic/heteroclinic orbits of the graph representative are shadowed by similar orbits.     

Partial classification of heteroclinic behaviour associated with the perturbation of hexagonal planforms

Physical systems often exhibit pattern-forming instabilities. Equivariant bifurcation theory is often used to investigate the existence and stability of spatially doubly periodic solutions with respect to the hexagonal lattice. Previous studies have focused on the six- and twelve-dimensional representation of the hexagonal lattice where the symmetry of the model is perfect. Here, perturbation of group orbits of translation-free axial planforms in the six- and twelve-dimensional representations is considered. This problem is studied via the abstract action of the symmetry group of the perturbation on the group orbit of the planform. A partial classification for the behaviour of the group orbits is obtained, showing the existence of homoclinic and heteroclinic cycles between equilibria.     

Canard solutions in planar piecewise linear systems with three zones

In this work, we analyze the existence and stability of canard solutions in a class of planar piecewise linear systems with three zones, using a singular perturbation theory approach. To this aim, we follow the analysis of the classical canard phenomenon in smooth planar slow–fast systems and adapt it to the piecewise-linear framework. We first prove the existence of an intersection between repelling and attracting slow manifolds, which defines a maximal canard, in a non-generic system of the class having a continuum of periodic orbits. Then, we perturb this situation and we prove the persistence of the maximal canard solution, as well as the existence of a family of canard limit cycles in this class of systems. Similarities and differences between the piecewise linear case and the smooth one are highlighted.     

Periodic orbits of oval billiards on surfaces of constant curvature

In this paper, we define and study the billiard problem on bounded regions on surfaces of constant curvature. We show that this problem defines a two-dimensional conservative and reversible dynamical system, defined by a Twist diffeomorphism, if the boundary of the region is an oval. Using these properties and defining good perturbations for billiards, we show that having only a finite number of nondegenerate periodic orbits for each fixed period is an open property for billiards on surfaces of constant curvature and a dense one on the hyperbolic plane. We finish this paper studying the stability of these nondegenerate orbits.     

一类高阶中立型微分方程周期解的存在性

本文研究一类高阶中立型泛函微分方程周期解的存在性,利用一些分析技巧和k-集压缩映射理论得到了该类方程至少存在一个周期解的两类充分条件.所得结果将现有关于常微分方程的结论推广到了泛函微分方程情形,同时减少或减弱了已有结果中的一些条件,从方程的形式和周期解的存在性条件两个方面推广和改进了文献中的相应工作.     

A codimension-two resonant bifurcation from a heteroclinic cycle with complex eigenvalues

Robust heteroclinic cycles between equilibria lose stability either through local bifurcations of their equilibria or through global bifurcations. This paper considers a global loss of stability termed a 'resonant' bifurcation. This bifurcation is usually associated with the birth or death of a nearby periodic orbit, and generically occurs in either a supercritical or subcritical manner. For a specific robust heteroclinic cycle between equilibria with complex eigenvalues we examine the codimension-two point that separates the supercritical and subcritical. We investigate the bifurcation structure and show the existence of further bifurcations of periodic orbits.     

一类泛函微分方程多个周期正解的存在性

本文利用Krasnoselskii锥映象不动点定理讨论了某类一阶泛函微分方程周期正解的存在性、非存在性与多解,给出了方程至少有一个解,有两个解或无解的若干充分条件。所得结果改进并推广了文献中的部分工作。     

含脉冲的随机延迟细胞神经网络的均方指数稳定性与周期解

本文主要研究一类带离散延迟和脉冲的随机细胞神经网络(SDCNNswI)的均方指数稳定性和周期解的存在性。首先,用庞加莱收缩理论分析了SDCNNswI的周期解存在条件;其次,用李雅谱诺夫函数、随机分析理论和Young不等式推出了几个定理,给出了保证SDCNNswl的周期解具有均方指数稳定性的几个充分条件,其中只包含SDCNNswI的几个控制参数,通过简单的代数方法即可验证。最后,通过两个例子说明了所提出准则的有效性。     

High-field magnetoresistance of AuGa

Measurements of the high-field magnetoresistance in AuGa indicate the existence of open orbits in [001]. The nearly free-electron model predicts open orbits in [001] and [100], but evidence for the existence of the latter only begins to appear at the highest fields used in this work.Work supported by the National Science Foundation.