Multilayered tori in a system of two coupled logistic maps

The Letter describes different mechanisms for the formation and destruction of tori that are formed as layered structures of several sets of interlacing manifolds, each with their associated stable and unstable resonance modes. We first illustrate how a three layered torus can arise in a system of two coupled logistic maps through period-doubling or pitchfork bifurcations of the saddle cycle on an ordinary resonance torus. We hereafter present two different scenarios by which a multilayered torus can be destructed. One scenario involves a cascade of period-doubling bifurcations of both the stable and the saddle cycles, and the second scenario describes a transition in which homoclinic bifurcations destroy first the two outer layers and thereafter also the inner layer of a three-layered torus. It is suggested that the formation of multilayered tori is a generic phenomenon in non-invertible maps.     

Coexisting tori and torus bubbling in non-smooth systems

Pulse modulated power electronic converters represent an important class of piecewise-smooth dynamical systems with a broad range of applications in modern power supply systems. The paper presents a detailed investigation of a number of unusual bifurcation phenomena that can occur in power converters with multilevel control. In the first example a closed invariant curve arises in a border-collision bifurcation as a period-6 saddle cycle collides with a stable fixed point of focus type and transforms it into an unstable focus point. The second example involves the formation of a structure of coexisting tori through the interplay between border-collision and global bifurcations. We examine the behavior of the system in the presence of two coexisting stable resonance tori and finally show how an existing torus can develop heteroclinic bubbles that connect the points of a stable resonance cycle with an external pair of saddle and focus cycles. The appearance of these structures is explained in terms of a sequence torus-birth bifurcations with pairs of stable and unstable tori folding one over the other.     

Novel routes to chaos through torus breakdown in non-invertible maps

The paper describes a number of new scenarios for the transition to chaos through the formation and destruction of multilayered tori in non-invertible maps. By means of detailed, numerically calculated phase portraits we first describe how three- and five-layered tori arise through period-doubling and/or pitchfork bifurcations of the saddle cycle on an ordinary resonance torus. We then describe several different mechanisms for the destruction of five-layered tori in a system of two linearly coupled logistic maps. One of these scenarios involves the destruction of the two intermediate layers of the five-layered torus through the transformation of two unstable node cycles into unstable focus cycles, followed by a saddle-node bifurcation that destroys the middle layer and a pair of simultaneous homoclinic bifurcations that produce two invariant closed curves with quasiperiodic dynamics along the sides of the chaotic set. Other scenarios involve different combinations of local and global bifurcations, including bifurcations that lead to various forms of homoclinic and heteroclinic tangles. We finally demonstrate that essentially the same scenarios can be observed both for a system of nonlinearly coupled logistic maps and for a couple of two-dimensional non-invertible maps that have previously been used to study the properties of invariant sets.     

神经元电活动不同节律模式的几种变化过程

利用神经元Chay模型,对实验中观察到的三种放电节律模式序列进行数值模拟,并应用余维1极限环分岔分析研究了其产生机理.首先考虑的是周期性放电模式的变化过程;其次,具有不同表象的一种超临界和一种亚临界倍周期簇放电序列产生并导致混沌现象的出现,然后以不同的方式转迁到逆超临界倍周期峰放电序列;最后研究无混沌的加周期簇放电序列,得出加周期分岔仅是一种与倍周期分岔密切相关的分岔现象.     

Study of a dynamical system with a strange attractor and invariant tori

A simple three-dimensional time-reversible system of ODEs with quadratic nonlinearities is considered in a recent paper by Sprott (2014). The author finds in this system, that has no equilibria, the coexistence of a strange attractor and invariant tori. The goal of this letter is to justify theoretically the existence of infinite invariant tori and chaotic attractors. For this purpose we embed the original system in a one-parameter family of reversible systems. This allows to demonstrate the presence of a Hopf-zero bifurcation that implies the birth of an elliptic periodic orbit. Thus, the application of the KAM theory guarantees the existence of an extremely complex dynamics with periodic, quasiperiodic and chaotic motions. Our theoretical study is complemented with some numerical results. Several bifurcation diagrams make clear the rich dynamics organized around a so-called noose bifurcation where, among other scenarios, cascades of period-doubling bifurcations also originate chaotic attractors. Moreover, a cross section and other numerical simulations are also presented to illustrate the KAM dynamics exhibited by this system.     

不连续导电模式DC-DC变换器的倍周期分岔机理研究

根据一般迭代映射的倍周期分岔定理,从数学上论证了电压型不连续导电模式(DCM) Boost和Buck变换器中倍周期分岔现象产生条件,由此揭示了DC-DC变换器中倍周期分岔现象发生的机理,完善了该类变换器倍周期分岔分析的理论和方法. 关键词： 倍周期分岔 迭代映射 Lyapunov 指数 施瓦茨导数     

Multi-frequency tori in wide-aperture lasers

A distributed model of wide-aperture laser based on Maxwell?Bloch equations in onedimensional approximation is considered. It is shown that an increase in the pumping parameter in the system gives rise to a cascade of bifurcations of periodic and quasi-periodic dynamic modes, as a result of which attractors in the form of three-frequency tori can be observed.     

Stochastic period-doubling bifurcation in biharmonic driven Dulling system with random parameter

Stochastic period-doubling bifurcation is explored in a forced Duffing system with a bounded random parameter as an additional weak harmonic perturbation added to the system. Firstly, the biharmonic driven Duffing system with a random parameter is reduced to its equivalent deterministic one, and then the responses of the stochastic system can be obtained by available effective numerical methods. Finally, numerical simulations show that the phase of the additional weak harmonic perturbation has great influence on the stochastic period-doubling bifurcation in the biharmonic driven Duffing system. It is emphasized that, different from the deterministic biharmonic driven Duffing system, the intensity of random parameter in the Duffing system can also be taken as a bifurcation parameter, which can lead to the stochastic period-doubling bifurcations.     

Bifurcation analysis and control of periodic solutions changing into invariant tori in Langford system

Bifurcation characteristics of the Langford system in a general form are systematically analysed, and nonlinear controls of periodic solutions changing into invariant tori in this system are achieved. Analytical relationship between control gain and bifurcation parameter is obtained. Bifurcation diagrams are drawn, showing the results of control for secondary Hopf bifurcation and sequences of bifurcations route to chaos. Numerical simulations of quasi-periodic tori validate analytic predictions.     

Bifurcations of tori and phase locking in a dissipative system of differential equations

In a system of ordinary differential equations, obtained through a seven-mode truncation of the plane incompressible Navier-Stokes equations, a two-dimensional torus undergoes first two period-doubling bifurcations and then a transition to a strange attractor. This strange attractor, of Liapunov dimension larger than three in a wide parameter interval, is characterized by a power spectrum which retains the two fundamental frequencies of the original torus superimposed on a broad, jagged background. As the Liapunov dimension goes down towards two, an interesting phenomenon of phase locking occurs, which gives rise to an alternation of chaotic and periodic behavior.     

竖直振动颗粒床中的倍周期运动

实验研究了竖直振动颗粒床中颗粒对容器底部的压力随振动强度的变化情况.发现压力随振动加速度的增加经历倍周期分岔，典型的分岔序列为：2P，4P，混沌，3P，6P，混沌，4P，8P，混沌.观察表明，伴随倍周期分岔现象，在颗粒床底部出现颗粒的聚集态.聚集态内颗粒密堆积在一起并作整体的上下运动.采用完全非弹性蹦球模型分析了颗粒对容器底的冲击力，并给出了倍周期分岔现象的一种解释. 关键词： 颗粒物质 混沌 倍周期分岔 非弹性碰撞     

The transition to explosive solitons and the destruction of invariant tori

We investigate the transition to explosive dissipative solitons and the destruction of invariant tori in the complex cubic-quintic Ginzburg-Landau equation in the regime of anomalous linear dispersion as a function of the distance from linear onset. Using Poncaré sections, we sequentially find fixed points, quasiperiodicity (two incommesurate frequencies), frequency locking, two torus-doubling bifurcations (from a torus to a 2-fold torus and from a 2-fold torus to a 4-fold torus), the destruction of a 4-fold torus leading to non-explosive chaos, and finally explosive solitons. A narrow window, in which a 3-fold torus appears, is also observed inside the chaotic region.     

基于Chebyshev多项式逼近的随机 van der Pol系统的倍周期分岔分析

应用 Chebyshev 多项式逼近法研究了谐和激励作用下具有随机参数的随机van der Pol系统 的倍周期分岔现象.随机系统首先被转化成等价的确定性系统，然后通过数值方法求得响应 ，借此探索了随机van der Pol系统丰富的随机倍周期分岔现象.数值模拟显示随机van der Pol 系统存在与确定性系统极为相似的倍周期分岔行为，但受随机因素的影响，又有与之不 同之处.数值结果表明，Chebyshev 多项式逼近是研究非线性系统动力学问题的一种新的有 效方法. 关键词： Chebyshev 多项式 随机van der Pol 系统 倍周期分岔     

The Criterion of Chaotic Pulsation in pulsating Stars

Up to now, the period-doubling bifurcation in variable star is identified through the existence of subharmonics in the poGer spectrum of light curves. Therefore the question is arisen: How to find a general criterion of period-doubling bifurcation in pulsating stars from theoretical analysis. It is discussed in this paper. We found that for the stellar pulsating mapping the bifurcation from a fixed point to a two-cycle occurs at parameter λ₁ = -0.63, i.e., adiabatic exponent Г₁ = 1.331.     

Local and global behavior near homoclinic orbits

We study the local behavior of systems near homoclinic orbits to stationary points of saddle-focus type. We explicitly describe how a periodic orbit approaches homoclinicity and, with the help of numerical examples, discuss how these results relate to global patterns of bifurcations.     

Period-doubling cascades of canards from the extended Bonhoeffer-van der Pol oscillator

This Letter investigates the period-doubling cascades of canards, generated in the extended Bonhoeffer-van der Pol oscillator. Canards appear by Andronov-Hopf bifurcations (AHBs) and it is confirmed that these AHBs are always supercritical in our system. The cascades of period-doubling bifurcation are followed by mixed-mode oscillations. The detailed two-parameter bifurcation diagrams are derived, and it is clarified that the period-doubling bifurcations arise from a narrow parameter value range at which the original canard in the non-extended equation is observed.     

含有界随机参数的双势阱Duffing-van der Pol系统的倍周期分岔

讨论简谐激励作用下含有界随机参数的双势阱Duffing-van der Pol系统的倍周期分岔现象.首先用Chebyshev 多项式逼近法将随机Duffing-van der Pol系统化成与其等价的确定性系统，然后通过等价确定性系统来探索该系统的倍周期分岔现象.数值模拟显示随机Duffing-van der Pol 系统与均值参数系统有着类似的倍周期分岔行为，同时指出，随机参数系统的倍周期分岔有其自身独有的特点.文中的主要数值结果表明Chebyshev 多项式逼近法是研究非线性随机参数系统动力学问题的一种有效方法. 关键词： Chebyshev多项式 随机Duffing-van der Pol系统 倍周期分岔     

Critical fluctuations of noisy period-doubling maps

We extend the theory of quasipotentials in dynamical systems by calculating, within a broad class of period-doubling maps, an exact potential for the critical fluctuations of pitchfork bifurcations in the weak noise limit. These far-from-equilibrium fluctuations are described by finite-size mean field theory, placing their static properties in the same universality class as the Ising model on a complete graph. We demonstrate that the effective system size of noisy period-doubling bifurcations exhibits universal scaling behavior along period-doubling routes to chaos.     

Properties of pulsating solitons in dissipative systems

<正>In this paper,a set of detailed numerical simulations of pulsating solitons in certain regions,where the pulsating solitons exist,have been carried out.The results show that the transformation between pulsating soliton and fronts can be realized through a series of period-doubling bifurcations,while there exist many kinds of special solutions.The complete transformation diagram has been obtained when the value of nonlinear gain varies within a definite range.The detailed analysis of the diagram reveals that the pulsating soliton experiences period-doubling bifurcations for smaller values of the nonlinear gain.For larger values of it,the pulsating solitons show chaotic behaviour and complex pulse splitting except for some special bifurcations.With the value of nonlinear gain increasing further,the pulse profiles resume pulsating,but the pulse energy is much higher than before and the pulse centre may move along the propagation direction.     

Robust tori in a double-waved Hamiltonian model

A Hamiltonian system perturbed by two waves with particular wave numbers can present robust tori, which are barriers created by the vanishing of the perturbed Hamiltonian at some defined positions. When robust tori exist, any trajectory in phase space passing close to them is blocked by emergent invariant curves that prevent the chaotic transport. Our results indicate that the considered particular solution for the two waves Hamiltonian model shows plenty of robust tori blocking radial transport.