Nonlinear dynamos: A complex generalization of the Lorenz equations

Plane nonlinear dynamo waves can be described by a sixth order system of nonlinear ordinary differential equations which is a complex generalization of the Lorenz system. In the regime of interest for modelling magnetic activity in stars there is a sequence of bifurcations, ending in chaos, as a stability parameter D (the dynamo number) is increased. We show that solutions undergo three successive Hopf bifurcations, followed by a transition to chaos. The system possesses a symmetry and can therefore be reduced to a fifth order system, with trajectories that lie on a 2-torus after the third bifurcation. As D is then increased, frequency locking occurs, followed by a sequence of period-doubling bifurcations that leads to chaos. This behaviour is probably caused by the Shil'nikov mechanism, with a (conjectured) homoclinic orbit when D is infinite.     

BIFURCATIONS AND CHAOS OF AN IMMERSED CANTILEVER BEAM IN A FLUID AND CARRYING AN INTERMEDIATE MASS

The concern of this work is the local stability and period-doubling bifurcations of the response to a transverse harmonic excitation of a slender cantilever beam partially immersed in a fluid and carrying an intermediate lumped mass. The unimodal form of the non-linear dynamic model describing the beam-mass in-plane large-amplitude flexural vibration, which accounts for axial inertia, non-linear curvature and inextensibility condition, developed in Al-Qaisia et al. (2000Shock and Vibration7 , 179-194), is analyzed and studied for the resonance responses of the first three modes of vibration, using two-term harmonic balance method. Then a consistent second order stability analysis of the associated linearized variational equation is carried out using approximate methods to predict the zones of symmetry breaking leading to period-doubling bifurcation and chaos on the resonance response curves. The results of the present work are verified for selected physical system parameters by numerical simulations using methods of the qualitative theory, and good agreement was obtained between the analytical and numerical results. Also, analytical prediction of the period-doubling bifurcation and chaos boundaries obtained using a period-doubling bifurcation criterion proposed in Al-Qaisia and Hamdan (2001 Journal of Sound and Vibration244, 453-479) are compared with those of computer simulations. In addition, results of the effect of fluid density, fluid depth, mass ratio, mass position and damping on the period-doubling bifurcation diagrams are studies and presented.     

Period-doubling bifurcations,chaos, phase-locking and devil's staircase in a Bonhoeffer-van der Pol oscillator

The effect of a periodic input current A₁ cos t in the Bonhoeffer-van der Pol oscillator along with a bias A₀ is investigated numerically. As the parameter A₁ is varied in the absence of bias by holding the other parameters at constant values, typical period-doubling bifurcation sequence is found to occur leading to chaotic motion in agreement with the Feigenbaum scenario. When the bias is switched on at the transition to chaos, frequency-locking is observed in the system. The frequency-locked intervals exhibit complete devil's staircase similar to the one observed in Belousov-Zhabotinsky reaction.     

Josephson结的动力学行为(Ⅰ)

本文对射频电流驱动下的包含干涉项电流的Josephson结方程进行了大量的数值研究工作。我们发现,对应于不同的参数范围,分别出现混沌行为、倍周期分岔序列、混沌带的反序列以及阵发混沌现象。我们也计算了2ⁿP序列的收敛因子δ_n和功率谱中2^k与2^k+1分频的平均峰高之比Φ(k)/Φ(k+1)。另外,还研究了周期解的对称性以及它与通向混沌途径的关系。 关键词：     

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.     

Fluctuations and simple chaotic dynamics

We describe the effects of fluctuations on the period-doubling bifurcation to chaos. We study the dynamics of maps of the interval in the absence of noise and numerically verify the scaling behavior of the Lyapunov characteristic exponent near the transition to chaos. As previously shown, fluctuations produce a gap in the period-doubling bifurcation sequence. We show that this implies a scaling behavior for the chaotic threshold and determine the associated critical exponent. By considering fluctuations as a disordering field on the deterministic dynamics, we obtain scaling relations between various critical exponents relating the effect of noise on the Lyapunov characteristic exponent. A rule is developed to explain the effects of additive noise at fixed parameter value from the deterministic dynamics at nearby parameter values.     

Dynamics of erbium-doped fibre laser with optical delay feedback and chaotic synchronization

The dynamical behaviour of the erbium-doped fibre single-ring laser with an optical delay feedback is discussed. Simulation shows that as the delay rate increases, the lasing light displays period-doubling which leads to chaos and via reverse period-doubling route returns from chaos to periodic. At a particular delay rate the intermittently chaotic route to chaos is also observed. The identical synchronization based on chaos in this ring laser is demonstrated by numerical simulation.     

Analysis of stochastic bifurcation and chaos in stochastic Duffing--van der Pol system via Chebyshev polynomial approximation

The Chebyshev polynomial approximation is applied to investigate the stochastic period-doubling bifurcation and chaos problems of a stochastic Duffing--van der Pol system with bounded random parameter of exponential probability density function subjected to a harmonic excitation. Firstly the stochastic system is reduced into its equivalent deterministic one, and then the responses of stochastic system can be obtained by numerical methods. Nonlinear dynamical behaviour related to stochastic period-doubling bifurcation and chaos in the stochastic system is explored. Numerical simulations show that similar to its counterpart in deterministic nonlinear system of stochastic period-doubling bifurcation and chaos may occur in the stochastic Duffing--van der Pol system even for weak intensity of random parameter. Simply increasing the intensity of the random parameter may result in the period-doubling bifurcation which is absent from the deterministic system.     

Some special features of the transition to chaos in the self-modulation of surface spin waves

The results of experimental investigations of the transition from regular to stochastic self-modulation of intense surface spin waves are presented. It is found that the transition to chaos follows the scenario of a sequence of period-doubling bifurcations. The fractal dimensions and the Kolmogorov entropy are determined for different regimes. The experiments are performed on an apparatus consisting of a microwave oscillator with a spin-wave delay line in the feedback circuit. Pis’ma Zh. éksp. Teor. Fiz. 66, No. 4, 243–246 (25 August 1997)     

Effects of noise on periodic orbits of the logistic map

Noise can induce an inverse period-doubling transition and chaos. The effects of noise on each periodic orbit of three different period sequences are investigated for the logistic map. It is found that the dynamical behavior of each orbit, induced by an uncorrelated Gaussian white noise, is different in the mergence transition. For an orbit of the period-six sequence, the maximum of the probability density in the presence of noise is greater than that in the absence of noise. It is also found that, under the same intensity of noise, the effects of uncorrelated Gaussian white noise and exponentially correlated colored (Gaussian) noise on the period-four sequence are different.      

Optical Bistability in V Medium by Squeezed Vacuum Input

In this paper we present the analytic expressions of partition lines in the parametric space of a piece-wise smooth mapping describing an electronic relaxation oscillator. The supercritical regions with permission of period-doubling bifurcation, prohibition of period-doubling bifurcation, and complete phase-locking are discussed. Among them the region, where period-doubling bifurcation is prohibited but chaos is permitted is reported for the first time to our knowledge. These kinds of phenomena and regions can be observed in a lot of similar systems.     

一个张弛振子的超临界子区域

解析导出了描述一种电子张弛振子的分段光滑圆映象在参数空间超临界区域内显示几种不同动力学行为的子区域分界线.这些子区域分别是：允许倍周期分岔发生的区域,禁止倍周期分岔发生的区域,以及完全锁相区域.类似的现象和子区域可以在许多分段光滑系统中观察到. 关键词：     

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     

Stability and chaotic behavior of a two-component Bose–Einstein condensate

We investigate the stability and chaotic behavior of a periodically driven Bose–Einstein condensate (BEC) with two hyperfine states. The effects of the population transfer K and relative energy fluctuation γ between the two hyperfine states are demonstrated analytically and numerically. The stability analysis shows that the steady-state relative population will appear the tuning-fork bifurcation, when the physical parameters are changed to some critical values. Meanwhile, the dependence of the macroscopic quantum self-trapping (MQST) on the initial conditions, the population transfer and the relative energy is revealed, and the stationary, periodical and chaotic MQSTs are found. Finally, we illustrate that the relative population oscillation can undergo a process from order to chaos, through a series of period-doubling bifurcations.     

Chaotic flow in a model for repeated yielding

Chaos exhibited by a model introduced in the context of repeated yielding is studied. The model shows an infinite sequence of period-doubling bifurcations with an exponent δ = 4.67 ± 0.1. The associated one-dimensional map and the projection of the strange attractor are also studied.     

Ballistic phonon transport through a Fibonacci array of acoustic nanocavities in a narrow constriction

We investigate the ballistic phonon transport through a Fibonacci array of acoustic nanocavities in a narrow constriction of a semiconductor nanowire at low temperatures. It is found that the transmission spectrum of such a system consists of quasiband gaps and narrow resonances caused by the coupling of phonon waves. Both phonon transmission and thermal conductance exhibit the similarity due to the Fibonacci sequence structure. The similarity is sensitive to the number n and parameters of nanocavities. The results are compared with those in a periodic acoustic nanocavities.     

广义的一维Fibonacci结构

本文提出一维k组元的Fibonacci结构,它包含k个无公度的基本长度,是具有两个基本长度的标准Fibonacci结构的自然推广,投影理论被用来处理它的X射线衍射花样及其指标化问题,理论模拟被用于验证投影理论的结果,值得注意的是,对于有限的链长,当k足够大时,它的衍射谱将趋于混沌。 关键词：     

Nonlinear oscillations and chaos in n-InSb

We present evidence for chaotic behavior in n-InSb. The Hall voltage exhibits a period-doubling route to chaos as the (non-ohmic) dc current is increased. The nonlinear oscillation and bifurcation processes are strongly influenced by irradiation with CO₂ laser radiation.     

Transverse elastic waves in Fibonacci superlattices

We study the transverse elastic waves propagating in 6-mm class hexagonal crystals forming Fibonacci superlattices. These are formed by repetitions of CdS and ZnO slabs in A and B constituent blocks following the Fibonacci sequence. We study the periodic superlattices formed by the infinite repetition of a given Fibonacci generation together with the finite Fibonacci generations having stress-free surfaces, in order to compare the effects introduced by the different boundary conditions. We have also considered the effects of piezoelectricity when all the interfaces are metallized and kept at a fixed potential. We use the surface Green function matching method forNnonequivalent interfaces to obtain the dispersion relations and the density of states of these systems. We have studied the influence of the increasing order of the Fibonacci generations on the dispersion relation of the transverse elastic modes. The Fibonacci spectrum is clearly seen even for low-order Fibonacci generations and is almost not modified by the piezoelectric coupling when the interfaces are metallized.     

Basin boundaries in asymmetric vibrations of a circular plate

In order to investigate further nonlinear asymmetric vibrations of a clamped circular plate under a harmonic excitation, we reexamine a primary resonance, studied by Yeo and Lee [Corrected solvability conditions for non-linear asymmetric vibrations of a circular plate, Journal of Sound and Vibration 257 (2002) 653-665] in which at most three stable steady-state responses (one standing wave and two traveling waves) are observed to exist. Further examination, however, tells that there exist at most five stable steady-state responses: one standing wave and four traveling waves. Two of the traveling waves lose their stability by Hopf bifurcation and have a sequence of period-doubling bifurcations leading to chaos. When the system has five attractors: three equilibrium solutions (one standing wave and two traveling waves) and two chaotic attractors (two modulated traveling waves), the basin boundaries of the attractors on the principal plane are obtained. Also examined is how basin boundaries of the modulated motions (quasi-periodic and chaotic motions) evolve as a system parameter varies. The basin boundaries of the modulated motions turn out to have the fractal nature.