The effect of reactor geometry on frontal polymerization spin modes

Using reactors of different sizes and geometries the dynamics of the frontal polymerization of 1,6-hexanediol diacrylate (HDDA) and pentaerythritol tetraacrylate (PETAC), with ammonium persulfate as the initiator were studied. For this system, the frontal polymerization exhibits complex behavior that depends on the ratio of the monomers. For a particular range of monomers concentration, the polymerization front becomes nonplanar, and spin modes appear. By varying the reactor diameter, we experimentally confirmed the expected shift of the system to a greater number of "hot spots" for larger diameters. For square test tubes a "zig-zag" mode was observed for the first time in frontal polymerization. We confirmed the viscosity-dependence of the spin mode instabilities. We also observed novel modes in cylinder-inside-cylinder reactors. Lastly, using a conical reactor with a continuously varying diameter, we observed what may be evidence for bistability depending on the direction of propagation. We discuss these finding in terms of the standard linear stability analysis for propagating fronts. (c) 2002 American Institute of Physics.     

Numerical modeling of self-propagating polymerization fronts: The role of kinetics on front stability

Frontal propagation of a highly exothermic polymerization reaction in a liquid is studied with the goal of developing a mathematical model of the process. As a model case we consider monomers such as methacrylic acid and n-butyl acrylate with peroxide initiators, although the model is not limited to these reactants and can be applied to any system with the similar basic polymerization mechanism. A three-step reaction mechanism, including initiation, propagation and termination steps, as well as a more simple one-step mechanism, were considered. For the one-step mechanism the loss of stability of propagating front was observed as a sequence of period doubling bifurcations of the front velocity. It was shown that the one-step model cannot account for less than 100% conversion and product inhomogeneities as a result of front instability, therefore the three-step mechanism was exploited. The phenomenon of superadiabatic combustion temperature was observed beyond the Hopf bifurcation point for both kinetic schemes and supported by the experimental measurements. One- and two-dimensional numerical simulations were performed to observe various planar and nonplanar periodic modes, and the results for different kinetic schemes were compared. It was found that stability of the frontal mode for a one-step reaction mechanism does not differ for 1-D and 2-D cases. For the three-step reaction mechanism 2-D solutions turned out to be more stable with respect to the appearance of nonplanar periodic modes than corresponding 1-D solutions. Higher Zeldovich numbers (i.e., higher effective activation energies or lower initial temperatures) are necessary for the existence of planar and nonplanar periodic modes in the 2-D reactor with walls than in the 1-D case. (c) 1997 American Institute of Physics.     

Single-head spin modes in frontal polymerization

The single-head spin mode is the first two-dimensional pattern of self-propagating thermal reaction fronts observed after the planar front loses stability. A "hot spot" is observed to propagate around the periphery of the front. The dynamics of the single-head frontal polymerization regime were studied experimentally, and two novel results were found. The "hot spot" was measured and found to be superadiabatic. The relationship between the rotational velocity and the propagating velocity was found. Experimental data were also compared with the theoretical results based on the linear stability analysis and found to be in a reasonable agreement. (c) 1998 American Institute of Physics.     

电流模式SEPIC变换器倍周期分岔现象研究

以SEPIC变换器中前置电感电流为控制对象,建立了SEPIC变换器断续电流模式下的离散时间模型, 并分析了不动点的稳定性.通过相轨图、功率谱图以及分岔图, 发现电路中存在一种特殊的倍周期分岔现象——电路的运行状态经历了1倍周期、 2倍周期、4倍周期再到2倍周期、4倍周期,最终进入混沌状态.实验结果与仿真结果相一致, 证实了SEPIC变换器在断续电流模式下存在这种倍周期分岔现象.     

Chaotic oscillation in diffusion flame induced by radiative heat loss

We numerically investigate the dynamic behavior of flame front instability in a diffusion flame caused by radiative heat loss from the viewpoint of nonlinear dynamics. As the Damköhler number increases at a high activation temperature, the dynamic behavior of the flame front undergoes a significant transition from a steady-state to high-dimensional deterministic chaos through the period-doubling cascade process known as the Feigenbaum transition. The existence of high-dimensional chaos in flame dynamics is clearly demonstrated using a sophisticated nonlinear time series analysis technique based on chaos theory.     

Modeling and visualization of quantum bifurcations in phase space

We investigate quantum-mechanical counterpart of a classical instability in a phase space by the numerical method of quantum trajectories with moving basis. As an application the model of coupled two oscillators driven by a monochromatic force in the presence of dissipation (intracavity second harmonic generation) is analyzed. The system of interest is characterized by two bifurcations leading to ranges of instability: the Hopf bifurcation which connects a steady state dynamics of the oscillatory modes to a self-pulsing temporal dynamics and the bifurcation of the period-doubling. The both two regimes are analyzed on the framework of the semiclassical phase trajectories and the Wigner functions of the oscillatory modes in phase space.     

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.     

Holes and chaotic pulses of traveling waves coupled to a long-wave mode

It is shown that localized traveling-wave pulses and holes can be stabilized by a coupling to a long-wave mode. Simulations of suitable real Ginzburg-Landau equations reveal a small parameter regime in which the pulses exhibit a breathing motion (presumably related to a front bifurcation), which subsequently becomes chaotic via period-doubling bifurcations.     

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     

Period-doubling bifurcation to alternans in paced cardiac tissue: crossover from smooth to border-collision characteristics

We investigate, both experimentally and theoretically, the period-doubling bifurcation to alternans in heart tissue. Previously, this phenomenon has been modeled with either smooth or border-collision dynamics. Using a modification of existing experimental techniques, we find a hybrid behavior: Very close to the bifurcation point, the dynamics is smooth, whereas further away it is border-collision-like. The essence of this behavior is captured by a model that exhibits what we call an unfolded border-collision bifurcation. This new model elucidates that, in an experiment, where only a limited number of data points can be measured, the smooth behavior of the bifurcation can easily be missed.     

Space charge dynamics in a semiconductor superlattice affected by titled magnetic field and heating

The transition between different modes of current oscillations in a semiconductor superlattice, from close-to-harmonic (near the generation onset) to relaxation oscillations, has been investigated. The transition type is shown to change with an increase in temperature. A period-doubling bifurcation is observed at low temperatures. With an increase in temperature, the period-doubling bifurcation is observed at increasingly larger values of the voltage across the superlattice. The doubling bifurcation ceases to be observed at voltages at which the generation of oscillations of the current through the semiconductor superlattice is suppressed.     

Dynamics and stabilization of peak current-mode controlled buck converter with constant current load

The discrete iterative map model of peak current-mode controlled buck converter with constant current load(CCL),containing the output voltage feedback and ramp compensation, is established in this paper. Based on this model the complex dynamics of this converter is investigated by analyzing bifurcation diagrams and the Lyapunov exponent spectrum. The effects of ramp compensation and output voltage feedback on the stability of the converter are investigated. Experimental results verify the simulation and theoretical analysis. The stability boundary and chaos boundary are obtained under the theoretical conditions of period-doubling bifurcation and border collision. It is found that there are four operation regions in the peak current-mode controlled buck converter with CCL due to period-doubling bifurcation and border-collision bifurcation. Research results indicate that ramp compensation can extend the stable operation range and transfer the operating mode, and output voltage feedback can eventually eliminate the coexisting fast-slow scale instability.     

Triad mode resonant interactions in suspended cables

A triad mode resonance, or three-wave resonance, is typical of dynamical systems with quadratic nonlinearities. Suspended cables are found to be rich in triad mode resonant dynamics. In this paper, modulation equations for cable's triad resonance are formulated by the multiple scale method. Dynamic conservative quantities, i.e., mode energy and Manley-Rowe relations, are then constructed. Equilibrium/dynamic solutions of the modulation equations are obtained, and full investigations into their stability and bifurcation characteristics are presented. Various bifurcation behaviors are detected in cable's triad resonant responses, such as saddle-node, Hopf, pitchfork and period-doubling bifurcations. Nonlinear behaviors, like jump and saturation phenomena, are also found in cable's responses. Based upon the bifurcation analysis, two interesting properties associated with activation of cable's triad resonance are also proposed, i.e., energy barrier and directional dependence. The first gives the critical amplitude of high-frequency mode to activate cable's triad resonance, and the second characterizes the degree of difficulty for activating cable's triad resonance in two opposite directions, i.e., with positive or negative internal detuning parameter.     

Period-doubling of multiple solitons in a passively mode-locked fiber laser

We report on the experimental observation of period-doubling of randomly distributed multiple solitons in a passively mode-locked fiber laser. We show that after a period-doubling bifurcation, the period of each soliton in the cavity becomes doubled. However, the intensity variation of the solitons is not necessarily synchronized, indicating that their period-doubling is unrelated. Numerical simulations verified the experimental observations.     

Spatial period-doubling in open flow

Coupled logistic lattices with asymmetric coupling in space, with a fixed boundary condition at the left end, are investigated. The system shows a period-doubling bifurcation to chaos as a lattice point goes downflow. In contrast with usual period-doubling in low-dimensional systems, (i) no scaling behavior has been found, (ii) low noise is important for the bifurcation structures. The system corresponds to a model for an open flow, which may be of use for the study of the onset of turbulence in pipe flows.     

Observation of stochastic resonance near a subcritical bifurcation

A hysteretic Subcritical period-doubling bifurcation is observed in the nonlinear strain dynamics of a magnetostrictive oscillator. The dynamic strain response of the magnetostrictive oscillator was observed with a high-resolution fiber optic interferometer. The effects of low-frequency modulation and band-limited stochastic fluctuations on such a bifurcation are investigated. Power spectral density measurements show that for an optimal value of externally injected noise the signal-to-noise ratio of a low-frequency modulation signal is enhanced by greater than 14 dB, thus indicating the first experimental observation of stochastic resonance near a bistable period-doubling bifurcation.     

Complex dynamics in the Oregonator model with linear delayed feedback

The Belousov-Zhabotinsky (BZ) reaction can display a rich dynamics when a delayed feedback is applied. We used the Oregonator model of the oscillating BZ reaction to explore the dynamics brought about by a linear delayed feedback. The time-delayed feedback can generate a succession of complex dynamics: period-doubling bifurcation route to chaos; amplitude death; fat, wrinkled, fractal, and broken tori; and mixed-mode oscillations. We observed that this dynamics arises due to a delay-driven transition, or toggling of the system between large and small amplitude oscillations, through a canard bifurcation. We used a combination of numerical bifurcation continuation techniques and other numerical methods to explore the dynamics in the strength of feedback-delay space. We observed that the period-doubling and quasiperiodic route to chaos span a low-dimensional subspace, perhaps due to the trapping of the trajectories in the small amplitude regime near the canard; and the trapped chaotic trajectories get ejected from the small amplitude regime due to a crowding effect to generate chaotic-excitable spikes. We also qualitatively explained the observed dynamics by projecting a three-dimensional phase portrait of the delayed dynamics on the two-dimensional nullclines. This is the first instance in which it is shown that the interaction of delay and canard can bring about complex dynamics.     

Experimental simulation of the Unruh effect on an NMR quantum simulator

A triad mode resonance, or three-wave resonance, is typical of dynamical systems with quadratic nonlinearities. Suspended cables are found to be rich in triad mode resonant dynamics. In this paper, modulation equations for cable’s triad resonance are formulated by the multiple scale method. Dynamic conservative quantities, i.e., mode energy and Manley-Rowe relations, are then constructed. Equilibrium/dynamic solutions of the modulation equations are obtained, and full investigations into their stability and bifurcation characteristics are presented. Various bifurcation behaviors are detected in cable’s triad resonant responses, such as saddle-node, Hopf, pitchfork and period-doubling bifurcations. Nonlinear behaviors, like jump and saturation phenomena, are also found in cable’s responses. Based upon the bifurcation analysis, two interesting properties associated with activation of cable’s triad resonance are also proposed, i.e., energy barrier and directional dependence. The first gives the critical amplitude of high-frequency mode to activate cable’s triad resonance, and the second characterizes the degree of difficulty for activating cable’s triad resonance in two opposite directions, i.e., with positive or negative internal detuning parameter.     

Oscillations, period doublings, and chaos in CO oxidation and catalytic mufflers

Early experimental observations of chaotic behavior arising via the period-doubling route for the CO catalytic oxidation both on Pt(110) and Ptgamma-Al(2)O(3) porous catalyst were reported more than 15 years ago. Recently, a detailed kinetic reaction scheme including over 20 reaction steps was proposed for the catalytic CO oxidation, NO(x) reduction, and hydrocarbon oxidation taking place in a three-way catalyst (TWC) converter, the most common reactor for detoxification of automobile exhaust gases. This reactor is typically operated with periodic variation of inlet oxygen concentration. For an unforced lumped model, we report results of the stoichiometric network analysis of a CO reaction subnetwork determining feedback loops, which cause the oscillations within certain regions of parameters in bifurcation diagrams constructed by numerical continuation techniques. For a forced system, numerical simulations of the CO oxidation reveal the existence of a period-doubling route to chaos. The dependence of the rotation number on the amplitude and period of forcing shows a typical bifurcation structure of Arnold tongues ordered according to Farey sequences, and positive Lyapunov exponents for sufficiently large forcing amplitudes indicate the presence of chaotic dynamics. Multiple periodic and aperiodic time courses of outlet concentrations were also found in simulations using the lumped model with the full TWC kinetics. Numerical solutions of the distributed model in two geometric coordinates with the CO oxidation subnetwork consisting of several tens of nonlinear partial differential equations show oscillations of the outlet reactor concentrations and, in the presence of forcing, multiple periodic and aperiodic oscillations. Spatiotemporal concentration patterns illustrate the complexity of processes within the reactor.     

基于Laguerre多项式逼近法的随机双势阱Duffing系统的分岔和混沌研究

应用Laguerre正交多项式逼近法研究了含有随机参数的双势阱Duffing系统的分岔和混沌行为.系统参数为指数分布随机变量的非线性动力系统首先被转化为等价的确定性扩阶系统，然后通过数值方法求得其响应.数值模拟结果的比较表明，含有随机参数的双势阱Duffing系统保持着与确定性系统相类似的倍周期分岔和混沌行为，但是由于随机因素的影响，在局部小区域内随机参数系统的动力学行为会发生突变. 关键词： 双势阱Duffing系统 指数分布概率密度函数 Laguerre多项式逼近 随机分岔