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A theoretical investigation of bifurcation structures of periodically forced oscillators is presented. In the plane of forcing frequency and amplitude, subharmonic entrainment occurs in v-shaped (Arnol'd) tongues, or entrainment bands, for small forcing amplitudes. These tongues terminate at higher forcing amplitudes. Between these two limits, individual tongues fit together to form a global bifurcation structure. The regime in which the forcing amplitude is much smaller than the amplitude of the limit cycle is first examined. Using the method of multiple time scales, expressions for solutions on the invariant torus, widths of Arnol'd tongues, and Liapunov exponents of periodic orbits are derived. Next, the regime of moderate to large forcing amplitudes is examined through studying a periodically forced Hopf bifurcation. In this case the forcing amplitude and the amplitude of the limit cycle can be of the same order of magnitude. From a study of the normal forms for this case, it is shown how Arnol'd tongues terminate and how complicated bifurcation structures are associated with strong resonances. Aspects of model and experimental chemical systems that show some of the phenomena predicted from the above theoretical results are mentioned. 相似文献
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Frequency-sensitive stochastic resonance in periodically forced and globally coupled systems 总被引:1,自引:0,他引:1
J. Xiao G. Hu H. Liu Y. Zhang 《The European Physical Journal B - Condensed Matter and Complex Systems》1998,5(1):133-138
A model of globally coupled bistable systems consisting of two kinds of sites, subject to periodic driving and spatially uncorrelated
stochastic force, is investigated. The extended system models the competing process of activators and suppressers. Analytical
computations for linear response of the system to the external periodic forcing is carried out. Noise-induced Hopf bifurcation
is revealed, and stochastic resonance, sensitively depending on the frequency of the external forcing, is predicted under
the Hopf bifurcation condition. Numerical simulations agree with the analytical predictions satisfactorily.
Received: 5 September 1997 / Revised: 13 May 1998 /
Accepted: 18 May 1998 相似文献
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We study a multimode semiconductor laser subject to a moderate optical feedback. The steady state is destabilized by either a simple Hopf bifurcation leading to in phase dynamics or by a degenerate Hopf bifurcation leading to antiphase dynamics. The degenerate bifurcation is also a source of multiple coexisting attractors. We show that a simple interpretation of the low frequency fluctuations in the multimode regime is provided by a chaotic itinerancy among the many coexisting unstable attractors produced by the degenerate Hopf bifurcation. 相似文献
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Experiments have yielded polarization self-modulation in vertical-cavity surface-emitting lasers (VCSELs) subject to a pi/2 polarization-rotating optical feedback. The phenomenon has been simulated numerically, but its bifurcation has never been explained. We show that polarization self-modulation results from a Hopf bifurcation mechanism that can be analyzed in terms of the laser feedback parameters. Our analysis predicts other bifurcations for low values of the feedback rate, which explain why more-complex time-dependent outputs have been observed as alternatives to polarization self-modulation. 相似文献
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Oscillatory Activities in Regulatory Biological Networks and Hopf Bifurcation 总被引:1,自引:0,他引:1 下载免费PDF全文
Exploiting the nonlinear dynamics in the negative feedback loop, we propose a statistical signal-response model to describe the different oscillatory behaviour in a biological network motif. By choosing the delay as a bifurcation parameter, we discuss the existence of Hopf bifurcation and the stability of the periodic solutions of model equations with the centre manifold theorem and the normal form theory. It is shown that a periodic solution is born in a Hopf bifurcation beyond a critical time delay, and thus the bifurcation phenomenon may be important to elucidate the mechanism of oscillatory activities in regulatory biological networks. 相似文献
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We review recent computational results for hexagon patterns in
non-Boussinesq convection. For sufficiently strong dependence of the
fluid parameters on the temperature we find reentrance of steady
hexagons, i.e. while near onset the hexagon patterns become unstable
to rolls as usually, they become again stable in the strongly
nonlinear regime. If the convection apparatus is rotated about a
vertical axis the transition from hexagons to rolls is replaced by a
Hopf bifurcation to whirling hexagons. For weak non-Boussinesq effects
they display defect chaos of the type described by the two-dimensional (2D)
complex Ginzburg–Landau equation. For stronger non-Boussinesq effects
the Hopf bifurcation becomes subcritical and localized bursting of the
whirling amplitude is found. In this regime the coupling of the
whirling amplitude to (small) deformations of the hexagon lattice
becomes important. For yet stronger non-Boussinesq effects this
coupling breaks up the hexagon lattice and strongly disordered states
characterized by whirling and lattice defects are obtained. 相似文献
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Lee SG Kim S 《Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics》1999,60(1):826-830
Recently, the phenomena of stochastic resonance (SR) have attracted much attention in the studies of the excitable systems under inherent noise, in particular, nervous systems. We study SR in a stochastic Hodgkin-Huxley neuron under Ornstein-Uhlenbeck noise and periodic stimulus, focusing on the dependence of properties of SR on stimulus parameters. We find that the dependence of the critical forcing amplitude on the frequency of the periodic stimulus shows a bell-shaped structure with a minimum at the stimulus frequency, which is quite different from the monotonous dependence observed in the bistable system at a small frequency range. The frequency dependence of the critical forcing amplitude is explained in connection with the firing onset bifurcation curve of the Hodgkin-Huxley neuron in the deterministic situation. The optimal noise intensity for maximal amplification is also found to show a similar structure. 相似文献
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分析一个简单二阶延迟系统的Hopf分支和混沌特性, 包括分支点、分支方向和分支周期解的稳定性, 解析求出退延迟情况下, 这个系统的相轨线方程; 通过数值计算并绘制分岔图, 揭示系统存在由倍周期通向混沌的道路; 利用单路线性组合信号, 反馈控制实现系统的部分完全同步; 利用主动-被动与线性反馈的联合, 实现系统的完全同步; 设计和搭建系统的电子实验线路, 并从实验中观测到与理论分析或数值计算相一致的结果.
关键词:
延迟非线性系统
电路实验
Hopf分支
混沌 相似文献
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M. Darwish 《The European Physical Journal D - Atomic, Molecular, Optical and Plasma Physics》2007,41(3):547-556
Theoretical investigations of dynamical behavior in optical
parametric oscillators (OPO) have generally assumed that the cavity
detunings of the interacting fields are controllable parameters.
However, OPOs are known to experience mode hops, where the system
jumps to the mode of lowest cavity detuning. We note that this
phenomenon significantly limits the range of accessible detunings
and thus may prevent instabilities predicted to occur above a
minimum detuning from being evidenced experimentally. As a simple
example among a number of instability mechanisms possibly affected
by this limitation, we discuss the Hopf bifurcation leading to
periodic behavior in the monomode mean-field model of a triply
resonant OPO and show that it probably can be observed only in very
specific setups. 相似文献
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We have modelled a continuously pumped Nd:YAG actively Q-switched by a variable interferometric mirror made up of a scanning Michelson or Fabry-Pérot mirror. We have characterised the three-mirror laser dynamics by using a bifurcation diagram constructed from the plot of peak power-enhancement factor as a function of mirror speed. One observes different chaotic windows separated by period-doubling bifurcations, and stable periodic regime. It is demonstrated that the best performance of the Q-switched laser is obtained rather for low than for high mirror speed (pulse width of 20 ns, and high peak power up to 400 times greater than the continuous emission). 相似文献
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The horizontal convection within a rectangular tank is numerically simulated. The flow is found to be unsteady at high Rayleigh numbers. There is a Hopf bifurcation of Ra from steady solutions to periodic solutions, and the critical Rayleigh number Rac is obtained to be Rac = 5.5377×10^8 for the middle plume forcing at Pr = 1, which is much larger than the value previously obtained. In addition, the unstable perturbations are always generated from the central jet, which implies that the onset of instability is due to velocity shear (shear instability) other than thermally dynamics (thermal instability). Finally, Paparella and Young's first hypotheses [J. Fluid Mech. 466 (2002) 205] about the destabilization of the flow is numerically proven, i.e. the middle plume forcing can lead to a destabilization of the flow. 相似文献
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A bifurcation sequence from a periodic to a quasiperiodic regime leading ultimately to a steady state, is reported in an experimental study of the Belousov-Zhabotinsky reaction. This sequence is understood in the frame of the interaction of two instabilities, namely a “hysteresis” and a Hopf bifurcation. 相似文献
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The stochastic resonance (SR) phenomenon induced by a multiplicative periodic signal in a logistic growth model with correlated
noises is studied by using the theory of signal-to-noise ratio (SNR) in the adiabatic limit. The expressions of the SNR are obtained. The effects of multiplicative noise intensity α and additive noise intensity D, and correlated intensity λ on the SNR are discussed respectively. It is found that the existence of a maximum in the SNR is the identifying characteristic of the SR phenomena. In comparison with the SR induced by additive periodic signal, some
new features are found: (1) When SNR as a function of λ for fixed ratio of α and D, the varying of α can induce a stochastic multi-resonance, and can induce a re-entrant transition of the peaks in SNR vs λ; (2) There exhibits a doubly critical phenomenon for SNR vs D and λ, i.e., the increasing of D (or λ) can induce the critical phenomenon for SNR with respect to λ (or D); (3) The doubly stochastic resonance effect appears when α and D are simultaneously varying in SNR, i.e., the increment of one noise intensity can help the SR on another noise intensity come forth.
相似文献
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We examine examples of weakly nonlinear systems whose steady states undergo a bifurcation with increasing forcing, such that a forced subsystem abruptly ceases to absorb additional energy, instead diverting it into an initially quiescent, unforced subsystem. We derive and numerically verify analytical predictions for the existence and behavior of such saturated states for a class of oscillator pairs. We also examine related phenomena, including zero-frequency response to periodic forcing, Hopf bifurcations to quasiperiodicity, and bifurcations to periodic behavior with multiple frequencies. 相似文献
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We investigate the Hopf bifurcation of the synchronous chaos in
coupled Lorenz oscillators. We find that the system undergoes a
phase transition along the Hopf instability of the synchronous
chaos. The phase transition makes the traveling wave component
with the phase difference φ(i)-φ(i+1)=2π/N between
adjacent sites unstable. The phase transition also plays a role to
relate the Hopf bifurcation with the co-dimension two bifurcation
of the synchronous chaos. 相似文献
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Hopf bifurcation control for a coupled nonlinear relative rotation system with time-delay feedbacks 下载免费PDF全文
This paper investigates the Hopf bifurcations resulting from time delay in a coupled relative-rotation system with time- delay feedbacks. Firstly, considering external excitation, the dynamical equation of relative rotation nonlinear dynamical system with primary resonance and 1:1 internal resonance under time-delay feedbacks is deduced. Secondly, the averaging equation is obtained by the multiple scales method. The periodic solution in a closed form is presented by a perturbation approach. At last, numerical simulations confirm that time-delay theoretical analyses have influence on the Hopf bifurcation point and the stability of periodic solution. 相似文献
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1.IntroductionLasersystemshavebccnkno`"ntounderg0stabiIitychangesunderccrtainconditi0ns.Inthesingle-m0de1asers,forexampIc,thereisasecondthresh0Idvalueabovewhichafurtherinstability,namely,sclf-pulsing,sctsin[1'2J.H0\\'cver,inthccaseof0nc-ph0tonlasersoftheclassB,n0self-pulsinginstabilityoccursf0rthefrcc-runningandatresonance.Instabili-tiesin'tx"o-ph0t0nlasersystc112shavcals0bceninvcstigatcdthe0rctically.Inrecentyears,two-phQtonlaser0scillationhasbccnobscrvedinaFabry-Per0tcavityfil1edx"ithrub… 相似文献