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.      

Unstable periodic orbits and noise in chaos computing

Different methods to utilize the rich library of patterns and behaviors of a chaotic system have been proposed for doing computation or communication. Since a chaotic system is intrinsically unstable and its nearby orbits diverge exponentially from each other, special attention needs to be paid to the robustness against noise of chaos-based approaches to computation. In this paper unstable periodic orbits, which form the skeleton of any chaotic system, are employed to build a model for the chaotic system to measure the sensitivity of each orbit to noise, and to select the orbits whose symbolic representations are relatively robust against the existence of noise. Furthermore, since unstable periodic orbits are extractable from time series, periodic orbit-based models can be extracted from time series too. Chaos computing can be and has been implemented on different platforms, including biological systems. In biology noise is always present; as a result having a clear model for the effects of noise on any given biological implementation has profound importance. Also, since in biology it is hard to obtain exact dynamical equations of the system under study, the time series techniques we introduce here are of critical importance.     

耦合不连续系统同步转换过程中的多吸引子共存

本文研究了耦合不连续系统的同步转换过程中的动力学行为, 发现由混沌非同步到混沌同步的转换过程中特殊的多吸引子共存现象. 通过计算耦合不连续系统的同步序参量和最大李雅普诺夫指数随耦合强度的变化, 发现了较复杂的同步转换过程: 临界耦合强度之后出现周期非同步态(周期性窗口); 分析了系统周期态的迭代轨道,发现其具有两类不同的迭代轨道: 对称周期轨道和非对称周期轨道, 这两类周期吸引子和同步吸引子同时存在, 系统表现出对初值敏感的多吸引子共存现象. 分析表明, 耦合不连续系统中的周期轨道是由于局部动力学的不连续特性和耦合动力学相互作用的结果. 最后, 对耦合不连续系统的同步转换过程进行了详细的分析, 结果表明其同步呈现出较复杂的转换过程.     

Controlling Spatiotemporal Chaos in Coupled Map Lattices to Periodic Orbits

Two methods are presented for controlling spatiotemporal chaotic motion in coupled map lattices to a kind of periodic orbit where the dynamical variables of all lattice sites are equal and act periodically as time evolves. Stability analysis of the periodic orbits is presented. We prove that especially the second controlling method can stabilize all the periodic orbits we concern. Basin of attraction and noise problem are discussed.     

Controlling Spatiotemporal Chaos in Coupled Map Lattices to Periodic Orbits

Two methods are presented for controlling spatiotemporal chaotic motion in coupled map lattices to a kind of periodic orbit where the dynamicM variables of all lattice sites are equM and act periodically as time evolves. Stability analysis of the periodic orbits is presented. We prove that especially the second controlling method can stabilize all the periodic orbits we concern. Basin of attraction and noise problem are discussed.     

状态反馈控制声光双稳系统的倍周期分岔和混沌

设计了一种动力学状态反馈(DSF)方法控制非线性混沌系统.介绍了DSF方法的控制原理,并用此方法控制声光双稳(AOB)系统的混沌,以此验证其有效性.仿真模拟显示,通过选择恰当的控制参数,有效地实现了声光双稳(AOB)系统中倍周期分岔的延迟控制和混沌吸引子中原不稳定周期轨道的稳定控制,同时,还可以将系统控制在2np、3mp 和2np×3mp这样其它任意所需的周期轨道上.     

Stochastic resonance in periodic potentials driven by colored noise

We studied the motion of an underdamped Brownian particle in a periodic potential subject to a harmonic excitation and a colored noise. The average input energy per period and the phase lag are calculated to quantify the phenomenon of stochastic resonance (SR). The numerical results show that most of the out-of-phase trajectories make a transition to the in-phase state as the temperature increases. And the colored noise delays the transitions between these two dynamical states. The each curve of the average input energy per period and the phase lag versus the temperature exist a mono peak and SR appears in this system. Moreover, the optimal temperature where the SR occurs becomes larger and the region of SR grows wider as the correlation time of colored noise increases.     

Feedback control of canards

We present a control mechanism for tuning a fast-slow dynamical system undergoing a supercritical Hopf bifurcation to be in the canard regime, the tiny parameter window between small and large periodic behavior. Our control strategy uses continuous feedback control via a slow control variable to cause the system to drift on average toward canard orbits. We apply this to tune the FitzHugh-Nagumo model to produce maximal canard orbits. When the controller is improperly configured, periodic or chaotic mixed-mode oscillations are found. We also investigate the effects of noise on this control mechanism. Finally, we demonstrate that a sensor tuned in this way to operate near the canard regime can detect tiny changes in system parameters.     

Existence and Dynamics of Bounded Traveling Wave Solutions to Getmanou Equation

In this paper, we study the existence and dynamics of bounded traveling wave solutions to Getmanou equations by using the qualitative theory of differential equations and the bifurcation method of dynamical systems. We show that the corresponding traveling wave system is a singular planar dynamical system with two singular straight lines, and obtain the bifurcations of phase portraits of the system under different parameters conditions. Through phase portraits, we show the existence and dynamics of several types of bounded traveling wave solutions including solitary wave solutions, periodic wave solutions, compactons, kink-like and antikink-like wave solutions. Moreover, the expressions of solitary wave solutions are given. Additionally, we confirm abundant dynamical behaviors of the traveling wave s olutions to the equation, which are summarized as follows: i) We confirm that two types of orbits give rise to solitary wave solutions, that is, the homoclinic orbit passing the singular point, and the composed homoclinic orbit which is comprised of two heteroclinic orbits and tangent to the singular line at the singular point of associated system. ii) We confirm that two types of orbits correspond to periodic wave solutions, that is, the periodic orbit surrounding a center, and the homoclinic orbit of associated system, which is tangent to the singular line at the singular point of associated system.     

Sequences of gluing bifurcations in an analog electronic circuit

We report on the experimental investigation of gluing bifurcations in the analog electronic circuit which models a dynamical system of the third order: Lorenz equations with an additional quadratic nonlinearity. Variation of one of the resistances in the circuit changes the coefficient at this nonlinearity and replaces the Lorenz route to chaos by a different scenario which leads, through the sequence of homoclinic bifurcations, from periodic oscillations of the voltage to the irregular ones. Every single bifurcation “glues” in the phase space two stable periodic orbits and creates a new one, with the doubled length: a sequence of such bifurcations results in the birth of the chaotic attractor.     

Noise destroys the coexisting of periodic orbits of a piecewise linear map

The effects of the Gaussian white noise and the Gaussian colored noise on the periodic orbits of period-5 (P-5) and period-6 (P-6) in their coexisting domain of a piecewise linear map are investigated numerically. The probability densities of some orbits are calculated. When the noise intensity is D=0.0001, only the orbits of P-5 exist, and the coexisting phenomenon is destroyed. On the other hand, the self-correlation time τ of the colored noise also affects the coexisting phenomenon. When τ_c<τ<τ_c', only the orbits of P-5 appear, and the stability of the orbits of P-5 is enhanced. However, when τ>τ_c' (τ_c and τ_c' are critical values), only the orbits of P-6 exist, and the stability of the orbits of P-6 is enhanced greatly. When τ<τ_c, the orbits of P-5 and P-6 coexist, but the stability of the orbits of P-5 is enhanced and that of P-6 is weakened with τ increasing.     

Steady-State Analysis of a Single-Mode Laser Driven by Colored Pump Noise with Cross-Correlation Between Real and Imaginary Parts of Quantum Noise

Applying the method of the unified colored noise approximation and phase lock, we study in this paper the stationary intensity distribution of the single-mode laser driven by colored pump noise with cross-correlation between the real and imaginary parts of the quantum noise. We present a thorough discussion of how the cross-correlation λq between the real and imaginary parts of the quantum noise and the self-correlation time τ of the pump noise determine the behaviors of the stationary distribution Qst(I), the mean (I）, and the variance λ2(0) of the laser intensity. It is shown that cross-correlation intensity λq of the complex quantum noise can induce a first-order-like transition. When the pump noise is colored noise (τ≠0), improving the pump parameters monotonously will make the curves of Qst(I) exhibit reentrant phase transition. The fluctuations of laser intensity are strongly influenced by λq and τ when the laser is operated near or below threshold. Especially when τ≠0, the heights of the peaks of the curves of λ2(0)-α0 and α3(0)-α0, (here a0 is the net gain coefficient) go up as λq increases. However the entire curves λ2(0)-α0 and λ3(0)-α0 are abruptly suppressed when λq = 1, in similarity to phase transition of stationary intensity distribution.     

Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics

In this paper we apply dynamical systems techniques to the problem of heteroclinic connections and resonance transitions in the planar circular restricted three-body problem. These related phenomena have been of concern for some time in topics such as the capture of comets and asteroids and with the design of trajectories for space missions such as the Genesis Discovery Mission. The main new technical result in this paper is the numerical demonstration of the existence of a heteroclinic connection between pairs of periodic orbits: one around the libration point L(1) and the other around L(2), with the two periodic orbits having the same energy. This result is applied to the resonance transition problem and to the explicit numerical construction of interesting orbits with prescribed itineraries. The point of view developed in this paper is that the invariant manifold structures associated to L(1) and L(2) as well as the aforementioned heteroclinic connection are fundamental tools that can aid in understanding dynamical channels throughout the solar system as well as transport between the "interior" and "exterior" Hill's regions and other resonant phenomena. (c) 2000 American Institute of Physics.     

THE UNIVERSAL TRANSITION OF PLATEAU STRUCTURE OF LYAPUNOV EXPONENTS

The universal transition of Lyapunov exponents between conservative limit and dissipa-tire limit of nonlinear dynamical system is studied. It is discovered numerically and proved analytically that for homogeneous dissipative two-dimensional maps, along the equal dissi-pation line in parameter space, the Lyapunov exponents of attractor orbits possess a plateau structure and strict symmetry about its plateau value, The ratios between the plateau width and the stable window width of period 1-4 orbits for Henon map are calculated. The result shows that the plateau structure of Lyapunov exponents remains invariant for the attractor orbits belonging to a period doubling bifurcation sequence. This fact reveals a new universal transition behavior between order and chaos when the dissipation of the dynamical system is weakened to zero.     

Correlated noise induced spatiotemporal coherence resonance in a square lattice network

The effects of additive correlated noise, which is composed of common Gaussian white noise and local Gaussian colored noise, on a square lattice network locally modelled by the Rulkov map are studied. We focus on the ability of noise to induce pattern formation in a resonant manner. It is shown that local Gaussian colored noise is able to induce pattern formation, which is more coherent at some noise intensity or correlation time, so it is able to induce spatiotemporal coherence resonance in the network. When common Gaussian white noise is applied in addition, it is seen that the correlated noise can induce coherent spatial structures at some intermediate noise correlation, while this is not the case for smaller and larger noise intensities. The mechanism of the observed spatiotemporal coherence resonance is discussed. It is also found that the correlation time of local colored noise has no evident effect on the optimal value of the noise strength for spatiotemporal coherence resonance in the network.     

A new hyperchaotic system and its linear feedback control

This paper reports a new hyperchaotic system by adding an additional state variable into a three-dimensional chaotic dynamical system, studies some of its basic dynamical properties, such as the hyperchaotic attractor, Lyapunov exponents, bifurcation diagram and the hyperchaotic attractor evolving into periodic, quasi-periodic dynamical behaviours by varying parameter k. Furthermore, effective linear feedback control method is used to suppress hyperchaos to unstable equilibrium, periodic orbits and quasi-periodic orbits. Numerical simulations are presented to show these results.     

Analytic study of chaos of the tent map: Band structures,power spectra,and critical behaviors

Chaotic behaviors of the tent map (a piecewise-linear, continuous map with a unique maximum) are studied analytically throughout its chaotic region in terms of the invariant density and the power spectrum. As the height of the maximum is lowered, successive band-splitting transitions occur in the chaotic region and accumulate to the transition point into the nonchaotic region. The timecorrelation function of nonperiodic orbits and their power spectrum are calculated exactly at the band-splitting points and in the vicinity of these points. The method of eigenvalue problems of the Frobenius-Perron operator is used. 2 ^m?1 critical modes, wherem = 1,2, 3, ..., are found which exhibit the critical slowing-down near the 2 ^m?1-band to 2 ^m -band transition point. After the transition these modes become periodic modes which represent the cycling of nonperiodic orbits among 2 ^m bands together with the periodic modes generated by the preceding band splittings. Scaling laws near the transition point into the nonchaotic region are investigated and a new scaling law is found for the total intensity of the periodic part of the spectrum.     

Stability transformation: a tool to solve nonlinear problems

We present an analysis of the properties as well as the diverse applications and extensions of the method of stabilisation transformation. This method was originally invented to detect unstable periodic orbits in chaotic dynamical systems. Its working principle is to change the stability characteristics of the periodic orbits by applying an appropriate global transformation of the dynamical system. The theoretical foundations and the associated algorithms for the numerical implementation of the method are discussed. This includes a geometrical classification of the periodic orbits according to their behaviour when the stabilisation transformations are applied. Several refinements concerning the implementation of the method in order to increase the numerical efficiency allow the detection of complete sets of unstable periodic orbits in a large class of dynamical systems. The selective detection of unstable periodic orbits according to certain stability properties and the extension of the method to time series are discussed. Unstable periodic orbits in continuous-time dynamical systems are detected via introduction of appropriate Poincaré surfaces of section. Applications are given for a number of examples including the classical Hamiltonian systems of the hydrogen and helium atom, respectively, in electromagnetic fields. The universal potential of the method is demonstrated by extensions to several other nonlinear problems that can be traced back to the detection of fixed points. Examples include the integration of nonlinear partial differential equations and the numerical determination of Markov-partitions of one-parametric maps.     

兴奋性和抑制性自反馈压制靠近Hopf分岔的神经电活动比较

突触输入刺激神经元产生的电活动,在神经编码中发挥着重要作用.通常认为,兴奋性输入增强电活动,抑制性输入压制电活动.本文选取可调节电流衰减速度的突触模型,研究了兴奋性自突触在亚临界Hopf分岔附近压制神经元电活动的反常作用,与抑制性自突触的压制作用进行了比较,并采用相位响应曲线和相平面分析解释了压制作用的机制.对于单稳的峰放电,快速和中速衰减的兴奋性自突触分别可以诱发频率降低的峰放电和混合振荡(峰放电与阈下振荡的交替),而中速和慢速衰减的抑制性自突触也可以分别诱发频率降低的峰放电和混合振荡.对于与静息共存的峰放电,除上述两种行为外,中速衰减的兴奋性和慢速衰减的抑制性自突触还可以诱发静息.兴奋性和抑制性自突触电流在不同的衰减速度下,分别作用在峰放电的不同相位,才能诱发同类压制行为.结果丰富了兴奋性突触压制电活动反常作用的实例,获得了兴奋性和抑制性自突触压制作用机制的不同,给出了调控神经放电的新手段.     

Statistical properties of intermittent diffusion in chaotic systems

Intermittent diffusion arises through tangent bifurcations from drifting periodic orbits in dynamical systems. We show the existence of infinite sequences of parameter values where intermittent diffusion sets in. These sequences are found to converge geometrically and their rate of convergence is determined. In continuous-time approximations we calculate the velocity autocorrelation function, its power spectrum, and the meansquare displacement. The spectrum exhibits excess noise (^–2) at low frequencies. The mean-square displacement grows liket ² below a crossover time.Intermittent diffusion occurs e.g. in driven Josephson junctions, where the excess noise can be detected easily. We show that quantities like the disorder parameter for the transition to intermittent chaos and the diffusion coefficient can be obtained directly from the voltage power spectrum.Work supported by Deutsche Forschungsgemeinschaft