On the realization of the Hunt-Ott strange nonchaotic attractor in a physical system

The object of investigation is a system consisting of two coupled nonautonomous van der Pol oscillators the characteristics frequencies of which differ by a factor of 2. The system is subjected to an external action in the form of slow periodic modulation of an oscillation-controlling parameter and also to an additional action at a frequency that is in an irrational relation with the modulation frequency. It is shown that the variation of the oscillation phase over a modulation period can be approximated by a 2D map on a torus that has a robust (structurally stable) Hunt-Ott strange nonchaotic attractor. Calculations of the quantitative characteristics of the attractor corresponding to the initial set of nonautonomous coupled oscillators (such as phase sensitivity exponent, structures and scaling of rational approximations, as well as Lyapunov exponents and their parameter dependence) confirm the presence of the Hunt-Ott strange nonchaotic attractor.     

Strange Nonchaotic Attractors in a Time-Delay System

A nonchaotic attractor is observed in an infinite-dimensional system which is related to optical bistability and described by a nonlinear time-delay differential equation. The observed nonchaotic attractor is characterized by the strange trajectory of attractor but with negative value for the largest Lyapunov exponent, as well as the Fourier power spectra.     

Conversion of a chaotic attractor into a strange nonchaotic attractor in an one dimensional map and BVP oscillator

In this paper we investigate numerically the possibility of conversion of a chaotic attractor into a nonchaotic but strange attractor in both a discrete system (an one dimensional map) and in a continuous dynamical system — Bonhoeffer—van der Pol oscillator. In these systems we show suppression of chaotic property, namely, the sensitive dependence on initial states, by adding appropriate i) chaotic signal and ii) Gaussian white noise. The controlled orbit is found to be strange but nonchaotic with largest Lyapunov exponent negative and noninteger correlation dimension. Return map and power spectrum are also used to characterize the strange nonchaotic attractor.     

准周期外力驱动下Lorenz系统的动力学行为

本文研究了准周期外力驱动下Lorenz系统的动力学行为，发现当外强迫的振幅达到某一个临界值时，系统的动力学行为将会发生根本性的变化，由此揭示了产生非混沌奇怪吸引子(Strange Nonchaotic Attractor, SNA)的一个新机制：准周期外强迫振幅的加大导致系统由奇怪的混沌吸引子转变为SNA，系统的相空间最终被压缩至一个准周期环上.并且本文的结果表明，外强迫的临界振幅与Lorenz系统Rayleigh数的大小成正比，而其受外强迫频率变化的影响并不大. 关键词： 准周期 Lorenz系统 非混沌奇怪吸引子     

基于收缩映射的奇异非混沌系统同步

提出一种基于收缩映射的奇异非混沌系统同步方案．通过利用一种混沌系统驱动另一种混沌系统产生出奇异非混沌吸引子,由于奇异非混沌吸引子的Ｌｙａｐｕｎｏｖ指数为负值,因而可有效抑制混沌系统对初始状态的敏感程度．为实现两个奇异非混沌吸引子的同步,文中采用收缩映射实现混沌驱动系统的快速同步．研究表明,该方案能够快速实现同步,并且有较强的鲁棒性,易于实现,可用于混沌保密通信 关键词：     

Nonchaotic random behaviour in the second order autonomous system

Evidence is presented for the nonchaotic random behaviour in a second-order autonomous deterministic system. This behaviour is different from chaos and strange nonchaotic attractor. The nonchaotic random behaviour is very sensitive to the initial conditions. Slight difference of the initial conditions will generate wholly different phase trajectories. This random behaviour has a transient random nature and is very similar to the coin-throwing case in the classical theory of probability. The existence of the nonchaotic random behaviour not only can be derived from the theoretical analysis, but also is proved by the results of the simulated experiments in this paper.     

Brain stem neuronal noise and neocortical “resonance”

We present a qualitative model and data in evidence for the selection and stabilization of neocortical brain-wave power spectral modes by slow periodic and fast noise driving by brain stem neurons. Unlike noise effects in a bistable potential, increasing noise amplitude via more brain stem neurons increases the measure on unstable manifolds trapped in the saddle-sinks of the neural membrane attractor andincreases dwell times. We suggest that the effect of noise in expanding dynamical systems such as the generalized neuronal membrane equations studied here may be analogous to that of many-frequency quasiperiodic driving which leads to the stabilization of the EEG as a strange, nonchaotic attractor.     

Intermittency and strange nonchaotic attractors in quasi-periodically forced circle maps

A possible mechanism for the creation of strange nonchaotic attractors close to the boundary of mode-locked tongues in a family of maps of the torus is described. This mechanism is based on the numerical observation that there are parameter values on the boundary of the mode-locked tongues at which the saddlenode bifurcation of invariant curves is not smooth, and assumptions about the nature of intermittency just outside the mode-locked tongues.     

A special type of attractor and transitions in a delayed system

We discuss strange nonchaotic attractors (SNAs) in addition to chaotic and regular attractors in a quasiperiodically driven system with time delays. A route and the associated mechanism are described for a special type of attractor called strange-nonchaotic-attractor-like (SNA-like) through T² torus bifurcation. The type of attractor can be observed in large parameter domains and it is easily mistaken for a true SNA judging merely from the phase portrait, power spectrum and the largest Lyapunov exponent. SNA-like attractor is not strange and has no phase sensitivity. Conditions for Neimark-Sacker bifurcation are obtained by theoretical analysis for the unforced system. Complicated and interesting dynamical transitions are investigated among the different tongues.     

非周期力在混沌控制中的双重功效

探讨了非周期力（有界噪声或混沌驱动力）在非线性动力系统混沌控制中的影响.以一类典型的含有五次非线性项的Duffing-van der Pol系统为范例,通过对系统的轨道、最大Lyapunov指数、功率谱幅值及Poincar截面的分析,发现适当幅值的有界噪声或混沌信号,一方面可以消除系统对初始条件的敏感依赖性,抑制系统的混沌行为,将系统的混沌吸引子转化为奇怪非混沌吸引子;另一方面也可以诱导系统的混沌行为,将系统的周期吸引子转化为混沌吸引子.从而揭示了非周期力在混沌控制中的双重功效：抑制混沌和诱导混沌. 关键词： 混沌控制 有界噪声 混沌驱动力     

Frequency locking by external force from a dynamical system with strange nonchaotic attractor

Usually, phase synchronization is studied in chaotic systems driven by either periodic force or chaotic force. In the present work, we consider frequency locking in chaotic Rössler oscillator by a special driving force from a dynamical system with a strange nonchaotic attractor. In this case, a transition from generalized marginal synchronization to frequency locking is observed. We investigate the bifurcation of the dynamical system and explain why generalized marginal synchronization can occur in this model.     

Strange—Nonchaotic—Attractor—Like Behaviors in Coupled Map SYstems

In a coupled map system,an attractor which seems to be strange nonchaotic attractor(SNA)is discovered for nonzero measure in parameter range,The attractor has nonpositive Lyapunov exponent(LE) and discrete structure.We call it strange-nonchaotic-attractor-like(SNA-like) behavior because the size of its size of its discrete structure decreases with the computing precision increasing and the true SNA does not change.The SNA-like behavior in the autonomous system is born when the truncation error of round-off is amplified to the size of the discrete part of the attractor during the long time interval of positive local LE.The SNA-like behavior is easily mistaken for a true SNA judging merely from the largest LE and the phase portrait in double precision computing.In non-autonomous system an SNA-like attractor is also found.     

双驱动布鲁塞尔振子的一些运动性质

以强迫布鲁塞尔振子方程为例,研究了外加第二驱动对振子运动性质的影响:某些有理频率比的第二驱动可以抑制方程的混沌运动,得到周期轨道;而某些无理频率比的外驱动则可以改变方程的运动性质,造成非混沌的奇怪吸引子。     

Nonlinear parametric effects and dynamic chaos in a nonautonomous oscillating system with a nonlinear capacitance

A mathematical model is constructed of a nonautonomous dynamic system containing a nonlinear capacitance and possessing a four-dimensional phase space. A numerical investigation is performed of branching processes and phenomena accompanying variations in the frequency and amplitude of an external force. The existence of complex dynamic processes that are a combination of a nonlinear force resonance and a parametric resonance is demonstrated. It is found that both a strange chaotic and a strange nonchaotic attractor exist in the phase space. It is shown that, in the case of a single-frequency external force, the latter attractor exhibits the property of roughness. The results of numerical calculations are confirmed by the results of laboratory experiments.     

Strange nonchaotic attractors in Harper maps

We study the existence of strange nonchaotic attractors (SNA) in the family of Harper maps. We prove that for a set of parameters of positive measure, the map possesses a SNA. However, the set is nowhere dense. By changing the parameter arbitrarily small amounts, the attractor is a smooth curve and not a SNA.     

Critical curves and coexisting attractors in a quasiperiodically forced delayed system

We study three critical curves in a quasiperiodically driven system with time delays, where occurrence of symmetry-breaking and symmetry-recovering phenomena can be observed. Typical dynamical tongues involving strange nonchaotic attractors (SNAs) can be distinguished. A striking phenomenon that can be discovered is multistability and coexisting attractors in some tongues surrounding by critical curves. The blowout bifurcation accompanying with on-off intermittency can also be observed. We show that collision of attractors at a symmetric invariant subspace can lead to the appearance of symmetry-breaking.     

Unpredictability of the Wada property in the parameter plane

We show a type of unpredictability of the Wada property in the parameter plane for fixed initial conditions. This property indicates a larger unpredictability of sensitive dependence on parameters except for the riddled parameter sets. We describe some numerical experiments giving evidences of the parameter Wada property for different types of attractors including strange nonchaotic attractors. A scaling exponent is used to characterize sensitive dependence on parameters. We present a qualitative explanation on the occurrence of the Wada property in the parameter plane.     

Strange Nonchaotic Attractor in a Nonautonomous Oscillatory System with Nonlinear Capacity

The mathematical model of a nonautonomous dynamic system with nonlinear capacity and four-dimensional phase space is numerically investigated. The behavior of the maximum characteristic Lyapunov parameter and attractor capacity, information, and correlation dimensions attendant to variations of the external force frequency is investigated. The existence of the strange nonchaotic attractor that retains the roughness property for a wide range of variations of the system parameters is established in the phase space. Results of field experiments that confirm the correctness of numerical calculations are presented.__________Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 89–93, March, 2005.     

Recurrences of strange attractors

The transitions from or to strange nonchaotic attractors are investigated by recurrence plot-based methods. The techniques used here take into account the recurrence times and the fact that trajectories on strange nonchaotic attractors (SNAs) synchronize. The performance of these techniques is shown for the Heagy-Hammel transition to SNAs and for the fractalization transition to SNAs for which other usual nonlinear analysis tools are not successful.      

Experimental observation of strange nonchaotic attractors in a driven excitable system

We report the observation of strange nonchaotic attractors in an electrochemical cell. The system parameters were chosen such that the system observable (anodic current) exhibits fixed point behavior or period one oscillations. These autonomous dynamics were thereafter subjected to external quasiperiodic forcing. Systematically varying the characteristics (frequency and amplitude) of the superimposed external signal; quasiperiodic, chaotic and strange nonchaotic behaviors in the anodic current were generated. The inception of strange nonchaotic attractors was verified using standard diagnostic techniques.