Intermittent Chaos Research Articles

Alfvén intermittency in space plasmas

The onset of Alfvén intermittent chaos in space plasmas is studied by numerically solving the derivative non-linear Schrödinger equation (DNLS) under the assumption of stationary Alfvén waves. We describe how the Alfvénic fluctuations of the magnetic field can evolve from periodic to chaotic behavior through a sequence of bifurcations as the plasma dissipation is varied. The collision of a chaotic attractor with an unstable periodic orbit leads to the generation of strongly chaotic behavior, in an event known as interior crisis. We also show that in the DNLS equation, chaotic attractors coexist with nonattracting chaotic sets responsible for transient chaotic behaviors. After the interior crisis point, a wide chaotic attractor can be decomposed into two coupled nonattracting chaotic sets, resulting in intermittent chaotic time series. Understanding transient chaos is a key to understand intermittency in space plasmas.

Intermittent chaos in nonlinear wave–wave interactions in space plasmas

There is increasing observational evidence of nonlinear wave–wave interactions in space and astrophysical plasmas. We first review a number of theoretical models of nonlinear wave–wave interactions which our group has developed in the past years. We next describe a nonlinear three-mode truncated model of Alfvén waves, involving resonant interactions of one linearly unstable mode and two linearly damped modes. We construct a bifurcation diagram for this three-wave model and investigate the phenomenon of intermittent chaos. The theoretical results presented in this paper can improve our understanding of intermittent time series frequently observed in space and astrophysical plasmas.

Novel co-operative phenomena observed in coupled LCR system

Novel co-operative phenomena in a coupled oscillation system have been demonstrated in both experiments and simulations. The system has 16 LCR nonlinear oscillators as subsystems. In experiments, the system has been constructed on a broadboard by using electrical circuits. The nonlinearity of every subsystem is realised by a varactor. The system takes various phases, such as turbulence, synchronised and partially synchronised phases. Co-operative phenomena takes place in partially synchronised phases. The system shows that a subsystem synchronises with another subsystem placed at a certain spatial distance while all subsystems behave chaotically. Moreover, an intermittent chaos with power law is found in boundary regions between synchronised and partially synchronised phases. This implies that the present intermittent chaos breaks synchronised states similarly to phenomena in globally coupled map and coupled map lattice. In order to verify the experimental results, SPICE simulations are carried out. Statistical analyses make clear that the system forms a spatial wave structure with a specific wavelength when the novel synchronised phenomenon occurs, and the intermittent chaos has close properties with the on-off intermittency.

Dynamic complexities in a mutual interference host–parasitoid model

The effect of parasitoid interference on a host–parasitoid model is investigated qualitatively. Mutual interference of parasitoids is a form of density dependence that is generally thought to have a stabilizing effect. Here we show that mutual interference not only can stabilize the dynamics but may strongly destabilize as well. Many forms of complex dynamics are observed, including pitchfork bifurcation with period-doubling cascade, attractor crises, chaotic bands with narrow or wide periodic windows, multiple attractors with fractal basins of attraction, intermittent chaos, and supertransient behavior. All these phenomena should be kept in mind when examining and interpreting the dynamics of populations.

Breaking a chaos-noise-based secure communication scheme

This paper studies the security of a secure communication scheme based on two discrete-time intermittently chaotic systems synchronized via a common random driving signal. Some security defects of the scheme are revealed: 1) The key space can be remarkably reduced; 2) the decryption is insensitive to the mismatch of the secret key; 3) the key-generation process is insecure against known/chosen-plaintext attacks. The first two defects mean that the scheme is not secure enough against brute-force attacks, and the third one means that an attacker can easily break the cryptosystem by approximately estimating the secret key once he has a chance to access a fragment of the generated keystream. Yet it remains to be clarified if intermittent chaos could be used for designing secure chaotic cryptosystems.

Space plasma dynamics: Alfvén intermittent chaos

The onset of Alfvén intermittent chaos in space plasmas is studied by numerically solving the derivative non-linear Schrödinger equation (DNLS) under the assumption of stationary Alfvén waves. We describe how the Alfvénic fluctuations of the magnetic field can evolve from periodic to chaotic behavior through a sequence of bifurcations as the dissipation is varied. The collision of a chaotic attractor with an unstable periodic orbit leads to the generation of strongly chaotic behavior, in an event known as interior crisis. We also show that in the DNLS equation, chaotic attractors coexist with non-attracting chaotic sets responsible for transient chaotic behavior. After the interior crisis point, a wide chaotic attractor can be decomposed into two coupled non-attracting chaotic sets, resulting in intermittent chaotic time series.

A stochastic digital oscillating system induced by the truncation error

It is generally believed that chaotic systems will collapse due to computer truncation error. In this paper we introduce a contrary example, in which a simple system is induced to complication or chaos completely by computer truncation. The system is defined as a cross-focus system in elliptical reflecting cavity map system. The theoretical solution is a limiting sequence and the computer solution is a stochastic digital oscillating system induced from the theoretical solution by truncation error. There is only a turn-round mechanism in the system theoretically. On the other hand, round-off induces the nonhyperbolic fixed points in the system to produce a new escape mechanism of intermittent chaos.

Chaos weak signal detecting algorithm and its application in the ultrasonic Doppler bloodstream speed measuring

At the present time, the ultrasonic Doppler measuring means has been extensively used in the human body's bloodstream speed measuring. The ultrasonic Doppler measuring means can achieve the measuring of liquid flux by detecting Doppler frequency shift of ultrasonic in the process of liquid spread. However, the detected sound wave is a weak signal that is flooded in the strong noise signal. The traditional measuring method depends on signal-to-noise ratio. Under the very low signal-to-noise ratio or the strong noise signal background, the signal frequency is not measured. This article studied on chaotic movement of Duffing oscillator and intermittent chaotic characteristic on chaotic oscillator of Duffing equation. In the light of the range of the bloodstream speed of human body and the principle of Doppler shift, the paper determines the frequency shift range. An oscillator array including many oscillators is designed according to it. The reflected ultrasonic frequency information can be ascertained accurately by the intermittent chaos quality of the oscillator. The signal-to-noise ratio of −26.5 dB is obtained by the result of the experiment. Compared with the tradition the frequency method compare, the dependence to signal-to-noise ratio is lowered consumedly. The measuring precision of the bloodstream speed is heightened.

Intermittent chaos driven by nonlinear Alfvén waves

Abstract. We investigate the relevance of chaotic saddles and unstable periodic orbits at the onset of intermittent chaos in the phase dynamics of nonlinear Alfvén waves by using the Kuramoto-Sivashinsky (KS) equation as a model for phase dynamics. We focus on the role of nonattracting chaotic solutions of the KS equation, known as chaotic saddles, in the transition from weak chaos to strong chaos via an interior crisis and show how two of these unstable chaotic saddles can interact to produce the plasma intermittency observed in the strongly chaotic regimes. The dynamical systems approach discussed in this work can lead to a better understanding of the mechanisms responsible for the phenomena of intermittency in space plasmas.

Dynamic complexities in a mutual interference host–parasitoid model

The effect of parasitoid interference on a host–parasitoid model is investigated qualitatively. Mutual interference of parasitoids is a form of density dependence that is generally thought to have a stabilizing effect. Here we show that mutual interference not only can stabilize the dynamics but may strongly destabilize as well. Many forms of complex dynamics are observed, including pitchfork bifurcation with period-doubling cascade, attractor crises, chaotic bands with narrow or wide periodic windows, multiple attractors with fractal basins of attraction, intermittent chaos, and supertransient behavior. All these phenomena should be kept in mind when examining and interpreting the dynamics of populations.

Complex dynamics in a permanent-magnet synchronous motor model*1

This paper characterizes the complex dynamics of the permanent-magnet synchronous motor (PMSM) model with a non-smooth-air-gap, extending the work on the smooth case studied elsewhere. The stability, the number of equilibrium points, and the pitchfork and Hopf bifurcations are analyzed by using bifurcation theory and the center manifold theorem. Numerical simulations not only confirm the theoretical analysis results but also show some more new results including the period-doubling bifurcation, cyclic fold bifurcation, single-scroll and double-scroll chaotic attractors, ribbon-chaotic attractor, as well as intermittent chaos that are different from those reported in the literature before. Moreover, analytical expressions of an approximate stability boundary are given, by computing the local quadratic approximation of the two-dimensional stable manifold at an order-2 saddle point. Combining the existing results with the new results reported in this paper, a fairly complete description of the complex dynamics of the PMSM model is now obtained.

Logarithmic periodicities in the bifurcations of type-I intermittent chaos.

The critical relations for statistical properties on saddle-node bifurcations are shown to display undulating fine structure, in addition to their known smooth dependence on the control parameter. A piecewise linear map with the type-I intermittency is studied and a log-periodic dependence is numerically obtained for the average time between laminar events, the Lyapunov exponent, and attractor moments. The origin of the oscillations is built in the natural probabilistic measure of the map and can be traced back to the existence of logarithmically distributed discrete values of the control parameter giving Markov partition. Reinjection and noise effect dependences are discussed and indications are given on how the oscillations are potentially applicable to complement predictions made with the usual critical exponents, taken from data in critical phenomena.

Study of intermittent bifurcations and chaos in boost PFC converters by nonlinear discrete models☆

Abstract This paper mainly deals with nonlinear phenomena like intermittent bifurcations and chaos in boost PFC converters under peak-current control mode. Two nonlinear models in the form of discrete maps are derived to describe precisely the nonlinear dynamics of boost PFC converters from two points of view, i.e., low- and high-frequency regimes. Based on the presented discrete models, both the evolution of intermittent behavior and the periodicity of intermittency are investigated in detail from the fast and slow-scale aspects, respectively. Numerical results show that the occurrence of intermittent bifurcations and chaos with half one line period is one of the most distinguished dynamical characteristics. Finally, we make some instructive conclusions, which prove to be helpful in improving the performances of practical circuits.

Complex dynamics in a permanent-magnet synchronous motor model☆

Abstract This paper characterizes the complex dynamics of the permanent-magnet synchronous motor (PMSM) model with a non-smooth-air-gap, extending the work on the smooth case studied elsewhere. The stability, the number of equilibrium points, and the pitchfork and Hopf bifurcations are analyzed by using bifurcation theory and the center manifold theorem. Numerical simulations not only confirm the theoretical analysis results but also show some more new results including the period-doubling bifurcation, cyclic fold bifurcation, single-scroll and double-scroll chaotic attractors, ribbon-chaotic attractor, as well as intermittent chaos that are different from those reported in the literature before. Moreover, analytical expressions of an approximate stability boundary are given, by computing the local quadratic approximation of the two-dimensional stable manifold at an order-2 saddle point. Combining the existing results with the new results reported in this paper, a fairly complete description of the complex dynamics of the PMSM model is now obtained.

Search of Many Good Solutions of QAP by Connected Hopfield NNs with Intermittency Chaos Noise

The method of adding chaos noise to the Hopfield neural network (NN) in application to combinatorial optimization problems has been proposed, and many researchers have suggested that adding intermittency chaos noise near the three-periodic window of the logistic map will yield the best performance. However, the Hopfield NN with chaos noise sometimes remains in a group of several solutions. In this study, some Hopfield NNs with intermittency chaos noise are connected like hierarchical networks in order to find good solutions of quadratic assignment problems (QAPs). It is confirmed that connected Hopfield NNs with intermittency chaos noise provide numerous nearly optimal solutions.

APPLYING RESONANT PARAMETRIC PERTURBATION TO CONTROL CHAOS IN THE BUCK DC/DC CONVERTER WITH PHASE SHIFT AND FREQUENCY MISMATCH CONSIDERATIONS

The buck converter has been known to exhibit chaotic behavior in a wide parameter range. In this paper, the resonant parametric perturbation method is applied to control chaos in a voltage-mode controlled buck converter. In particular, the effects of phase shift and frequency mismatch in the perturbing signal are studied. It is shown that the control power can be significantly reduced if the perturbation is applied with an appropriate phase shift. Moreover, when frequency mismatch is inevitable, intermittent chaos occurs, but effective control can still be accomplished at the expense of raising the control power. Analysis, simulations and experimental measurements are presented to provide theoretical and practical evidences for the proposed control method.

Intermittent chaos in a model of financial markets with heterogeneous agents

In this paper we study the price dynamics in a simple model of financial markets with heterogeneous-agents. We concentrate on how increase in the total number of active traders influences on fluctuations of asset prices. We find that a curious route to chaos is observed when the total number of active traders who participate into trading increases. Particularly we show that intermittent chaos [Berge et al., Order within Chaos, John Wiley and Sons, New York, 1984] of price fluctuations is observed as the total number of trader increases.

Explanation of appearance and characteristics of intermittency chaos of the impact oscillator

Chaotic motion of an intermittency type of the impact oscillator appears near segments of saddle-node stability boundaries of subharmonic motions with two different impacts in motion period, which is n multiple ( n⩾3) of excitation period. Chaotic motion arises due to an additional impact, which interrupts the process of instability. It is proved and shown by numerical simulations of the system motion. More detail characteristics of the intermittency chaos are evaluated. Described phenomena present a non-usual example, when transition cross special segments of saddle-node stability boundaries of subharmonic impact motions is reversible.

Spatial and Temporal Stabilities of Flow in a Natural Circulation Loop: Influences of Thermal Boundary Condition

In a natural circular loop, the thermal convection demonstrates various spatial patterns and temporal instabilities. Problem consists in determining them with respects to thermal boundary conditions. To this end a multiple scales analysis is applied which resembles the inherent characteristic of the pattern formation in the Rayleigh-Be´nard convection. A three-dimensional nonlinear model is proposed by incorporating the flow modes derived along the analysis. The differences of thermal boundary condition are reflected by a coefficient δ. For small δ, numerical solution to the model shows that only temporal instability exists and Lorenz chaos is possible, otherwise, for large values both spatial and temporal instabilities occur leading to cellular flow and intermittency chaos. The model predicted some additional phenomena opening for experimental observation. It seems significant that this study proposes an algorithm for the control of flow stability and distribution by varying the thermal boundary condition.

Ensemble dynamics of intermittency and power-law decay

A spectral decomposition of the Frobenius–Perron operator is constructed for one-dimensional maps with intermittent chaos, using the method of coherent states. A technique using the spectral density function is applied to the the well-known cusp map, which generates weak type-II intermittency. Higher-order corrections are obtained to the leading 1/t long-time behavior of the x−x autocorrelation.