Intermittent Chaos Research Articles

Study on Nonlinear Phenomena in Buck-Boost Converter with Switched-Inductor Structure

The switched-inductor structure can be inserted into a traditional Buck-Boost converter to get a high voltage conversion ratio. Nonlinear phenomena may occur in this new converter, which might well lead the system to be unstable. In this paper, a discrete iterated mapping model is established when the new Buck-Boost converter is working at continuous conduction current-controlled mode. On the basis of the discrete model, the bifurcation diagrams and Poincare sections are drawn and then used to analyze the effects of the circuit parameters on the performances. It can be seen clearly that various kinds of nonlinear phenomena are easy to occur in this new converter, including period-doubling bifurcation, border collision bifurcation, tangent bifurcation, and intermittent chaos. Value range of the circuit parameters that may cause bifurcations and chaos are also discussed. Finally, the time-domain waveforms, phase portraits, and power spectrum are obtained by using Matlab/Simulink, which validates the theoretical analysis results.

Study on weak signal detection method with Duffing oscillators

By numerical simulations we find that the Duffing state transition from big-cycle motion to chaotic motion is more suited to weak signal detection. Based on this, in this paper, we propose a new weak periodic signal detection method from the angle of theoretical analysis. The principle of the proposed method is introduced and its feasibility is analyzed. Then, the proposed method and the traditional method are compared and analyzed both in transition zone effect and detection probability, and then a contrast simulation is carried out. The analysis and simulation results both indicate, the Duffing oscillator state transition from big-cycle motion to chaotic motion is affected less by transition zone under the same condition, based on which the proposed detection method possesses better detection capability. The simulation data also show that weak signal detection with Duffing oscillator must be based on unilateral state transition. The frequency difference detection based on intermittent chaos is suited only for the case where the signal-to-noise ratio of the detected signal is relatively high.

CHAOTIC REGIMES, POST-BIFURCATION DYNAMICS, AND COMPETITIVE MODES FOR A GENERALIZED DOUBLE HOPF NORMAL FORM

We discuss the post-bifurcation dynamics of the general double Hopf normal form, which allows us to study two intermittent routes to chaos (routes following either (i) subcritical or (ii) supercritical Hopf or double Hopf bifurcations). In particular, the route following supercritical bifurcations is somewhat subtle. Such behavior following repeated Hopf bifurcations is well-known and widely observed, including the classical Ruelle–Takens and quasiperiodic routes to chaos. However, it has not, to the best of our knowledge, been considered in the context of the double Hopf normal form, although it has been numerically observed and tracked in the post-double-Hopf regime. We then apply the method of competitive modes to verify parameter regimes for which the double Hopf normal form exhibits chaotic behavior. Such an analysis is useful, as it allows us to potentially identify specific parameter regimes for which the system may exhibit strange or irregular behavior, something which would be extremely difficult otherwise in a system with so many parameters. Indeed, it is conjectured that for parameter sets with two of the square-mode frequencies competitive or nearly competitive, chaotic behavior is likely to be observed in the system. We apply the method of competitive modes to two representative cases where intermittent chaos is found, and the competitive mode analysis there seems to verify the occurrence of chaos in each of the two types of regimes.

Complex mixed-mode oscillations in a Bonhoeffer–van der Pol oscillator under weak periodic perturbation

In this paper, we elucidate the extremely complicated bifurcation structure of a weakly driven relaxation oscillator by focusing on chaos, and notably, on complex mixed-mode oscillations (MMOs) generated in a simple dynamical model. Our model uses the Bonhoeffer–van der Pol (BVP) oscillator subjected to a weak periodic perturbation near a subcritical Andronov–Hopf bifurcation (AHB). The mechanisms underlying the chaotic dynamics can be explained using an approximate one-dimensional map. The MMOs that appear in this forced dynamical model may be more sophisticated than those appearing in three-variable slow–fast autonomous dynamics because the approximate one-dimensional mapping of the dynamics used herein is a circle map, whereas the one-dimensional first-return map that is derived from the three-variable slow–fast autonomous dynamics is usually a unimodal map. In this study, we generate novel bifurcations such as an MMO-incrementing bifurcation (MMOIB) and intermittently chaotic MMOs. MMOIBs trigger an MMO sequence that, upon varying a parameter, is followed by another type of MMO sequence. By constructing a two-parameter bifurcation diagram, we confirmed that MMOIBs occur successively. According to our numerical results, MMOIBs are often observed between two neighboring MMOs. Numerically, MMOIBs may occur as many times as desired. We also derive the universal constant of the associated successive MMOIBs. The existence of the universal constant suggests that MMOIBs could occur infinitely many times. Furthermore, intermittently chaotic MMOs appear in this dynamical circuit. The intermittently chaotic MMOs relate to a type of intermittent chaos that resembles MMOs at first glance, but includes rare bursts over a long time interval. Complex intermittently chaotic MMOs of various types are observed, and we clarify that the intermittently chaotic MMOs are generated by crisis-induced intermittency.

Weak Signal Detection Based on the Nonlinear Dynamic Model

In order to obtain new ways of weak signal detection, we analyzed the motion states of nonlinear dynamic model Duffing oscillator in the case of different amplitude of input signals by solving the Duffing equation and expounded the basic principles of weak signal detection based on Duffing oscillator phase-change characteristics, further illustrated the relationship between signal detection accuracy and detection time by the experimental, researched the impact on signal detection coming from Gaussian white noise and also pointed out how to use intermittent chaos state to implement weak signal detection. The results showed that Duffing oscillator can be effectively detect the slight changes of input signal in the strong noises background, so as to achieve the purpose of weak signal detection. Compared with existing methods, it could greatly improve the detection results.

Dynamics of current controlled switching converters under wide circuit parameter variation

With current-controlled buck-boost converter used as an example, through a detailed description of the switch state of the switching converter under wide circuit parameter variation, such as input voltage and load resistance variation, two inductor current borders in the current controlled switching converter are derived and an accurate discrete-time model is established. The validation of the discrete-time model is verified by a piecewise-linear model. Based on the discrete-time model, the complex dynamical behaviors existing in switching converter, such as period-double bifurcation, border-collision bifurcation, robust chaos and intermittent chaos, etc., are revealed. By formulating the Jacobian, the maximum Lyapunov exponent and the movement trajectories of eigvalues with the variations of circuit parameters are obtained. By utilizing the parameter-space maps, the operation-state regions corresponding to circuit parameter regions are estimated. Finally, an experimental setup is implemented, the corresponding observation results are consistent with those of theory analyses. In this paper the dynamics theory in switching converters is investigated systematically; the analysis methods and research results are helpful for designing and controlling switching converters.

Nonlinear dynamics of 1550 nm vertical-cavity surface-emitting laser with polarization- preserved optical feedback

Based on the framework of the spin-flip model, the nonlinear dynamics of 1550 nm vertical-cavity surface-emitting laser (VCSEL) subject to polarization-preserved optical feedback is theoretically investigated. The results show that for a free-running 1550 nm-VCSEL, the current value for polarization switching (PS) is affected seriously by the internal parameters; with the increase of the gain anisotropy coefficient γa, the corresponding current for PS increases. Due to the introduction of the polarization-preserved optical feedback, the dominant polarized mode (Y polarized mode) will display different dynamical states for different injection currents while the other mode (X polarized mode) may be excited. As a result, the average output powers of the two polarized modes increase with fluctuation with the increase of current. For different feedback times, the dynamic states of 1550 nm-VCSEL with polarization- preserved optical feedback maybe route into chaos via different evolution paths such as period doubling bifurcation, quasi-periodic bifurcation and intermittent chaos.

Multiscale Analysis of Biological Data by Scale-Dependent Lyapunov Exponent

Physiological signals often are highly non-stationary (i.e., mean and variance change with time) and multiscaled (i.e., dependent on the spatial or temporal interval lengths). They may exhibit different behaviors, such as non-linearity, sensitive dependence on small disturbances, long memory, and extreme variations. Such data have been accumulating in all areas of health sciences and rapid analysis can serve quality testing, physician assessment, and patient diagnosis. To support patient care, it is very desirable to characterize the different signal behaviors on a wide range of scales simultaneously. The Scale-Dependent Lyapunov Exponent (SDLE) is capable of such a fundamental task. In particular, SDLE can readily characterize all known types of signal data, including deterministic chaos, noisy chaos, random 1/fα processes, stochastic limit cycles, among others. SDLE also has some unique capabilities that are not shared by other methods, such as detecting fractal structures from non-stationary data and detecting intermittent chaos. In this article, we describe SDLE in such a way that it can be readily understood and implemented by non-mathematically oriented researchers, develop a SDLE-based consistent, unifying theory for the multiscale analysis, and demonstrate the power of SDLE on analysis of heart-rate variability (HRV) data to detect congestive heart failure and analysis of electroencephalography (EEG) data to detect seizures.

Investigation of tangent bifurcation in voltage mode controlled DCM boost converters

The voltage mode controlled boost converters contain plenty of bifurcation phenomena. Tangent bifurcation is one of these phenomena. The investigation of the tangent bifurcation in voltage mode controlled boost converters operating in discontinuous conduction mode (DCM) is performed. The third iterative map is derived according to the first iterative map. Based on the tangent bifurcation theorem, the conditions of producing the tangential bifurcation in the DCM boost converters are deduced mathematically. The computer simulations are performed to capture the effects of some chosen parameters on the tangent bifurcation behavior of the system. The results show that the variation of the feedback factor leads to tangent bifurcation and intermittent chaos phenomena. Experimental results show that the system does exhibit tangent bifurcation under the particular operating conditions, thus validating the theoretical analysis and the simulation results.

Entropy measures for biological signal analyses

Entropies are among the most popular and promising complexity measures for biological signal analyses. Various types of entropy measures exist, including Shannon entropy, Kolmogorov entropy, approximate entropy (ApEn), sample entropy (SampEn), multiscale entropy (MSE), and so on. A fundamental question is which entropy should be chosen for a specific biological application. To solve this issue, we focus on scaling laws of different entropy measures and introduce an ensemble forecasting framework to find the connections among them. One critical component of the ensemble forecasting framework is the scale-dependent Lyapunov exponent (SDLE), whose scaling behavior is found to be the richest among all the entropy measures. In fact, SDLE contains all the essential information of other entropy measures, and can act as a unifying multiscale complexity measure. Furthermore, SDLE has a unique scale separation property to aptly deal with nonstationarity and characterize high-dimensional and intermittent chaos. Therefore, SDLE can often be the first choice for exploratory studies in biology. The effectiveness of SDLE and the ensemble forecasting framework is illustrated by considering epileptic seizure detection from EEG.

Unified Classification of Operation-State Regions for Switching Converters with Ramp Compensation

With the variation of circuit parameters, the operation-state regions of current-mode-controlled switching dc-dc converters can shift among stable period region, robust chaos region in continuous conduction mode, and intermittent chaos region in discontinuous conduction mode. This paper presents a unified approach to the operation-states analysis of switching dc-dc converters with ramp compensation. The piecewise map model of the converters is first derived. With the bifurcation analyses, two boundary classification equations of the orbit state shifting are then obtained. Finally, the operation-state regions are well classified. To verify the theoretical analysis results, 2-D bifurcation diagrams are simulated and experimentations with current-mode-controlled buck converter are conducted. It is revealed that regular and irregular (chaotic or intermittent) operation states can be generated depending on circuit parameters or control of ramp-compensation current.

Research of Noise Induced Intermittent Chaos in Power Converter

This paper analyzes the working process of current-mode controlled Boost converter and finds that the system will produce the physical phenomenon of intermittency chaos when the converter works in the period windows of chaos zone and external noise reaches certain intensity. It can observe the phenomenon of noise-induced intermittent chaos and it also analyzes the relationship between noise intensity and induced intermittent chaos, the relationship between the noise intensity threshold and circuit parameters in resulting intermittent chaos from the perspective of numerical simulation. The research has an important theoretical and practical significance to nonlinear science and it can provide a reliable theoretical reference for the stable design of switching power converters.

电光双稳态系统的混沌特性分析

The stability of the electrical-optical bistable system is theoretically analyzed and the stability of the fixed points is researched by diagrammatized mode.Through the method of numerical caculation,the concrete positions of the bifurcation points are caculated,the ways of chaos generation are studied which are intermittent chaos and period doubling bifurcations.Bifurcation diagram of the system state evolving with parameter and largest Lyapunov exponent changing with parameter of the system all show that the analysis is reasonable in theory.

GLOBAL LOW DIMENSIONAL SEISMIC CHAOS IN THE HELLENIC REGION

In this study, we present results concerning seismogenesis in the Hellenic region (land and sea of Greece), applying nonlinear analysis to an earthquake time series. The model of the dripping faucet is used as a physical interpretation of the seismic process and the construction of inter-event seismic time series. Geometrical and dynamical characteristics estimated in the reconstructed state space support the low dimensional, chaotic character of the global seismic process in the Hellenic region. The method of stochastic surrogate data was employed to the exclusion of "pseudo chaos" caused by the nonlinear distortion of a purely stochastic process. These results are in agreement with general theoretical models concerning distributed driven threshold dynamics applied to the case of seismic processes. Moreover, the observed global character of low dimensionality and chaoticity over such a complex system of faults supports the hypothesis that seismogenesis is characterized by spatiotemporal intermittent chaos throughout the Hellenic region.

Global bifurcations in externally excited autoparametric systems

We consider an autoparametric system consisting of an oscillator coupled with an externally excited subsystem. The oscillator and the subsystem are in one-to-one internal resonance. The excited subsystem is in primary resonance. The method of second-order averaging is used to obtain a set of autonomous equations of the second-order approximations to the externally excited system with autoparametric resonance. The Šhilnikov-type homoclinic orbits and chaotic dynamics of the averaged equations are studied in detail. The global bifurcation analysis indicates that there exist the heteroclinic bifurcations and the Šhilnikov-type homoclinic orbits in the averaged equations. The results obtained above mean the existence of the amplitude-modulated chaos for the Smale horseshoe sense in the externally excited system with autoparametric resonance. Furthermore, a detailed bifurcation analysis of the dynamic (periodic and chaotic) solutions of the averaged equations is presented. Nine branches of dynamic solutions are found. Two of these branches emerge from two Hopf bifurcations and the other seven are isolated. The limit cycles undergo symmetry-breaking, cyclic-fold and period-doubling bifurcations, whereas the chaotic attractors undergo attractor-merging and boundary crises. Simultaneous occurrence of the limit cycle and chaotic attractors, homoclinic orbits, intermittency chaos and homoclinic explosions are also observed.

Complex nonlinear dynamics and controlling chaos in a Cournot duopoly economic model

Complex nonlinear economic dynamics in a Cournot duopoly model proposed by M. Kopel is studied in detail in this work. By utilizing the topological horseshoe theory proposed by Yang XS, the authors detect the topological horseshoe chaotic dynamics in the Cournot duopoly model for the first time, and also give the rigorous computer-assisted verification for the existence of horseshoe. In the process of the proof, the topological entropy of the Cournot duopoly model is estimated to be bigger than zero, which implies that this economic system definitely exhibits chaos. In particular, the authors observe two different types of economic intermittencies, including the Pomeau–Manneville Type-I intermittency arising near a saddle-node bifurcation, and the crisis-induced attractor widening intermittency caused by the interior crisis, which lead to the appearance of intermittency chaos. The authors also observe the transient chaos phenomenon which leads to the destruction of chaotic attractors. All these intermittency phenomena will help us to understand the similar dynamics observed in the practical stock market and the foreign exchange market. Besides, the Nash-equilibrium profits and the chaotic long-run average profits are analyzed. It is numerically demonstrated that both firms can have higher profits than the Nash-equilibrium profits, that is to say, both of the duopolists could be beneficial from a chaotic market. The controlled Cournot duopoly model can make one firm get more profit and reduce the profit of the other firm, and control the system to converge to an equilibrious state, where the two duopolists share the market equally.

Determination of the parameters of unknown signals based on intermittent chaos

We use the extreme frequency sensitivity of Duffing’s equation to produce intermittent chaos（we call it “breather”） and propose a new method to quantitatively detect the parameters of unknown weak periodic signals. The theoretical analysis and instance simulation have proved its feasibility. We also put forward a way to improve the detection results and enhance the accuracy.

Novel Ultrasonic Dispersion of Carbon Nanotubes

A double ultrasonic source has been shown to dramatically increase dispersion efficiency of carbon nanotubes. Thermal measurements of dispersing fluid only show temperature rises commensurate with the power levels of the two ultrasonic sources; which is validated by predictions of statistical energy analysis (SEA) based on wave superposition principles. In this paper, nonlinear wave resonance concepts have been proposed to contain explanations for the dramatic increase in dispersion performance, and more specifically, the effect of intermittency chaos. Such a hypothesis was made because of the similarity between the pressure wave pattern in the double sonication system and sliding charge density wave with an A.C. electric field, which was cited to exhibit intermittency behavior.

OBSERVABILITY OF CHAOS AND CYCLES IN ECOLOGICAL SYSTEMS: LESSONS FROM PREDATOR–PREY MODELS

We examine and assess deterministic chaos as an observable. First, we present the development of model ecological systems. We illustrate how to apply the Kolmogorov theorem to obtain limits on the parameters in the system, which assure the existence of either stable equilibrium point or stable limit cycle behavior in the phase space of two-dimensional (2D) dynamical systems. We also illustrate the method of deriving conditions using the linear stability analysis. We apply these procedures on some basic existing model ecological systems. Then, we propose four model ecological systems to study the dynamical chaos (chaos and intermittent chaos) and cycles. Dynamics of two predation and two competition models have been explored. The predation models have been designed by linking two predator–prey communities, which differ from one another in one essential way: the predator in the first is specialist and that in the second is generalist. The two competition models pertain to two distinct competition processes: interference and exploitative competition. The first competition model was designed by linking two predator–prey communities through inter-specific competition. The other competition model assumes that a cycling predator–prey community is successfully invaded by a predator with linear functional response and coexists with the community as a result of differences in the functional responses of the two predators. The main criterion behind the selection of these two model systems for the present study was that they represent diversity of ecological interactions in the real world in a manner which preserves mathematical tractability. For investigating the dynamic behavior of the model systems, the following tools are used: (i) calculation of the basin boundary structures, (ii) performing two-dimensional parameter scans using two of the parameters in the system as base variables, (iii) drawing the bifurcation diagrams, and (iv) performing time series analysis and drawing the phase space diagrams. The results of numerical simulation are used to distinguish between chaotic and cyclic behaviors of the systems.The conclusion that we obtain from the first two model systems (predation models) is that it would be difficult to capture chaos in the wild because ecological systems appear to change their attractors in response to changes in the system parameters quite frequently. The detection of chaos in the real data does not seem to be a possibility as what is present in ecological systems is not robust chaos but short-term recurrent chaos. The first competition model (interference competition) shares this conclusion with those of predation ones. The model with exploitative competition suggests that deterministic chaos may be robust in certain systems, but it would not be observed as the constituent populations frequently execute excursions to extinction-sized densities. Thus, no matter how good the data characteristics and analysis techniques are, dynamical chaos may continue to elude ecologists. On the other hand, the models suggest that the observation of cyclical dynamics in nature is the most likely outcome.

Spatiotemporal intermittency and chaotic saddles in the regularized long-wave equation

Transition to intermittent spatiotemporal chaos is studied in the regularized long-wave equation, a nonlinear model of shallow water waves. A mechanism for the onset of on-off spatiotemporal intermittency is explored. In this mechanism, the coupling of two chaotic saddles triggers random switching between phases of laminar and bursty behaviors. The average time between bursts as a function of the control parameter follows a power law typical of crisis transitions in chaotic systems. The degree of spatiotemporal disorder in the observed fluid patterns is quantified by means of the time-averaged spectral entropy for both chaotic attractors and chaotic saddles. The implications of these results to other fluid systems are discussed.