Intermittent Chaos Research Articles

NON-LINEAR DYNAMICS OF GEAR-PAIR SYSTEMS WITH PERIODIC STIFFNESS AND BACKLASH

The present work investigates dynamics of a gear-pair system involving backlash and time-dependent mesh stiffness. In addition, the system is under the action of external excitation, caused by torsional moments and gear geometry errors. First, the equation of motion is established in a strongly non-linear form. Then, the emphasis is laid on a specific forcing frequency range, corresponding to conditions of simultaneous fundamental parametric resonance and principal external resonance. For these conditions, several types of periodic steady state response are identified and determined by employing suitable methodologies, including techniques applicable to piecewise linear systems and to oscillators with time-periodic coefficients. Moreover, these methodologies are complemented by appropriate procedures revealing the stability properties of the located periodic solutions. In the second part of the work, numerical results are presented. These results verify the validity and effectiveness of the new analytical methodology and provide information on the gear-pair dynamics. First, series of typical response diagrams are obtained, illustrating the effect of the mesh stiffness variation, the damping and the forcing parameters on the gear-pair periodic response. These response diagrams are accompanied by results obtained with direct integration of the equation of motion. In this way, it is demonstrated that for some parameter combinations, the dynamical system examined can exhibit more complicated and irregular response, including crises and intermittent chaos.

Mathematical modeling of complex oscillatory phenomena during CO oxidation over Pd zeolite catalysts

A mathematical model, which simulates the complicated dynamic behavior experimentally observed during CO oxidation over Pd zeolite catalysts is presented. It describes the coupling of reaction rate oscillations, generated by various parts of the inhomogeneous catalytic layer through the gas phase. It can be shown, that the resulting dynamic behavior depends upon the difference between natural frequencies of local oscillators and the strength of coupling, which is defined mostly by the degree of conversion. Chaotic behavior could be identified under the condition of weak coupling for local oscillators with widely different natural frequencies. In the range of strong coupling the phenomenon of phase death has been obtained. A special type of intermittency chaos (“on–off” chaos) was observed in a small region of parameters under the conditions of strong coupling.

Fractal analysis of a circulating flow field with two different velocity laws

The turbulent shear flow around a rotating cylinder in a quiescent flow is a simple case of a rotating turbulent flow field, where centrifugal force works. Two different power-law mean-velocity distributions exist in this flow field. One is U∝1/ r and the other is U∝1/ r 2, where r is a distance from the surface of a cylinder. The behavior of chaos and fractal properties for this complex flow field are investigated. The former concerns the dynamic property in fluid flow and the latter is useful to characterize complex flow patterns or the distribution of turbulence quantities. From the instantaneous velocity signal, we defined the iso-velocity set, and its fractal property was investigated both in the U∝1/ r and U∝1/ r 2 region. The instantaneous Reynolds stress is found to be a key factor in this fractal property as conceived for the flat plate boundary layer. The intermittency chaos was applied to investigate the turbulent and non-turbulent distribution in the outer region. A simple one-dimensional model could be useful to identify turbulent and non-turbulent distributions even in this complex flow.

Control of Intermittent Chaos Caused by Applying a Pulse

The control of intermittent chaos caused by the current-driven ion acoustic instability is attempted. It is found that when a positive pulse is applied to one of two mesh grids which are installed in a plasma, the instability is suddenly suppressed. On the other hand, when a negative pulse is applied, the system changes from chaotic orbits to periodic orbits while maintaining the instability, showing that the chaotic state caused by the instability is well controlled by applying a negative pulse.

The application of chaotic oscillators to weak signal detection

In this paper, the authors introduce a signal detection scheme based on the bifurcation behavior of the driven Duffing oscillator. Chaotic systems are sensitive to certain signals and immune to noise at the same time, the properties of which demonstrate their potential application in weak signal detection. Starting from the analysis of the intermittent chaotic motion occurring in the detecting process, they put forward a new frequency-locking principle based on the periodic characteristic of the intermittent chaos. Then, an exposition is made on how to use an array of the oscillators to detect the weak signals of unknown frequency.

Three-body dynamics: intermittent chaos with strange attractor

We have studied the structure of chaos in three-body dynamics using the concept of intermittency, implying that violent states of a system alternate in time with quasi-regular states producing together a non-stationary and evolving pattern of unpredictable behaviour. Computer simulations are produced to demonstrate explicitly sporadic short violent bursts in quasi-regular hierarchical states of the systems. This is seen both in orbits and in the long time series generated by the system. The time series prove to be similar in shape to what is observed in various physical experiments with laboratory chaotic systems when they reveal the so-called type-III intermittency. The new effective methods of time series analysis enable us to discover a strange attractor with a fractal dimension slightly above 2. This shows that three-body dynamics has the same intrinsic qualitative structure and quantitative measure of chaos as the widely known chaotic system, the Lorenz attractor.

Intermittent chaos in three-body dynamics

Chaotic behaviour of dynamical three-body systems is characterized by the fact that their evolution consists typically of sequences of transient states that change from one to another in an unpredictable manner. Some of these states are fairly regular ones; for instance, a hierarchical state with a very close binary and a remote third body. Other ones are more chaotic; for instance, a state of interplay in which each body interacts with two others with more or less equal intensity. The alternation of these states is similar to the phenomenon of intermittency discovered first in the ocean flows, where laminar currents alternate with areas of developed turbulence. Onset of chaos and the structure of intermittency in the dynamics of three-body systems is the focus of the present paper. General physical considerations on the nature of chaos in nonlinear systems are applied to the analysis of the three-body dynamics given by analytical and computer models. Onset of chaos and the nature of unpredictability in a typical elementary act of three-body interaction is studied with analytical methods. High sensitivity of the systems to small changes in control parameters is also demonstrated and a critical mass for transition to chaos is found. Computer simulations and homology mapping are used to clarify why and how the three-body systems are sensitive to fine details of initial conditions. Physical characteristics of local dynamical instability, which is the fundamental mechanism of the onset of chaos in dynamical systems, are estimated. Rapid divergence of two initially close trajectories in phase space is demonstrated. Stochasization in an ensemble of systems is studied with the use of homology mapping. A strange attractor with fractal dimension slightly above 2 is discovered in the long time series generated by the system in the state of intermittent chaos. General manifestations of weak low-dimensional intermittent chaos in three-body dynamics are finally summarized.

Population Floors and the Persistence of Chaos in Ecological Models

Chaotic dynamics have been observed in a wide range of population models. Here we describe the effects of perturbing several of these models so as to introduce a non-zero minimum population size. This perturbation generally reduces the likelihood of observing chaos, in both discrete and continuous time models. The extent of this effect depends on whether chaos is generated through period-doubling, quasiperiodicity, or intermittence. Chaos reached via the quasiperiodic route is more robust against the perturbation than period-doubling chaos, whilst the inclusion of a population floor in a model exhibiting intermittent chaos may increase the frequency of population bursts although these become non-chaotic.

Effect of Noise on the Low Flow Rate Chaos in the Belousov-Zhabotinsky Reaction

We investigate systematically the effect of Gaussian white noise (type P noise) on the low flow rate chaos in both systems far from and close to bifurcation points using the three-variables ODE of the Belousov-Zhabotinsky reaction developed by Gyorgyi and Field. When the noise is added to the chaos with the bifurcation parameter far from bifurcation points, the chaos trajectories are slightly scattered. However, in the chaos having the bifurcation parameter near bifurcation points, it happens that topological entropy is constant but the Lyapunov exponent decreases. We have found that this phenomenon, named “noise-induced order”, appears in intermittent chaos with the internal structure of m -periodic oscillation, and that “noise-induced order” is caused by an increase in the length of laminar region and the subsequent change of the invariant density.

Intermittent chaos in a mutually coupled PLL's system

Novel intermittent chaos is experimentally observed in a mutually coupled phase-locked loop system, which is constructed mainly of two commercial phase-locked loops used as nonlinear oscillators and two op amps used as couplers. This intermittent chaos has similar statistical properties as the on-off intermittency.

Intermittent chaos in dense gaseouslike media driven by coupling cross thermodiffusive effects

Intermittent chaotic behavior induced by a thermodiffusive coupling is investigated by analysis of a five-mode (Lorenz-like) truncated model describing a binary, diluted solution which behaves as a gaslike system. The obtained nonlinear equations coherently agree with the marginal stability locus point when the absence of a stationary state for the mass transfer is considered. For a wide range of reduced Rayleigh number values $r$, we show that the truncated model exhibits the Pomeau-Manneville intermittency route to chaos when the control (Soret-based) phenomenological coefficient $〈\ensuremath{\delta}〉$ approaches some critical values $〈\ensuremath{\delta}{〉}_{c}.$ Numerical simulations evaluating the generalized Lyapunov exponent against the control parameter displacement close to the intermittency threshold, $L(1)$ vs $(〈\ensuremath{\delta}{〉}_{c}\ensuremath{-}〈\ensuremath{\delta}〉)$, are reported inside the chaotic region. The achieved results agree reasonably with theoretical predictions.

Critical scaling and type-III intermittent chaos in isolated rabbit resistance arteries

We have shown that spontaneous oscillations in flow in rabbit ear resistance arteries may sometimes exhibit behavior typical of type-III Pomeau-Manneville intermittency. The average number of oscillations per laminar length $〈n〉$ was related to a bifurcation parameter \ensuremath{\varepsilon} according to power-law scaling of the form $〈n〉\ensuremath{\sim}{\ensuremath{\varepsilon}}^{\ensuremath{-}\ensuremath{\beta}}.$ The critical exponent \ensuremath{\beta} was estimated as $\ensuremath{\sim}0.80,$ which is within the range reported for type-III intermittent chaos in nonbiological systems.

Torus Destruction and Chaos–Chaos Intermittency in a Commodity Distribution Chain

The destruction of two-dimensional tori T2 and the transitions to chaos are studied numerically in a high-dimensional model describing the decision making behavior of human subjects in a simulated managerial environment (the beer production-distribution model). Two different routes from quasiperiodicity to chaos can be distinguished. Intermittency transitions between chaotic and hyperchaotic attractors are characterized, and transients in which the "system pursues the ghost" of a vanished hyperchaotic attractor are studied.

Controlling spatial chaos in metapopulations with long-range dispersal

We propose two methods to control spatial chaos in an ecological metapopulation model with long-range dispersal. The metapopulation model consists of local populations living in a patchily distributed habitat. The habitat patches are arranged in a one-dimensional array. In each generation, density-dependent reproduction occurs first in each patch. Then individuals disperse according to a Gaussian distribution. The model corresponds to a chain of coupled oscillators with long-range interactions. It exhibits chaos for a broad range of parameters. The proposed control methods are based on the method described by Güémez and Matías for single difference equations. The methods work by adjusting the local population sizes in a selected subset of all patches. In the first method (pulse control), the adjustments are made periodically at regular time intervals, and consist of always removing (or adding) a fixed proportion of the local populations. In the second method (wave control), the adjustments are made in every generation, but the proportion of the local population that is affected by the control changes sinusoidally. As long as dispersal distances are not too low, these perturbations can drive chaotic metapopulations to cyclic orbits whose period is a multiple of the control period. we discuss the influence of the magnitude of the pulses and wave amplitudes, and of the number and the distribution of controlled patches on the effectiveness of control. When the controls start to break down, interesting dynamic phenomena such as intermittent chaos can be observed.

Controlling spatial chaos in metapopulations with long-range dispersal

We propose two methods to control spatial chaos in an ecological metapopulation model with long-range dispersal. The metapopulation model consists of local populations living in a patchily distributed habitat. The habitat patches are arranged in a one-dimensional array. In each generation, density-dependent reproduction occurs first in each patch. Then individuals disperse according to a Gaussian distribution. The model corresponds to a chain of coupled oscillators with long-range interactions. It exhibits chaos for a broad range of parameters. The proposed control methods are based on the method described by Gülémez and Matías for single difference equations. The methods work by adjusting the local population sizes in a selected subset of all patches. In the first method (pulse control), the adjustments are made periodically at regular time intervals, and consist of always removing (or adding) a fixed proportion of the local populations. In the second method (wave control), the adjustments are made in every generation, but the proportion of the local population that is affected by the control changes sinusoidally. As long as dispersal distances are not too low, these perturbations can drive chaotic metapopulations to cyclic orbits whose period is a multiple of the control period. We discuss the influence of the magnitude of the pulses and wave amplitudes, and of the number and the distribution of controlled patches on the effectiveness of control. When the controls start to break down, interesting dynamic phenomena such as intermittent chaos can be observed.

Stabilization mechanisms of short waves in stratified gas–liquid flow

Interfacial waves grow in a cocurrent, stratified gas–liquid flow by extracting energy from the main flow. The most unstable mode typically has a wavelength comparable to or less than the liquid depth. Experiments show that these short waves can saturate at small amplitude with no generation of long-wave or transverse modes. By decomposing the typical Stuart–Landau analysis into three components, it is found that saturation usually occurs by cubic self-interaction of the fundamental mode but quadratic resonant interaction with the first overtone is also possible. Interaction with mean flow modes is usually much less important. Experiments confirm the predictions of weakly nonlinear theory. The measured overtone is found to be O(|A1|2) and is phase-locked with the fundamental except near a 1–2 resonance point where the fundamental and the overtone have comparable speeds. Near this resonance, the amplitudes are of the same order and the phase angle between them is observed to jump irregularly as predicted by modern dynamical systems theory for intermittent chaos near a heteroclinic cycle. The phase and magnitude of the overtone interaction specify the shape, chaotic dynamics and symmetry of the waves across resonance which are analyzed and confirmed experimentally.

Chaos Induced Transition

To study the coherent nature of chaos, two models are proposed. Model 1 is a simple nonlinear system [Formula: see text] and Model 2 is a linear harmonic oscillator [Formula: see text], which are driven by a chaotic force f(t). The chaotic force f(t) is defined by [Formula: see text] for nτ < t ≤ (n + 1)τ(n = 0, 1, 2, …), where yn+1 is a chaotic sequence of a map F(y; r) with the bifurcation parameter r: yn+1 = F(yn; r) (-0.5 ≤ yn ≤ 0.5) and ŷn = yn - < y0>. In Model 1 the relaxation process of this system and the τ- and r-dependence of the stationary distribution of x are discussed. It is shown that the small change of the bifurcation parameter r causes the drastic change of the stationary distribution. In Model 2, resonance phenomena are investigated near the period 3 window of the logistic map, in particular, in the intermittent chaos region and the period doubling region. The theoretical results are shown to be in a good agreement with numerical ones, which have been done for the logistic map as F(y; r).

Self-organized marginal stability resulting from inconsistency between fuzzy logic and deterministic logic: an application to biological systems

Complex systems in which internal agents (observers) interact with each other with finite velocity of information propagation cannot be described with a single consistent logic. We have proposed the bootstrapping system of cellular automata for describing such complex systems using two types of complementary logic: Boolean and non-Boolean. We extend this in this paper to a system of time-discrete continuous maps using fuzzy logic in place of non-Boolean logic. Fuzziness implies the intrinsic ambiguity of internal measurement. The bootstrapping system evolves, changing the dynamics perpetually, so that the discrepancy between the two types of complementary logic may be minimized. The equilibration force defined from the strength of discrepancy forms a landscape for self-organization which is similar to the fitness landscape for evolution. Though they appear similar, the former is derived from the internal dynamics. The goal of evolution, when applied to the map of the Belousov-Zabochinsky reaction, is demonstrated to be near the border between periodicity and chaos. The behavior depends on the degree of fuzziness and the extent of noise. When fuzziness increases too much, the system becomes unstable. Near the boundary, it exhibits intermittent chaos with a background of 1/ f noise.

間欠性カオス写像を用いた遺伝的アルゴリズムのパラメータ動的制御

間欠性カオス写像を用いた遺伝的アルゴリズムのパラメータ動的制御

Map-Based Analysis of Noise-Induced Convergence in Chaos

The mechanism and condition of appearance of order in chaos with Gaussian white noise is investigated based on the Poincare return map constructed from the three-variables ODE of the Belousov-Zhabotinsky reaction. When the noise is added to the chaos having the bifurcation parameter near period-three oscillation, it occasionally happens that topological entropy is constant but the Kolmogorov entropy decreases. Analysis on three-times iterated map reveals that this phenomenon, named “noise-induced convergence”, is caused by an increase of the length of laminar phase and the subsequent change of the invariant density. The same phenomenon is observed in m-times iterated map of the chaos of the logistic map. We consider that “noise-induced convergence” is characteristic of intermittent chaos.