type-III Intermittency Research Articles

Fog Intermittency and Critical Behavior

The intermittency of fog occurrence (the switching between fog and no-fog) is a key stochastic feature that plays a role in its duration and the amount of moisture available. Here, fog intermittency is studied by using the visibility time series collected during the month of July 2022 on Sable Island, Canada. In addition to the visibility, time series of air relative humidity and turbulent kinetic energy, putative variables akin to the formation and breakup conditions of fog, respectively, are also analyzed in the same framework to establish links between fog intermittency and the underlying atmospheric variables. Intermittency in the time series is quantified with their binary telegraph approximations to isolate clustering behavior from amplitude variations. It is shown that relative humidity and turbulent kinetic energy bound many stochastic features of visibility, including its spectral exponent, clustering exponent, and the growth of its block entropy slope. Although not diagnostic, the visibility time series displays features consistent with Pomeau–Manneville Type-III intermittency in its quiescent phase duration PDF scaling (−3/2), power spectrum scaling (−1/2), and signal amplitude PDF scaling (−2). The binary fog time series exhibits properties of self-organized criticality in the relation between its power spectrum scaling and quiescent phase duration distribution.

Study of type-III intermittency in the Landau–Lifshitz-Gilbert equation

We have studied a route of chaos in the dissipative Landau–Lifshitz-Gilbert equation representing the magnetization dynamics of an anisotropic nanoparticle subjected to a time-variant magnetic field. This equation presents interesting chaotic dynamics. In the parameter space, for some forcing frequency and magnetic strength of the applied field, one observes a transition from a regular periodic behavior to chaotic dynamics. The chaotic dynamics, close to the bifurcation, are characterized by type-III intermittency. Long epochs of quasi-regular dynamics followed by turbulent bursts. The characterization of the intermittencies has been done through four different techniques. The first method is associated with the computation of the Lyapunov exponents that characterize the chaotic regime. The second and third methods are associated with the statistics of the duration of the laminar epochs prior to a turbulent burst. The fourth method is associated with the subharmonic instability present in those laminar epochs and quantified through a Poincaré section method. At the end of the manuscript, we compare the result obtained by the different techniques and discuss the methods’ limitations.

Intermittent transition to chaos in vibroimpact system

Abstract Chaotic behaviour of dynamical systems, their routes to chaos, and the intermittency in particular are interesting and investigated subjects in nonlinear dynamics. The studying of these phenomena in non-smooth dynamical systems is of the special scientists’ interest. In this paper we study the type-III intermittency route to chaos in strongly nonlinear non-smooth discontinuous 2-DOF vibroimpact system. We apply relatively new mathematical tool – continuous wavelet transform CWT – for investigation this phenomenon. We show that CWT applying allows to detect and determine the chaotic motion and the intermittency with great confidence and reliability, gives the possibility to demonstrate intermittency route to chaos, to distinguish and analyze the laminar and turbulent phases.

Multiple Scenarios of Transition to Chaos in the Alternative Splicing Model

We have investigated the scenarios of transition to chaos in the mathematical model of a genetic system constituted by a single transcription factor-encoding gene, the expression of which is self-regulated by a feedback loop that involves protein isoforms. Alternative splicing results in the synthesis of protein isoforms providing opposite regulatory outcomes — activation or repression. The model is represented by a differential equation with two delayed arguments. The possibility of transition to chaos dynamics via all classical scenarios: a cascade of period-doubling bifurcations, quasiperiodicity and type-I, type-II and type-III intermittencies, has been numerically demonstrated. The parametric features of each type of transition to chaos have been described.

Reinjection Probability Density for Type-III Intermittency With Noise and Lower Boundary of Reinjection

In this paper, we extend a methodology developed recently to study type-III intermittency considering different values of the noise intensity and the lower boundary of reinjection (LBR). We obtain accurate analytic expressions for the reinjection probability density (RPD). The proposed RPD has a piecewise definition depending on the nonlinear behavior, the LBR value, and the noise intensity. The new RPD is a sum of exponential functions with exponent α + 2, where α is the exponent of the noiseless RPD. The theoretical results are verified with the numerical simulations.

Theory of intermittency applied to classical pathological cases

The classical theory of intermittency developed for return maps assumes uniform density of points reinjected from the chaotic to laminar region. Though it works fine in some model systems, there exist a number of so-called pathological cases characterized by a significant deviation of main characteristics from the values predicted on the basis of the uniform distribution. Recently, we reported on how the reinjection probability density (RPD) can be generalized. Here, we extend this methodology and apply it to different dynamical systems exhibiting anomalous type-II and type-III intermittencies. Estimation of the universal RPD is based on fitting a linear function to experimental data and requires no a priori knowledge on the dynamical model behind. We provide special fitting procedure that enables robust estimation of the RPD from relatively short data sets (dozens of points). Thus, the method is applicable for a wide variety of data sets including numerical simulations and real-life experiments. Estimated RPD enables analytic evaluation of the length of the laminar phase of intermittent behaviors. We show that the method copes well with dynamical systems exhibiting significantly different statistics reported in the literature. We also derive and classify characteristic relations between the mean laminar length and main controlling parameter in perfect agreement with data provided by numerical simulations.

Laminar length and characteristic relation in Type-I intermittency

A method to investigate systems showing Type-I intermittency phenomenon is presented. This method is an extension of the procedure we have recently established to study the Type-II and Type-III intermittencies. With this approach, new accurate analytical expressions for the reinjection and the laminar phase length probability densities are obtained. The new theoretical formulas are tested by numerical computation, showing an excellent agreement between analytical models and numerical results. In addition, our method fully generalizes the well-known classical characteristic relations, in such a way that it properly characterizes those systems showing Type-I intermittency.

Effect of noise on the reinjection probability density in intermittency

This paper analyzes the effect of noise in type-II and type-III intermittency, with respective local Poincaré maps of xn+1=(1+ε)xn+(1-ε)xn3 and xn+1=-(1+ε)xn-axn3 (a>0). We develop a method to obtain the noisy reinjection probability density (NRPD), which basically consists in extending a recent procedure used to derive the noiseless reinjection probability density (RPD). Our approach also provides information to accurately describe the noiseless system. We also derive the probability density of the laminar length. Our analytical results show a good agreement with numerical simulations. Finally, we have also found that, for large values of the instability parameter ε, the characteristic relations approach the associated ones to the noiseless intermittency. However, for low values of the instability parameter ε, the characteristic relations reach a saturation level that depends on the NRPD.

Reinjection probability density in type-III intermittency

A method to obtain the reinjection probability density (RPD) for systems showing type-III intermittency phenomenon is presented. The method emerges as a natural extension of a recent procedure we established to derive the RPD for type-II intermittency. A new two-parameter class of reinjection probability densities is presented to describe a broad class of type-III intermittency maps. Our RPD expression also provides information about the lower bound of the reinjection (LBR). The new characteristic relation εβ depends on the LBR and it has a critical exponent β such that −1<β<−0.5. The corresponding analytical duration probability densities are also derived in agreement with the numerical computation. The main classical results can be extracted from our RPD as particular cases.

Precipitation, dynamical intermittency, and sporadic randomness

While rainfall intermittency is a dynamical phenomenon, little progress has been made in the literature on the link between rainfall intermittency and atmospheric dynamics. We present the basic dynamical models of intermittency that are phenomenologically most similar to rainfall: Pomeau–Manneville Type-III and On–Off. We then illustrate each type with both a 1-D iterative map and a corresponding stochastic process stressing the appearance of these dynamics in high-dimensional (stochastic) systems as opposed to low-dimensional chaotic systems. We show that the pdf of rainfall intensities, the pdf of “laminar phases” (periods of zero rainfall intensity), and the spectrum of the rainfall series all have power-law behavior that is broadly consistent with intermittency in the classic types. Using a seasonal analysis, we find that summer convective rainfall at daily and sub-daily scales seems consistent with features of Type-III intermittency. The correspondence with Type-III intermittency and a preliminary entropic analysis further suggest that rainfall may be an example of sporadic randomness, blending deterministic and stochastic components.

Lyapunov exponent for type-III intermittent chaos

The behaviour of the Lyapunov exponent near the transition to chaos via type-III intermittency is numerically determined for a generic map showing this transition. A critical exponent β expressing the scaling of the Lyapunov exponent as a function of both, the reinjection probability and the nonlinearity of the map is calculated. It is found that the critical exponent varies on the interval 0<β<1. This extends earlier predictions for the scaling behaviour of the Lyapunov exponent in type-III intermittency.

Spatiotemporal dynamics of shear induced bands en route to rheochaos

We show experimentally that the route to rheochaos in shear rate relaxation measurements is via Type-III intermittency and mixed mode oscillations in the shear-thinning wormlike micellar system of cetyltrimethylammonium tosylate in the presence of salt sodium chloride. Depolarised small angle light scattering measurements performed during flow show that scattered intensity temporally follows the shear rate/stress dynamics and portrays the crucial role played by nematic ordering. Direct visualization of the gap of the Couette cell, illuminated by an unpolarised laser sheet, in the (vorticity, velocity gradient) plane shows that the spatiotemporal dynamics of the shear induced structures is closely related to the temporal behaviour of shear rate/stress fluctuations.

Tuning Rheochaos by Temperature in Wormlike Micelles

We investigate the critical role played by the mean micellar length during the route to rheochaos for wormlike micellar gels of surfactant cetyltrimethylammonium tosylate in the presence of salt sodium chloride that show coupling of flow to concentration fluctuations. To this end, we have carried out stress/shear rate relaxation experiments at a fixed shear rate/stress but at different temperatures to take the sample through the route to rheochaos. We see the type-II intermittency route to rheochaos in stress relaxation measurements and the type-III intermittency route to rheochaos in shear rate relaxation measurements. We have also carried out linear rheology measurements at different temperatures to estimate the mean micellar length (-)L, the reptation time tau(rep), and the breaking time tau(break). It is shown that (-)L changes by approximately 58%, as the sample goes through the route to rheochaos.

Critical behavior of the Lyapunov exponent in type-III intermittency

The critical behavior of the Lyapunov exponent near the transition to robust chaos via type-III intermittency is determined for a family of one-dimensional singular maps. Critical boundaries separating the region of robust chaos from the region where stable fixed points exist are calculated on the parameter space of the system. A critical exponent β expressing the scaling of the Lyapunov exponent is calculated along the critical curve corresponding to the type-III intermittent transition to chaos. It is found that β varies on the interval 0 ⩽ β < 1/2 as a function of the order of the singularity of the map. This contrasts with earlier predictions for the scaling behavior of the Lyapunov exponent in type-III intermittency. The variation of the critical exponent β implies a continuous change in the nature of the transition to chaos via type-III intermittency, from a second-order, continuous transition to a first-order, discontinuous transition.

Experimental observation of characteristic relations of type-III intermittency in the presence of noise in a simple electronic circuit.

We investigate the characteristic relations of type-II and -III intermittencies in the presence of noise. The theoretically predicted characteristic relation is that <l> approximately exp[/epsilon/(2)] for a negative regime of epsilon and <l> approximately epsilon(-nu) for the positive regime of epsilon (1/2</=nu<1), where <l> is the average laminar length and (1+epsilon) is the slope of the local Poincaré map around the tangent point. We experimentally confirm these relations in a simple electronic circuit.

Shil'nikov homoclinic chaos is intimately related to type-III intermittency in isolated rabbit arteries: role of nitric oxide.

We provide experimental evidence for the existence of Shil'nikov homoclinic chaos in the fluctuations in flow which can be observed in isolated perfused rabbit ear arteries, and establish a close association between homoclinicity and type-III Pomeau-Manneville intermittent behavior. The transition between the homoclinic scenario and type-III intermittency is clarified by a mathematical model of the arterial smooth muscle cell. Simulations of the effects of nitric oxide (NO) by the vascular endothelium on these patterns of behavior closely match experimental observations.

Averages and critical exponents in type-III intermittent chaos.

The natural measure in a map with type-III intermittent chaos is used to define critical exponents for the average of a variable from a dynamical system near bifurcation. Numerical experiments were done with maps and verify the analytical predictions. Physical experiments to test the usefulness of such exponents to characterize the nonlinearity at bifurcations were done in a driven electronic circuit with diode as a nonlinear element. Two critical exponents were determined at the same bifurcation: one from the fitting of the average voltage across the diode and the other one from the average length of the laminar phase events. Both values are consistent with the predictions of a type-III intermittency of cubic nonlinearity. The averages of variables in intermittent chaotic systems is a technique complementary to the measurements of laminar phase histograms, to identify the nonlinear mechanisms. The average exponents may have a broad application in ultrafast chaotic phenomena.

Bifurcation structure of a model of bursting pancreatic cells

One- and two-dimensional bifurcation studies of a prototypic model of bursting oscillations in pancreatic β-cells reveal a squid-formed area of chaotic dynamics in the parameter plane, with period-doubling bifurcations on one side of the arms and saddle-node bifurcations on the other. The transition from this structure to the so-called period-adding structure is found to involve a subcritical period-doubling bifurcation and the emergence of type-III intermittency. The period-adding transition itself is not smooth but consists of a saddle-node bifurcation in which ( n+1)-spike bursting behavior is born, slightly overlapping with a subcritical period-doubling bifurcation in which n-spike bursting behavior loses its stability.

Intermittency transitions to strange nonchaotic attractors in a quasiperiodically driven duffing oscillator

Different mechanisms for the creation of strange nonchaotic attractors (SNAs) are studied in a two-frequency parametrically driven Duffing oscillator. We focus on intermittency transitions in particular, and show that SNAs in this system are created through quasiperiodic saddle-node bifurcations (type-I intermittency) as well as through a quasiperiodic subharmonic bifurcation (type-III intermittency). The intermittent attractors are characterized via a number of Lyapunov measures including the behavior of the largest nontrivial Lyapunov exponent and its variance, as well as through distributions of finite-time Lyapunov exponents. These attractors are ubiquitous in quasiperiodically driven systems; the regions of occurrence of various SNAs are identified in a phase diagram of the Duffing system.

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.