Deterministic Chaotic Behavior Research Articles

A NEW TYPE OF DETERMINISTIC CHAOS: DETERMINISTIC CHAOS OF THE SECOND KIND

Abstract. Currently, the phenomena of deterministic (Dynamic) chaos are widely used in science and technology. Therefore, now the urgent task is to study new phenomena and new manifestations of deterministic chaos. This paper examines a fundamentally new manifestation of deterministic chaos (Deterministic chaos of the second kind). In this regard, previously known manifestations of deterministic chaos should be called manifestations of chaos of the first kind (The previously known type of deterministic chaos, in our terminology, is chaos of the first kind). Deterministic chaos of the first kind is chaos that is directly related to the deterministic - chaotic behavior of the dynamic variables of a given dynamic system. At the same time, our work examines the characteristic features of deterministic chaos of the first kind and methods for generating such chaos. In our work, we considered a fundamentally different type of manifestation of dynamic chaos, namely deterministic chaos of the second kind. Deterministic chaos of the second kind is not directly related to the deterministic chaotic behavior of the dynamic variables of a dynamic system and can only exist in characteristics that are not directly related to the dynamic variables. That is, dynamic chaos of the second kind arises only in second-order manifestations: For example, in the Fourier spectrum. In a more general case, this could be any other type of spectrum (Not Fourier) and even another characteristic different from the spectrum. The paper provides two examples of deterministic chaos of the second kind. The work also proposes a general technique for generating time series demonstrating deterministic chaos of the second kind*. It is very important that dynamic chaos in the examples given in this work manifests itself not in the time series themselves (Which are completely random) but in the Fourier spectrum of these time series. Thus, the original series itself (The original signal itself) can be random, but at the same time, the spectrum of this series, if considered as another time series, can exhibit deterministic chaotic behavior. Thus, for the first time we were able to build a bridge between completely random systems and deterministic chaotic systems. That is, in our work we were the first to show that the same system can be considered both random (From the point of view of the behavior of its variables) and at the same time as deterministic-chaotic (From the point of view of its spectrum or other similar characteristics). Goal of the work. Study of a newtype of deterministic chaos discovered by the authors (Deterministic chaos of the second kind) and the study of its properties and methods for generating this type of chaos.

Research Spotlights

What surprising patterns do fluid dynamics and neurobiological networks have in common? In this issue's first Research Spotlights article, “Bump Attractors and Waves in Networks of Leaky Integrate-and-Fire Neurons,” a connection is made between the waves at the onset of pipe turbulence and the localized traveling waves in spiking neural networks. Authors Daniele Avitabile, Joshua L. Davis, and Kyle Wedgwood consider a so-called “discrete integrate-and-fire model” to describe the dynamics of a connected neuron firing via a system of piecewise linear ordinary differential equations. Constructing a solution to this system effectively amounts to determining firing times of the neurons. One can then witness “bump” patterns, in this case characterized by a core of neurons that fire frequently, while neurons away from the core do not fire. Such a model's deterministic chaotic behavior can then be studied around a firing set and used to construct waves and analyze their stability. Simulations and analysis demonstrate oscillatory and nested branching of these waves, and such behavior can accumulate in a bump attractor and wandering wave. The authors show that such complex spatiotemporal patterns, and their conditions for stability, are not unlike those witnessed in the transition to turbulence in a pipe. The paper concludes with a challenge to future researchers: can further dynamical systems characterizations of such event-driven models uncover links between localized waves and bumps in complex spatial networks? The second Research Spotlights article addresses the problem of identifying a system from noisy observations of dynamical data. In “Learning Dynamical Systems with Side Information” authors Amir Ali Ahmadi and Bachir El Khadir present a way to incorporate system knowledge to improve the learning process beyond what is possible using data alone. Such knowledge, or “side information,” arises when one has contextual information about the unknown dynamics without knowing the dynamics precisely. Examples of side information covered by the authors' framework include known trajectories (e.g., when one has knowledge of some equilibrium points), group symmetry, and coordinatewise properties (e.g., nonnegativity or monotonicity). Restricting attention to a particular class of functions to be learned and particular side information allows the learning process to be posed as a semidefinite optimization formulation amenable to efficient solution. This approach is demonstrated numerically on systems modeling the diffusion of a contagion, behavior of a simple pendulum, and time evolution of a cancerous tumor, as well as on the classical Lorenz system. In each case, imposing side information via semidefinite programming allows the authors to learn the unknown dynamics from far fewer observations than otherwise possible and provides a potential road map for addressing even broader classes of systems and side information.

A New Robust Method to Investigate Dynamic Instability of FTV for the Double Tripod Industrial Driveshafts in the Principal Parametric Resonance Region

The present work aims to design a robust method to detect and certify the deterministic chaos or ergodic process for the forced torsional vibrations (FTV) of a double tripod industrial driveshaft (DTID) in transition through the principal parametric resonance region (PPRR) which is considered by the researchers in the field as one of the most important resonance regions for the systems having parametric excitations. The DTID’s model for FTV considers the following effects: nonuniformities of inertial characteristics of the DTID’s elements, the harmonic torque excitation induced by the asynchronous electrical motor used for a heavy-duty grain mill, and the harmonic reaction torque generated by different granulation of the substance needed to be milled. Based on these aspects, a model of the FTV for the DTID was designed which was a modified, physically consistent model already used by the authors to investigate the FTV of automotive driveshafts (homokinetic transmission). For the DTID elements, the dynamic instability for nonstationary FTV in the PPRR using time–history analysis (THA) was analyzed—THA represents the phase portraits. Time–history analysis is a detection method for possible chaotic dynamic behavior for the nonstationary FTV (NFTV) in transition through PPRR. If this dynamic behavior was seen, a new robust method LEA–PM was created to certify and confirm the deterministic chaos for the NFTV of DTID. The new method, LEA–PM, is composed of the Lyapunov exponent’s approach (LEA) coupled with the Poincaré Map (PM) applied to the global system of differential equations that describe the FTV of DTID in the PPRR. This new robust method, which embeds LEA and PM, LEA–PM, establishes if the mechanical system has a deterministic chaotic dynamic behavior (strange attractor) or an ergodic dynamic process in this resonant region. LEA represents a new method that includes not only the maximal Lyapunov exponent method (MLEM) but also new mathematical criteria that is “the sum of all Lyapunov exponents has to be negative” which, coupled with MLEM, indicates the presence of deterministic chaos (strange attractors). THA–LEA–PM had been used for the NFTV of DTID computing the phase portraits, the Lyapunov exponents, and representing the Poincaré Maps of the NFTV for the DTID’s elements in transition through PPRR, founding deterministic chaos or ergodic dynamic behavior. Based on the obtained results, numerical simulations revealed the pitting manifestations of the DTID’s elements, typical for the geared systems transmission, mentioned recently in experimental data research for the homokinetic transmissions. Using the new robust method, THA–LEA–PM (time–history analysis coupled with LEA–PM) can be used in future research for chaotic dynamic analysis of DTID’s NFTV transition through superharmonic resonances, subharmonic resonances, combination resonances, and internal resonances. Time–history analysis as a detection method for chaos and LEA–PM as a certifying method for deterministic chaos can be integrated as a design tool for DTID’s FTV control of the homokinetic transmission.

On the chaotic nature of random telegraph noise in unipolar RRAM memristor devices

Random telegraph noise (RTN) owns its very name to its assumed stochastic nature. In this paper, we follow up previous works that questioned this stochastic nature, and we investigate this assumption using experimentally measured noise coming from properly biased Ni/HfO2 unipolar Resistive RAM memristor nanodevices. We have used established, well–known tools from nonlinear theory to examine the current–noise temporal series. Evaluation results show that this series appears to exhibit not a stochastic, but a deterministic chaotic behavior, also demostrating interesting fractal characteristics in 2D and 3D phase space projections. The presented results clearly advocate for a strong component of complex (chaotic) fluctuation of deterministic origin, instead of a typical (fully stochastic) RTN. This result could pave the path for an enhanced understanding of the mechanisms behind RTN emergence, as well as improve its noise models.

Chaotic Dynamics and Stability of Liposomal Nanosystems

Background: Natural and living systems are dynamical systems that demonstrate complex behavior, which appears to be deterministic chaotic, characterized and governed by entropy increase and loss of information throughout their entire lifespan. Lipidic nanoparticles, such as liposomes, as artificial biomembranes, have long been considered appropriate models for studying various membrane phenomena that cell systems exhibit. By utilizing these models, we can better comprehend cellular functions, stability, as well as factors that might alter the cell physiology, leading to severe disease states. In addition, liposomes are well-established drug and vaccine delivery nanosystems, which are present in the market, playing a significant role; therefore, due to their importance, issues concerning their effectiveness and stability are research topics that are constantly investigated and updated. Methods: In this study, the emergent deterministic chaotic behavior of liposomes is described, while evaluation in accordance to their colloidal physical stability, by utilizing established nonlinear dynamics tools, is presented. Two liposomes of different composition and physical stability were developed and a chaotic evaluation on the time series of their size and polydispersity was conducted. Results: The utilized models revealed instability, loss of information and order loss for both liposomes in due time, with important differentiations. An initial interpretation of the results is apposed, whereas the foundations for further investigating possible exploitation of the demonstrated nonlinearity and adaptability of artificial biomembranes is laid, with projection on biosystems. Conclusion: The present approach is expected to impact the application of lipidic nanoparticles and liposomes in various crucial fields, such as drug and vaccine delivery, providing useful information for both the academia and industry.

Phenomenological Model of Nonlinear Dynamics and Deterministic Chaotic Gas Migration in Bentonite: Experimental Evidence and Diagnostic Parameters

Understanding gas migration in compacted clay materials, e.g., bentonite and claystone, is important for the design and performance assessment of an engineered barrier system of a radioactive waste repository system, as well as many practical applications. Existing field and laboratory data on gas migration processes in low-permeability clay materials demonstrate the complexity of flow and transport processes, including various types of instabilities, caused by nonlinear dynamics of coupled processes of liquid–gas exchange, dilation, fracturing, fracture healing, etc., which cannot be described by classical models of fluid dynamics in porous media. We here show that the complexity of gas migration processes can be explained using a phenomenological concept of nonlinear dynamics and deterministic chaos theory. To do so, we analyzed gas pressure and gas influx (i.e., input) and outflux (i.e., output), recorded during the gas injection experiment in the compact Mx80-D bentonite sample, and calculated a set of the diagnostic parameters of nonlinear dynamics and chaos, such a global embedding dimension, a correlation dimension, an information dimension, and a spectrum of Lyapunov exponents, as well as plotted 2D and 3D pseudo-phase-space strange attractors, based on the univariate influx and outflux time series data. These results indicate the presence of phenomena of low-dimensional deterministic chaotic behavior of gas migration in bentonite. In particular, during the onset of gas influx in the bentonite core, before the breakthrough, the development of gas flow pathways is characterized by the process of chaotic gas diffusion. After the breakthrough, with inlet-to-outlet movement of gas, the prevailing process is chaotic advection. During the final phase of the experiment, with no influx to the sample, the relaxation pattern of gas outflux is resumed back to a process of chaotic diffusion. The types of data analysis and a proposed phenomenological model can be used to establish the basic principles of experimental data-gathering, modeling predictions, and a research design.

Fractals and discrete dynamics associated to prime numbers

Abstract On this work, it is shown that fractal theory can be applied in number theory to analyze the distribution of prime numbers. Based on the similarities between the distribution of prime numbers’ powers for each natural number and generalized Cantor’s sets, some symmetric properties are analyzed and used to propose a set of discrete dynamics that allow visualizing the recursive and symmetric properties of prime numbers, which leads to the proposition of an approximation of a fractal version of the distribution function of prime numbers. Then, with this set of discrete dynamics, it is proposed a prime numbers’ sieve algorithm that seems to describe the growth of a fractal and its fractal dimension is computed, suggesting that the fractal and chaotic behavior on the distribution of prime numbers emerges from: 1) the increase of the memory or dimension of the discrete dynamics, that introduces new frequencies of oscillation each stage and 2) the symmetry break due to the exponential growth of the length of the symmetries with respect to the domain of applicability of each discrete dynamics. Therefore, although chaotic, the distribution of prime numbers has a deterministic chaotic behavior, with symmetries and harmonics, that follows the classical path to chaos, i.e. from periodic to quasi periodic to chaos.

Deterministic chaos and forecasting in Amazon?s share prices

Research background: The application of non-linear analysis and chaos theory modelling on financial time series in the discipline of Econophysics.
 Purpose of the article: The main aim of the article is to identify the deterministic chaotic behavior of stock prices with reference to Amazon using daily data from Nasdaq-100.
 Methods: The paper uses nonlinear methods, in particular chaos theory modelling, in a case study exploring and forecasting the daily Amazon stock price.
 Findings & Value added: The results suggest that the Amazon stock price time series is a deterministic chaotic series with a lot of noise. We calculated the invariant parameters such as the maxi-mum Lyapunov exponent as well as the correlation dimension, managed a two-days-ahead forecast through phase space reconstruction and a grouped data handling method.

Вычислительный метод поиска особых точек на плоскости комплексного времени для исследования детерминировано-хаотических систем (на примере системы Э. Лоренца)

The general scheme of search and recognition (identification) of singular points of the solution of dynamic systems is present. Under singular points we understand not a features of the phase flow, but the poles of the functions of the solution components for analytical continuation into the plane of complex time. Pole orders may be different for different components. As examples for calculation, precisely known system exhibiting deterministic chaotic behavior – E. Lorentz system, and also indicates a general scheme for matching of the differential dynamic system solution with the special integer sequence (quantization).The introduction describes a method for searching for singular points on the plane of complex time, which is a specific variant of numerical integration with the optimal choice of the direction of the next step. Optimization is the minimization of the number of steps, that is, ultimately – the computational time. The second part of the paper describes an algorithm for searching for singular points of the layer closest to the real time axis, and this algorithm is numerically implemented for the Lorentz system. The third (final) part of the paper describes a possible application of the method for constructing quantum-mechanical models of multielectronic systems.

Determining the chaotic behaviour of copper prices in the long-term using annual price data

Mineral commodity prices are influenced by economic, technological, psychological, and geopolitical factors. Stochastic approaches, and time series and econometric techniques have been used to represent the dynamics of mineral commodity markets and predict prices. However, these techniques cannot provide a comprehensive representation of market dynamics because they do not recognise the relationship between these factors over time, and they are unable to capture both the evolution and the cumulative effects of these factors on prices. Stability of motion and chaos theories can detect sensitivity to initial conditions, and therefore the evolutionary patterns allowing a proper understanding and representation of mineral commodity market dynamics. Most of the techniques used to assess chaos require a colossal amount of data, so the use of small data sets to assess chaos has been largely criticised. Nevertheless, by definition, the dynamics of a chaotic system remain at different scales owing to its self-organisation features that exhibit ordered patterns in the absence of codes or rules. Therefore, any deterministic chaotic behaviour of mineral commodity prices can be captured by using small data sets if a detailed qualitative and quantitative analysis are carried out. This paper examines the chaotic behaviour of annual copper prices between 1900 and 2015. To do so, we combine chaos theory, stability of motion and statistical techniques to reconstruct the long-term dynamics of copper prices. First, we examine the time dependency and the presence of a strange attractor by a visual analysis of the time series and phase space reconstruction based on Takens’ theorem and determine embedding parameters. Then we examine the dynamic characteristics of the system which assesses its complexity and regularity patterns to measure the system’s entropy. Finally, we calculate the largest Lyapunov exponent λ to assess the sensitivity to initial conditions and determine chaotic behaviour supported by a surrogate test. We find that annual copper prices have a chaotic behaviour embedded in a high-dimensional space and short time delay. The study suggests that copper prices exhibit only a single state of low prices, which fluctuate through transitional periods of high prices. It challenges the assertion that metal markets have fluctuated over four major super cycles and debate the adequacy of stochastic and econometric models for representing mineral commodity market behaviour.This study recommends that the use of chaotic behaviour improves our understanding of mineral commodity markets and narrows the data searching, processing and monitoring requirements for forecasting. Therefore, it improves the performance of traditional techniques for selecting key factors that influence the market dynamics, and may also be used to select the most suitable algorithm for forecasting prices.

Chaos and periodicity in solar wind speed: cycle 23

The solar wind speed time series data from 1st January, 1997 to 28th October, 2003 has been pre-processed using simple exponential smoothing, discrete wavelet transform for denoising to investigate the underneath dynamics of it. Recurrence plot and recurrence quantification analysis has revealed that the time series is non-stationary one with deterministic chaotic behavior. The Hilbert-Huang Transform has been used in search of the underlying periods of the data series. Present investigation has revealed the periods of 21 days, 32.5 days, 43.6 days, 148.86 days, 180.7 days, 355.5 days, 403.2 days, 413.6 days, 490.72 days, 729.6 days, 1086.76 days, 1599.4 days and 1892.6 days.

A Study of the Dynamic Behaviour of Fine Particulate Matter in Santiago, Chile

We present here a study about the limitations found when trying to develop an accurate atmospheric particulate matter forecasting model based on real data, and evidence that the time series of fine particulate matter concentration exhibit deterministic chaotic behavior. We have calculated the Lyapunov exponents of PM2.5 time series obtained from measurements from four monitoring stations located in the city of Santiago, Chile, in recent years. Values obtained for the largest Lyapunov exponents turned out to be positive and ranging between 0.3 and 0.5 which, according to the theory of chaos, is a condition for the presence of deterministic chaos and random behavior in time series. Given the shape of decay of autocorrelation functions and values of correlation dimension and Hurst exponents, random behavior can be discarded: we therefore conclude that the series are chaotic and very sensitive to initial conditions. The study presented here can be replicated in other mid-sized cities that present similar situations to the city of Santiago, where complexity of topography, meteorology and seasonal trends favor the generation of high concentration episodes of atmospheric particulate matter and where a reliable air quality forecasting model may be important for environmental management.

Dynamics, stability, and statistics on lattices and networks

These lectures aim at surveying some dynamical models that have been widely explored in the recent scientific literature as case studies of complex dynamical evolution, emerging from the spatio-temporal organization of several coupled dynamical variables. The first message is that a suitable mathematical description of such models needs tools and concepts borrowed from the general theory of dynamical systems and from out-of-equilibrium statistical mechanics. The second message is that the overall scenario is definitely reacher than the standard problems in these fields. For instance, systems exhibiting complex unpredictable evolution do not necessarily exhibit deterministic chaotic behavior (i.e., Lyapunov chaos) as it happens for dynamical models made of a few degrees of freedom. In fact, a very large number of spatially organized dynamical variables may yield unpredictable evolution even in the absence of Lyapunov instability. Such a mechanism may emerge from the combination of spatial extension and nonlinearity. Moreover, spatial extension allows one to introduce naturally disorder, or heterogeneity of the interactions as important ingredients for complex evolution. It is worth to point out that the models discussed in these lectures share such features, despite they have been inspired by quite different physical and biological problems. Along these lectures we describe also some of the technical tools employed for the study of such models, e.g., Lyapunov stability analysis, unpredictability indicators for “stable chaos,” hydrodynamic description of transport in low spatial dimension, spectral decomposition of stochastic dynamics on directed networks, etc.

The comparative study of chaoticity and dynamical complexity of the low-latitude ionosphere, over Nigeria, during quiet and disturbed days

Abstract. The deterministic chaotic behavior and dynamical complexity of the space plasma dynamical system over Nigeria are analyzed in this study and characterized. The study was carried out using GPS (Global Positioning System) TEC (Total Electron Content) time series, measured in the year 2011 at three GPS receiver stations within Nigeria, which lies within the equatorial ionization anomaly region. The TEC time series for the five quietest and five most disturbed days of each month of the year were selected for the study. The nonlinear aspect of the TEC time series was obtained by detrending the data. The detrended TEC time series were subjected to various analyses for phase space reconstruction and to obtain the values of chaotic quantifiers like Lyapunov exponents, correlation dimension and also Tsallis entropy for the measurement of dynamical complexity. The observations made show positive Lyapunov exponents (LE) for both quiet and disturbed days, which indicates chaoticity, and for different days the chaoticity of the ionosphere exhibits no definite pattern for either quiet or disturbed days. However, values of LE were lower for the storm period compared with its nearest relative quiet periods for all the stations. The monthly averages of LE and entropy also show no definite pattern for the month of the year. The values of the correlation dimension computed range from 2.8 to 3.5, with the lowest values recorded at the storm period of October 2011. The surrogate data test shows a significance of difference greater than 2 for all the quantifiers. The entropy values remain relatively close, with slight changes in these values during storm periods. The values of Tsallis entropy show similar variation patterns to those of Lyapunov exponents, with a lot of agreement in their comparison, with all computed values of Lyapunov exponents correlating with values of Tsallis entropy within the range of 0.79 to 0.81. These results show that both quantifiers can be used together as indices in the study of the variation of the dynamical complexity of the ionosphere. The results also show a strong play between determinism and stochasticity. The behavior of the ionosphere during these storm and quiet periods for the seasons of the year are discussed based on the results obtained from the chaotic quantifiers.

NONLINEARITY AND CHAOS IN 8B SOLAR NEUTRINO FLUX SIGNALS FROM SUDBURY NEUTRINO OBSERVATORY

The Sudbury neutrino observatory (SNO) detects 8 B solar neutrino fluxes from both the D2O and Salt detector. In the present analysis we have taken into consideration the flux data from 2nd November, 1999 to 27th May, 2001 from the D2O detector and that from 26th July, 2001 to 28th August, 2003 from the Salt detector. We have applied Delay Vector Variance analysis, 0-1 test, correlation dimension analysis, largest Lyapunov exponent method, recurrence plot and recurrence quantification analysis to explore the complexity and chaosity in these two time series. Present study reveals deterministic chaotic behaviour of these two signals which in turn suggests that long-term forecasting is not possible for these two signals but short-term forecasting can be made provided the model for the process dynamics is known to us.

A study on chaotic behaviour of equatorial/low latitude ionosphere over Indian subcontinent, using GPS-TEC time series

The deterministic chaotic behaviour of ionosphere, over Indian subcontinent falling under equatorial/low latitude region, −0.3 to 22.19°N (geomagnetic), was studied using GPS-TEC time series. The values of Lyapunov exponent are low at Thiruvananthapuram and Agatti (−0.30 and 2.38°N, geomagnetic, respectively), and thereafter increase through Bangalore and Hyderabad (4.14 and 8.54°N, geomagnetic, respectively), and attain maximum at Mumbai (10.09°N, geomagnetic), which is near/at the edge of an anomaly crest. The values of correlation dimension computed for TEC time series are in the range 3.1–3.6, which indicate that equatorial/low latitude ionosphere can be described with four variables. Entropy values estimated for TEC time series show no appreciable latitudinal variabilites. The values of non-linear prediction error exhibit a trough, around the latitude sector, 4.14–16.15°N (Geomagnetic). Based on the values of the above quantifiers, the features of chaotic behaviour of equatorial/low latitude ionosphere are briefly discussed.

Chaotic behaviour of interplanetary magnetic field under various geomagnetic conditions

In the present study, the deterministic chaotic behaviour of interplanetary magnetic field (IMF) under various geomagnetic conditions of low and high solar active periods was analyzed, using the time series of IMF | B| and B z , by employing chaotic quantifiers like, Lyapunov exponent, Tsallis entropy, correlation dimension, and non-linear prediction error. We have investigated whether the chaotic behaviour of interplanetary magnetic field would modify, when it produces major geomagnetic storms, and how it depends on the phase of solar activity. The yearly average values of Lyapunov exponent for the time series of IMF | B| and B z , show solar flux dependence, whereas those values of entropy, correlation dimension and non-linear prediction error had no significant solar flux dependence. The yearly average values of entropy for quiet periods are higher compared to those values for major storm periods belonging to low/high solar active conditions, for both the time series | B| and B z .

Comparison of chaotic aspects of magnetosphere under various physical conditions using AE index time series

Abstract. The deterministic chaotic behaviour of magnetosphere was analyzed, using AE index time series. The significant chaotic quantifiers like, Lyapunov exponent, spatio-temporal entropy and nonlinear prediction error for AE index time series under various physical conditions were estimated and compared. During high solar activity (1991), the values of Lyapunov exponent for AE index time series representing quiet conditions (yearly mean = 0.5±0.1 min−1) have no significant difference from those values for corresponding storm conditions (yearly mean = 0.5±0.17 min−1). This implies that, for the cases considered here, geomagnetic storms may not be an additional source to increase or decrease the deterministic chaotic aspects of magnetosphere, especially during high solar activity. During solar minimum period (1994), the seasonal mean value of Lyapunov exponent for AE index time series belong to quiet periods in winter (0.7±0.11 min−1) is higher compared to corresponding value of storm periods in winter (0.36±0.09 min−1). This may be due to the fact that, stochastic part, which is Dst dependent could be more prominent during storms, thereby increasing fluctuations/stochasticity and reducing determinism in AE index time series during storms. It is observed that, during low solar active period (1994), the seasonal mean value of entropy for time series representing storm periods of equinox is greater than that for quiet periods. However, significant difference is not observed between storm and quiet time values of entropy during high solar activity (1991), which is also true for nonlinear prediction error for both low and high solar activities. In the case of both high and low solar activities, the higher standard deviations of yearly mean Lyapunov exponent values for AE index time series for storm periods compared to those for quiet periods might be due to the strong interplay between stochasticity and determinism during storms. It is inferred that, the external driving forces, mainly due to solar wind, make the solar-magnetosphere-ionosphere coupling more complex, which generates many active degrees of freedom with various levels of coupling among them, under various physical conditions. Hence, the superposition of a large number of active degrees of freedom can modify the stability/instability conditions of magnetosphere.

Differences in daytime and nighttime ionospheric deterministic chaotic behavior: GPS total electron content analyses

A study on the differences in deterministic chaotic behavior of midlatitude ionosphere during local daytime and nighttime, using GPS total electron content (TEC), under various geophysical conditions, is conducted. The noninteger value of correlation dimension, the positive value of Lyapunov exponent, the power spectra in broadband, etc., and the implementation of surrogate data test strongly indicate the chaoticity of the underlying dynamics of ionosphere. The values of correlation dimension obtained for daytime and nighttime TEC of a major storm period in winter measured at 35.1°N are 3.25 ± 0.03 and 2.35 ± 0.05, respectively, which indicate the fractal dimension of the attractor. The values of Lyapunov exponent are estimated as 0.1387 × 10−2 s−1 and 0.0979 × 10−2 s−1 for the above time series during daytime and nighttime, respectively. It is observed that at the lower sector (32.1–38.1°N), Lyapunov exponents of daytime are higher than those of nighttime, during a major storm period. The values of nonlinear prediction error during nighttime are higher than those of daytime for all the latitudes, except at 32.1°N, where the trend is opposite. During winter quiet times, the values of Lyapunov exponent of nighttime at the lower sector (32.1–38.1°N) have higher values than those of daytime, where as at the upper section (41.1–44.1°N), both have the same values. During equinoctial quiet times also, the above trend exists, by exhibiting higher values of Lyapunov exponent for nighttime compared to those of daytime, especially at the upper latitude sector (38.1–44.1°N) of the present study. The values of nonlinear prediction error, during quiet times in winter and equinox, generally show higher values at daytimes, compared to those at nighttimes. However, during summer, the opposite trend is observed, with higher values of the above parameter at daytime compared to those at nighttime, except at 32.1°N. This is a case study in a broad sense and the results presented here are based on the observations made for each event. The possible causes of chaotic aspects of ionosphere, based on the values of above quantifiers under various geophysical conditions, are discussed.

Differences in magnetic storm and quiet ionospheric deterministic chaotic behavior: GPS total electron content analyses

In the present study we have investigated whether deterministic chaotic behavior is exhibited by the GPS total electron content (TEC) fluctuations at midlatitude ionosphere, under various geophysical conditions, using nonlinear aspects. Local slopes of the logarithms of correlation sum for the time series of TEC representing the winter storm and quiet periods exhibit a clear plateau, indicating a fractal dimension of the attractor of the system (5.86 ± 0.09 for quiet time and 4.15 ± 0.03 for storm time, at 35°N). The observed positive values of the Lyapunov exponent (for quiet and storm times at 35.1°N, the values are 0.1695 × 10−2 s−1 and 0.1167 × 10−2 s−1, respectively) are another important criterion for chaoticity of the system during the above periods. To further confirm this, we have employed the surrogate data test. On the basis of the significance of difference of the original data and surrogates for various aspects, the surrogate data test rejects the null hypothesis that the time series of TEC during storm and quiet times represent a linear stochastic process. Also, we compared the chaotic behavior of TEC during disturbed and quiet periods, under different seasons and latitudes. For all seasons, the values of the Lyapunov exponent during quiet times are greater than those during storm periods for the latitude sector, 35°–41°N. The nonlinear aspects presented here imply that geomagnetic disturbances influence the stability/instability conditions of ionosphere due to the superposition of various active degrees of freedom generated by an external stochastic driver (solar wind), and could alter the inherent dynamics of the system, if the coupling is powerful.