Positive Maximal Lyapunov Exponent Research Articles

Domestic and Global Causes for Exchange Rate Volatility: Evidence From Turkey

Understanding which factors influence exchange rate movements is important for understanding economic development, trade patterns, investment decisions, and for designing economic policies. In this study, we allow for monetary, domestic and global financial variables to assess the relevant importance of each of the variables to exchange rate volatility in the case of Turkey. The paper investigates the dynamics of exchange rate volatility of the Turkish lira in a complementary perspective by employing both Generalized autoregressive conditional heteroskedasticity (GARCH) method and Lyapunov exponents over the period from March 1, 2019 to November 11, 2021. Firstly, decomposing the impact of domestic and global financial variables, the results of the GARCH model indicate that the exchange rate volatility is affected by Volatility index (VIX) and Credit default swaps (CDS). This result suggests that the exchange rate shocks experienced are mainly caused by global factors, therefore policymakers should focus on volatility spillovers caused by global financial markets. Secondly, detected positive maximal Lyapunov exponent shows that complexity in foreign exchange markets has been increased, market expectations lead to multiple-equilibria and diverging volatility eventually will generate recurrent spikes in currency value. These complementary findings have important implications for interventions of Central banks and preventing systemic risks, as well as portfolio and risk management. JEL Classification: C32, G14, G15.

Conjoined Lorenz twins-a new pseudohyperbolic attractor in three-dimensional maps and flows.

We describe new types of Lorenz-like attractors for three-dimensional flows and maps with symmetries. We give an example of a three-dimensional system of differential equations, which is centrally symmetric and mirror symmetric. We show that the system has a Lorenz-like attractor, which contains three saddle equilibrium states and consists of two mirror-symmetric components that are adjacent at the symmetry plane. We also found a discrete-time analog of this "conjoined-twins" attractor in a cubic three-dimensional Hénon map with a central symmetry. We show numerically that both attractors are pseudohyperbolic, which guarantees that each orbit of the attractor has a positive maximal Lyapunov exponent, and this property is preserved under small perturbations. We also describe bifurcation scenarios for the emergence of the attractors in one-parameter families of three-dimensional flows and maps possessing the symmetries.

Dynamics of Time-Delayed Optoelectronic Oscillators With Nonlinear Amplifiers and Its Potential Application to Random Numbers Generation

Time-delayed dynamical systems generally feature smooth nonlinear transfer functions in the feedback loop, such as polynomial or sinusoidal functions. As a consequence, the complexity of their dynamical behavior mainly originates from the time-delay. In this paper, we explore the opposite case where the nonlinear transfer function is complex ( cos <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> (sinh)), and therefore, non-smooth. We perform a bifurcation analysis of the system, and evidence that this novel type of time-delayed system can display a chaotic behavior characterized by positive maximum Lyapunov exponent and quasi-maximal entropy. The high entropy behavior of the system combined with post-processing are used to generate random numbers for small values of the feedback gain with an overall bit rate up to 1.478 Gb/s. Our theoretical results are in excellent agreement with experimental measurements, performed with an optoelectronic oscillator involving a complex transfer function designed ad hoc.

Capturing the continuous complexity of behaviour in Caenorhabditis elegans

Animal behaviour is often quantified through subjective, incomplete variables that mask essential dynamics. Here, we develop a maximally predictive behavioural-state space from multivariate measurements, in which the full instantaneous state is smoothly unfolded as a combination of short-time posture sequences. In the off-food behaviour of the roundworm Caenorhabditis elegans, we discover a low-dimensional state space dominated by three sets of cyclic trajectories corresponding to the worm’s basic stereotyped motifs: forward, backward and turning locomotion. We find similar results in the on-food behaviour of foraging worms and npr-1 mutants. In contrast to this broad stereotypy, we find variability in the presence of locally unstable dynamics with signatures of deterministic chaos: a collection of unstable periodic orbits together with a positive maximal Lyapunov exponent. The full Lyapunov spectrum is symmetric with positive, chaotic exponents driving variability balanced by negative, dissipative exponents driving stereotypy. The symmetry is indicative of damped–driven Hamiltonian dynamics underlying the worm’s movement control. Animal behaviour is characterized by repeated movements which can be difficult to analyse quantitatively. Here, the authors apply a data-driven framework based on theory of dynamical systems to characterize nematode behaviour and explain its complexity through deterministic chaotic dynamics.

Chaos Detection and Optimal Control in a Cannibalistic Prey–Predator System with Harvesting

This paper deals with a stage-structured predator–prey system which incorporates cannibalism in the predator population and harvesting in both population. The predator population is categorized into two divisions; adult predator and juvenile predator. The adult predator and prey species are harvested via hypothesis of catch-per-unit-effort, whereas juveniles are safe from being harvested. Mathematically, the dynamic behavior of the system such as existing conditions of equilibria with their stability is studied. The global asymptotic stability of prey-free equilibrium point and nonzero equilibrium point, if they exist, is proved by considering respective Lyapunov functions. The system undergoes transcritical and Hopf–Andronov bifurcations. The impacts of predator harvesting rate and prey harvesting rate on the system are analyzed by taking them as bifurcation parameters. The route to chaos is discussed by showing maximum Lyapunov exponent to be positive with sensitivity dependence on the initial conditions. The chaotic behavior of the system is confirmed by positive maximum Lyapunov exponent and non-integer Kaplan–Yorke dimension. Numerical simulations are executed to probe our theoretic findings. Also, the optimal harvesting policy is studied by applying Pontryagin’s maximum principle. Harvesting effort being an emphatic control instrument is considered to protect prey–predator population, and preserve them also through an optimal level.

Effects of Inducing Gamma Oscillations in Hippocampal Subregions DG, CA3, and CA1 on the Potential Alleviation of Alzheimer's Disease-Related Pathology: Computer Modeling and Simulations.

The aim of this study was to evaluate the possibility of the gamma oscillation function (40–130 Hz) to reduce Alzheimer’s disease related pathology in a computer model of the hippocampal network dentate gyrus, CA3, and CA1 (DG-CA3-CA1) regions. Methods: Computer simulations were made for a pathological model in which Alzheimer’s disease was simulated by synaptic degradation in the hippocampus. Pathology modeling was based on sequentially turning off the connections with entorhinal cortex layer 2 (EC2) and the dentate gyrus on CA3 pyramidal neurons. Gamma induction modeling consisted of simulating the oscillation provided by the septo-hippocampal pathway with band frequencies from 40–130 Hz. Pathological models with and without gamma induction were compared with a control. Results: In the hippocampal regions of DG, CA3, and CA1, and jointly DG-CA3-CA1 and CA3-CA1, gamma induction resulted in a statistically significant improvement in terms of increased numbers of spikes, spikes per burst, and burst duration as compared with the model simulating Alzheimer’s disease (AD). The positive maximal Lyapunov exponent was negative in both the control model and the one with gamma induction as opposed to the pathological model where it was positive within the DG-CA3-CA1 region. Gamma induction resulted in decreased transfer entropy in accordance with the information flow in DG → CA3 and CA3 → CA1. Conclusions: The results of simulation studies show that inducing gamma oscillations in the hippocampus may reduce Alzheimer’s disease related pathology. Pathologically higher transfer entropy values after gamma induction returned to values comparable to the control model.

Computer Model of Synapse Loss During an Alzheimer's Disease-Like Pathology in Hippocampal Subregions DG, CA3 and CA1-The Way to Chaos and Information Transfer.

The aim of the study was to compare the computer model of synaptic breakdown in an Alzheimer’s disease-like pathology in the dentate gyrus (DG), CA3 and CA1 regions of the hippocampus with a control model using neuronal parameters and methods describing the complexity of the system, such as the correlative dimension, Shannon entropy and positive maximal Lyapunov exponent. The model of synaptic breakdown (from 13% to 50%) in the hippocampus modeling the dynamics of an Alzheimer’s disease-like pathology was simulated. Modeling consisted in turning off one after the other EC2 connections and connections from the dentate gyrus on the CA3 pyramidal neurons. The pathological model of synaptic disintegration was compared to a control. The larger synaptic breakdown was associated with a statistically significant decrease in the number of spikes (R = −0.79, P < 0.001), spikes per burst (R = −0.76, P < 0.001) and burst duration (R = −0.83, P < 0.001) and an increase in the inter-burst interval (R = 0.85, P < 0.001) in DG-CA3-CA1. The positive maximal Lyapunov exponent in the control model was negative, but in the pathological model had a positive value of DG-CA3-CA1. A statistically significant decrease of Shannon entropy with the direction of information flow DG->CA3->CA1 (R = −0.79, P < 0.001) in the pathological model and a statistically significant increase with greater synaptic breakdown (R = 0.24, P < 0.05) of the CA3-CA1 region was obtained. The reduction of entropy transfer for DG->CA3 at the level of synaptic breakdown of 35% was 35%, compared with the control. Entropy transfer for CA3->CA1 at the level of synaptic breakdown of 35% increased to 95% relative to the control. The synaptic breakdown model in an Alzheimer’s disease-like pathology in DG-CA3-CA1 exhibits chaotic features as opposed to the control. Synaptic breakdown in which an increase of Shannon entropy is observed indicates an irreversible process of Alzheimer’s disease. The increase in synapse loss resulted in decreased information flow and entropy transfer in DG->CA3, and at the same time a strong increase in CA3->CA1.

Comparison of Variations of TEC at Disturbed and Quiet Time Using Nonlinear Dynamics Tools

AbstractWe analyze the underlying dynamics at three different latitudes of ionosphere through total electron content‐series for 4 months. The global dynamics reconstructed from the data is shown to be deterministic with low dimensional chaotic attractors at all the locations, and the dynamics is seen to be less complex at the midlatitude station with smaller dimension of correlation (CD) and maximum Lyapunov exponent (MLE). The influence of geomagnetic storms on the ionospheric dynamics is studied, by analyzing the local dynamics during disturbed and quiet periods, reconstructed from segments of total electron content‐series of shorter duration. With positive MLEs and smaller CDs, the local dynamics is seen to be deterministic and chaotic at the low latitude, midlatitude, and high latitude during both the disturbed and quiet days. However, it is interesting to observe that the dynamics is less chaotic and complex during the disturbed days of the entire period of observation compared to the adjacent quiet days at the three locations. This is further evidenced by sample predictions made using nonlinear tools, yielding better predictions during storms. The smaller CD value during disturbed periods indicates that the geomagnetic activities tend to make a few of the variables dominant enough to control the essential dynamics of the system. Further, the MLEs of the disturbed days are found to increase nearly in a linear fashion from beginning to end of the observation period at all the locations considered. This is an indication of the strong influence of geomagnetic activities on the ionosphere and might have caused the restructuring of the ionospheric dynamics.

Variety of strange pseudohyperbolic attractors in three-dimensional generalized Hénon maps

In the present paper we focus on the problem of the existence of strange pseudohyperbolic attractors for three-dimensional diffeomorphisms. Such attractors are genuine strange attractors in that sense that each orbit in the attractor has positive maximal Lyapunov exponent and this property is robust, i.e., it holds for all close systems. We restrict attention to the study of pseudohyperbolic attractors that contain only one fixed point. Then we show that three-dimensional maps may have only 5 different types of such attractors, which we call the discrete Lorenz, figure-8, double-figure-8, super-figure-8, and super-Lorenz attractors. We find the first four types of attractors in three-dimensional generalized Hénon maps of form x̄=y, ȳ=z, z̄=Bx+Az+Cy+g(y,z), where A,B and C are parameters (B is the Jacobian) and g(0,0)=g′(0,0)=0.

Synthetic Modeling of Autonomous Learning with a Chaotic Neural Network

We investigate the possible role of intermittent chaotic dynamics called chaotic itinerancy, in interaction with nonsupervised learnings that reinforce and weaken the neural connection depending on the dynamics itself. We first performed hierarchical stability analysis of the Chaotic Neural Network model (CNN) according to the structure of invariant subspaces. Irregular transition between two attractor ruins with positive maximum Lyapunov exponent was triggered by the blowout bifurcation of the attractor spaces, and was associated with riddled basins structure. We secondly modeled two autonomous learnings, Hebbian learning and spike-timing-dependent plasticity (STDP) rule, and simulated the effect on the chaotic itinerancy state of CNN. Hebbian learning increased the residence time on attractor ruins, and produced novel attractors in the minimum higher-dimensional subspace. It also augmented the neuronal synchrony and established the uniform modularity in chaotic itinerancy. STDP rule reduced the residence time on attractor ruins, and brought a wide range of periodicity in emerged attractors, possibly including strange attractors. Both learning rules selectively destroyed and preserved the specific invariant subspaces, depending on the neuron synchrony of the subspace where the orbits are situated. Computational rationale of the autonomous learning is discussed in connectionist perspective.

Melt index prediction by aggregated RBF neural networks trained with chaotic theory

Melt index (MI) is considered as one of the most significant parameter to determine the quality and the grade of the practical polypropylene polymerization products. In this paper, the chaotic property of the melt index series is put forward for the first time, where its correlation dimension D2 is obtained to be 1.57, and the maximum Lyapunov exponent is 0.143. Through a fractional dimension and a positive maximum Lyapunov exponent, we demonstrate that the random nature of melt index can be explained as a chaotic phenomenon. Then, the phase space of the melt index series are reconstructed based on the obtained chaotic characteristics, and a novel RBF prediction model for MI prediction (RBF-chaos) are therefore further set up naturally to characterized its strong nonlinear and correlated relationships under chaotic theory. Furthermore, the detailed comparative researches between the RBF-chaos model and the other forecast models reported in the open literatures are carried out, and the research results show that the proposed chaos based neural network approach is superior to the previous models without considering chaotic characteristics.

Characterization of chaotic multiscale features on the time series of melt index in industrial propylene polymerization system

The chaos characteristics of melt index have been first explored, and the Hilbert–Huang transform method and time delay embedding method are applied to multiscale dynamic analysis on the time series of the melt index (MI) in the propylene polymerization industry. The research results show that the embedding delay is 2, the embedding dimension is 5, the correlation dimension D2 is 1.57, and the maximum Lyapunov exponent is 0.143 for the melt index series, which provide clear evidence of chaotic multiscale features in the propylene polymerization process. Three intrinsic mode functions (IMFs) are decomposed from the melt index time series; the presence of non-integer fractal correlation dimension and positive finite maximum Lyapunov exponent are found in some IMF components. The PP melt index series are divided into two chaotic signals, a determined signal and a random signal respectively, and its complexity is therefore reduced. Furthermore, the coupling of subscale structures of the propylene polymerization is explored with the dimension of interaction dynamics and a robust algorithm for detecting interdependence. It is found that IMF(2) is the main driver in the coupling system of IMF(1)and IMF(2). All these provide a guideline for studying propylene polymerization process with chaotic multiscale theory and may offer more candidate tools to model and control propylene polymerization system in the future.

Predictability and chaotic nature of daily streamflow

The predictability of a chaotic series is limited to a few future time steps due to its sensitivity to initial conditions and the exponential divergence of the trajectories. Over the years, streamflow has been considered as a stochastic system. In this study, the chaotic nature of daily streamflow is investigated using autocorrelation function, Fourier spectrum, correlation dimension method (Grassberger-Procaccia algorithm) and false nearest neighbour method. Embedding dimensions of 6-7 obtained, indicate the possible presence of low-dimensional chaotic behaviour. The predictability of the system is estimated by calculating the system's Lyapunov exponent. A positive maximum Lyapunov exponent of 0.167 indicates that the system is chaotic and unstable with a maximum predictability of only 6 days. These results give a positive indication towards considering streamflow as a low dimensional chaotic system than as a stochastic system. Prediction is done using local polynomial method for a range of embedding dimensions and delay times. The uncertainty in the chaotic streamflow series is reasonably captured through the ensemble approach using local polynomial method.

Predictability and Chaotic Nature of Daily Streamflow

The predictability of a chaotic series is limited to a few future time steps due to its sensitivity to initial conditions and the exponential divergence of the trajectories. Over the years, streamflow has been considered as a stochastic system. In this study, the chaotic nature of daily streamflow is investigated using autocorrelation function, Fourier spectrum, correlation dimension method (Grassberger-Procaccia algorithm) and false nearest neighbour method. Embedding dimensions of 6-7 obtained, indicate the possible presence of low-dimensional chaotic behaviour. The predictability of the system is estimated by calculating the system’s Lyapunov exponent. A positive maximum Lyapunov exponent of 0.167 indicates that the system is chaotic and unstable with a maximum predictability of only 6 days. These results give a positive indication towards considering streamflow as a low dimensional chaotic system than as a stochastic system. Prediction is done using local polynomial method for a range of embedding dimensions and delay times. The uncertainty in the chaotic streamflow series is reasonably captured through the ensemble approach using local polynomial method.

TIME-EVOLVING STATISTICS OF CHAOTIC ORBITS OF CONSERVATIVE MAPS IN THE CONTEXT OF THE CENTRAL LIMIT THEOREM

We study chaotic orbits of conservative low-dimensional maps and present numerical results showing that the probability density functions (pdfs) of the sum of N iterates in the large N limit exhibit very interesting time-evolving statistics. In some cases where the chaotic layers are thin and the (positive) maximal Lyapunov exponent is small, long-lasting quasi-stationary states (QSS) are found, whose pdfs appear to converge to q-Gaussians associated with nonextensive statistical mechanics. More generally, however, as N increases, the pdfs describe a sequence of QSS that pass from a q-Gaussian to an exponential shape and ultimately tend to a true Gaussian, as orbits diffuse to larger chaotic domains and the phase space dynamics becomes more uniformly ergodic.

Statements on chaos control designs, including a fractional order dynamical system, applied to a “MEMS” comb-drive actuator

In this work, we deal with a micro electromechanical system (MEMS), represented by a micro-accelerometer. Through numerical simulations, it was found that for certain parameters, the system has a chaotic behavior. The chaotic behaviors in a fractional order are also studied numerically, by historical time and phase portraits, and the results are validated by the existence of positive maximal Lyapunov exponent. Three control strategies are used for controlling the trajectory of the system: State Dependent Riccati Equation (SDRE) Control, Optimal Linear Feedback Control, and Fuzzy Sliding Mode Control. The controls proved effective in controlling the trajectory of the system studied and robust in the presence of parametric errors.

Interactions between two neural populations: A mechanism of chaos and oscillation in neural mass model

Neural mass model developed by Lopes da Silva et al. is able to describe limit cycle behavior in Electroencephalography (EEG) of alpha rhythm and exhibit complex dynamics between cortical areas. In this work, we extend Grimbert and Faugeras's work to study the dynamical behavior caused by interaction of cortical areas. The model is developed with the coupling of two neural populations. We show that various attractors, including equilibrium points, periodic solutions and chaotic strange attractors, could coexist in different ways with different value of the connectivity parameters. The main findings are that: (1) The stable equilibrium points only appear with a small value of the parameter. (2) While the alpha activities always exist for both two populations with proper initial conditions. Interestingly, the coexistence of the multiple alpha-to-epileptic activities implies the multiple coupling ways for these activities in phase. Two neuronal populations with epileptic activities could interact with multiple rhythms depending on their connectivity. (3) For particular interest, chaotic behaviors are identified in four regions divided by the connectivity parameter with the positive maximal Lyapunov exponent. The four types of chaotic attractors have their own structures, but all of them are related to the epileptic activities.

Multiscale dynamic analysis of blast furnace system based on intensive signal processing

In this paper, the Hilbert-Huang transform method and time delay embedding method are applied to multiscale dynamic analysis on the time series of silicon content in hot metal collected from a medium-sized blast furnace with the inner volume of 2500 m3. The results provide clear evidence of multiscale features in blast furnace ironmaking process. Ten intrinsic mode functions (IMFs) are decomposed from the silicon content time series; the presence of noninteger fractal dimension, positive finite Kolmogorov entropy, and positive finite maximum Lyapunov exponent are found in some IMF components. In addition, the coupling of subscale structures of blast furnace system is studied using the dimension of interaction dynamics and a robust algorithm for detecting interdependence. It is found that IMF(3) is the main driver in the coupling system IMF(2) and IMF(3) while for the coupling system IMF(3) and IMF(4) neither subsystem can act as the driver. All these provide a guideline for studying blast furnace ironmaking process with multiscale theory and methods, and may open way for more candidate tools to model and control blast furnace system in the future.

Forecasting weekly freight rates for one-year time charter 65 000 dwt bulk carrier, 1989–2008, using nonlinear methods

This paper forecast/predicted the one-year time charter weekly freight rates earned by a 65 000 dwt bulk carrier using 996 weeks of data from 1989 to 2008. First, the need and the importance, but also the futility, of forecasting is discussed in shipping. This is a volatile industry that can be easily likened to the roulette. The introduction is followed by a literature review that has examined the principal recent works in this area and presented a critique of earlier works. Most of the research studied dealt with the shipping industry per se. Since the methods used are considered as a departure from the classical Random Walk, a comprehensive section of the paper is devoted to the methodology of nonlinear, chaotic and deterministic methods. The relevant time series have been transformed into stationary ones, as this is the proper practice (using first logarithmic differences). The time series were tested for randomness (identically and independently distributed) and for long-term correlation using BDS statistic. The methods used were: Rescaled Range Analysis and the related Hurst Exponent; Power Spectrum Analysis; V-statistic and BDS Statistic (using software MATLAB 5.3 and NLTSA V.2.0/2000). The analysis of the data was presented in three separate sections. The relevant ‘attractor’ of the system has been graphically shown. System's dimension has been calculated, which was found to be non-integer, fractal and equal to 3.95. This finding permitted us to proceed to forecasting, as this is a case of a low dimensional chaos (3.95 < 10 dimensions). In order for the predictions to be robust, the prediction horizon allowed was found equal to 8.24 weeks, as indicated by the positive maximum Lyapunov exponent (0.12 rounded). Then NLTSA software was used to make prediction inside- and forecasting outside- the sample, using by selection nonlinear Principal Components and Kernel Density Estimation methods.

Stochastic vagal modulation of cardiac pacemaking may lead to erroneous identification of cardiac “chaos”

Fluctuations in the time interval between two consecutive R-waves of electrocardiogram during normal sinus rhythm may result from irregularities in the autonomic drive of the pacemaking sinoatrial node (SAN). We use a biophysically detailed mathematical model of the action potentials of rabbit SAN to quantify the effects of fluctuations in acetylcholine (ACh) on the pacemaker activity of the SAN and its variability. Fluctuations in ACh concentration model the effect of stochastic activity in the vagal parasympathetic fibers that innervate the SAN and produce varying rates of depolarization during the pacemaker potential, leading to fluctuations in cycle length (CL). Both the estimated maximal Lyapunov exponent and the noise limit of the resultant sequence of fluctuating CLs suggest chaotic dynamics. Apparently chaotic heart rate variability (HRV) seen in sinus rhythm can be produced by stochastic modulation of the SAN. The identification of HRV data as chaotic by use of time series measures such as a positive maximal Lyapunov exponent or positive noise limit requires both caution and a quantitative, predictive mechanistic model that is fully deterministic.