Minimum Embedding Research Articles

Chaotic behaviour of an electrical analogue to the mechanical double pendulum

In this paper the analogy between a mechanical double pendulum and an oscillating electrical system is presented. Instead of using analytic equations, we used the MultiSim circuit simulation environment in order to reproduce and interpret the response of the electrical oscillator. The electrical double pendulum presents a chaotic regime which is studied quantitatively by means of state space reconstruction. For this purpose the optimal delay time is calculated and the minimum embedding dimension is found with the method of False Nearest Neighbors.

Minimum embedding of Steiner triple systems into [formula omitted]-designs II

A ( K 4 − e ) -design of order v + w embeds a given Steiner triple system if there is a subset of v points on which the graphs of the design induce the blocks of the original Steiner triple system. It has been established that w ≥ v / 3 , and that when equality is met, such a minimum embedding of an STS( v ) exists, except when v = 15 . Equality only holds when v ≡ 15 , 27 ( mod 30 ) . One natural question is: What is the smallest order w such that some STS ( v ) can be embedded into a ( K 4 − e ) -design of order v + w ? We solve the problem for 7 of the 10 congruence classes modulo 30.

Hydrogen Diffusion Behavior in Perfect <EM>δ</EM>-Pu Metal

The diffusion behavior of hydrogen atom in perfect δ-Pu metal was studied with density functional theory (DFT) and periodical model. The stablest site for hydrogen is the octahedral interstitial site. A single hydrogen atom has the minimum embedding energy about -3.12 and -2.22 eV in the octahedral interstitial site of fcc δ-Pu at non-spin polarization and spin-polarization levels, respectively. The embedding energy in tetrahedral interstitial site is little larger than that in tetrahedral interstitial site. Hydrogen atom in δ-Pu crystal most preferably diffuses along the path linked with different interstitials. The diffusion barrier along the path from octahedral interstitial site to tetrahedral interstitial site is 1.06 eV. The diffusion barrier along the reverse path is 0.38 eV. In addition, the diffusion barrier along the path from tetrahedral interstitial site to tetrahedral interstitial site is 1.83 eV, and the path from octahedral interstitial site to octahedral interstitial site corresponds to the highest diffusion barrier.

Characterization of chaos in air pollutants: A Volterra–Wiener–Korenberg series and numerical titration approach

The present study attempts to provide an insight into the chaotic nature of air pollutants by applying the recent developments in the field of nonlinear dynamics. The Volterra–Wiener–Korenberg (VWK) series approach by Barahona and Poon [1996. Detection of nonlinear dynamics in short, noisy time series. Nature 381, 215–217] has been used to investigate the nonlinearity of O 3, NO, NO 2 and CO time series at two urban stations, namely—Hohenpeissenberg and Jungfraujoch. Nonlinearity has been detected in NO 2 and CO time series at both the stations. The numerical titration technique [Poon, C., Barahona, M., 2001. Titration of chaos with added noise. Proceedings of the National Academy of Sciences 98, 7107–7112] reveals that the dynamics of NO 2 and CO are indeed governed by deterministic chaos. Cao's method [Cao, L., 1997. Practical method for determining the minimum embedding dimension of a scalar time series. Physica D 110, 43–50] to determine the minimum embedding dimension further reveals that probably the dynamics of both NO 2 and CO time series are manifestations of high-dimensional chaos. It is interesting to note that similar chaotic characteristic of NO 2 and CO time series have been observed at both the sites indicating a possible universality in their chaotic nature in the ambient urban environment.

Minimum embedding of Steiner triple systems into [formula omitted]-designs I

A ( K 4 - e ) -design on v + w points embeds a Steiner triple system (STS) if there is a subset of v points on which the graphs of the design induce the blocks of a STS. It is established that w ⩾ v / 3 , and that when equality is met that such a minimum embedding of an STS ( v ) exists, except when v = 15 .

Time series analysis in a single transistor chaotic circuit

An externally triggered single transistor chaotic circuit is presented and its operation is simulated in order to reveal the presence of chaos. Time series analysis is performed following Grassberger and Procaccia’s method while correlation and minimum embedding dimension are, respectively, calculated. As shown, the distribution of the time intervals between successive transitions exhibits a frequency dependent power law. Moreover, this power law and the associated phase portraits indicate that the considered system crisis is accompanied by an intermittent behaviour of the corresponding orbits.

Total curvature, ropelength and crossing number of thick knots

AbstractWe first study the minimum total curvature of a knot when it is embedded on the cubic lattice. Let $\mathcal{K}$ be a knot or link with a lattice embedding of minimum total curvature $\tau(\mathcal{K})$ among all possible lattice embeddings of $\mathcal{K}$. We show that there exist positive constants c1 and c2 such that $c_1\sqrt{Cr(\mathcal{K})}\le \tau(\mathcal{K})\le c_2 Cr(\mathcal{K})$ for any knot type $\mathcal{K}$. Furthermore we show that the powers of $Cr(\mathcal{K})$ in the above inequalities are sharp hence cannot be improved in general. Our results and observations show that lattice embeddings with minimum total curvature are quite different from those with minimum or near minimum lattice embedding length. In addition, we discuss the relationship between minimal total curvature and minimal ropelength for a given knot type. At the end of the paper, we study the total curvatures of smooth thick knots and show that there are some essential differences between the total curvatures of smooth thick knots and lattice knots.

Minimum embedding of [formula omitted]-designs into [formula omitted

A triple system of order v + w and index λ embeds a P 3 -design on v points if there is a subset of v points on which the K 3 -blocks induce the blocks of a P 3 -design. In this paper is settled the minimum embedding of a P 3 -design of order v into a TS ( v + w , λ ) .

Enhancement of Chinese speech based on nonlinear dynamics

Based on recently observed nonlinear dynamic features of human speech, the local projection (LP) method, originally developed for noisy chaotic time series, is generalized and adapted to the enhancement of Chinese speech. The analysis of minimum embedding dimensions estimated by the false nearest neighbor algorithm shows that all the basic phonemes and syllables in Chinese can be faithfully embedded in some low-dimensional phase space. Over-embedding is applied to reconstruct the dynamics of continuous speech in some extended phase space of higher dimension, thus solving the problem of nonstationarity in continuous speech. A generalization of the LP method, named the local subspace method, is presented for speech enhancement in the phase space. It is demonstrated that, the local subspace method is essentially an extension of the well-known linear subspace technique in the local phase space, and the LP method is the least square case of this generalization. Noise reduction is then carried out in the local phase space. Results show that the LP method, with 2 or 3 iterations, achieves better performances than the local subspace method. For both isolated and continuous speech with additive white noise, experiments show the superiority of the LP method over two other popular algorithms.

Underactuated robot dynamic modelling and control based on differential geometry

The second order nonholonomic constraint underac- tuated robot dynamic modeling and control are investigated by applying configuration manifold minimum embedding model. So the dynamic equation is reduced and the basis for further study on underactuated robot's dynamic modeling and control is found.The existence of passive joints is expressed by one homogeneous linear equation.And solution structure for dynamic equation based on minimum embedding model is posed.Another homogeneous linear equation associated with actuated joints and control inputs is presented by introducing computed torque.And one underdetermined linear equation set related with control inputs and theirs solutions is formulated by analyzing the relationship between control inputs and solutions of the two equations.Improved dynamic equation of underactuated robot is developed by solving the equations.The minimum embedding model control is performed by full actuated robot control law.And the underactuated robot joint space control is realized.Finally,the above principle feasibility and effectiveness are illustrated with a planar two bar underactuated robot.

Efficient algorithms for minimum congestion hypergraph embedding in a cycle

The minimum congestion hypergraph embedding in a cycle (MCHEC) problem is to embed the hyperedges of a hypergraph as paths in a cycle with the same node set such that the maximum congestion (the maximum number of paths that use any single edge in the cycle) is minimized. The MCHEC problem has many applications, including optimizing communication congestions in computer networks and parallel computing. The problem is NP-hard. In this paper, we give a 1.8-approximation algorithm for the MCHEC problem. This improves the previous 2-approximation results. Our algorithm has the optimal time complexity O(mn) for a hypergraph with m hyperedges and n nodes. We also propose an algorithm which finds an embedding with the optimal congestion L* for the MCHEC problem in O(n(nL*)/sup L*/) time. This improves the previous O((mn)/sup L*+1/) time algorithm.

Non-Linear Analyses of Heart Rate Variability During Heavy Exercise and Recovery in Cyclists

We investigated the time course of RR interval variability during exercise and subsequent 50 minutes of recovery in seven well-trained male cyclists who performed an exercise with 3 successive 8 min stages at 40 %, 70 % and 90 % of their maximal oxygen uptake. The goal of the study was to check whether the decrease in the amplitude of heart rate variability during heavy exercise was accompanied by changes in the chaotic structure of the fluctuations. Heart rate variability was analysed in the temporal and frequency domain using traditional tools and using non-linear methods (Largest Lyapunov Exponent, Detrended Fluctuation Analysis, Minimum Embedding Dimension). When compared to rest, variability at the heaviest exercise intensity was significantly lower (RR: 0.94 +/- 0.22 vs. 0.34 +/- 0.01 ms; SDRR: 0.11 +/- 0.04 vs. 0.01 +/- 0.00 ms) due to a decrease in both LF (2101 +/- 1450 vs. 0.14 +/- 0.09 ms (2) . Hz (-1)) and HF spectral energy (1148 +/- 1126 vs. 7.88 +/- 9.24 ms (2) . Hz (-1)). Non-linear analyses showed that heart rate variability remained chaotic whatever the exercise intensity (the largest Lyapunov exponent was positive at 90 % of the maximal oxygen uptake), with a fractal organisation that tended towards white noise (DFA value close to 0.5) during heavy exercise. During recovery, temporal and spectral variables came back to their rest values within about 30 minutes following an exponential pattern. Non-linear analyses revealed that heartbeat dynamics were disorganised at the beginning of recovery, and involved more regulating systems than at rest, even after 50 minutes of recovery. We concluded that, during heavy exercise, heart rate variability was mainly influenced by other factors than autonomous nervous system, and suggest that mechanical or neurological couplings between the cardiac, locomotor and respiratory systems could play an important part in the observed changes.

EFFECT OF NOISE ON THE AVERAGED FALSE NEIGHBORS METHOD APPLIED TO SIMULATED AND EXPERIMENTAL CHAOTIC TIMES SERIES

We examine the influence of noise on the averaged false neighbors method (AFN), which was proposed by L. Cao to identify the deterministic nature and estimate the minimum embedding dimension of time series. In a numerical experiment, we add a raising amount of noise to time series from numerical simulation of well known chaotic models and to experimentally recorded time series. We find that the AFN method acknowledges the deterministic nature of the underlying process of these noisy times series. For the assessment of the embedding, sensitivity to noise generally decreases when the dimension of the underlying dynamical process increased.

Time series analysis in chaotic diode resonator circuit

Abstract A diode resonator chaotic circuit is presented. Multisim is used to simulate the circuit and show the presence of chaos. Time series analysis performed by the method proposed by Grasberger and Procaccia. The correlation and minimum embedding dimension ν and mmin, respectively, were calculated. Also the corresponding Kolmogorov entropy was calculated.

A 1.5 approximation algorithm for embedding hyperedges in a cycle

The problem of minimum congestion hypergraph embedding in a cycle (MCHEC) is to embed the hyperedges of a hypergraph as adjacent paths around a cycle, such that the maximum congestion over any physical link in the cycle is minimized. The problem is NP-complete in general, but solvable in polynomial time when all hyperedges contain exactly two vertices. In this paper, we first formulate the problem as an integer linear program (ILP). Then, a solution with approximation bound of 1.5(opt + 1) is presented by using a clockwise (2/3)-rounding algorithm, where opt denotes the optimal value of maximum congestion. To our knowledge, this is the best approximation bound known for the MCHEC problem.

Analysis of heart rhythm variability by linear and non-linear dynamics methods

Spectral analysis of heart rhythm variability is a noninvasive method to study cardiovascular autonomic control. Nonlinear methods of analysis of heart rhythm variability may provide an additional information on properties of RR interval dynamics, which cannot be revealed by linear methods. The aim of this study was to investigate and to compare some parameters of deterministic chaos, tractal and spectral properties of heart rhythm variability in healthy subjects. Sixty healthy subjects, who participated in this study, were divided in four different age groups: children (< 15 years old), young (15-24 years), adults (25-39 years) and middle-aged (40-61 years). We analyzed the heart period variability extracted from a 5-minute electrocardiograms recordings in supine rest. Short-term fractal scaling exponent, sample entropy, minimum embedding dimension and the largest Lyapunov exponent along with the spectral measures (low-frequency and high-frequency power) were determined. Heart frequency, short-term fractal scaling exponent, and the largest Lyapunov exponent did not differ between the tested groups (p > 0.05). Middle-aged subjects had the lower low-frequency power as compared to the children (p < 0.001) and the young (p < 0.05), and the lower high-frequency power as compared to the other three groups (p < 0.001). Middle-aged also had a significantly lower power of high frequency and sample entropy as compared to the other three groups (p < 0.001). Children had lower values of minimum embedding dimension compared to the middle aged (p < 0.001). Significant negative correlation between the age of the tested subjects, and the low-and, high-frequency power and sample entropy was found. The obtained results suggested that healthy aging was associated with significant alteration in heart rhythm dynamics reflected on a higher regularity and lower variability of heart rhythm time-series. Significant decrease in a high-frequency power with aging suggested that reduction in parasympathetic activity was the basic cause of these changes.

Determining the minimum embedding dimension of nonlinear time series based on prediction method

Determining the embedding dimension of nonlinear time series plays an important role in the reconstruction of nonlinear dynamics. The paper first summarizes the current methods for determining the embedding dimension. Then, inspired by the fact that the optimum modelling dimension of nonlinear autoregressive (NAR) prediction model can characterize the embedding feature of the dynamics, the paper presents a new idea that the optimum modelling dimension of the NAR model can be taken as the minimum embedding dimension. Some validation examples and results are given and the present method shows its advantage for short data series.

Oscillations in the α-Si/Si(p)/Si(n) device: embedding- and correlation dimensions

Electrical oscillations appear in the NDR-region of the current controlled I– U characteristic of the α-Si/Si(p)/Si(n) device. The corresponding attractor is characterized by its correlation- and minimum embedding dimension, ν and m min, respectively. As the voltage U BE, applied in the Si(p)-region increases, m min remains constant, equal to 2, and ν increases obtaining always non-integer values, lower than m min.

Measures of LLE of heart rate in different frequency bands: a possible measure of relative vagal and sympathetic activity

In this study, we investigated spectral measures and measures of nonlinear dynamics and chaos of heart rate (HR) time series in 10 normal controls and 10 patients with panic disorder (a condition associated with anxiety) during awake and sleep periods and 28 normal control subjects and 42 age-matched patients with panic disorder, in supine and standing postures. We obtained minimum embedding dimension (MED) and largest Lyapunov exponent (LLE) of unfiltered (UF) and filtered HR time series. We digitally filtered the series into very low-frequency series (VLF: <0.04 Hz ) using a low-pass filter, low-frequency series (LF: 0.04– 0.15 Hz ) using a band-pass filter and high-frequency (HF: 0.15– 0.5 Hz ) series using a high-pass filter. MED quantifies system's complexity and LLE, chaos and predictability. During sleep, there was a significant increase in LLE of UF and HF series in normal subjects. There was also a highly significant increase in LLE ( p<0.00001) in standing posture, due mainly to an increase in LLE-LF. In fact, there was a significant decrease in LLE-HF during standing posture. There was a significant decrease in LF/HF ratios of LLE during sleep ( p=0.003). While there was a highly significant increase in spectral LF/HF ratio up on standing, there was a less significant increase in the ratio of LLE (LF/HF). During sleep and supine posture, though LF/HF ratios of spectral power did not differ significantly between controls and patients with panic disorder, the ratio of LLE (LF/HF) was very significantly higher in patients ( p=0.0005), probably reflecting higher relative sympathetic activity. HF-LLE was also significantly lower in patients with panic disorder. The ratio of LLE (LF/UF) was also significantly higher ( p=0.00001) in patients with panic disorder. In other words, the net contribution of LLE-LF to the UF series was significantly more in patients with anxiety. These findings throw new light on the contribution of various frequencies in a given time series toward the UF set of data and the importance of using band analysis of nonlinear measures. These may be used as more effective clinical tools to investigate certain aspects of autonomic function such as sympathovagal interaction in addition to the most commonly used spectral powers in different bands and spectral LF/HF ratio.

Combining global and multi-scale features in a description of the solar wind-magnetosphere coupling

Abstract. The solar wind-magnetosphere coupling during substorms exhibits dynamical features in a wide range of spatial and temporal scales. The goal of our work is to combine the global and multi-scale description of magnetospheric dynamics in a unified data-derived model. For this purpose we use deterministic methods of nonlinear dynamics, together with a probabilistic approach of statistical physics. In this paper we discuss the mathematical aspects of such a combined analysis. In particular we introduce a new method of embedding analysis based on the notion of a mean-field dimension. For a given level of averaging in the system the mean-filed dimension determines the minimum dimension of the embedding space in which the averaged dynamical system approximates the actual dynamics with the given accuracy. This new technique is first tested on a number of well-known autonomous and open dynamical systems with and without noise contamination. Then, the dimension analysis is carried out for the correlated solar wind-magnetosphere database using vBS time series as the input and AL index as the output of the system. It is found that the minimum embedding dimension of vBS - AL time series is a function of the level of ensemble averaging and the specified accuracy of the method. To extract the global component from the observed time series the ensemble averaging is carried out over the range of scales populated by a high dimensional multi-scale constituent. The wider the range of scales which are smoothed away, the smaller the mean-field dimension of the system. The method also yields a probability density function in the reconstructed phase space which provides the basis for the probabilistic modeling of the multi-scale dynamical features, and is also used to visualize the global portion of the solar wind-magnetosphere coupling. The structure of its input-output phase portrait reveals the existence of two energy levels in the system with non-equilibrium dynamical features such as hysteresis which are typical for non-equilibrium phase transitions. Further improvements in space weather forecasting tools may be achieved by a combination of the dynamical description for the global component and a statistical approach for the multi-scale component.Key words. Magnetospheric physics (solar wind– magnetosphere interactions; storms and substorms) – Space plasma physics (nonlinear phenomena)