Characteristics Of Attractor Research Articles

Sharp changes in fractal basin of attraction in passive dynamic walking

A passive dynamic walker is a mechanical system that walks down a slope without any control, and gives useful insights into the dynamic mechanism of stable walking. This system shows specific attractor characteristics depending on the slope angle due to nonlinear dynamics, such as period-doubling to chaos and its disappearance by a boundary crisis. However, it remains unclear what happens to the basin of attraction. In our previous studies, we showed that a fractal basin of attraction is generated using a simple model over a critical slope angle by iteratively applying the inverse image of the Poincaré map, which has stretching and bending effects. In the present study, we show that the size and fractality of the basin of attraction sharply change many times by changing the slope angle. Furthermore, we improved our previous analysis to clarify the mechanisms for these changes and the disappearance of the basin of attraction based on the stretching and bending deformation in the basin formation process. These findings will improve our understanding of the governing dynamics to generate the basin of attraction in walking.

Imbricated chaos in rough potential [formula omitted]-Rayleigh-Duffing oscillator

In this paper, a new ϕ6 - Rayleigh–Duffing system with dual potential is examined. Research is being done on the effect of roughness on forced dynamical systems. To evaluate the performance of the system parameters, the resonance oscillations are investigated using the multiple scale method. Both equilibrium points and local stability are calculated, while abrupt changes in the topological characteristics of attractors’ phase portraits are treated as bifurcation-Lyapunov parameters.

Lateral nonlinear vibration of the camshaft considering the structural clearance and load force

To reveal the influence of variations in dynamic characteristics on the working performance of the camshaft, the lateral nonlinear vibration of the camshaft considering the structural clearance and load force is studied. By using the Magnus series expansion method, the response of lateral nonlinear vibration is solved, and the numerical computation scheme applied to the global evolution characteristics of attractors and their basins is developed, which solves the problems of inadequate accuracy and long-term numerical unreliability in numerical solutions and simulations. By analyzing the attractors and basins, single-parameter bifurcation, and bi-parameter dynamic characteristics, the influence of different parameters on the nonlinear dynamic characteristics of the camshaft system is revealed. The results show that as the structural clearance increases, the attraction basins of both periodic and chaotic attractors exhibit tongue-shaped fractal features, and the camshaft system changes from the coexistence of two types of period-3 attractors to chaotic attractors, which reveals the mechanism of global vibration instability of the camshaft system. In addition, when the camshaft system evolves from the flutter to forced vibration conditions, the period-doubling bifurcation behavior occurs, and the parameter ranges for stable operation of the system increase. More importantly, the bi-parameter dynamic characteristics confirm that the parameter ranges for the periodic motion of the system are enlarged when the load force increases, and there exist multi-parameter optimal matching solutions for the stable motion of the camshaft system. The research results reveal the evolution mechanism of the lateral nonlinear vibration of the camshaft, which can provide a theoretical reference for suppressing the vibration and improving the working performance.

Effect of temporal resolution on the reproduction of chaotic dynamics via reservoir computing.

Reservoir computing is a machine learning paradigm that uses a structure called a reservoir, which has nonlinearities and short-term memory. In recent years, reservoir computing has expanded to new functions such as the autonomous generation of chaotic time series, as well as time series prediction and classification. Furthermore, novel possibilities have been demonstrated, such as inferring the existence of previously unseen attractors. Sampling, in contrast, has a strong influence on such functions. Sampling is indispensable in a physical reservoir computer that uses an existing physical system as a reservoir because the use of an external digital system for the data input is usually inevitable. This study analyzes the effect of sampling on the ability of reservoir computing to autonomously regenerate chaotic time series. We found, as expected, that excessively coarse sampling degrades the system performance, but also that excessively dense sampling is unsuitable. Based on quantitative indicators that capture the local and global characteristics of attractors, we identify a suitable window of the sampling frequency and discuss its underlying mechanisms.

Learning unseen coexisting attractors.

Reservoir computing is a machine learning approach that can generate a surrogate model of a dynamical system. It can learn the underlying dynamical system using fewer trainable parameters and, hence, smaller training data sets than competing approaches. Recently, a simpler formulation, known as next-generation reservoir computing, removed many algorithm metaparameters and identified a well-performing traditional reservoir computer, thus simplifying training even further. Here, we study a particularly challenging problem of learning a dynamical system that has both disparate time scales and multiple co-existing dynamical states (attractors). We compare the next-generation and traditional reservoir computer using metrics quantifying the geometry of the ground-truth and forecasted attractors. For the studied four-dimensional system, the next-generation reservoir computing approach uses ∼ 1.7 × less training data, requires 10 × shorter "warmup" time, has fewer metaparameters, and has an ∼ 100 × higher accuracy in predicting the co-existing attractor characteristics in comparison to a traditional reservoir computer. Furthermore, we demonstrate that it predicts the basin of attraction with high accuracy. This work lends further support to the superior learning ability of this new machine learning algorithm for dynamical systems.

A 6D Fractional-Order Memristive Hopfield Neural Network and its Application in Image Encryption

This paper proposes a new memristor model and uses pinched hysteresis loops (PHL) to prove the memristor characteristics of the model. Then, a new 6D fractional-order memristive Hopfield neural network (6D-FMHNN) is presented by using this memristor to simulate the induced current, and the bifurcation characteristics and coexistence attractor characteristics of fractional memristor Hopfield neural network is studied. Because this 6D-FMHNN has chaotic characteristics, we also use this 6D-FMHNN to generate a random number and apply it to the field of image encryption. We make a series of analysis on the randomness of random numbers and the security of image encryption, and prove that the encryption algorithm using this 6D-FMHNN is safe and sensitive to the key.

Shape Characteristic Analysis of Chaotic Attractors for Leakage Current Signal of Ice-Covered Insulator

The change and development of leakage current (LC) during the flashover process of insulators are closely related to the discharge process. Characteristic quantities of LC play an important role in the identification and prediction of ice-covered flashover development stages. In this paper, the feature extraction method for LC of ice-covered insulator is discussed based on the shape characteristics of chaotic attractor. Firstly, the power spectrum and the largest Lyapunov exponent are used to analyze the LC of ice-covered insulator qualitatively and quantitatively, which confirmed positive chaotic features during the flashover process. On this basis, the LC is reconstructed in phase space. To solve the problem that the attractor in the high-dimensional space cannot be observed in the phase space reconstruction, the principal vector analysis method is used to project the high-dimensional reconstruction vector into the three-dimensional space to reduce the dimension. Finally, the evolution law of chaotic attractor in each stage of LC under different pollution degree is analyzed and discussed. The results show that the LC of ice-covered insulator has obvious characteristics of attractor shape at different stages of pollution, and is sensitive to the degree of pollution and each stage of development. So the shape of the attractor of LC has important application value for the identification of pollution flashover state and prediction of pollution flashover process of ice-covered insulators.

Predicting attractor characteristics using Lyapunov exponents in a laser with injected signal.

Coexisting attractors are studied in a single-mode coherent model of a laser with an injected signal. We report that every attractor has a unique Lyapunov exponent (LE) pattern that is choreographed by the subtle variations in the attractor's dynamics and circumscribed by a common Lyapunov spectral pattern that begins and ends with two-zero LEs. Lyapunov spectra form symmetric-like and asymmetric bubbles; the former foreshadows an attractor's proximity to the cusp of an eminent change in dynamics and the latter indicates the presence of a bifurcation. We show that the peak values of the asymmetric bubbles are always associated with two-zero LEs; in fact, they are allied inseparably in forecasting period-doubling episodes. The two-zero LEs' predictor of torus dynamics is refined to include the convergence of three LEs to a triplet of zeros as a precursor to the two-zero spectra. We report that the long-standing two-zero LEs' signature is a necessary but not sufficient condition for predicting attractors and their dynamic conditions. The evolution of the attractor volume as a function of the injected signal is compared to the spectral formation of the attractor; we report slope changes and points of inflections in the volume trajectory where spectral changes indicate dynamic changes. Attractor viability is tested preliminarily by including random low-level noise in the frequency of the injected signal.

Neuromorphic Implementation of Attractor Dynamics in a Two-Variable Winner-Take-All Circuit with NMDARs: A Simulation Study.

Neural networks configured with winner-take-all (WTA) competition and N-methyl-D-aspartate receptor (NMDAR)-mediated synaptic dynamics are endowed with various dynamic characteristics of attractors underlying many cognitive functions. This paper presents a novel method for neuromorphic implementation of a two-variable WTA circuit with NMDARs aimed at implementing decision-making, working memory and hysteresis in visual perceptions. The method proposed is a dynamical system approach of circuit synthesis based on a biophysically plausible WTA model. Notably, slow and non-linear temporal dynamics of NMDAR-mediated synapses was generated. Circuit simulations in Cadence reproduced ramping neural activities observed in electrophysiological recordings in experiments of decision-making, the sustained activities observed in the prefrontal cortex during working memory, and classical hysteresis behavior during visual discrimination tasks. Furthermore, theoretical analysis of the dynamical system approach illuminated the underlying mechanisms of decision-making, memory capacity and hysteresis loops. The consistence between the circuit simulations and theoretical analysis demonstrated that the WTA circuit with NMDARs was able to capture the attractor dynamics underlying these cognitive functions. Their physical implementations as elementary modules are promising for assembly into integrated neuromorphic cognitive systems.

A Watermarking Technique with Chaotic Fractional S-Box Transformation

In this paper, the system of non-linear ordinary differential equations which defines a continuous-time dynamical system that shows the fractal characteristics of attractor is used to construct chaotic S-box. In this new digital watermarking technique the priority is of importance of robustness along with chaos to create confusion. The inclusion of chaos along with watermarking in frequency domain ensures robustness. As we have proposed a frequency domain watermarking as which we embed watermark into the low or middle frequencies, these changes will be spread all over the image. The strength of fractional S-box is evaluated with the help of bit independence criterion, nonlinearity analysis, strict avalanche criterion, linear approximation probability and differential approximation probability. Additionally, some security analyses in the form of correlation, contrast, energy, entropy, homogeneity, mean square error and peak signal to noise ratio are performed for validity of proposed watermarking scheme. The confidence measure after these analyses indicates the survival aligned with malicious attacks like noise, cropping and compression.

Boiling water reactor instability analysis using attractor characteristics

The main purpose of this work is to develop an alternative technique to analyze the neutron flux signal with the aim of determining boiling water reactor (BWR) stability or instability. Another benefit would be to demonstrate the intrinsic stability or even reversibility of the BWR neutron flux and hence, of power.The methodology is based on the attractor morphology analysis. After calculating the time delay and embedded dimension, a tridimensional attractor is constructed. The morphological analysis is based on the attractor center position and the orbit radius value.Analysis results show that all BWRs analyzed have the same attractor shape, meaning the attractor is a system indicator. The orbit radius is a good stability indicator. It is demonstrated that the limit orbit is not reached in any studied case. Reversibility is justified and intrinsic stability is demonstrated.

A Generalized Stability Theorem for Discrete-Time Nonautonomous Chaos System with Applications

Firstly, this study introduces a definition of generalized stability (GST) in discrete-time nonautonomous chaos system (DNCS), which is an extension for chaos generalized synchronization. Secondly, a constructive theorem of DNCS has been proposed. As an example, a GST DNCS is constructed based on a novel 4-dimensional discrete chaotic map. Numerical simulations show that the dynamic behaviors of this map have chaotic attractor characteristics. As one application, we design a chaotic pseudorandom number generator (CPRNG) based on the GST DNCS. We use the SP800-22 test suite to test the randomness of four 100-key streams consisting of 1,000,000 bits generated by the CPRNG, the RC4 algorithm, the ZUC algorithm, and a 6-dimensional CGS-based CPRNG, respectively. The numerical results show that the randomness performances of the two CPRNGs are promising. In addition, theoretically the key space of the CPRNG is larger than 21116. As another application, this study designs a stream avalanche encryption scheme (SAES) in RGB image encryption. The results show that the GST DNCS is able to generate the avalanche effects which are similar to those generated via ideal CPRNGs.

Maximizing Sensitivity Vector Fields: A Parametric Study

Sensitivity vector fields (SVFs) have proven to be an effective method for identifying parametric variations in dynamical systems. These fields are constructed using information about how a dynamical system's attractor deforms under prescribed parametric variations. Once constructed, they can be used to quantify any additional variations from the nominal parameter set as they occur. Since SVFs are based on attractor deformations, the geometry and other qualities of the baseline system attractor impact how well a set of SVFs will perform. This paper examines the role attractor characteristics and the choices made in SVF construction play in determining the sensitivity of SVFs. The use of nonlinear feedback to change a dynamical system with the intent of improving SVF sensitivity is explored. These ideas are presented in the context of constructing SVFs for several dynamical systems.

Preservation of dynamic properties in qualitative modeling frameworks for gene regulatory networks

Mathematical modeling often helps to provide a systems perspective on gene regulatory networks. In particular, qualitative approaches are useful when detailed kinetic information is lacking. Multiple methods have been developed that implement qualitative information in different ways, e.g., in purely discrete or hybrid discrete/continuous models. In this paper, we compare the discrete asynchronous logical modeling formalism for gene regulatory networks due to R. Thomas with piecewise affine differential equation models. We provide a local characterization of the qualitative dynamics of a piecewise affine differential equation model using the discrete dynamics of a corresponding Thomas model. Based on this result, we investigate the consistency of higher-level dynamical properties such as attractor characteristics and reachability. We show that although the two approaches are based on equivalent information, the resulting qualitative dynamics are different. In particular, the dynamics of the piecewise affine differential equation model is not a simple refinement of the dynamics of the Thomas model

Polynomial level-set method for attractor estimation

In this study, we present a polynomial level-set method for attractor estimation. This method uses the sub-level representation of sets. The problem of flowing these sets under the advection map of a dynamic system is converted to a semi-definite program, which is used to compute the coefficients of the polynomials. The required storage space for describing the result is much less than the mesh-based methods. The characteristics of attractors are used in the algorithm formulations so that the associated numerical error can be reduced. We further address the related problems of constraining the degree of the polynomials. Various numerical examples are used to show the effectiveness of the advection approach.

Size effect on serrated yielding in Al-Cu polycrystals

In this paper, through a systematic statistical analysis of the characteristics of the serrated yielding, including the stress drop amplitude and the drop duration, etc., the difference between the results of thinner samples (1mm and 2mm) and those of thicker samples (3mm), has been reported. The size effect in Portevin-Le Chrtelier (PLC) effect also manifests itself in the difference of PLC types of jerky flow of 1mm or 2mm thick samples and those of 3mm thick samples under the same experimental conditions. Moreover, the geometrical characteristics of attractor of the former series in phase space are described by using methods of nonlinear time series analysis, and chaotic behavior is proved to exist in these series through the application of principal component analysis. The distribution of stress drops of the latter is found to follow a power-law distribution, which is referred to as the signature of self-organized criticality phenomenon.

Static mental representations in recurrent neural networks for the control of dynamic behavioural sequences

What enables an organism to perform behaviour we would call cognitive and adaptive, like language? Here, it is argued that an essential prerequisite is the ability to build up mental representations of external situations to uncouple the behaviour from direct environmental control. Such representations can be realized by building up cell assemblies. The recurrent neural network presented to cope with this task has been used for generation of action but can also be utilized as a basis for mental representations due to its attractor characteristics. In this context, a new learning algorithm (Dynamic Delta Rule) is proposed, which leads to a self-organized weight distribution yielding stable states on the one hand and which, on the other hand, only activates subpopulations of larger networks that code for the respective situation. In a second step, ways are shown of how the static information of these internal models can be transformed into time-dependent behavioural sequences.

Effect of Attractor Construction Methods to Fractal Dimension for Heart Sound Analysis

Strange attractor can be constructed from time series data such as heart sound. In the areas of the recognition and diagnosis of abnormal heart sounds, signal presentation method is very useful because good features can be detected from good presentation. This paper examines efficiency in diagnosing abnormal heart sounds of the two different methods for constructing attractor. Nine different heart sounds from typical clinical conditions were used for this study. The first method was constructing attractors using original heart sounds, and the second was modifying the original sounds by autocorrelation and they were then applied to the orignal sounds as to cross correlation checks. Attractors could be constructed using signals generated by these methods, and values of fractal dimensions would then be calculated which has been a well known method to measure characteristics of attractors. The results showed that the second method appeared to provide more efficient way to correctly classify abnormal heart sounds.

Probabilistic Stability of an Atmospheric Model to Various Amplitude Perturbations

Every forecast should include an estimate of its likely accuracy, as a measure of predictability. A new measure, the first passage time (FPT), which is defined as the time period when the model error first exceeds a predetermined criterion (i.e., the tolerance level), is proposed here to estimate model predictability. A theoretical framework is developed to determine the mean and variance of FPT. The low-order Lorenz atmospheric model is taken as an example to show the robustness of using FPT as a quantitative measure for prediction skill. Both linear and nonlinear perspectives of forecast errors are analytically investigated using the self-consistent Nicolis model. The mean and variance of FPT largely depends on the ratio between twice the maximum Lyapunov exponent (σ) and the intensity of attractor fluctuations (q2), λ = 2σ/q2. Two types of predictability are found: λ > 1 referring to low predictability and λ < 1 referring to high predictability. The mean and variance of FPT can be represented by the e-folding timescales in the low-predictability range, but not in the high-predictability range. The transition between the two predictability ranges is caused by the variability of the attractor characteristics along the reference trajectory.

Strange attractor characteristics of earthquakes in Shillong Plateau and adjoining regions

Strange attractor dimensions were determined for two concentric areas of about 220 km (2°) and 440 km (4°) radii around WWSSN station at Shillong based on earthquake data during the period 1964 to 1992. Existence of deterministic chaos for both the areas was supported by the same fractal dimension of 6.1 and positive value of the largest Lyapunov exponent. The results have provided justification for delineating precursory seismicity patterns based on earthquakes up to epicentral distance of 440 km on different fault systems.Conspicuous higher value of the strange attractor dimension in Hindukush, Himalaya and northeast India than that in Koyna region in the peninsular India suggests a new criteria for distinction between interplate and intraplate earthquakes. Differences with subduction zone in Pacific vis‐à‐vis stress drop in great earthquakes (M≥8) and implications of the chaotic approach in earthquake hazard analysis have also been brought out.