Return Map Research Articles

Numerical Analysis of a Novel 3D Chaotic System with Period-Subtracting Structures

In this work, we present a novel three-dimensional chaotic system with only two cubic nonlinear terms. Dynamical behavior of the system reveals a period-subtracting bifurcation structure containing all [Formula: see text]th-order ([Formula: see text]) periods that are found in the dynamical evolution of the novel system concerning different values of parameters. The new system could be evolved into different states such as point attractor, limit cycle, strange attractor and butterfly strange attractor by changing the parameters. Also, the system is multistable, which implies another feature of a chaotic system known as the coexistence of numerous spiral attractors with one limit cycle under different initial values. Furthermore, bifurcation analysis reveals interesting phenomena such as period-doubling route to chaos, antimonotonicity, periodic solutions, and quasi-periodic motion. In the meantime, the existence of periodic solutions is confirmed via constructed Poincaré return maps. In addition, by studying the influence of system parameters on complexity, it is confirmed that the chaotic system has high spectral entropy. Numerical analysis indicates that the system has a wide variety of strong dynamics. Finally, a message coding application of the proposed system is developed based on periodic solutions, which indicates the importance of studying periodic solutions in dynamical systems.

Measuring chaos in the Lorenz and Rössler models: Fidelity tests for reservoir computing.

This study focuses on the qualitative and quantitative characterization of chaotic systems with the use of a symbolic description. We consider two famous systems, Lorenz and Rössler models with their iconic attractors, and demonstrate that with adequately chosen symbolic partition, three measures of complexity, such as the Shannon source entropy, the Lempel-Ziv complexity, and the Markov transition matrix, work remarkably well for characterizing the degree of chaoticity and precise detecting stability windows in the parameter space. The second message of this study is to showcase the utility of symbolic dynamics with the introduction of a fidelity test for reservoir computing for simulating the properties of the chaos in both models' replicas. The results of these measures are validated by the comparison approach based on one-dimensional return maps and the complexity measures.

Devising a new filtration method and proof of self-similarity of electromyograms

The main attention is paid to the analysis of electromyogram (EMG) signals using Poincaré plots (PP). It was established that the shapes of the plots are related to the diagnoses of patients. To study the fractal dimensionality of the PP, the method of counting the coverage figures was used. The PP filtration was carried out with the help of Haar wavelets. The self-similarity of Poincaré plots for the studied electromyograms was established, and the law of scaling was used in a fairly wide range of coverage figures. Thus, the entire Poincaré plot is statistically similar to its own parts. The fractal dimensionalities of the PP of the studied electromyograms belong to the range from 1.36 to 1.48. This, as well as the values of indicators of Hurst exponent of Poincaré plots for electromyograms that exceed the critical value of 0.5, indicate the relative stability of sequences. The algorithm of the filtration method proposed in this research involves only two simple stages: Conversion of the input data matrix for the PP using the Jacobi rotation. Decimation of both columns of the resulting matrix (the so-called "lazy wavelet-transformation", or double downsampling). The algorithm is simple to program and requires less machine time than existing filters for the PP. Filtered Poincaré plots have several advantages over unfiltered ones. They do not contain extra points, allow direct visualization of short-term and long-term variability of a signal. In addition, filtered PPs retain both the shape of their prototypes and their fractal dimensionality and variability descriptors. The detected features of electromyograms of healthy patients with characteristic low-frequency signal fluctuations can be used to make clinical decisions.

Symbolic Encoding of Periodic Orbits and Chaos in the Rucklidge System

To describe and analyze the unstable periodic orbits of the Rucklidge system, a so-called symbolic encoding method is introduced, which has been proven to be an efficient tool to explore the topological properties concealed in these periodic orbits. In this work, the unstable periodic orbits up to a certain topological length in the Rucklidge system are systematically investigated via a proposed variational method. The dynamics in the Rucklidge system are explored by using phase portrait analysis, Lyapunov exponents, and Poincaré first return maps. Symbolic encodings of the periodic orbits with two and four letters based on the trajectory topology in the phase space are implemented under two sets of parameter values. Meanwhile, the bifurcations of the periodic orbits are explored, significantly improving the understanding of the dynamics of the Rucklidge system. The multiple-letter symbolic encoding method could also be applicable to other nonlinear dynamical systems.

Stress Classification by Multimodal Physiological Signals Using Variational Mode Decomposition and Machine Learning.

In this pandemic situation, importance and awareness about mental health are getting more attention. Stress recognition from multimodal sensor based physiological signals such as electroencephalogram (EEG) and electrocardiography (ECG) signals is a very cost-effective way due to its noninvasive nature. A dataset, recorded during the mental arithmetic task, consisting of EEG + ECG signals of 36 participants is used. It contains two categories of performance, namely, “Good” (nonstressed) and “Bad” (stressed) (Gupta et al. 2018 and Eraldeír et al. 2018). This paper presents an effective approach for the recognition of stress marker at frontal, temporal, central, and occipital lobes. It processes the multimodality physiological signals. The variational mode decomposition (VMD) strategy is used for data preprocessing and for the decomposition of signals into various oscillatory mode functions. Poincare plots (PP) are derived from the first eight variational modes and features from these plots have been extracted such as mean, area, and central tendency measure of the elliptical region. The statistical significance of the extracted features with p < 0.5 has been performed using the Wilcoxson test. The multilayer perceptron (MPLN) and Support Vector Machine (SVM) algorithms are used for the classification of stress and nonstress categories. MLPN has achieved the maximum accuracies of 100% for frontal and temporal lobes. The suggested method can be incorporated in noninvasive EEG signal processing based automated stress identification systems.

Lock-in phenomenon of vortex shedding in flows excited with two commensurate frequencies: a theoretical investigation pertaining to combustion instability

An analytical investigation is performed to study the dynamics of vortex shedding behaviour during two commensurate frequency velocity excitations, with an emphasis on the phenomenon of lock-in. We attempt to theoretically study the dynamical features of lock-in under two-frequency excitations and contrast the behaviour with single-frequency excitation. We employ an existing low-order model to characterise the vortex shedding process behind a step/bluff-body. The continuous-time domain model is transformed to a nonlinear dynamical map that relates time instances of successive vortex shedding. Further, these time instances are converted to phase instances, involving which criteria for a generic p : q phase lock-in is obtained. Four parameters are involved: amplitude and frequency of the two excitation components termed as primary and secondary. Bifurcations occurring are investigated using return maps. The inclusion of secondary excitation leads to the existence of two orders of lock-in within a single lock-in boundary. Furthermore, our results indicate that secondary excitation can be used as a control in order to tailor the 1:1 lock-in region formed by the primary excitation. Finally, analytical expressions are obtained to identify lock-in boundaries and their salient geometrical features. Interesting features such as the occurrence of bistability and change in the order of lock-in are observed, which can be explored further with future experiments.

Dissecting a Resonance Wedge on Heteroclinic Bifurcations

This article studies routes to chaos occurring within a resonance wedge for a 3-parametric family of differential equations acting on a 3-sphere. Our starting point is an autonomous vector field whose flow exhibits a weakly attracting heteroclinic network made by two 1-dimensional connections and a 2-dimensional separatrix between two equilibria with different Morse indices. After changing the parameters, while keeping the 1-dimensional connections unaltered, we concentrate our study in the case where the 2-dimensional invariant manifolds of the equilibria do not intersect. We derive the first return map near the ghost of the attractor and we reduce the analysis of the system to a 2-dimensional map on the cylinder. Complex dynamical features arise from a discrete-time Bogdanov-Takens singularity, which may be seen as the organizing center by which one can obtain infinitely many attracting tori, strange attractors, infinitely many sinks and non-trivial contracting wandering domains. These dynamical phenomena occur within a structure that we call resonance wedge. As an application, we may see the "classical" Arnold tongue as a projection of a resonance wedge. The results are general, extend to other contexts and lead to a fine-tuning of the theory.

A micromechanics-based fractional frictional damage model for quasi-brittle rocks

For quasi-brittle rocks, a novel micromechanics-based fractional friction-damage coupling model is presented by combining the fractional plasticity theory and the micromechanical formulations for cracked solids. The plastic deformation is stemmed directly from the frictional sliding along microcracks while the damage is related to the initiation and propagation of microcracks. These two dissipation processes are inherently coupled. By applying the stress-fractional plasticity operation and the covariant transformation technique, a fractional operation instead of plastic potential is proposed to capture the non-orthogonal plastic flow. In addition, an energy release rate based damage criterion is adopted to describe material degradation. For numerical implementation, a novel explicit return mapping (ERM) integration algorithm is put forward. The predictive performance of the model is verified by comparisons with the associated model (fractional order α=1) and laboratory observations from literatures. Moreover, the effectiveness of the ERM algorithm is assessed numerically.

Linear and nonlinear benchmarks between the CLT code and the M3D-C1 code for the 2/1 resistive tearing mode and the 1/1 resistive kink mode

The linear and nonlinear benchmarks between the CLT code and the M3D-C1 code for the 2/1 resistive tearing mode and the 1/1 resistive kink mode are presented. CLT is an explicit finite difference code, while M3D-C1 is an implicit finite element code. Although the implementations of CLT and M3D-C1 are totally different, we find that the simulation results of the resistive-kink mode and the m/n=2/1 tearing mode from M3D-C1 and CLT are almost the same, including the linear and nonlinear growth rates, the mode structures, the nonlinear saturation levels, the Poincare plots, and the scaling laws. This confirms that the nonlinear results for the 1/1 resistive-kink mode and 2/1 tearing mode are accurate and reliable.

Nonlinear Oscillations of Particle-Reinforced Electro-Magneto-Viscoelastomer Actuators

Abstract This work presents the dynamic modeling and analysis of a particle-reinforced and pre-stressed electro-magneto-viscoelastic plate actuator. The actuator belongs to a smart actuator category and is made of an electro-magneto-active polymer filled with a particular volume fraction of suitable fillers. An energy-based electro-magneto-viscoelastic model is developed to predict the actuator response and interrogate the impact of particle reinforcement on the dynamic oscillations of a pre-stressed condition of the actuator. An Euler–Lagrange equation of motion is implemented to deduce the governing dynamic equation of the actuator. The findings of the model solutions provide preliminary insights on the alteration of the nonlinear behavior of the actuator driven by DC and AC dynamic modes of actuation. It is observed that the enrichment in the particle reinforcement characterized by the amount of fillers strengthens the polymer and depleted the associated level of deformation. Also, the depletion in the intensity of oscillation and enhancement in the frequency of excitation is perceived with an increase in the particle reinforcement. In addition, the time-history response, Poincare plots, and phase diagrams are also plotted to assess the stability, periodicity, beating phenomenon, and resonant behavior of the actuator. In general, the current study provides initial steps toward the modern actuator designs for various futuristic applications in the engineering and medical field.

The effect of principal component analysis in the diagnosis of congestive heart failure via heart rate variability analysis.

In this study, we investigated the effect of principal component analysis (PCA) in congestive heart failure (CHF) diagnosis using various machine learning algorithms from 5-min HRV data. The extracted 59 heart rate variability (HRV) features consist of statistical time-domain measures, frequency-domain measures (power spectral density estimations from Fourier transform and Lomb-Scargle methods), time-frequency HRV measures (Wavelet transform), and nonlinear HRV measures (Poincare plot, symbolic dynamics, detrended fluctuation analysis, and sample entropy). All these HRV features are the classifiers' inputs. We repeated the study ten times using the first one to the first 10 principal components from PCA instead of all HRV features. Nine different classifiers, namely logistic regression, Naive Bayes, k-nearest neighbors, decision tree, AdaBoost, support vector machines, stochastic gradient descent, random forest, and artificial neuronal network (multilayer perceptron) are examined. The proposed study results in the 100% accuracy, 100% specificity, and 100% sensitivity after utilizing PCA (with the first eight principal components) using the Random Forest classifier where the maximum classifier performances are the 86% accuracy, 79% specificity, and 86% sensitivity before PCA. In conclusion, PCA is beneficial in the diagnosis of patients with CHF. In addition, we experienced the online Python-based visual machine learning tool, Orange, which can implement well-known machine learning algorithms.

Human Activity Classification Based on Angle Variance Analysis Utilizing the Poincare Plot

We propose a single sensor-based activity classification method where the Poincare plot was introduced to analyze the variance of the angle between acceleration vector with gravity calculated from the raw accelerometer data for human activity classification. Two datasets named ‘Human Activity Recognition’ and ‘MHealth dataset’ were used to develop the model to classify activity from low to vigorous intensity activities and posture estimation. Short-term and long-term variability analyzing the property of the Poincare plot was used to classify activities according to the vibrational intensity of body movement. Commercially available Actigraph’s activity classification metric ‘count’ resembled value was used to compare the feasibility of the proposed classification algorithm. In the case of the HAR dataset, laying, sitting, standing, and walking activities were classified. Poincare plot parameters SD1, SD2, and SDRR of angle in the case of angle variance analysis and the mean count of X-, Y-, and Z-axis were fitted to a support vector machine (SVM) classifier individually and jointly. The variance- and count-based methods have 100% accuracy in the static–dynamic classification. Laying activity classification has 100% accuracy from other static conditions in the proposed method, whereas the count-based method has 98.08% accuracy with 10-fold cross-validation. In the sitting–standing classification, the proposed angle-based algorithm shows 88% accuracy, whereas the count-based approach has 58% accuracy with a support vector machine classifier with 10-fold cross-validation. In the classification of the variants of dynamic activities with the MHealth dataset, the accuracy for angle variance-based and count-based methods is 100%, in both cases, for fivefold cross validation with SVM classifiers.

Fast growth of the number of periodic points arising from heterodimensional connections

We consider $C^{r}$-diffeomorphisms ($1 \leq r \leq +\infty$) of a compact smooth manifold having two pairs of hyperbolic periodic points of different indices which admit transverse heteroclinic points and are connected through a blender. We prove that, by giving an arbitrarily $C^{r}$-small perturbation near the periodic points, we can produce a periodic point for which the first return map in the center direction coincides with the identity map up to order $r$, provided the transverse heteroclinic points satisfy certain natural conditions involving higher derivatives of their transition maps in the center direction. As a consequence, we prove that $C^{r}$-generic diffeomorphisms in a small neighborhood of the diffeomorphism under consideration exhibit super-exponential growth of number of periodic points. We also give examples which show the necessity of the conditions we assume.

Modeling Lithospheric Deformation Using a Compressible Visco‐Elasto‐Viscoplastic Rheology and the Effective Viscosity Approach

AbstractDeformations of the colder regions of the lithosphere mainly occur in the frictional regime. In geodynamic models, frictional plastic deformations are often highly localized (shear bands) and are used as proxies for faults. However, capturing the generation and evolution of shear bands in geodynamic models is troublesome. Indeed, mesh dependency and lack of convergence affect, to some extent, the results of geodynamic models. Here we extend the most common plasticity implementation used in geodynamic codes (effective viscosity approach [EVA]) to include the combined effects of elastic compressibility, plastic dilatancy, strain softening, and viscoplasticity. The latter acts as a regularization that cures most of the known issues of geodynamic models related to frictional plasticity. Using regularized models based on the M2Di MATLAB routines, we show that volumetric elasto‐plastic deformations can significantly impact crustal‐scale shear banding. We also show that the artificial overstress caused by viscoplasticity can be mitigated by employing power‐law models. Furthermore, we demonstrate that plasticity algorithms common in geodynamics (based on the EVA) can be as accurate as those obtained with algorithms typically used in engineering (return mapping with a consistent tangent operator). Finally, we show examples of long‐term tectonic deformations using the state‐of‐the art geodynamic code MDoodz. They indicate that viscoplastic regularization can be used efficiently to obtain reliable simulations in geodynamics.

Ordered intricacy of Shilnikov saddle-focus homoclinics in symmetric systems.

Using the technique of Poincaré return maps, we disclose an intricate order of subsequent homoclinic bifurcations near the primary figure-8 connection of the Shilnikov saddle-focus in systems with reflection symmetry. We also reveal admissible shapes of the corresponding bifurcation curves in a parameter space. Their scalability ratio and organization are proven to be universal for such homoclinic bifurcations of higher orders. Two applications with similar dynamics due to the Shilnikov saddle-foci are used to illustrate the theory: a smooth adaptation of the Chua circuit and a 3D normal form.

Mixed-mode oscillations from a constrained extended Bonhoeffer-van der Pol oscillator with a diode.

An extended Bonhoeffer-van der Pol (BVP) oscillator is a circuit that is naturally extended to a three-variable system from a two-variable BVP oscillator. A BVP oscillator is known to exhibit a canard explosion, and the extended BVP oscillator generates mixed-mode oscillations (MMOs). In this work, we considered a case study where the nonlinear conductor in the extended BVP oscillator includes an idealized diode. The idealized case corresponds to a degenerate case where one of the parameters tends to infinity, and circuit dynamics are represented using a constrained equation, and at the expense of the model's naturalness, i.e., in a case in which the solutions of the dynamics are defined only forward in time, the Poincaré return maps are constructed as one-dimensional (1D). Using these 1D return maps, we explain various phenomena, such as simple MMOs and MMO-incrementing bifurcations. In this oscillator, there exists a small amplitude oscillation, which emerges as a consequence of supercritical Hopf bifurcation, and there exists large relaxation oscillation which appears via canard explosion by changing the bifurcation parameter. Between these small and large amplitude oscillations, the MMO bifurcations exhibit asymmetric Farey trees. Furthermore, these theoretical results were verified using laboratory measurements and experiments.

An ordinary state-based peridynamic elastoplastic 2D model consistent with J2 plasticity

A new ordinary state-based peridynamic model (OSB-PD) in 2D consistent with J2 plasticity using a novel decomposition for force and extension states is presented. A novel strategy for testing the consistency of an OSB-PD formulation for elastoplasticity is introduced. In contrast with other similar models, the new elasto-plastic OSB-PD model is objective and works for large rotations. Two rate-independent yield functions equivalent to J2 plasticity with associated flow rules are formulated based on force states and on the strain energy density. An efficient return mapping algorithm for peridynamic formulations in plasticity is proposed. The model is verified against results obtained with Abaqus for the classical formulation using two examples, one featuring plastic deformations under large rotations. The evolution of the plastic zone computed with the new model is found to match that from the classical model computed with Abaqus. With a simple damage model, a demonstrative plane stress example for shear localization and full rupture captures the same failure mechanism as observed experimentally: an initial almost symmetrical shear banding, evolves into a single, predominant shear band, followed by full rupture.

Gaussian process-based kernel as a diagnostic model for prediction of type 2 diabetes mellitus risk using non-linear heart rate variability features

The main objective of the study was to develop a low-cost, non-invasive diagnostic model for the early prediction of T2DM risk and validation of this model on patients. The model was designed based on the machine learning classification technique using non-linear Heart rate variability (HRV) features. The electrocardiogram of the healthy subjects (n = 35) and T2DM subjects (n = 100) were recorded in the supine position for 15min, and HRV features were extracted. The significant non-linear HRV features were identified through statistical analysis. It was found that Poincare plot features (SD1 and SD2) can differentiate the T2DM subject data from healthy subject data. Several machine learning classifiers, such as Linear Discriminant Analysis (LDA), Quadratic Discriminant Analysis, Naïve Bayes, and Gaussian Process Classifier (GPC), have classified the data based on the cross-validation approach. A GP classifier was implemented using three kernels, namely radial basis, linear, and polynomial kernel, considering the ability to handle the non-linear data. The classifier performance was evaluated and compared using performance metrics such as accuracy(AC), sensitivity(SN), specificity(SP), precision(PR), F1 score, and area under the receiver operating characteristic curve(AUC). Initially, all non-linear HRV features were selected for classification, but the specificity of the model was the limitation. Thus, only two Poincare plot features were used to design the diagnostic model. Our diagnostic model shows the performance using GPC based linear kernel as AC of 92.59%, SN of 96.07%, SP of 81.81%, PR of 94.23%, F1 score of 0.95, and AUC of 0.89, which are more extensive compared to other classification models. Further, the diagnostic model was deployed on the hardware module. Its performance on unknown/test data was validated on 65 subjects (healthy n = 15 and T2DM n = 50). Considering the desirable performance of the diagnostic model, it can be used as an initial screening test tool for a healthcare practitioner to predict T2DM risk.

Usefulness of blinking duration variability (BDV) in discriminating emotional states

Numerous studies show that various measures of eye blinking event including blinking rate, duration, blinking rate variability etc. has a strong correlation with mental processes. However, the blinking duration variability (BDV) in connection with different emotional states has not been explored in details. Therefore, this present study investigates the usefulness of BDV to discriminate different emotional states. In this purpose, SEED-IV database has been considered, in which 15 participants were presented with four types of emotional (neutral, sad, fear and happy) video clips in three different sessions and blinking number along with duration of each blinking (in ms) have been recorded simultaneously. Next, for each subject, the BDV signals have been generated for each type of emotional states and considered them as time series data. Now, keeping the analogy with the heart rate variability (HRV) or blinking rate variability (BRV) analysis, several time series analysis techniques, such as statistical, geometrical, sample entropy and recurrence plot have been employed. The results show that the sample frequency from histogram analysis, short-term BDV from Poincare plot, sample entropy and recurrence rate from recurrence plot are very much suitable to discriminate different emotional states, especially among neutral, negative (sad and fear) and positive (happy). Such findings reveal that the BDV could be a suitable measure to differentiate different basic emotional states. Hence, this study could enrich the domain of ocular event based emotion recognition and analysis.

Nonlinear Constitutive Model of Rock Joint in Geological Structure and Application of Implicit Return Mapping Algorithm

Nonlinear Constitutive Model of Rock Joint in Geological Structure and Application of Implicit Return Mapping Algorithm