Narrow Results

In this paper, we study the synchronization of three Piecewise Linear (PWL) Rössler systems that exhibit coexistence of two attractors, coupled in linear configuration. In particular, we analyze the four linear configurations of the network motifs. We analyze the interaction of the systems by varying the unidirectional or bidirectional coupling strengths depending on the case under study, in order to characterize the conditions under which complete synchronization between the three PWL Rössler systems are obtained. We have found, by means of the error functions, the range of values of the coupling strengths where the different types of synchronization occurs. The numerical results show that the interaction of various multistable chaotic systems causes different types of synchronization and very complex dynamics between the systems.

This study aims to solve the nonlinear Pochhammer–Chree ($\mathbb{N}\mathbb{P}ℂ$) equation to understand its physical implications and establish connections with other nonlinear evolution equations, particularly in plasma dynamics. By using the Khater II ($𝕂$hat. II) method for analytical solutions and validating these solutions numerically with the variational iteration method, this study offers a detailed understanding of the equation’s behavior. The results demonstrate the effectiveness of these methods in accurately modeling the system, highlighting the importance of combining analytical and numerical approaches for reliable solutions. This research significantly advances the field of nonlinear dynamics, especially in plasma physics, by employing multidisciplinary methods to tackle complex physical processes. Moreover, the $\mathbb{N}\mathbb{P}ℂ$ equation is relevant in various physical contexts beyond plasma dynamics such as optical and quantum fields. In optics, it models the propagation of nonlinear waves in fiber optics, where similar nonlinear evolution equations describe wave interactions. In quantum field theory, the $\mathbb{N}\mathbb{P}ℂ$ equation helps in understanding the behavior of quantum particles and fields under nonlinear effects, making it a versatile tool for studying complex phenomena across different domains of physics.

In this study, we examine the multifractal characteristics and nonlinear interactions between the major energy commodities (crude oil, natural gas, heating oil) and the traditional and green bond markets. Using multifractal detrended cross-correlation analysis (MFDCCA) in conjunction with the maximum overlap discrete wavelet transform (MODWT) and nonlinear Granger causality tests, we discover complex dynamics and potential causality within these markets. The results of the study show strong multifractal cross-correlations between green bonds and traditional bonds with energy commodities. Green bonds are found to have a more pronounced multifractal nature compared to traditional bonds, suggesting different market dynamics in response to energy commodity price changes. This strategy advances the understanding of market movements and provides insights for risk mitigation and formulating investment plans. Our findings offer novel insights for investors, policymakers, and academics, highlighting the intricate relationships that exist between energy commodities and financial markets, with implications for portfolio diversification, risk management, and sustainable finance strategies.

This study explores the incorporation of flexoelectricity and size-dependence effects in electricity harvesting from a novel nanostructure. This nanostructure, featuring a cylindrical-shaped rigid tip mass, is stimulated at the base with axial pulsating excitation. Designed with a nanobeam material, the energy harvester accounts for structural nonlinearity, flexoelectricity, and the presence of an off-center rigid mass, operating under multiple excitations. The eigenanalysis is employed to delve into the dynamic characteristics of the system, providing engineers with insights to identify safe operating regions and operational constraints. Additionally, perturbation techniques are utilized to estimate steady-state voltage and power characteristics, with a focus on maximizing voltage and power generation. The study underscores the impact of size dependence on nanoscale design, flexoelectricity, asymmetric tip mass, and various excitation parameters within a constrained operating range. Analysis reveals how saddle-node and pitchfork bifurcations can disrupt energy harvesting performance and suggests strategies for mitigating these issues by adjusting operational parameters. Moreover, the study emphasizes the significant role of the off-center rigid tip mass in energy estimation compared to conventional microstructure-based energy harvesters. Analytical findings are rigorously validated against numerical results under different resonant scenarios, showcasing the accuracy of the analytical approach in capturing nonlinear sources and optimizing energy harvesting from nanostructures.

The motion of a pitching airfoil at low Reynolds number and high angle of attack involves nonlinear dynamics and unstable characteristics, making it a fascinating area for research. In this study, the precise modeling of this complex fluid flow phenomenon is addressed. By combining manifold learning and time delay embedding theory, stable nonlinear low-dimensional manifolds are successfully extracted from the complex high-dimensional spatiotemporal data of the fluid system, simplifying the representation of intricate physical behaviors in space. Subsequently, utilizing the Least Absolute Shrinkage and Selection Operator (LASSO) model, we establish a set of ordinary differential equations governing the flow field, thereby automatically detecting and simplifying nonlinear terms to enhance understanding of the essence of the flow field. This research provides an efficient and accurate approach for the analysis and control of nonlinear dynamic systems.

Multivariate models in the framework of artificial neural network have been constructed for systems where time series data of several variables is known. The models have been tested using computer generated data for the Lorenz and Henon systems. They are found to be robust and give accurate short term predictions. Analysis of the models is able to throw some light on theoretical questions related to multivariate "embedding" and removal of redundancy in the embedding.

A problem of self-organized structure storage and retrieving is considered. The novel model of nonlinear dynamical system for multi-unit structures memorizing is proposed and its properties are investigated in this paper. The multi-unit system consists of N points in the two-dimensional Euclidean space, and its dynamics is defined by a potential function, that is translation and rotation invariant relatively points coordinates. The nonlinear potential function allows to compose attractors corresponding to the proper configurations of points, while these configurations are memorized by the system. Any initial form flows to one of the attractors independently from possible rotations and spatial shifts. The system can memorize and successfully restore up to N-2 required configurations. The characteristics of structure retrieving in the presence of beginning form distortion are considered, and the stability of method is shown even in the case of high level noise. The proposed approach could be helpful to design physical, technical and informational objects with the desired self-assembling properties.

To determine the attractor dimension of chaotic dynamics, the box-counting method has the difficulty in getting accurate estimates because the boxes are not weighted by their relative probabilities. We present a new method to minimize this difficulty. The local box-counting method can be quite effective in determining the attractor dimension of high-order chaos as well as low-order chaos.

Analytical and numerical results on the ordering role of external random fluctuations in excitable systems are presented. Our study focuses on a simple model for excitable systems. Regular waves are created and sustained out of noise when the system is forced with random perturbations. Explicit results for the generation and dynamics of rings and targets are presented.

In this work, we study the stochastic resonance phenomenon in a bistable nonlinear dynamical system in the presence of an uncorrelated noise source whose distribution decays asymptotically as P(ξ) ∝ 1/ξ^2α. We investigate the influence of the decay exponent α on the transition rate and on the optimal noise intensity giving the maximum signal-to-noise ratio when a weak periodic signal is superposed to the external noise. We find that the transition rate achieves a maximum for a finite decay exponent α. However, the optimal noise intensity for stochastic resonance depicts a monotonic power-law correction relative to the usual behavior of nonlinear dynamical systems driven by Gaussian noises.

Recently, synchronization in the Burridge–Knopoff model has been found to depend on the initial conditions. Here we report the existence of three modes of oscillations of the system of three blocks. In one of the modes, two lateral blocks are synchronized. In the second mode, the central block moves with almost constant velocity, i.e., it does not stick. Two lateral blocks do stick and they move in opposite phases. In the third mode, the blocks oscillate with aperiodic amplitude. The lateral blocks move in opposite phases and their frequency is lower than the one for the central block. The mode selected by the system depends on the initial conditions. Numerical results indicate that there is no modes in the phase space.

Dynamics of complex systems is studied by first considering a chaotic time series generated by Lorenz equations and adding noise to it. The trend (smooth behavior) is separated from fluctuations at different scales using wavelet analysis and a prediction method proposed by Lorenz is applied to make out of sample predictions at different regions of the time series. The prediction capability of this method is studied by considering several improvements over this method. We then apply this approach to a real financial time series. The smooth time series is modeled using techniques of non linear dynamics. Our results for predictions suggest that the modified Lorenz method gives better predictions compared to those from the original Lorenz method. Fluctuations are analyzed using probabilistic considerations.

We systematically carried out oil–gas–water three-phase flow experiments for measuring the time series of flow signals. We first investigate flow pattern behaviors from the energy and frequency point of view and find that different flow patterns exhibit different flow behaviors. In order to quantitatively characterize dynamic behaviors underlying different oil–gas–water three-phase flow patterns, we infer and analyze visibility graphs (complex networks) from signals measured under different flow conditions. The results indicate that the combination parameters of network degree are sensitive to the transition among different flow patterns, which can be used to distinguish different flow patterns and quantitatively characterize nonlinear dynamics of the three-phase flow. In this regard, visibility graph can be a useful tool for characterizing the nonlinear dynamic behaviors underlying different oil–gas–water three-phase flow patterns.

Taking the steam turbine cracked rotor system coupling multi-fault as an example, this paper reports systematic numerical experiments exploring two-parameter periodic complexification cascades. Specifically, we report high-resolution phase diagrams predicting and describing how the periodic sequences unfold over a quite extended range of parameter planes. Surprisingly, such diagrams reveal that the distribution of periodic sequences either agrees well with the characteristics of “eye” of chaos, or is characterized by the derived Farey sum tree and the bidirectional symmetrical Stern–Brocot sum tree. These cascades show that the frequent occurrence of transient multi-period motion is a generic feature for the large rotating machinery system with multi-fault. And the paper obtains a cluster of sequences of the Farey sum tree and the Stern–Brocot sum tree specially suitable for the large nonlinear and nonautonomous systems, which can provide a powerful reference for failure prediction and parameter matching.

We use the logistic equation to model the dynamics of the GDP and the trade of the six countries with the highest GDP in the world, namely, USA, China, Japan, Germany, UK and India. From the modeling of the economic data, which are made available by the World Bank, we predict the maximum values of the growth of GDP and trade, as well as the duration over which exponential growth can be sustained. We set up the correlated growth of GDP and trade as the phase solutions of an autonomous second-order dynamical system. GDP and trade are related to each other by a power law, whose exponent seems to differentiate the six national economies into two types. Under conducive conditions for economic growth, our conclusions have general validity.

A study on the master-slave synchronization scheme between Rayleigh–Duffing and Duffing oscillators is presented. We analyze the elastic and dissipative couplings and a combination of both. We compare the results to explore which coupling is more effective to achieve synchronization between both oscillators. The numerical results demonstrate that for the elastic or dissipative coupling at best there is complete synchronization in only one state of the slave system. However, it was also observed that depending on which oscillator acts as the master system and the coupling used, there may be partial or complete synchronization for large values of the coupling strength. When the combination of both couplings is used, there always exists complete synchronization for the two states of the slave system.

In this work, we study the synchronization in the network motifs of three piecewise Rössler systems that exhibit coexistence of two attractors, coupled in ring configuration, in particular a bidirectional auxiliary coupling. For this configuration, we analyze the interaction of the systems by varying the coupling strengths, in order to characterize the conditions under which complete synchronization between the three systems can be achieved. The numerical results prove that the interaction of various multistable chaotic systems causes different types of synchronization and very complex dynamics between the systems is achieved. We obtain complete synchronization between the three systems and find the range of values of the coupling strengths where the distinct types of synchronization occur.

We present a numerical study of the logistic map dynamics with time dependent parameter values whose evolution is in turn governed by a deterministic cellular automaton. We find an unusual type of noise which presents a surprising internal structure yielding a multiple band attractor.

We consider the scattering of solitons and antisolitons in a class of models in (2+1) dimensions. We point out that although in general the interaction forces between solitons and antisolitons are attractive, in some σ models they depend on the relative orientation between the solitons and antisolitons in their internal space. We discuss the scattering properties of such solitons and antisolitons.

A hybrid approach, incorporating concepts of nonlinear dynamics in artificial neural networks (ANN), is proposed to model a time series generated by complex dynamic systems. We introduce well-known features used in the study of dynamic systems — time delay τ and embedding dimension d — for ANN modeling of time series. These features provide a theoretical basis for selecting the optimal size for the number of neurons in the input layer. The main outcome of the new approach for such problems is that to a large extent it defines the ANN architecture, models the time series and gives good prediction. As a consequence, we have an integrated and systematic data-driven scheme for modeling time series data. We illustrate our method by considering computer generated periodic and chaotic time series. The ANN model developed gave excellent quality of fit for the training and test sets as well as for iterative dynamic predictions for future values of the two time series. Further, computer experiments were conducted by introducing Gaussian noise of various degrees in the two time series, to simulate real world effects. We find that up to a limit introduction of noise leads to a smaller network with good generalizing capability.