Behavior Of Nonlinear Dynamical Systems Research Articles

Innovative coupling of s-stage one-step and spectral methods for non-smooth solutions of nonlinear problems

The behavior of nonlinear dynamical systems arising in mathematical physics through numerical tools is a challenging task for researchers. In this context, an efficient semi-spectral method is proposed and applied to observe the robust solutions for the mathematical physics problems. Firstly, the space variable is approximated by the Vieta-Lucas polynomials and then the s-stage one-step method is applied to discretize the temporal variable which transfers the problem in the form Cn+1=Cn+Δtϕ(x,t,Cn,F(un)). Novel operational matrices of integer order are developed to replace the spatial derivative terms presented in the discussed problem. Related theorems are included in the study to validate the approach mathematically. The proposed semi-spectral schemes convert the considered nonlinear problem to a system of linear algebraic equations which is easier to tackle. We also accomplish an investigation on the error bound and convergence to confirm the mathematical formulation of the computational algorithm. To show the accuracy and effectiveness of the suggested computational method numerous test problems, such as the advection-diffusion problem, generalized Burger-Huxley, sine-Gordon, and modified KdV–Burgers’ equations are considered. An inclusive comparative examination demonstrates the currently suggested computational method in terms of credibility, accuracy, and reliability. Moreover, the coupling of the spectral method with the fourth-order Runge-Kutta method seems outstanding to handle the nonlinear problem to examine the precise smooth and non-smooth solutions of physical problems. The computational order of convergence (COC) is computed numerically through numerous simulations of the proposed schemes. It is found that the proposed schemes are in exponential order of convergence in the spatial direction and the COC in the temporal direction validates the studies in the literature.

Extrapolating tipping points and simulating non-stationary dynamics of complex systems using efficient machine learning

Model-free and data-driven prediction of tipping point transitions in nonlinear dynamical systems is a challenging and outstanding task in complex systems science. We propose a novel, fully data-driven machine learning algorithm based on next-generation reservoir computing to extrapolate the bifurcation behavior of nonlinear dynamical systems using stationary training data samples. We show that this method can extrapolate tipping point transitions. Furthermore, it is demonstrated that the trained next-generation reservoir computing architecture can be used to predict non-stationary dynamics with time-varying bifurcation parameters. In doing so, post-tipping point dynamics of unseen parameter regions can be simulated.

The fusion of quantum chaos and nonlinear dynamics in securing image encryption

In the rapidly evolving landscape of digital communication, the security of image data transmits paramount importance. This paper introduces a novel encryption framework that integrates the principles of quantum chaos and nonlinear dynamics to fortify image encryption processes. Recognizing the vulnerabilities inherent in conventional encryption methods, our research seeks to harness the unpredictable nature of quantum chaos alongside the complex behavior of nonlinear dynamical systems to generate robust encryption keys that are highly sensitive to initial conditions. Utilizing a mathematical model that combines the unpredictability of quantum chaotic systems with the intricate patterns of nonlinear dynamics, we develop an encryption algorithm that demonstrates superior resistance to both brute-force attacks and statistical analysis. The algorithm’s efficacy is evaluated through a series of rigorous tests, including entropy analysis, correlation coefficient assessment, and resistance to known-plaintext and chosen-plaintext attacks.

A novel approach to investigate the stability analysis and the dynamics of reaction–diffusion SVIR epidemic model

Understanding the behaviour of nonlinear dynamical systems is crucial in epidemiological modelling. Stability analysis is one of the important concepts in assessing the qualitative behaviour of such systems. This technique has been widely implemented on deterministic models involving ordinary differential equations (ODEs). Nevertheless, the application of stability analysis to distinct complex systems involving partial differential equations (PDEs) has been infrequent. In this paper, an in-depth investigation is conducted on the stability analysis and the dynamics of a reaction–diffusion SVIR model with partial immunity. Firstly, numerical simulation is performed for the proposed PDE model under several realistic scenarios to demonstrate the crucial factors that may affect the qualitative dynamics of the epidemiological model. Our numerical simulation findings highlight the occurrence of disease-free equilibrium when the transmission rates are sufficiently low, preventing the spread of the disease. However, an endemic situation is observed when the transmission rates are moderate (or high) enough to sustain ongoing transmission, leading to a stable endemic steady state. Other epidemiological factors such as vaccination and treatment (or recovery) strategies also play important roles in controlling disease outbreaks. The transition between disease-free equilibrium (DFE) and endemic equilibrium (EE), resulting from switching stability between equilibria, is of particular interest in epidemiology, due to its implications for public health interventions and strategies. This insight helps to inform decision-making processes regarding the control and prevention of diseases. To examine the stability of steady states in this PDE model, we employ a novel technique called numerical range approaches, which relies on the application of operator theory. Typically, discretising differential operators in the PDE model leads to spectral complexity and a wide range of eigenvalues. Our analysis has demonstrated that employing the numerical range (NR) and the cubic numerical range (CNR) techniques make it feasible to estimate each individual eigenvalue in the spectral range. For the DFE (respectively, EE) scenario, the spectrum of eigenvalues for the PDE system agrees with CNR estimates in which the eigenvalues are real-valued and bounded above by zero (respectively, complex conjugates pairs with negative real parts); these numerical estimations have also been verified analytically. Despite the challenges posed, the use of numerical range techniques provides a good estimation of the eigenvalues spectrum, thereby overcoming the difficulties associated with discretisation, and providing a more comprehensive understanding of the dynamics of the epidemiological system.

Dynamics of Newtonian Liquids with Distinct Concentrations Due to Time-Varying Gravitational Acceleration and Triple Diffusive Convection: Weakly Non-Linear Stability of Heat and Mass Transfer

One of the practical methods for examining the stability and dynamical behaviour of non-linear systems is weakly non-linear stability analysis. Time-varying gravitational acceleration and triple-diffusive convection play a significant role in the formation of acceleration, inducing some dynamics in the industry. With an emphasis on the natural Rayleigh–Bernard convection, more is needed on the significance of a modulated gravitational field on the heat and mass transfer due to triple convection focusing on weakly non-linear stability analysis. The Newtonian fluid layers were heated, salted and saturated from below, causing the bottom plate’s temperature and concentration to be greater than the top plate’s. In this study, the acceleration due to gravity was assumed to be time-dependent and comprised of a constant gravity term and a time-dependent gravitational oscillation. More so, the amplitude of the modulated gravitational field was considered infinitesimal. The case in which the fluid layer is infinitely expanded in the x-direction and between two concurrent plates at z=0 and z=d was considered. The asymptotic expansion technique was used to retrieve the solution of the Ginzburg–Landau differential equation (i.e., a system of non-autonomous partial differential equations) using the software MATHEMATICA 12. Decreasing the amplitude of modulation, Lewis number, Rayleigh number and frequency of modulation has no significant effect on the Nusselt number proportional to heat-transfer rates (Nu), Sherwood number proportional to mass transfer of solute 1 (Sh1) and Sherwood number proportional to mass transfer of solute 2 (Sh2) at the initial time. The crucial Rayleigh number rises in value in the presence of a third diffusive component. The third diffusive component is essential in delaying the onset of convection.

Nonlinear features of photoplethysmography signals for Non-invasive blood pressure estimation

ObjectiveContinuous monitoring of blood pressure (BP) plays an essential role in the prognosis and prevention of hypertension and related cardiovascular diseases. Moreover, the ever-increasing demand for portable continuous health monitoring systems coupled with promising capabilities of photoplethysmography (PPG) sensors for developing easy-to-use, portable wearable devices have motivated many researchers toward applying PPG signals for non-invasive health monitoring. Nonlinear nature of BP and proven capability of the Poincaré plot for analyzing dynamic behavior of nonlinear systems motivated us to use this powerful tool for BP estimation. This study aims to explore the dynamical behavior of PPG signals to assist feature extraction for machine-learning (ML) algorithms to estimate BP. We proposed a Poincaré-based feature extraction method for BP estimation with no need to extract precise local points on the PPG signal. MethodsFirst, a Poincaré plot of 10-s segments of the PPG signal was prepared. Then, three distinct time series were retrieved from Poincaré mapping of PPG signals; following this, numerous indices were extracted from the three time series. The F-Test feature selection method was applied to pick the most effective features. Finally, the picked features were fed into different ML algorithms for the estimation of BP. ResultsThe proposed method was validated using a subset of the Medical Information Mart for Intensive Care II (MIMIC-II) database containing ambulatory blood pressure (ABP) and PPG records. The performance evaluation was carried out in terms of mean absolute error (MAE), standard deviation (STD), and Pearson’s correlation coefficient. According to the results, application of the Gaussian process regression (GPR) led to the best performance for both systolic and diastolic BP (0.79 ± 3.08 and 1.38 ± 4.53 mmHg, respectively). Satisfying the criteria for Advancement of Medical Instrumentation (AAMI), it was rated as Grade A for systolic and diastolic BP measurements under terms of British Hypertension Society (BHS) standards, which confirms the efficiency of the Poincaré-based features for BP estimation. ConclusionThe results justify the usefulness of the proposed Poincaré-based features as a powerful tool in BP estimation scenarios.

A New Nonlinear Recurrent Type-2 Fuzzy Model to Identify the Behavior of Nonlinear Dynamic Systems

A New Nonlinear Recurrent Type-2 Fuzzy Model to Identify the Behavior of Nonlinear Dynamic Systems

PiSL: Physics-informed Spline Learning for data-driven identification of nonlinear dynamical systems

Nonlinear dynamics is ubiquitous in nature and commonly seen in many science and engineering disciplines. Distilling analytical expressions that govern the behavior of nonlinear dynamical systems from limited data is vital but remains challenging. To tackle this fundamental issue, we propose a novel Physics-informed Spline Learning (PiSL) framework to discover parsimonious governing equations for nonlinear dynamics, based on sparsely sampled noisy data. Specifically, splines are employed to locally interpolate the dynamics and perform analytical differentiation, feeding the discovery of underlying equations in form of either a linear interpolation of candidate terms or a symbolic-activated neural network model. The physics residual in turn informs the spline learning. The synergy between splines and discovered governing equations produces great robustness against high-level data sparsity and noise. Subsequently, a hybrid sparsity-promoting alternating direction optimization strategy is developed for fine-tuning the coefficients with a sparsity enforcement approach to obtain a parsimonious structure of discovered governing equations. The effectiveness and supremacy of the proposed PiSL architectures have been demonstrated by several numerical and experimental examples, in comparison with two state-of-the-art methods serving as baselines.

Autoregulation of excitation-Ca2+ signaling-contraction and arrhythmogenesisin cardiomyocyte under mechanical load.

The heart pumps blood into circulation against the mechanical load from vascular resistance and actively regulates cardiomyocyte contraction to compensate for load changes. We conducted Cell-in-Gel experiments to decipher the mechano-chemo-electro-transduction (MCET) mechanisms that transduce the mechanical load during cell contraction to regulate the excitation-Ca2+ signaling-contraction (E-C) coupling. Our experimental studies reveal that MCET increases the L-type Ca2+ current and action potential duration, which increases the Ca2+ entry during the cardiac cycle; increases the diastolic SR Ca2+ content, the systolic SR Ca2+ release, and the Ca2+ uptake; and increases the cytosolic Ca2+ transient, resulting in MCET-Ca2+ gain. The MCET-Ca2+ gain enhances contractility in response to increased load. Hence, MCET provides feedback loops in the E-C coupling system, which may enable autoregulation of the dynamic system when mechanical load changes. Based on experimental data, we developed quantitative models to incorporate the MCET feedback loops into the E-C coupling system to study the nonlinear dynamic system's behavior. Model simulations show that autoregulation of E-C coupling contractility naturally arises from the MCET feedback, enabling the cardiomyocyte to autoregulate contractility in compensatory response to load changes. However, excessively high load causes SR Ca2+ overload, spontaneous Ca2+ release, and arrhythmogenic activities. These model predictions are verified by experimental data. In conclusion, our experimental and modeling studies reveal an Autoregulatory E-C coupling dynamic system in the cardiomyocytes contracting under mechanical load. Deciphering the autoregulatory E-C couplingdynamics and MCET mechanisms will be essential for understanding how the heart autoregulates contractility in compensatory response to physiological load changes to maintain cardiac output, and why pathological loading (i.e., hypertension, TAC pressure-overload) leads to arrhythmias and heart failure.

Failing parametrizations: what can go wrong when approximating spectral submanifolds

Invariant manifolds provide useful insights into the behavior of nonlinear dynamical systems. For conservative vibration problems, Lyapunov subcenter manifolds constitute the nonlinear extension of spectral subspaces consisting of one or more modes of the linearized system. Conversely, spectral submanifolds represent the spectral dynamics of non-conservative, nonlinear problems. While finding global invariant manifolds remains a challenge, approximations thereof can be simple to acquire and still provide an effective framework for analyzing a wide variety of problems near equilibrium solutions. This approach has been successfully employed to study both the behavior of autonomous systems and the effects of non-autonomous forcing. The current computation strategies rely on a parametrization of the invariant manifold and the reduced dynamics thereon via truncated power series. While this leads to efficient recursive algorithms, the problem itself is ambiguous, since it permits the use of various approaches for constructing the reduced system to which the invariant manifold is conjugated. Although this ambiguity is well known, it is rarely discussed and usually resolved by an ad hoc choice of method, the effects of which are mostly neglected. In this contribution, we first analyze the performance of three popular approaches for constructing the conjugate system: the graph style parametrization, the normal form parametrization, and the normal form parametrization for “near resonances.” We then show that none of them is always superior to the others and discuss the potential benefits of tailoring the parametrization to the analyzed system. As a means for illustrating the latter, we introduce an alternative strategy for constructing the reduced dynamics and apply it to two examples from the literature, which results in a significantly improved approximation quality.

Information geometry theory of bifurcations? A covariant formulation.

The conventional local bifurcation theory (CBT) fails to present a complete characterization of the stability and general aspects of complex phenomena. After all, the CBT only explores the behavior of nonlinear dynamical systems in the neighborhood of their fixed points. Thus, this limitation imposes the necessity of non-trivial global techniques and lengthy numerical solutions. In this article, we present an attempt to overcome these problems by including the Fisher information theory in the study of bifurcations. Here, we investigate a Riemannian metrical structure of local and global bifurcations described in the context of dynamical systems. The introduced metric is based on the concept of information distance. We examine five contrasting models in detail: saddle-node, transcritical, supercritical pitchfork, subcritical pitchfork, and homoclinic bifurcations. We found that the metric imposes a curvature scalar R on the parameter space. Also, we discovered that R diverges to infinity while approaching bifurcation points. We demonstrate that the local stability conditions are recovered from the interpretations of the curvature R, while global stability is inferred from the character of the Fisher metric. The results are a clear improvement over those of the conventional theory.

A data-driven approach for approximating non-linear dynamic systems using LSTM networks

The analysis of sensor data in nonlinear dynamical systems plays a fundamental role in a variety of modern engineering problems like Fatigue Analysis and Predictive Maintenance. In many applications, the accurate approximation of sensor signals supports and complements experimental efforts in order to accelerate the development process and conserve resources.Virtual sensing techniques aim to replace physical sensors in a system by using the data from available sensors to estimate additional unknown quantities of interest. Forward prediction estimates the system response for a given drive signal during the commission of component test rigs. Data-driven approaches can be convenient tools for forward prediction and virtual sensing, as they only require a sufficiently large dataset of the desired input and output quantities for the purpose of model parametrization. The presented approach explores two hybrid modeling strategies, which combine Frequency Response Function models with Long Short-Term Memory networks to approximate the behavior of non-linear dynamic systems with multiple input and output channels.The proposed method utilizes short subsequences of signals to carry out model training and prediction. Long sequence estimations are generated by combining the individual subsequence predictions using a windowing technique.Our approach is tested on a large, non-linear experimental dataset, obtained from a servo hydraulic fatigue test bench. To enable a comprehensive evaluation of the model quality, both hybrid modeling approaches are compared on virtual sensing and forward prediction tasks using multiple error metrics relevant to fatigue analysis.

Globalism, controlled chaos and values

Consideration of the program of globalism and the concept of controlled chaoswas carried out on the basis of a study of the works of A. Toffler and S. Mann. This article analyzes the terminological basis of the proposed concepts. It is proved that the development of information communications testifies to the evolutionarity of processes on the world stage, which cannot be considered a feature of a revolutionary change in the type of civilization. The criticism of the Western attitude about the positioning of the newest society as purely informational, practically unrelated to industrial achievements, is given. It is shown that in reality the information boom is fully provided by the material production of the "second wave", within the framework of the development of which the possibility of solving various pressing problems, including the problem of energy resources, is visible. Special attention in the article is devoted to the criticism of the concept of «controlled chaos» from the perspective of synergetic methodology, the real threats associated with misunderstanding of the behaviour of nonlinear dynamical systems in the bifurcation mode are outlined. The implementation of the globalism program uses many techniques aimed at standardizing values. In this regard, the Kantian narrative of absolute value.

Signal processing of nonlinear dynamic systems

The paper considers Hermite polynomials that act as a self-similar basis for the decomposition of functions in phase space. It is shown that the equations of behavior of nonlinear dynamical systems are simplified. It is also noted that the wavelet decomposition over Hermite polynomials reduces the number of approximation coefficients and improves the quality of approximation.

Data-driven inference of high-accuracy isostable-based dynamical models in response to external inputs.

Isostable reduction is a powerful technique that can be used to characterize behaviors of nonlinear dynamical systems using a basis of slowly decaying eigenfunctions of the Koopman operator. When the underlying dynamical equations are known, previously developed numerical techniques allow for high-order accuracy computation of isostable reduced models. However, in situations where the dynamical equations are unknown, few general techniques are available that provide reliable estimates of the isostable reduced equations, especially in applications where large magnitude inputs are considered. In this work, a purely data-driven inference strategy yielding high-accuracy isostable reduced models is developed for dynamical systems with a fixed point attractor. By analyzing steady-state outputs of nonlinear systems in response to sinusoidal forcing, both isostable response functions and isostable-to-output relationships can be estimated to arbitrary accuracy in an expansion performed in the isostable coordinates. Detailed examples are considered for a population of synaptically coupled neurons and for the one-dimensional Burgers' equation. While linear estimates of the isostable response functions are sufficient to characterize the dynamical behavior when small magnitude inputs are considered, the high-accuracy reduced order model inference strategy proposed here is essential when considering large magnitude inputs.

A geometric interpretation of maximal Lyapunov exponent based on deviation curvature

In this study, we discuss a relationship between the behavior of nonlinear dynamical systems and geometry of a system of second-order differential equations based on the Jacobi stability analysis. We consider how a maximal Lyapunov exponent is related to the geometric quantities. As a result of a theoretical investigation, the maximal Lyapunov exponent can be represented by a nonlinear connection and a deviation curvature. Thus, this means that the Jacobi stability given by the sign of the deviation curvature affects the change of the maximal Lyapunov exponent. Additionally, for an equation of nonlinear pendulum, we numerically confirm the theoretical results. We observe that a change of the maximal Lyapunov exponent is related to a change of an average deviation curvature. These results indicate that the deviation curvature and Jacobi stability are essential for considering the change of maximal Lyapunov exponent.

Curry-Yorke route to shearless attractors and coexistence of attractors in dissipative nontwist systems.

The routes to chaos play an important role in predictions about the transitions from regular to irregular behavior in nonlinear dynamical systems, such as electrical oscillators, chemical reactions, biomedical rhythms, and nonlinear wave coupling. Of special interest are dissipative systems obtained by adding a dissipation term in a given Hamiltonian system. If the latter satisfies the so-called twist property, the corresponding dissipative version can be called a "dissipative twist system." Transitions to chaos in these systems are well established; for instance, the Curry-Yorke route describes the transition from a quasiperiodic attractor on torus to chaos passing by a chaotic banded attractor. In this paper, we study the transitions from an attractor on torus to chaotic motion in dissipative nontwist systems. We choose the dissipative standard nontwist map, which is a non-conservative version of the standard nontwist map. In our simulations, we observe the same transition to chaos that happens in twist systems, known as a soft one, where the quasiperiodic attractor becomes wrinkled and then chaotic through the Curry-Yorke route. By the Lyapunov exponent, we study the nature of the orbits for a different set of parameters, and we observe that quasiperiodic motion and periodic and chaotic behavior are possible in the system. We observe that they can coexist in the phase space, implying in multistability. The different coexistence scenarios were studied by the basin entropy and by the boundary basin entropy.

Dynamical analysis on a bacteria-phages model with delay and diffusion

Delay and diffusion have important significance in enriching the dynamic behavior of nonlinear dynamical systems. In this paper, we study the effects of delay and diffusion on the dynamics of a bacteria-phages model. By analyzing the stability of equilibria and the existence of Hopf bifurcation, we obtain the following results: (i) Under certain condition, only diffusion cannot contribute to the Turing instability, which is different from the general results that the diffusion can destabilize the model and lead model to produce the Turing instability; (ii) Delay or the combination of diffusion and delay can destabilize the model and lead model to generate the Hopf bifurcation and patterns. Moreover, the formulae for determining the properties of Hopf bifurcation are derived. Numerical simulations are carried out to reveal the effects of delay or the combination of diffusion and delay on the stability and Hopf bifurcation for the model. The results obtained are of significance to predict the coexistence of bacteria and phages and to find the appropriate time to implement the phage therapy.

Affine linear parameter‐varying embedding of non‐linear models with improved accuracy and minimal overbounding

In this study, automated generation of linear parameter-varying (LPV) state-space models to embed the dynamical behaviour of non-linear systems is considered, focusing on the trade-off between scheduling complexity and model accuracy and the minimisation of the conservativeness of the resulting embedding. The LPV state-space model is synthesised with affine scheduling dependency, while the scheduling variables themselves are non-linear functions of the state and input variables of the original system. The method allows to generate complete or approximative embedding of the non-linear system model and also it can be used to minimise the complexity of existing LPV embeddings. The capabilities of the method are demonstrated on simulation examples and also in an empirical case study where the first-principle motion model of a three degrees of freedom (3DOF) control moment gyroscope is converted by the proposed method to an LPV model with low scheduling complexity. Using the resulting model, a gain-scheduled controller is designed and applied on the gyroscope, demonstrating the efficiency of the developed approach.

Hopf Bifurcations in Complex Multiagent Activity: The Signature of Discrete to Rhythmic Behavioral Transitions.

Most human actions are composed of two fundamental movement types, discrete and rhythmic movements. These movement types, or primitives, are analogous to the two elemental behaviors of nonlinear dynamical systems, namely, fixed-point and limit cycle behavior, respectively. Furthermore, there is now a growing body of research demonstrating how various human actions and behaviors can be effectively modeled and understood using a small set of low-dimensional, fixed-point and limit cycle dynamical systems (differential equations). Here, we provide an overview of these dynamical motor primitives and detail recent research demonstrating how these dynamical primitives can be used to model the task dynamics of complex multiagent behavior. More specifically, we review how a task-dynamic model of multiagent shepherding behavior, composed of rudimentary fixed-point and limit cycle dynamical primitives, can not only effectively model the behavior of cooperating human co-actors, but also reveals how the discovery and intentional use of optimal behavioral coordination during task learning is marked by a spontaneous, self-organized transition between fixed-point and limit cycle dynamics (i.e., via a Hopf bifurcation).