Analysis Of Nonlinear Dynamical Systems Research Articles

Methods for bifurcation and recurrent analysis of nonlinear dynamical systems on memristive circuit example

Methods for bifurcation and recurrent analysis of nonlinear dynamical systems on memristive circuit example

Van der Waals universe with adiabatic matter creation

A FRWL cosmological model with perfect fluid comprising of van der Waals gas and dust has been studied in the context of dynamical analysis of a three-component autonomous non-linear dynamical system for the particle number density n, the Hubble parameter H, and the temperature T. Perfect fluid isentropic particle creation at rate proportional to an integer power alpha of H has been incorporated. The existence of a global first integral allows the determination of the temperature evolution law and hence the reduction of the dynamical system to a two-component one. Special attention is paid to the cases of alpha = 2 and alpha = 4 and these are illustrated with numerical examples. The global dynamics is comprehensively studied for different choices of the values of the physical parameters of the model. Trajectories in the (n, H) phase space are identified for which temporary inflationary regime exists.

On the spectral properties of nonsingular matrices that are strictly sign-regular for some order with applications to totally positive discrete-time systems

A matrix is called strictly sign-regular of order k (denoted by S S R k ) if all its k × k minors are non-zero and have the same sign. For example, totally positive matrices, i.e., matrices with all minors positive, are S S R k for all k . Another important subclass are those that are S S R k for all odd k . Such matrices have interesting sign variation diminishing properties, and it has been recently shown that they play an important role in the analysis of certain nonlinear cooperative dynamical systems. In this paper, the spectral properties of nonsingular matrices that are S S R k for a specific value k are studied. One of the results is that the product of the first k eigenvalues is real and of the same sign as the k × k minors, and that linear combinations of certain eigenvectors have specific sign patterns. It is then shown how known spectral properties for matrices that are S S R k for several values of k can be derived from these results. Using these theoretical results, the notion of a totally positive discrete-time system (TPDTS) is introduced. This may be regarded as the discrete-time analogue of the important notion of a totally positive differential system. It is shown that TPDTSs can be applied to prove that certain time-varying nonlinear dynamical systems entrain to periodic excitations.

Characteristic, completion or matching timescales? An analysis of temporary boundaries in enzyme kinetics

Scaling analysis exploiting timescale separation has been one of the most important techniques in the quantitative analysis of nonlinear dynamical systems in mathematical and theoretical biology. In the case of enzyme catalyzed reactions, it is often overlooked that the characteristic timescales used for the scaling the rate equations are not ideal for determining when concentrations and reaction rates reach their maximum values. In this work, we first illustrate this point by considering the classic example of the single-enzyme, single-substrate Michaelis–Menten reaction mechanism. We then extend this analysis to a more complicated reaction mechanism, the auxiliary enzyme reaction, in which a substrate is converted to product in two sequential enzyme-catalyzed reactions. In this case, depending on the ordering of the relevant timescales, several dynamic regimes can emerge. In addition to the characteristic timescales for these regimes, we derive matching timescales that determine (approximately) when the transitions from transient to quasi-steady-state kinetics occurs. The approach presented here is applicable to a wide range of singular perturbation problems in nonlinear dynamical systems.

Non-stationary response of MDOF dynamical systems under combined Gaussian and Poisson white noises by the generalized cell mapping method

A block matrix analysis procedure of the generalized cell mapping (GCM) method is proposed in this paper. The proposed method solves the storage problem in the response analysis of nonlinear stochastic dynamical system with the GCM method, and makes it possible to compute the non-stationary and stationary probability density functions (PDFs) of multi-degree-of-freedom (MDOF) nonlinear systems without using supercomputing. Two examples of two-degree-of-freedom systems under external or parametric excitations of combined Gaussian and Poisson white noises are presented to demonstrate the efficiency of the proposed method. Monte Carlo (MC) simulations are used to verify the accuracy of the solutions obtained with the proposed method.

Estimation of time-variant system reliability of nonlinear randomly excited systems based on the Girsanov transformation with state-dependent controls

The problem of time-variant system reliability analysis of nonlinear dynamical systems subjected to random excitations is considered. The governing equations are formulated as a set of Ito’s stochastic differential equations. Subsequently, a Monte Carlo simulation strategy, which incorporates Girsanov’s transformation- based variance reduction step, is developed. The novel element of the work lies in the formulation of state-dependent Girsanov’s control forces for estimating the system reliability. The study considers failure modes arranged in series, parallel, or composite configurations. Illustrative examples include studies on a 5-dof Duffing’s system and an inelastic frame subjected to multi-support, non-stationary, Gaussian excitations. The numerical results demonstrate significant variance reduction achieved in estimating low probabilities of failure.

Discrete differential operators for periodic and two-periodic time functions

PurposeTo identify the properties of novel discrete differential operators of the first- and the second-order for periodic and two-periodic time functions.Design/methodology/approachThe development of relations between the values of first and second derivatives of periodic and two-periodic functions, as well as the values of the functions themselves for a set of time instants. Numerical tests of discrete operators for selected periodic and two-periodic functions.FindingsNovel discrete differential operators for periodic and two-periodic time functions determining their first and the second derivatives at very high accuracy basing on relatively low number of points per highest harmonic.Research limitations/implicationsReduce the complexity of creation difference equations for ordinary non-linear differential equations used to find periodic or two-periodic solutions, when they exist.Practical implicationsApplication to steady-state analysis of non-linear dynamic systems for solutions predicted as periodic or two-periodic in time.Originality/valueIdentify novel discrete differential operators for periodic and two-periodic time functions engaging a large set of time instants that determine the first and second derivatives with very high accuracy.

Identification and robustness analysis of nonlinear hybrid dynamical system of genetic regulation in continuous culture

In this paper, we present a framework to infer the possible transmembrane transport of intracellular substances. Considering four key enzymes, a modified fourteen-dimensional nonlinear hybrid dynamic system is established to describe the microbial continuous culture with enzyme-catalytic and genetic regulation. A novel quantitative definition of biological robustness is proposed to characterize the system's resilience when system parameters were perturbed. It not only considers the expectation of system output data after parameter disturbance but also considers the influence of the variance of these data. In this way, the definition can be used as an objective function of the system identification model due to the lack of data on the concentration of intracellular substances. Then, we design a parallel computing method to solve the system identification model. Numerical results indicate that the most likely transmembrane mode of transport is active transport coupling with passive diffusion for glycerol and 1, 3-propanediol.

A subdomain synthesis method for global analysis of nonlinear dynamical systems based on cell mapping

Unaffordable memory requirement becomes a bottleneck for the cell mapping method when global analysis is carried out in a chosen domain of the state space with a specific computer in order to depict complex invariant sets with high resolution, and/or in high dimensions. This may also be often encountered even though the subdivision technique is adopted. Thus a subdomain synthesis method for cell mapping is proposed in this paper in order to release this difficulty. The proposed method first divides the chosen domain into smaller subdomains with affordable memory requirements, then carries the cell mapping analysis out on each subdomain independently meanwhile records the message on inter-subdomain mappings, finally recovers the global structure in the complete chosen domain based on the dynamical information revealed in the subdomains. The method can effectively avoid the loss of the invariant sets in the chosen domain that are unavoidably sliced into pieces by the state space partition without prior knowledge on them. Furthermore, both the subdivision technique and the parallel computing based on graphical processing units can also be incorporated into the proposed method that enhances the computation efficiency. Three examples of application on global analysis of nonlinear dynamical systems are presented to demonstrate the basic idea and the potential of the proposed method.

Parameter converting method for bifurcation analysis of nonlinear dynamical systems

For detecting behavior of a dynamical system, bifurcation analysis is necessary with respect to change in parameters of system. In this work, based on the solution of ordinary differential equations from initial value and parameters, a simple method is presented, which can efficiently reveal different bifurcations of system. In addition to its simplicity, this method does not required to have deep physical and mathematical understanding of the problem, and because of its high precision and the speed of solutions, does not need to reduce the order of models in many complex problems or problems with high degrees of freedom. This method is named parameter converting method (PCM), which has two steps. In the first step the parameter is varied as a function of time and in the second step, time is expressed as inverse of this assumed function. With this method bifurcation and amplitude-frequency diagrams and hidden attractors of some complex dynamics will be analyzed and the sensitivity of the multi potential well systems to initial conditions is studied. With this algorithm, a simple way to find the domain of high-energy orbit in bistable systems is obtained.

Identifiability analysis of nonlinear dynamical system

Identifiability analysis of nonlinear dynamical system

Effect of indoor wheelchair curling training on trunk control of person with chronic spinal cord injury: a randomised controlled trial.

Randomised, controlled, single-blind crossover design study. Effect of indoor wheelchair curling training on trunk control of a person with chronic spinal cord injury (SCI). SCI Centre of Balgrist University Hospital in Zurich, Switzerland. The trunk control of 13 subjects was assessed by the modified functional reach test (MFRT) and nonlinear dynamic systems analysis (NDSA) before and after eight indoor curling training sessions and compared to everyday life over 4 weeks. The attendance rate was 95% during the training sessions. There were no adverse events. Neither the MRFT nor the NDSA showed any significant differences in the sitting stability. The subjects subjectively reported improved physical feeling, an increase in their trunk control and strength; 39% of the participants wanted to continue the training. With subjective improvements and no adverse events, indoor wheelchair curling training is a safe physical activity for people with SCI. Wheelchair curling offers a suitable alternative to sports already used in rehabilitation and in recreational activities of a person with SCI. In order to be able to draw more unambiguous conclusions from the training method for trunk control and to answer the question of the study unambiguously, the number of subjects would have to be greater and the test methods should provide more exact and specific measurements.

Differential Privacy and Qualitative Privacy Analysis for Nonlinear Dynamical Systems

Abstract In this paper, we pursue privacy analysis of nonlinear dynamical systems from two different aspects. As a quantitative criterion for privacy of “mechanisms” in the form of data-generating processes, the concept of differential privacy was proposed in computer science and has been applied to linear dynamical systems. In this paper, we further extend this concept to nonlinear dynamical systems and show that the incrementally input-to-output stable system is always differentially private. In fact, differential privacy evaluates the privacy level of the scenario involving the least private information. Therefore, based on differential privacy, it is difficult to study exactly what kind of information is protected. To address this problem, we proceed with qualitative analysis of privacy in terms of input observability for nonlinear systems. In particular, we provide a necessary and sufficient condition for input observability, revealing an impossibility result on protecting private information.

Analysis of the temporal self-organizational phenomena observed during the electrodissolution of Zn in sulfuric acid solutions

This work aims to analyze the high-frequency current oscillations observed during the electrodissolution of zinc in sulfuric acid solutions. The current oscillations are emerged within a fixed potential region, ΔΕosc in the characteristic polarization curve, I = f(E) of the Zn | x M H2SO4 system at proper concentrations of the H2SO4. The ΔΕosc appears in the limiting current region under precipitation/dissolution conditions of the ZnSO4.yH2O across the interface Zn | x M H2SO4, during the electrodissolution of Zn. The oscillation characteristics (frequency, amplitude, waveform, induction time needed for their emergence) are changing by varying certain parameters, such as the sulfuric acid concentration, the angular rotational speed of the Zn disc-electrode and a variable external resistance added to the solution resistance. The time series, I = f(t) are recorded under potentiostatic conditions within the ΔΕosc and, using techniques for the analysis of non-linear dynamical systems, are characterized as chaotic. We have proposed construction of recurrence plots and the recurrence quantification analysis as appropriate tools for characterizing current oscillations far from bifurcation points and unraveling dynamical transitions of the electrochemical system as the applied potential is varied.

Generalizing Koopman Theory to Allow for Inputs and Control.

We develop a new generalization of Koopman operator theory that incorporates the e ects of inputs and control. Koopman spectral analysis is a theoretical tool for the analysis of nonlinear dynamical systems. Moreover, Koopman is intimately connected to dynamic mode decomposition (DMD), a method that discovers coherent, spatio-temporal modes from data, connects local-linear analysis to nonlinear operator theory, and importantly creates an equation-free architecture for the study of complex systems. For actuated systems, standard Koopman analysis and DMD are incapable of producing input-output models; moreover, the dynamics and the modes will be corrupted by external forcing. Our new theoretical developments extend Koopman operator theory to allow for systems with nonlinear input-output characteristics. We show how this generalization is rigorously connected to a recent development called dynamic mode decomposition with control. We demonstrate this new theory on nonlinear dynamical systems, including a standard susceptible-infectious-recovered model with relevance to the analysis of infectious disease data with mass vaccination (actuation).

Heuristic Determination of Resolving Controls for Exact and Approximate Controllability of Nonlinear Dynamic Systems

Dealing with practical control systems, it is equally important to establish the controllability of the system under study and to find corresponding control functions explicitly. The most challenging problem in this path is the rigorous analysis of the state constraints, which can be especially sophisticated in the case of nonlinear systems. However, some heuristic considerations related to physical, mechanical, or other aspects of the problem may allow coming up with specific hierarchic controls containing a set of free parameters. Such an approach allows reducing the computational complexity of the problem by reducing the nonlinear state constraints to nonlinear algebraic equations with respect to the free parameters. This paper is devoted to heuristic determination of control functions providing exact and approximate controllability of dynamic systems with nonlinear state constraints. Using the recently developed approach based on Green’s function method, the controllability analysis of nonlinear dynamic systems, in general, is reduced to nonlinear integral constraints with respect to the control function. We construct parametric families of control functions having certain physical meanings, which reduce the nonlinear integral constraints to a system of nonlinear algebraic equations. Regimes such as time-harmonic, switching, impulsive, and optimal stopping ones are considered. Two concrete examples arising from engineering help to reveal advantages and drawbacks of the technique.

A Review on the Nonlinear Dynamical System Analysis of Electrocardiogram Signal.

Electrocardiogram (ECG) signal analysis has received special attention of the researchers in the recent past because of its ability to divulge crucial information about the electrophysiology of the heart and the autonomic nervous system activity in a noninvasive manner. Analysis of the ECG signals has been explored using both linear and nonlinear methods. However, the nonlinear methods of ECG signal analysis are gaining popularity because of their robustness in feature extraction and classification. The current study presents a review of the nonlinear signal analysis methods, namely, reconstructed phase space analysis, Lyapunov exponents, correlation dimension, detrended fluctuation analysis (DFA), recurrence plot, Poincaré plot, approximate entropy, and sample entropy along with their recent applications in the ECG signal analysis.

Stability analysis of distributed-order nonlinear dynamic systems

ABSTRACTThe problem of asymptotic stability analysis of equilibrium points in nonlinear distributed-order dynamic systems with non-negative weight functions is considered in this paper. The Lyapunov direct method is extended to be used for this stability analysis. To this end, at first, a discretisation scheme with convergence property is introduced for distributed-order dynamic systems. Then, on the basis of this tool, Lyapunov theorems are proved for asymptotic stability analysis of equilibrium points in distributed-order systems. As the order weight function assumed for the distributed-order systems is general enough, the results are applicable to a wide range of nonlinear distributed-order systems such as fractional-order systems with multiple fractional derivatives. To verify the applicability of the obtained results, these results are applied for the stability analysis of a distributed-order diffusion system and control of a fractional-order Lorenz system with multiple fractional derivatives.

An adaptive cubature formula for efficient reliability assessment of nonlinear structural dynamic systems

Abstract Extreme value distribution (EVD) evaluation is a critical topic in reliability analysis of nonlinear structural dynamic systems. In this paper, a new method is proposed to obtain the EVD. The maximum entropy method (MEM) with fractional moments as constraints is employed to derive the entire range of EVD. Then, an adaptive cubature formula is proposed for fractional moments assessment involved in MEM, which is closely related to the efficiency and accuracy for reliability analysis. Three point sets, which include a total of 2 d 2 + 1 integration points in the dimension d, are generated in the proposed formula. In this regard, the efficiency of the proposed formula is ensured. Besides, a “free” parameter is introduced, which makes the proposed formula adaptive with the dimension. The “free” parameter is determined by arranging one point set adjacent to the boundary of the hyper-sphere which contains the bulk of total probability. In this regard, the tail distribution may be better reproduced and the fractional moments could be evaluated with accuracy. Finally, the proposed method is applied to a ten-storey shear frame structure under seismic excitations, which exhibits strong nonlinearity. The numerical results demonstrate the efficacy of the proposed method.

An alternative approach for stability analysis of discrete time nonlinear dynamical systems

The paper endeavours to solve the problem of the necessary and sufficient conditions for testing asymptotic stability of the equilibrium state without using a positive definite or semi-definite Lyapunov function for time-invariant nonlinear discrete-time dynamical systems. The solution is based on the concept of the -functions introduced in this paper. As a result, new necessary and sufficient conditions for asymptotic stability of such systems and an estimation or the exact determination of the asymptotic stability domain of the state are established. Examples are worked out to illustrate the results.