Analysis Of Nonlinear Dynamical Systems Research Articles

Stochastic sensitivity analysis: theory and numerical algorithms

A problem of the analysis of nonlinear dynamic systems forced by the random disturbances is considered. For the description of probabilistic distributions around the deterministic attractors, a theory of the quasipotential and its approximations based on the stochastic sensitivity functions (SSF) technique is suggested. Constructive methods for the computation of SSF for the equilibria and limit cycles are presented and discussed. We show how SSF can be used for the visual description of probabilistic distributions in the form of confidence domains.

A generalized bilinear amplitude and frequency approximation for piecewise-linear nonlinear systems with gaps or prestress

Analysis of piecewise-linear nonlinear dynamical systems is critical for a variety of civil, mechanical, and aerospace structures that contain gaps or prestress that are caused by cracks, delamination, joints or interfaces among components. Recently, a technique referred to as bilinear amplitude approximation (BAA) was developed to estimate the response of bilinear systems that have no gap or prestress. The method is based on an idea that the dynamics of a bilinear system can be treated as a combination of linear responses in two time intervals both of which the system behaves as a distinct linear system: (1) the open state and (2) the closed or sliding state. Both geometric and momentum constraints are then applied as compatibility conditions between the states to couple the linear vibrational response for each time interval. In order to estimate the response for more general cases where there are either gaps or prestress in the system, a generalized BAA method is proposed in this paper. The new method requires inclusion of contact stiffness and damping to model contact behavior in the sliding state, and new equilibrium positions for each state to establish proper coordinates. The new method also finds the bilinear frequency of the system, which cannot be computed using the bilinear frequency approximation method previously developed since that method is only accurate for the zero gap and no prestress case. The generalized BAA method is demonstrated on a single degree of freedom system, a three degree of freedom system, and a cracked cantilever beam model for various gap sizes and prestress levels.

Shape Distributions of Nonlinear Dynamical Systems for Video-Based Inference.

This paper presents a shape-theoretic framework for dynamical analysis of nonlinear dynamical systems which appear frequently in several video-based inference tasks. Traditional approaches to dynamical modeling have included linear and nonlinear methods with their respective drawbacks. A novel approach we propose is the use of descriptors of the shape of the dynamical attractor as a feature representation of nature of dynamics. The proposed framework has two main advantages over traditional approaches: a) representation of the dynamical system is derived directly from the observational data, without any inherent assumptions, and b) the proposed features show stability under different time-series lengths where traditional dynamical invariants fail. We illustrate our idea using nonlinear dynamical models such as Lorenz and Rossler systems, where our feature representations (shape distribution) support our hypothesis that the local shape of the reconstructed phase space can be used as a discriminative feature. Our experimental analyses on these models also indicate that the proposed framework show stability for different time-series lengths, which is useful when the available number of samples are small/variable. The specific applications of interest in this paper are: 1) activity recognition using motion capture and RGBD sensors, 2) activity quality assessment for applications in stroke rehabilitation, and 3) dynamical scene classification. We provide experimental validation through action and gesture recognition experiments on motion capture and Kinect datasets. In all these scenarios, we show experimental evidence of the favorable properties of the proposed representation.

Bifurcation tracking by Harmonic Balance Method for performance tuning of nonlinear dynamical systems

The aim of this paper is to provide an efficient frequency-domain method for bifurcation analysis of nonlinear dynamical systems. The proposed method consists in directly tracking the bifurcation points when a system parameter such as the excitation or nonlinearity level is varied. To this end, a so-called extended system comprising the equation of motion and an additional equation characterizing the bifurcation of interest is solved by means of the Harmonic Balance Method coupled with an arc-length continuation technique. In particular, an original extended system for the detection and tracking of Neimark-Sacker (secondary Hopf) bifurcations is introduced. By applying the methodology to a nonlinear energy sink and to a rotor-stator rubbing system, it is shown that the bifurcation tracking can be used to efficiently compute the boundaries of stability and/or dynamical regimes, i.e., safe operating zones.

Visual Analysis of Nonlinear Dynamical Systems: Chaos, Fractals, Self-Similarity and the Limits of Prediction

Nearly all nontrivial real-world systems are nonlinear dynamical systems. Chaos describes certain nonlinear dynamical systems that have a very sensitive dependence on initial conditions. Chaotic systems are always deterministic and may be very simple, yet they produce completely unpredictable and divergent behavior. Systems of nonlinear equations are difficult to solve analytically, and scientists have relied heavily on visual and qualitative approaches to discover and analyze the dynamics of nonlinearity. Indeed, few fields have drawn as heavily from visualization methods for their seminal innovations: from strange attractors, to bifurcation diagrams, to cobweb plots, to phase diagrams and embedding. Although the social sciences are increasingly studying these types of systems, seminal concepts remain murky or loosely adopted. This article has three aims. First, it argues for several visualization methods to critically analyze and understand the behavior of nonlinear dynamical systems. Second, it uses these visualizations to introduce the foundations of nonlinear dynamics, chaos, fractals, self-similarity and the limits of prediction. Finally, it presents Pynamical, an open-source Python package to easily visualize and explore nonlinear dynamical systems’ behavior.

Hyper-chaotic analysis and adaptive multi-switching synchronization of a novel asymmetric non-linear dynamical system

In this paper, a new 4D autonomous asymmetric hyperchaotic system obtained from a three dimensional autonomous chaotic system is introduced. We analyse the hyperchaotic properties of the new system such as dissipation, equilibrium, Lyapunov exponent, stability, time series, phase portraits, Poincare map and bifurcation diagram. Furthermore, the multi-switching synchronization of the new asymmetric hyperchaotic system is analysed by using the adaptive control strategy. Based on the theory of Lyapunov stability, we sketch the input controllers and updating laws of distinct switching, and it is extended to examine the synchronization problems of response and drive systems with distinct combinations. Brief theoretical analysis and numerical results are presented to show the dynamical behaviour of the new 4D hyperchaotic system.

Analysis of the Stochastic Excitement in a Model of Flow Reactor

Mathematical modeling and analysis of nonlinear dynamical systems in the presence of random perturbations attracts attention of researchers from various fields of science. It is well known that noise not only blurs the attractors of deterministic models but it can also lead to qualitative changes in the stochastic dynamics due to nonlinearity of these models. The interplay of nonlinearity and stochasticity results in such phenomena as noise-induced transitions [1, 2], stochastic resonance [3,4], noise-induced order [5], chaos [6] and stochastic excitability [7]. These phenomena are found in mechanics, physics, population and neural dynamics. Typically, the system is extremely sensitive to noise near bifurcation points. An analysis of the effects of noise on the oscillatory systems near the soft and hard Hopf bifurcations attracted attention of many researchers (see, e.g., [8–12]). Mathematical models of chemical kinetics [13] have always been a source of new problems in nonlinear dynamics. The classical example is the model of the Belousov-Zhabotinsky reaction [14]. Among the models of chemical kinetics, models of chemical flow reactor with ideal mixing [15] play an important practical role. One of the models of chemical flow reactor is considered in this paper. The phenomenon of stochastic excitability is demonstrated and analyzed for this model. There are well developed analytical methods for deterministic models. The analysis of the stochastic dynamics of nonlinear systems faces serious difficulties. Indeed, the probability density function that satisfies the Fokker-Planck-Kolmogorov equation gives a full probabilistic description of stochastic regimes. The direct use of this equation is very difficult even in two-dimensional case. In this situation,

IISS and ISS dissipation inequalities: preservation and interconnection by scaling

In analysis and design of nonlinear dynamical systems, (nonlinear) scaling of Lyapunov functions has been a central idea. This paper proposes a set of tools to make use of such scalings and illustrates their benefits in constructing Lyapunov functions for interconnected nonlinear systems. First, the essence of some scaling techniques used extensively in the literature is reformulated in view of preservation of dissipation inequalities of integral input-to-state stability (iISS) and input-to-state stability (ISS). The iISS small-gain theorem is revisited from this viewpoint. Preservation of ISS dissipation inequalities is shown to not always be necessary, while preserving iISS which is weaker than ISS is convenient. By establishing relationships between the Legendre–Fenchel transform and the reformulated scaling techniques, this paper proposes a way to construct less complicated Lyapunov functions for interconnected systems.

Assessment of drug-induced arrhythmic risk using limit cycle and autocorrelation analysis of human iPSC-cardiomyocyte contractility

Cardiac safety assays incorporating label-free detection of human stem-cell derived cardiomyocyte contractility provide human relevance and medium throughput screening to assess compound-induced cardiotoxicity. In an effort to provide quantitative analysis of the large kinetic datasets resulting from these real-time studies, we applied bioinformatic approaches based on nonlinear dynamical system analysis, including limit cycle analysis and autocorrelation function, to systematically assess beat irregularity. The algorithms were integrated into a software program to seamlessly generate results for 96-well impedance-based data. Our approach was validated by analyzing dose- and time-dependent changes in beat patterns induced by known proarrhythmic compounds and screening a cardiotoxicity library to rank order compounds based on their proarrhythmic potential. We demonstrate a strong correlation for dose-dependent beat irregularity monitored by electrical impedance and quantified by autocorrelation analysis to traditional manual patch clamp potency values for hERG blockers. In addition, our platform identifies non-hERG blockers known to cause clinical arrhythmia. Our method provides a novel suite of medium-throughput quantitative tools for assessing compound effects on cardiac contractility and predicting compounds with potential proarrhythmia and may be applied to in vitro paradigms for pre-clinical cardiac safety evaluation.

Variability features associated with ozone column and surface UV irradiance observed over Svalbard from 2008 to 2014

The present report shortly summarises the conclusions about the ozone column and solar UV irradiance variability that were achieved through the analysis of the data obtained from the narrow-band filter radiometer UV-RAD, operating at Ny-Alesund from 2008 to 2014. The polar summer gives the opportunity to register the solar irradiance 24 h per day during several months that provides comparatively long continuous time series as regards for the short- (diurnal) and medium-term (monthly) variations. To exclude the hypothesis about the artificial nature of the large amplitudes registered in the short-term ozone column variations, which can be due to the measurement or methodological errors, they were related to the corresponding variations in the solar UV radiation. In addition, these oscillations were studied using the methods developed for the analysis of non-linear dynamical systems that revealed a complex chaotic interaction between the ozone column and five other atmospheric factors. This approach leads to the conclusion that the short-term variations can be predicted for 10–20 h if a long history is available. The effect of a sporadic phenomenon, such as the ozone depletion event, occurred over Arctic in the spring of 2011, which can be considered an impulse perturbation of the medium-term ozone variations, on the mid-latitude ozone column and surface solar UV irradiance has been studied by analysing the data collected from six surface stations.

On analysis of nonlinear dynamical systems via methods connected with $$\lambda $$ λ -symmetry

This study focuses on the analysis of the first integrals, the integrating factors and the solutions for some classes of nonlinear dynamical systems represented by a mass–spring–damper model. The study consists of the applications of local–nonlocal transformations and the extended Prelle–Singer approach by considering the relation with Lie symmetry groups and \(\lambda \)-symmetry. In addition, the mathematical relations between these methods are presented, and time-dependent and time-independent first integrals and the corresponding exact solutions of nonlinear dynamical systems such as general Morse oscillator equation, the path equation, the harmonic oscillator equation and the displaced harmonic oscillator equation as special cases of Lienard-type equations are investigated. Furthermore, the general forms of Lienard-type equations including general nonlinear damping and nonlinear spring functions are studied.

Input-to-State Stability, Integral Input-to-State Stability, and <inline-formula> <tex-math notation="LaTeX">${\cal L}_{2} $</tex-math></inline-formula>-Gain Properties: Qualitative Equivalences and Interconnected Systems

Input-to-state stability (ISS) and ${\cal L}_{2} $ -gain are well-known robust stability properties that continue to find wide application in the analysis and control of nonlinear dynamical systems and their interconnections. We investigate the relationship between ISS-type and ${\cal L}_{2} $ -gain properties, demonstrating several qualitative equivalences between these two properties via coordinate transformations. In particular, we investigate and clearly specify the relationship between ISS and linear ${\cal L}_{2}$ -gain, and between integral ISS and nonlinear ${\cal L}_{2} $ -gain. This latter relationship raises an interesting question as it is known that cascade and feedback interconnections of integral ISS systems are not generally well-behaved, whereas we demonstrate that such interconnections for systems with nonlinear ${\cal L}_{2} $ -gain are well-behaved. We subsequently present several new sufficient conditions for the stability of interconnected systems derived by exploiting the derived qualitative equivalences.

Point group identification algorithm in dynamic response analysis of nonlinear stochastic systems

The point group identification (PGI) algorithm is proposed to determine the representative point sets in response analysis of nonlinear stochastic dynamic systems. The PGI algorithm is employed to identify point groups and their feature points in an initial point set by combining subspace clustering analysis and the graph theory. Further, the representative point set of the random-variate space is determined according to the minimum generalized F-discrepancy. The dynamic responses obtained by incorporating the algorithm PGI into the probability density evolution method (PDEM) are compared with those by the Monte Carlo simulation method. The investigations indicate that the proposed method can reduce the number of the representative points, lower the generalized F-discrepancy of the representative point set, and also ensure the accuracy of stochastic structural dynamic analysis.

Bal: A library for the brute-force analysis of dynamical systems

This paper describes the functionality and usage of bal, a C/C++ library with a Python front-end for the brute-force analysis of continuous-time dynamical systems described by ordinary differential equations (ODEs). bal provides an easy-to-use wrapper for the efficient numerical integration of ODEs and, by detecting intersections of the trajectory with appropriate Poincaré sections, allows to classify the asymptotic trajectory of a dynamical system for bifurcation analysis. Some examples of application are discussed, concerning two-dimensional bifurcation diagrams, Lyapunov exponents and finite-time Lyapunov exponents, basins of attraction, simulation of switching ODE systems, and integration with AUTO, a software package for continuation analysis. Program summaryProgram title: BAL (Library)Catalogue identifier: AEYY_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEYY_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 65256No. of bytes in distributed program, including test data, etc.: 438133Distribution format: tar.gzProgramming language: C/C++, Python.Computer: PCs and Workstations.Operating system: Any Unix based system such as Linux and Mac OSX.RAM: Variable, depending on the number of dimensions of the dynamical system and duration of the simulation.Classification: 4.3, 4.12.External routines: CVODE, Boost, HDF5Nature of problem: The numerical analysis of continuous-time nonlinear dynamical systems often requires the computation of a large number of solutions of the system, for varying parameter sets, and the subsequent classification of the steady state solution.Solution method: This library provides a convenient interface to the library CVODE for the integration of sets of differential equations and at the same time implements all the machinery required for integrating and classifying the behaviour of the system for varying parameter sets and initial conditions. A high-level Python interface makes the usage of the library amenable also to users who are not familiar with the C/C++ programming languages.

A simple chaotic and hyperchaotic time-delay system: design and electronic circuit implementation

The present paper reports the design, analysis and experimental implementation of a simple first-order nonlinear time-delay dynamical system, which is capable of showing chaotic and hyperchaotic oscillations even at low values of intrinsic time delay. The system consists of a nonlinearity that has a closed-form mathematical function, which enables one to derive exact stability and bifurcation conditions. A detailed stability and bifurcation analyses reveal that a limit cycle is born via a supercritical Hopf bifurcation. Also, we observe both mono-scroll and double-scroll chaotic and hyperchaotic attractors even for a small value of time delay. The complexity of the system is characterized by bifurcation diagram and Lyapunov exponent spectrum. We implement the system in an electronic circuit, and a data acquisition (DAQ) system with LabView environment is used to visualize the experimental bifurcation diagram along with the other characteristic diagrams, namely time series waveform, phase plane plots and frequency spectra. Experimental observations agree well with the analytical and numerical results.

Finding zeros of nonlinear functions using the hybrid parallel cell mapping method

Analysis of nonlinear dynamical systems including finding equilibrium states and stability boundaries often leads to a problem of finding zeros of vector functions. However, finding all the zeros of a set of vector functions in the domain of interest is quite a challenging task. This paper proposes a zero finding algorithm that combines the cell mapping methods and the subdivision techniques. Both the simple cell mapping (SCM) and generalized cell mapping (GCM) methods are used to identify a covering set of zeros. The subdivision technique is applied to enhance the solution resolution. The parallel implementation of the proposed method is discussed extensively. Several examples are presented to demonstrate the application and effectiveness of the proposed method. We then extend the study of finding zeros to the problem of finding stability boundaries of potential fields. Examples of two and three dimensional potential fields are studied. In addition to the effectiveness in finding the stability boundaries, the proposed method can handle several millions of cells in just a few seconds with the help of parallel computing in graphics processing units (GPUs).

Time-stepping approach for solving upper-bound problems: Application to two-dimensional Rayleigh-Bénard convection.

An alternative computational procedure for numerically solving a class of variational problems arising from rigorous upper-bound analysis of forced-dissipative infinite-dimensional nonlinear dynamical systems, including the Navier-Stokes and Oberbeck-Boussinesq equations, is analyzed and applied to Rayleigh-Bénard convection. A proof that the only steady state to which this numerical algorithm can converge is the required global optimal of the relevant variational problem is given for three canonical flow configurations. In contrast with most other numerical schemes for computing the optimal bounds on transported quantities (e.g., heat or momentum) within the "background field" variational framework, which employ variants of Newton's method and hence require very accurate initial iterates, the new computational method is easy to implement and, crucially, does not require numerical continuation. The algorithm is used to determine the optimal background-method bound on the heat transport enhancement factor, i.e., the Nusselt number (Nu), as a function of the Rayleigh number (Ra), Prandtl number (Pr), and domain aspect ratio L in two-dimensional Rayleigh-Bénard convection between stress-free isothermal boundaries (Rayleigh's original 1916 model of convection). The result of the computation is significant because analyses, laboratory experiments, and numerical simulations have suggested a range of exponents α and β in the presumed Nu∼Pr(α)Ra(β) scaling relation. The computations clearly show that for Ra≤10(10) at fixed L=2√[2],Nu≤0.106Pr(0)Ra(5/12), which indicates that molecular transport cannot generally be neglected in the "ultimate" high-Ra regime.

Parallel Cell Mapping Method for Global Analysis of High-Dimensional Nonlinear Dynamical Systems1

The cell mapping methods were originated by Hsu in 1980s for global analysis of nonlinear dynamical systems that can have multiple steady-state responses including equilibrium states, periodic motions, and chaotic attractors. The cell mapping methods have been applied to deterministic, stochastic, and fuzzy dynamical systems. Two important extensions of the cell mapping method have been developed to improve the accuracy of the solutions obtained in the cell state space: the interpolated cell mapping (ICM) and the set-oriented method with subdivision technique. For a long time, the cell mapping methods have been applied to dynamical systems with low dimension until now. With the advent of cheap dynamic memory and massively parallel computing technologies, such as the graphical processing units (GPUs), global analysis of moderate- to high-dimensional nonlinear dynamical systems becomes feasible. This paper presents a parallel cell mapping method for global analysis of nonlinear dynamical systems. The simple cell mapping (SCM) and generalized cell mapping (GCM) are implemented in a hybrid manner. The solution process starts with a coarse cell partition to obtain a covering set of the steady-state responses, followed by the subdivision technique to enhance the accuracy of the steady-state responses. When the cells are small enough, no further subdivision is necessary. We propose to treat the solutions obtained by the cell mapping method on a sufficiently fine grid as a database, which provides a basis for the ICM to generate the pointwise approximation of the solutions without additional numerical integrations of differential equations. A modified global analysis of nonlinear systems with transient states is developed by taking advantage of parallel computing without subdivision. To validate the parallelized cell mapping techniques and to demonstrate the effectiveness of the proposed method, a low-dimensional dynamical system governed by implicit mappings is first presented, followed by the global analysis of a three-dimensional plasma model and a six-dimensional Lorenz system. For the six-dimensional example, an error analysis of the ICM is conducted with the Hausdorff distance as a metric.

Polynomial optimization with applications to stability analysis and control - Alternatives to sum of squares

In this paper, we explore the merits of various algorithms for solving polynomial optimization and optimization of polynomials, focusing on alternatives to sum of squares programming. While we refer to advantages and disadvantages of Quantifier Elimination, Reformulation Linear Techniques, Blossoming and Groebner basis methods, our main focus is on algorithms defined by Polya's theorem, Bernstein's theorem and Handelman's theorem. We first formulate polynomial optimization problems as verifying the feasibility of semi-algebraic sets. Then, we discuss how Polya's algorithm, Bernstein's algorithm and Handelman's algorithm reduce the intractable problem of feasibility of semi-algebraic sets to linear and/or semi-definite programming. We apply these algorithms to different problems in robust stability analysis and stability of nonlinear dynamical systems. As one contribution of this paper, we apply Polya's algorithm to the problem of $H_\infty$ control of systems with parametric uncertainty. Numerical examples are provided to compare the accuracy of these algorithms with other polynomial optimization algorithms in the literature.

A Data–Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition

The Koopman operator is a linear but infinite dimensional operator that governs the evolution of scalar observables defined on the state space of an autonomous dynamical system, and is a powerful tool for the analysis and decomposition of nonlinear dynamical systems. In this manuscript, we present a data driven method for approximating the leading eigenvalues, eigenfunctions, and modes of the Koopman operator. The method requires a data set of snapshot pairs and a dictionary of scalar observables, but does not require explicit governing equations or interaction with a "black box" integrator. We will show that this approach is, in effect, an extension of Dynamic Mode Decomposition (DMD), which has been used to approximate the Koopman eigenvalues and modes. Furthermore, if the data provided to the method are generated by a Markov process instead of a deterministic dynamical system, the algorithm approximates the eigenfunctions of the Kolmogorov backward equation, which could be considered as the "stochastic Koopman operator" [1]. Finally, four illustrative examples are presented: two that highlight the quantitative performance of the method when presented with either deterministic or stochastic data, and two that show potential applications of the Koopman eigenfunctions.