Nonlinear Physical Systems Research Articles

Identification of closed-loop non-linear systems with structure optimization

Experiments to obtain input/output data for system identification of some systems can be conducted only with the controllers in action. Estimating an open-loop system model using closed-loop data offers many difficulties, particularly when noise is present in the measurements. In this paper, a new indirect non-linear system identification technique using a non-linear auto regressive moving average with exogenous input (NARMAX) structure and employing input/output data obtained from closed-loop experiments is described for the first time. Furthermore, a new method to estimate the linear multivariable open-loop transfer function from closed-loop data using an algebraic equation set is presented. The effectiveness of the proposed linear and non-linear identification approaches is illustrated by simulation studies on a non-linear physical system.

Fast local reduced basis updates for the efficient reduction of nonlinear systems with hyper-reduction

Projection-based model reduction techniques rely on the definition of a small dimensional subspace in which the solution is approximated. Using local subspaces reduces the dimensionality of each subspace and enables larger speedups. Transitions between local subspaces require special care and updating the reduced bases associated with each subspace increases the accuracy of the reduced-order model. In the present work, local reduced basis updates are considered in the case of hyper-reduction, for which only the components of state vectors and reduced bases defined at specific grid points are available. To enable local reduced basis updates, two comprehensive approaches are proposed. The first one is based on an offline/online decomposition. The second approach relies on an approximated metric acting only on those components where the state vector is defined. This metric is computed offline and used online to update the local bases. An analysis of the error associated with this approximated metric is then conducted and it is shown that the metric has a kernel interpretation. Finally, the application of the proposed approaches to the model reduction of two nonlinear physical systems illustrates their potential for achieving large speedups and good accuracy.

A new data assimilation method based on dual-number theory

In gradient computations of the variational data assimilation (VDA) by the adjoint method, in order to overcome a lot of shortcomings such as low accuracy, difficult implementation, and great complexity, etc., a novel data assimilation method is proposed based on the dual-number theory. The important advantages are that the coding of adjoint models and reverse integrations are not necessary any more, and the values of cost functional and its corresponding gradient vectors can be attained simultaneously only by one forward computation in dual-number space. Furthermore, the accuracy of gradient can be close to the computer machine precision without other error sources. The paper is organised as follows. Firstly, the dual-number theory and algorithm rules are introduced. Then, the issues of gradient analysis and computation in VDA are transformed into the processes of calculating the cost functional numerically in dual-number space, and the gradient vectors can be obtained at the same time in an easy, efficient and accurate way. Secondly, the new algorithm for data assimilation in nonlinear physical systems is developed by combining accurate gradient information from the dual-number method with classical optimization algorithm. Thirdly, numerical experiments on sensitivity analysis for an ENSO nonlinear air-sea coupled oscillator are implemented, and the results are presented to demonstrate the important advantages of the dual-number method in the calculation of derivative information. Finally, numerical simulations for data assimilation are carried out respectively for the typical Lorenz 63 chaotic systems, the specific humidity evolving equation with physical “on-off” process at a single grid point, and a parabolic partial differential equation. Some conclusions can be drawn from the numerical experiments. The newly proposed method may be suited to many kinds of optimization problems with ordinary or partial differential equations as constraints, such as data assimilation, parameter estimation, inverse problems, sensitivity analysis etc. Results show that the new method can reconstruct the initial conditions or parameters of a nonlinear dynamical system very conveniently and accurately. Its another advantage is being very easy to implement with a high accuracy in gradient computation, so it is robust in the process of numerical optimization. The estimated initial states or parameters are convergent to real value in the cost of no more computations, when there are noises in the observations. But many tests are still needed to demonstrate the validity and advantages of the new data assimilation method, especially in more complex and realistic numerical prediction models of atmosphere and ocean.

ИОНОСФЕРНЫЕ ЭФФЕКТЫ ГЕОКОСМИЧЕСКОЙ БУРИ 13-14 НОЯБРЯ 2012 г.

The results of studying the F region and topside ionosphere response to the strong magnetic storm of November 13–14, 2012 (K p max=6+) are presented. The observations are carried out by the Kharkiv incoherent scatter radar. The magnetic storm was accompanied by the ionospheric storm with sign-variable phases. This storm was featured by the presence of two positive and one negative phases of disturbance. Key words : geospace storm, ionosphere storm phases, ionosphere, incoherent scattering Manuscript submitted: 14.02.2014 Radio phys. radio astron. 2014, 19(2): 170-180 REFERENCES 1. CHERNOGOR, L. F. and DOMNIN , I. F., 2014. Physics of geocosmic storms . Kharkiv: V. N. Karazin Kharkiv National University Publ. (in Russian). 2. CHERNOGOR, L. F., GRIGORENKO, Ye. I., LYSENKO, V. N., and TARAN, V. I., 2007. Dynamic processes in the ionosphere during magnetic storms from the Kharkov in coherent scatter radar observations. Int. J. Geomagn. Aeron . vol. 7, no. 3, id. GI3001. 3. BUONSANTO, M. J., 1999. Ionospheric Storms: A Review. Space Sci. Rev . vol. 88, no. 3–4, pp. 563–601. doi: https://doi.org/10.1023/A:1005107532631 4. MIKHAILOV, A. V. and FOSTER, J. C., 1997. Daytime thermosphere above Millstone Hill during severe geomagnetic storms. J. Geophys. Res: Space Phys . vol. 102, is. A8, pp. 17275–17282. 5. DANILOV, A. D., 2001. F2 region response to geomagnetic disturbances. J. Atmos. Solar-Terr. Phys . vol. 63,Is. 5, pp. 441–449. 6. DOMNIN, I. F., EMELYANOV, L. Ya., KATSKO, S. V., LYASHENKO, M. V., and CHERNOGOR, L. F., 2013. Ionospheric effects of a magnetic storm November 13-15, 2012 In: Proc. 13-th Ukrainian Conf. On space research . Evpatoria, Ukraine. p. 52 (in Russian). 7. KATSKO, S. V., 2013. Geocosmic storm on November 14, 2012: the results of observations on the Kharkov non-coherent scattering radar. In: Sb. Tez. Doc. Conf. ION-2013 "Remote radio sounding of the ionosphere" . Small Lighthouse, Crimea, Ukraine. p. 46 (in Russian). 8. CHERNOGOR , L. F., 2003. Physics of Earth, Atmosphere, and Geospace from the Standpoint of System Paradigm. Radio Phys. Radio Astron . vol. 8, no. 1, pp. 59–106 (in Russian). 9. CHERNOGOR, L. F., 2006. Earth–atmosphere–ionosphere–magnetosphere as an open dynamic nonlinear physical system (Part 1). Nelinejnyj mir . vol. 4, no. 12, pp. 655–697 (in Russian). 10. BARKHATOV, N. A. and BARKHATOVA , O. M., 2012. The identification of classes of ionospheric disturbances from long-term data on the critical frequency of the F2 layer. Geomagnetism and Aeronomy . vol. 52, no. 4, pp. 510–518 (in Russian). 11. MIKHAILOV, A. V., SKOBLIN, M. G., and FORSTER , M., 1995. Daytime F2-layer positive storm effect at middle and lower latitudes. Ann. Geophys . vol. 13, no. 5, pp. 532–540. DOI: https://doi.org/10.1007/s00585-995-0532-y 12. DANILOV, A. D. and MOROZOVA , L. D., 1985. Ionospheric storms in the F2 region. Morphology and physics (review). Geomagnetism and aeronomy . vol. 25, no. 5, pp. 705–721 (in Russian). 13. KRINBERG, I. A. and TASHCHILIN , A. V., 1984. Ionosphere and Plasmasphere . Moscow: Nauka Publ. (in Russian). 14. DANILOV, A. D., MOROZOVA, L. D., AND MIRMOVICH , E. G., 1985. On the possible nature of the positive phase of ionospheric storms. Geomagnetism and Aeronomy . vol. 25, no. 5, pp. 768–772 (in Russian). 15. CHERNOGOR, L. F., 2007. Earth–atmosphere–ionosphere–magnetosphere as an open dynamic nonlinear physicalsystem (Part 2). Nelinejnyj mir . vol. 5, no. 4, pp. 225-246 (in Russian). 16. CHERNOGOR , L. F., 2011. The Earth – atmosphere – geospace system: main properties and processes. Int. J. Remote Sens . vol. 32, no. 11, pp. 3199–3218. DOI: https://doi.org/10.1080/01431161.2010.541510

A generalized tool for accurate time-domain separation of excited modes in spin-torque oscillators

We propose and develop an advanced signal processing technique that, combined with micromagnetic simulations, is able to deeply describe the non-stationary behavior of spin-torque oscillators, both in terms of time domain and spatial distribution of the magnetization dynamics. The Hilbert-Huang Transform is used for the identification of the time traces of each oscillation in a multimode excitation and enhanced with masking signals and the Ensemble Empirical Mode Decomposition. We emphasize that the technique developed here is general and can be used for any physical non-linear system in the presence of multimode dynamical excitation or intermittence.

F-Deformed Squeezed Vacuum and First Excited States, Their Superposition and Corresponding Nonclassical Properties

In this paper, we introduce two new classes of nonlinear squeezed states that we name as f-deformed squeezed vacuum state |ξ, f〉even and f-deformed squeezed first excited state |ξ, f〉odd, which according to their production processes, essentially include only even and odd bases of Fock space, respectively. In the continuation, we introduce the superposition of these two distinct nonlinear squeezed states with a respective phase φ. Then, some of the criteria which imply the nonclassicality of the states, such as Mandel parameter, second-order correlation function, quadrature squeezing, amplitude-squared squeezing, Husimi and Wigner–Weyl quasi-distribution functions, are numerically examined. At last, by considering a well-known nonlinearity function associated with a nonlinear physical system, we present our results which outcome from the numerical calculations. It is shown that, the introduced f-deformed states can reveal high nonclassical features.

Complex solitary wave dynamics, pattern formation and chaos in the gain–loss nonlinear Schrödinger equation

A numerical exploration of a gain–loss nonlinear Schrödinger equation was carried out utilizing over 180 000 core hours to conduct more than 10 000 unique simulations in an effort to characterize the model's six dimensional parameter space. The study treated the problem in full generality, spanning a minimum of eight orders of magnitude for each of three linear and nonlinear gain terms and five orders of magnitude for higher order nonlinearities. The gain–loss nonlinear Schrödinger equation is of interest as a model for spin wave envelopes in magnetic thin film active feedback rings and analogous driven damped nonlinear physical systems. Bright soliton trains were spontaneously driven out of equilibrium and behaviors stable for tens of thousands of round trip times were numerically identified. Nine distinct complex dynamical behaviors with lifetimes on the order of ms were isolated as part of six identified solution classes. Numerically located dynamical behaviors include: (i) low dimensional chaotic modulations of bright soliton trains; (ii) spatially symmetric/asymmetric interactions of solitary wave peaks; (iii) dynamical pattern formation and recurrence; (iv) steady state solutions; and (v) intermittency. Simulations exhibiting chaotically modulating bright soliton trains were found to qualitatively match previous experimental observations. Ten new dynamical behaviors, eight demonstrating long lifetimes, are predicted to be observable in future experiments.

A nonlinear causality measure in the frequency domain: Nonlinear partial directed coherence with applications to EEG

Frequency domain Granger causality measures have been proposed and widely applied in analyzing rhythmic neurophysiological and biomedical signals. Almost all these measures are based on linear time domain regression models, and therefore can only detect linear causal effects in the frequency domain. A frequency domain causality measure, the partial directed coherence, is explicitly linked with the frequency response function concept of linear systems. By modeling the nonlinear relationships between time series using nonlinear models and employing corresponding frequency-domain analysis techniques (i.e., generalized frequency response functions), a new nonlinear partial directed coherence method is derived. The advantages of the new method are illustrated via a numerical example of a nonlinear physical system and an application to electroencephalogram signals from a patient with childhood absence epilepsy. The new method detects both linear and nonlinear casual effects between bivariate signals in the frequency domain, while the existing measures can only detect linear effects. The proposed new method has important advantages over the classical linear measures, because detecting nonlinear dependencies has become more and more important in characterizing functional couplings in neuronal and biological systems.

A procedure to linearize a class of non-linear systems modelled by bond graphs

A procedure is proposed to obtain the linearization of a class of non-linear physical systems using bond graphs. Also, a junction structure of a non-linear bond graph considering linearly dependent and independent state variables is described. From the junction structure of the non-linear bond graph a procedure to build a linearized bond graph is presented. Finally, an example of a Programmable Universal Manipulation Arm (PUMA) manipulator is given.

Enhanced Multistage Homotopy Perturbation Method: Approximate Solutions of Nonlinear Dynamic Systems

We introduce a new approach called the enhanced multistage homotopy perturbation method (EMHPM) that is based on the homotopy perturbation method (HPM) and the usage of time subintervals to find the approximate solution of differential equations with strong nonlinearities. We also study the convergence of our proposed EMHPM approach based on the value of the control parameterhby following the homotopy analysis method (HAM). At the end of the paper, we compare the derived EMHPM approximate solutions of some nonlinear physical systems with their corresponding numerical integration solutions obtained by using the classical fourth order Runge-Kutta method via the amplitude-time response curves.

Uniqueness and stability of the simultaneous detection of the nonlinear restoring and excitation of a forced nonlinear oscillation

This paper presents a new inverse problem for the simultaneous identification of the restoring characteristic and the excitations of a nonlinear physical system. An original mathematical formalism for the identification is developed based on the notion of a zero-crossing time and a new function, called the J-function. We analyze the developed formalism to prove that the inverse problem presented has a unique solution and the stability properties. That is, it is mathematically shown that the nonlinear restoring and excitation of a forced nonlinear oscillation can be detected in a unique and stable manner.

On the effect of intermittency of turbulence on the parabolic relation between skewness and kurtosis in magnetized plasmas

This paper is aimed to contribute to the scientific discussions that have been triggered by the experimental observation of a quadratic relation between the kurtosis and skewness of turbulent fluctuations present in fusion plasmas and other nonlinear physical systems. In this paper, we offer a general statistical model which attributes the observed K=aS2+b relation to the varying intermittency of the experimental signals. The model is a two random variable model constructed to catch the essential intermittent feature of the real signal. One of the variables is the amplitude of the underlying intermittent event (e.g., turbulent structure) while the other is connected to the intermittency level of the system. This simple model can attribute physical meaning to the a and b coefficients, as they characterize the spatio-temporal statistics of intermittent events. By constructing a particle-conserving Gaussian model for the underlying coherent structures, the experimentally measured a and b coefficients could be adequately reproduced.

FINITE SYMMETRY GROUP AND COHERENT SOLITON SOLUTIONS FOR THE BROER–KAUP–KUPERSHMIDT SYSTEM

A modified CK direct method is generalized to find finite symmetry groups of nonlinear mathematical physics systems. For the (2 + 1)-dimensional Broer–Kaup–Kupershmidt (BKK) system, both the Lie point symmetry and the non-Lie symmetry groups are obtained by this method. While using the traditional Lie approach, one can only find the Lie symmetry groups. Furthermore, abundant localized structures of the BKK equation are also obtained from the non-Lie symmetry group.

Fractional Fourier Detection of L´evy Flights: Application to Hamiltonian Chaotic Trajectories

A signal processing method designed for the detection of linear (coherent) behaviors among random fluctuations is presented. It is dedicated to the study of data recorded from nonlinear physical systems. More precisely the method is suited for signals having chaotic variations and sporadically appearing regular linear patterns, possibly impaired by noise. We use time-frequency techniques and the Fractional Fourier transform in order to make it robust and easily implementable. The method is illustrated with an example of application: the analysis of chaotic trajectories of advected passive particles. The signal has a chaotic behavior and encounter L\'evy flights (straight lines). The method is able to detect and quantify these ballistic transport regions, even in noisy situations.

Passivity-based control of AC drives: theory for the user and application examples

Passivity-based control is a methodology to design (possibly nonlinear) controllers for nonlinear physical systems modifying its total energy function and damping characteristics. In this article, the basic principles of this technique, and some of the recent results of application for high performance AC drives, are briefly reviewed. Since the target audience of this article is the drives community, the material is presented emphasising its immediate practical applicability, with minimal mathematical details. However, it is underscored that in all applications, a rigorous stability analysis for the full nonlinear dynamics has been reported in the literature.

Computational models of signalling networks for non-linear control

Artificial signalling networks (ASNs) are a computational approach inspired by the signalling processes inside cells that decode outside environmental information. Using evolutionary algorithms to induce complex behaviours, we show how chaotic dynamics in a conservative dynamical system can be controlled. Such dynamics are of particular interest as they mimic the inherent complexity of non-linear physical systems in the real world. Considering the main biological interpretations of cellular signalling, in which complex behaviours and robust cellular responses emerge from the interaction of multiple pathways, we introduce two ASN representations: a stand-alone ASN and a coupled ASN. In particular we note how sophisticated cellular communication mechanisms can lead to effective controllers, where complicated problems can be divided into smaller and independent tasks.

Applicability of artificial neural network in hydraulic experiments using a new sewer overflow screening device

During wet weather conditions, sewer overflows to receiving water bodies raise serious environmental, aesthetic and public health problems. These issues trigger the need the most appropriate device/system for a particular installation, especially at unmanned remote locations. A new sewer overflow device consists of a rectangular tank and a sharp crested weir with a series of vertical combs is presented. A series of laboratory tests to determine trapping efficiencies for common sewer solids were conducted for different flow conditions, number of combs layers and spacing of combs. To overcome physical limitations inherent in laboratory studies such as significant cost and time. Artificial neural model was adopted as it has the capacity to accurately predict the outcome of complex, non-linear physical systems with relatively poorly understood physicochemical processes. A series of laboratory tests were conducted with 55 different sets of data. Forty-seven sets of experimental data are used with 60% for training, 20% each for testing and validation of the model. A separate validation data sets were used to judge the overall performance of the trained network. The model can successfully predict the experimental results with more than 90% accuracy with an average absolute percentage error of around 7%.

Applicability of Artificial Neural Network in Hydraulic Experiments Using a New Sewer Overflow Screening Device

During wet weather conditions, sewer overflows to receiving water bodies raise serious environmental, aesthetic and public health problems. These issues trigger the need the most appropriate device/system for a particular installation, especially at unmanned remote locations. A new sewer overflow device consists of a rectangular tank and a sharp crested weir with a series of vertical combs is presented. A series of laboratory tests to determine trapping efficiencies for common sewer solids were conducted for different flow conditions, number of combs layers and spacing of combs. To overcome physical limitations inherent in laboratory studies such as significant cost and time. Artificial neural model was adopted as it has the capacity to accurately predict the outcome of complex, non-linear physical systems with relatively poorly understood physicochemical processes. A series of laboratory tests were conducted with 55 different sets of data. Forty-seven sets of experimental data are used with 60% for training, 20% each for testing and validation of the model. A separate validation data sets were used to judge the overall performance of the trained network. The model can successfully predict the experimental results with more than 90% accuracy with an average absolute percentage error of around 7%.

A new data assimilation method using complex-variable differentiation

A new method for data assimilation is proposed using complex-variable differentiation (CVD), which can be used to estimate the initial conditions of the nonlinear physical system governed by the following equation:. Firstly, the gradient analysis of cost function in variational data assimilation is transformed into function numerical computation in complex domain, and the value of gradient is computed more efficiently and exactly. Secondly, the new algorithm of data assimilation is developed by combining an accurate gradient information from CVD with the classical optimization method. Finally, numerical simulations of typical chaotic systems and a humidity evolution equation with physical on-off process show that the new data assimilation method can reconstruct initial conditions of the nonlinear dynamical system very conveniently and accurately.

Geometric phases of nonlinear coherent and squeezed states: a new approach

Recently, geometric phases of nonlinear coherent and squeezed states have been investigated by Yang et al (2011 J. Phys. B: At. Mol. Opt. Phys. 44 075502). In this paper, after a modification of Yang et al's derivation procedure, we deduced the non-cyclic and non-unitary geometric phases of nonlinear coherent and nonlinear squeezed states with a new approach which is consistent with the nonlinear coherent states theory. Indeed, the most important distinguishable feature of our presentation lies in the fact that we employed the (nonlinear) Hamiltonian which appropriately generates the time evolution of nonlinear coherent and squeezed states. It is established that this modification directly affects the resultant geometric phases of the states under consideration. As a physical realization, we applied our formalism to a particular nonlinear physical system corresponding to the centre-of-mass motion of a trapped ion coherent state. Then, the variation of the geometric phases of the considered nonlinear states in terms of time, intensity of radiation field, squeezing and Lamb–Dicke parameters are investigated. Finally, we discuss our numerical results.