Sparse Identification of Nonlinear Duffing Oscillator From Measurement Data

In this paper we aim to apply an adaptation of the recently developed technique of sparse identification of nonlinear dynamical systems on a Duffing experimental setup with cubic feedback of the output. The Duffing oscillator described by nonlinear differential equation which demonstrates chaotic behavior and bifurcations, has received considerable attention in recent years as it arises in many real-world engineering applications. Therefore its identification is of interest for numerous practical problems. To adopt the existing identification method to this application, the optimization process which identifies the most important terms of the model has been modified. In addition, the impact of changing the amount of regularization parameter on the mean square error of the fit has been studied. Selection of the true model is done via balancing complexity and accuracy using Pareto front analysis. This study provides considerable insight into the employment of sparse identification method on the real-world setups and the results show that the developed algorithm is capable of finding the true nonlinear model of the considered application including a nonlinear friction term.Comment: 6 pages, 8 figures, conference pape

A preliminary study into emergent behaviours in a lattice of interacting nonlinear resonators and oscillators

Future sensor arrays will be composed of interacting nonlinear components with complex behaviours with no known analytic solutions. This paper provides a preliminary insight into the expected behaviour through numerical and analytical analysis. Specically, the complex behaviour of a periodically driven nonlinear Duffing resonator coupled elastically to a van der Pol oscillator is investigated as a building block in a 2D lattice of such units with local connectivity. An analytic treatment of the 2-device unit is provided through a two-time-scales approach and the stability of the complex dynamic motion is analysed. The pattern formation characteristics of a 2D lattice composed of these units coupled together through nearest neighbour interactions is analysed numerically for parameters appropriate to a physical realisation through MEMS devices. The emergent patterns of global and cluster synchronisation are investigated with respect to system parameters and lattice size

FORCED NONLINEAR OSCILLATOR IN A FRACTAL SPACE

A critical hurdle of a nonlinear vibration system in a fractal space is the inefficiency in modelling the system. Specifically, the differential equation models cannot elucidate the effect of porosity size and distribution of the periodic property. This paper establishes a fractal-differential model for this purpose, and a fractal Duffing-Van der Pol oscillator (DVdP) with two-scale fractal derivatives and a forced term is considered as an example to reveal the basic properties of the fractal oscillator. Utilizing the two-scale transforms and He-Laplace method, an analytic approximate solution may be attained. Unfortunately, this solution is not physically preferred. It has to be modified along with the nonlinear frequency analysis, and the stability criterion for the equation under consideration is obtained. On the other hand, the linearized stability theory is employed in the autonomous arrangement. Consequently, the phase portraits around the equilibrium points are sketched. For the non-autonomous organization, the stability criteria are analyzed via the multiple time scales technique. Numerical estimations are designed to confirm graphically the analytical approximate solutions as well as the stability configuration. It is revealed that the exciting external force parameter plays a destabilizing role. Furthermore, both of the frequency of the excited force and the stiffness parameter, execute a dual role in the stability picture

Zayıf sinyal tespit uygulamalarına yönelik yeni kaotik sistem geliştirme yaklaşımı

06.03.2018 tarihli ve 30352 sayılı Resmi Gazetede yayımlanan “Yükseköğretim Kanunu İle Bazı Kanun Ve Kanun Hükmünde Kararnamelerde Değişiklik Yapılması Hakkında Kanun” ile 18.06.2018 tarihli “Lisansüstü Tezlerin Elektronik Ortamda Toplanması, Düzenlenmesi ve Erişime Açılmasına İlişkin Yönerge” gereğince tam metin erişime açılmıştır.Bu tez çalışmasında, zayıf sinyal tespit uygulamaları için özgün kaotik sistem geliştirmeye yönelik yeni bir yaklaşım sunulmuştur. Önerilen kaos tabanlı zayıf sinyal tespit yöntemi literatürdeki standart zayıf sinyal tespit yöntemlerinden farklıdır. Bu yolla, farklı frekans değerlerinde tespit yapmaya uygun iki yeni kaotik sistem bulunmuştur. Tezde, yeni geliştirilen yöntem kullanılarak elde edilen iki özgün kaotik osilatör tanıtılmıştır. Bu sistemler basit yapılı olup, parametrik çeşitliliğe ve yüksek uygulanabilme kapasitesine sahiptir. Yeni sistemlerin dinamik karakteristikleri detaylı olarak incelenmiştir. Bununla beraber, Duffing-Holmes, Van Der Pol ve iki hiperkaotik Lorenz sisteminin de dinamik karakteristikleri detaylı olarak incelenmiştir. İlk olarak, sistemlerin Lyapunov metodu ile analizleri yapılmıştır. Sistemlerin durumu ile sürülme teriminin genliği arasındaki ilişki Lyapunov üstellerinin incelenmesi ile ortaya çıkarılmıştır. Kaotik sistemlerin dinamik davranışları bu yolla gözlemlenmiştir. İkinci olarak, kaotik sistemlerin kritik eşik değeri çatallaşma analizi yapılarak tespit edilmiştir. Tanjant çatallaşma noktası adı verilen bu nokta, güçlü gürültü altındaki zayıf sinyal bilgisinin tespiti için en uygun noktadır. Bununla beraber, yeni kaotik sistemlerin elektronik devreleri tasarlanarak benzetimleri de yapılmıştır. Son olarak önerilen sistemlerin zayıf sinyal tespit uygulamaları yapılmıştır. Benzetim sonuçları, sistemlerin yüksek doğrulukta ve düşük değerli sinyal gürültü oranı (SGO) ile zayıf sinyal tespiti yapabildiğini göstermiştir. Bununla beraber bu sistemler, yüksek frekans değerlerinde de tespit yapabilmektedir. Duffing-Holmes, Van Der Pol ve iki hiperkotik Lorenz sisteminin de zayıf sinyal tespit uygulamaları yapılmıştır. Matlab-Simulink® ve OrCAD-PSpice® ortamlarında gerçekleştirilen benzetim çalışmalarının sonuçları, çalışılan sistemlerin teorik analizlerinin doğru olduğunu göstermiştir. Yeni yöntemle geliştirilen özgün kaotik sistemler, endüstriyel metal malzemeleri tahribatsız muayene eden cihazlar, metal dedektörler, elektromanyetik akustik transdüserler gibi cihazların zayıf yankı sinyallerinin tespitinde kullanılabilecek potansiyel sistemlerdir.In this thesis, a new approach to improve novel chaotic systems for weak signal detection applications is presented. The new weak signal detection method based on chaos is different from standart weak signal detection metod in the literature. Two novel chaotic systems, which are suitable for high level weak signal detection applications, are improved by this way. In the thesis, two novel sinusoidal attractors, which are improved by the new metod, are presented. These new systems have simple stracture, parametric variety and high applicability. Dynamic characteristics of the novel systems are studied detailed. Furthermore, dynamic characteristics of Duffing-Holmes, Van Der Pol and two hyperchaotic Lorenz systems are also studied. Firstly, the relationship between the system state and amplitude of the forcing term is defined by examining the Lyapunov exponents of the systems. Dynamical behaviors of these chaotic systems are observed by this way. Secondly, the critical threshold values of these chaotic systems are determined by the bifurcation analysis. This critical value named as critical bifurcation point is a suitable one to detect weak signal which is submerged in strong noise. Furthermore, electronic circuits of the novel chaotic attractors are designed and simulated. Finally, weak signal detection applications of the novel systems are studied. Simulation results indicate that these novel systems can detect weak signal with high detection accuracy and low signal to noise ratio (SNR). These systems can also detect weak signal in high frequencies. Weak signal detection applications of Duffing-Holmes, Van Der Pol and two hyperchaotic Lorenz systems are also studied. Matlab-Simulink® and OrCAD-PSpice® simulation results prove the correctness of the theorycal analysis of studied systems. These improved novel systems are potential sistems to detect weak echo signals, which are non-destructive detection devices of industrial metal materials, metal detectors and electromagnetic acustic transducers

Connectionist Learning Based Numerical Solution of Differential Equations

It is well known that the differential equations are back bone of different physical systems. Many real world problems of science and engineering may be modeled by various ordinary or partial differential equations. These differential equations may be solved by different approximate methods such as Euler, Runge-Kutta, predictor-corrector, finite difference, finite element, boundary element and other numerical techniques when the problems cannot be solved by exact/analytical methods. Although these methods provide good approximations to the solution, they require a discretization of the domain via meshing, which may be challenging in two or higher dimension problems. These procedures provide solutions at the pre-defined points and computational complexity increases with the number of sampling points.In recent decades, various machine intelligence methods in particular connectionist learning or Artificial Neural Network (ANN) models are being used to solve a variety of real-world problems because of its excellent learning capacity. Recently, a lot of attention has been given to use ANN for solving differential equations. The approximate solution of differential equations by ANN is found to be advantageous but it depends upon the ANN model that one considers. Here our target is to solve ordinary as well as partial differential equations using ANN. The approximate solution of differential equations by ANN method has various inherent benefits in comparison with other numerical methods such as (i) the approximate solution is differentiable in the given domain, (ii) computational complexity does not increase considerably with the increase in number of sampling points and dimension of the problem, (iii) it can be applied to solve linear as well as non linear Ordinary Differential Equations (ODEs) and Partial Differential Equations (PDEs). Moreover, the traditional numerical methods are usually iterative in nature, where we fix the step size before the start of the computation. After the solution is obtained, if we want to know the solution in between steps then again the procedure is to be repeated from initial stage. ANN may be one of the ways where we may overcome this repetition of iterations. Also, we may use it as a black box to get numerical results at any arbitrary point in the domain after training of the model.Few authors have solved ordinary and partial differential equations by combining the feed forward neural network and optimization technique. As said above that the objective of this thesis is to solve various types of ODEs and PDEs using efficient neural network. Algorithms are developed where no desired values are known and the output of the model can be generated by training only. The architectures of the existing neural models are usually problem dependent and the number of nodes etc. are taken by trial and error method. Also, the training depends upon the weights of the connecting nodes. In general, these weights are taken as random number which dictates the training. In this investigation, firstly a new method viz. Regression Based Neural Network (RBNN) has been developed to handle differential equations. In RBNN model, the number of nodes in hidden layer may be fixed by using the regression method. For this, the input and output data are fitted first with various degree polynomials using regression analysis and the coefficients involved are taken as initial weights to start with the neural training. Fixing of the hidden nodes depends upon the degree of the polynomial.We have considered RBNN model for solving different ODEs with initial/boundary conditions. Feed forward neural model and unsupervised error back propagation algorithm have been used for minimizing the error function and modification of the parameters (weights and biases) without use of any optimization technique. Next, single layer Functional Link Artificial Neural Network (FLANN) architecture has been developed for solving differential equations for the first time. In FLANN, the hidden layer is replaced by a functional expansion block for enhancement of the input patterns using orthogonal polynomials such as Chebyshev, Legendre, Hermite, etc. The computations become efficient because the procedure does not need to have hidden layer. Thus, the numbers of network parameters are less than the traditional ANN model. Varieties of differential equations are solved here using the above mentioned methods to show the reliability, powerfulness, and easy computer implementation of the methods. As such singular nonlinear initial value problems such as Lane-Emden and Emden-Fowler type equations have been solved using Chebyshev Neural Network (ChNN) model. Single layer Legendre Neural Network (LeNN) model has also been developed to handle Lane-Emden equation, Boundary Value Problem (BVP) and system of coupled ordinary differential equations. Unforced Duffing oscillator and unforced Van der Pol-Duffing oscillator equations are solved by developing Simple Orthogonal Polynomial based Neural Network (SOPNN) model. Further, Hermite Neural Network (HeNN) model is proposed to handle the Van der Pol-Duffing oscillator equation. Finally, a single layer Chebyshev Neural Network (ChNN) model has also been implemented to solve partial differential equations

Partial periodic oscillation: an interesting phenomenon for a system of three coupled unbalanced damped Duffing oscillators with delays

In this paper, a system of two coupled damped Duffing resonators driven by a van der Pol oscillator with delays is studied. Some sufficient conditions to ensure the periodic and partial periodic oscillations for the system are established. Computer simulation is given to demonstrate our result

Analysis of bilinear oscillators under harmonic loading using nonlinear output frequency response functions

In this paper, the new concept of Nonlinear Output Frequency Response Functions (NOFRFs) is extended to the harmonic input case, an input-independent relationship is found between the NOFRFs and the Generalized Frequency Response Functions (GFRFs). This relationship can greatly simplify the application of the NOFRFs. Then, beginning with the demonstration that a bilinear oscillator can be approximated using a polynomial type nonlinear oscillator, the NOFRFs are used to analyze the energy transfer phenomenon of bilinear oscillators in the frequency domain. The analysis provides insight into how new frequency generation can occur using bilinear oscillators and how the sub-resonances occur for the bilinear oscillators, and reveals that it is the resonant frequencies of the NOFRFs that dominate the occurrence of this well-known nonlinear behaviour. The results are of significance for the design and fault diagnosis of mechanical systems and structures which can be described by a bilinear oscillator model

Parameter identification in nonlinear systems using PD controllers as penalty functions

The identification of parameters in nonlinear systems using a partial set of experimental measurements is considered in this paper. The estimation of these parameters introduces an optimization problem. For parameter estimation, the use of gradient-based optimizers often converges to a local minimum rather than the global optimum. To overcome the local convergence of the parameters, a PD controller algorithm is implemented for estimation. The addition of a morphing parameter with a proportional-derivative controller (PD) to the system equation transforms the objective function into convex, and the optimization is performed using a gradient-based optimizer. To illustrate the nonlinear parameter estimation using the present approach, a numerical example of Van der Pol-Duffing oscillator is presented. A comparative analysis is then carried out with global optimization methods, such as genetic algorithm (GA) and particle swarm optimization (PSO) techniques. The numerical results confirm that the PD controller algorithm is superior in terms of computational effort and convergence efficiency. © 2020, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved

Estimation of coupling between oscillators from short time series via phase dynamics modeling: limitations and application to EEG data

We demonstrate in numerical experiments that estimators of strength and directionality of coupling between oscillators based on modeling of their phase dynamics [D.A. Smirnov and B.P. Bezruchko, Phys. Rev. E 68, 046209 (2003)] are widely applicable. Namely, although the expressions for the estimators and their confidence bands are derived for linear uncoupled oscillators under the influence of independent sources of Gaussian white noise, they turn out to allow reliable characterization of coupling from relatively short time series for different properties of noise, significant phase nonlinearity of the oscillators, and non-vanishing coupling between them. We apply the estimators to analyze a two-channel human intracranial epileptic electroencephalogram (EEG) recording with the purpose of epileptic focus localization.Comment: 22 pages, 7 figures, the paper is to be published in Chaos, 2005, vol.15, issue 2, see http://chaos.aip.org