Simulation Of Nonlinear Systems Research Articles

Design of a reservoir for cloud-enabled echo state network with high clustering coefficient

Reservoir computing (RC) is considered as a suitable alternative for descending gradient methods in recursive neural networks (RNNs) training. The echo state network (ESN) is a platform for RC and nonlinear system simulation in the cloud environment with many external users. In the past researches, the highest eigenvalue of reservoir connection weight (spectral radius) was used to predict reservoir dynamics. Some researchers have illustrated; the characteristics of scale-free and small-world can improve the approximation capability in echo state networks; however, recent studies have shown importance of the infrastructures such as clusters and the stability criteria of these reservoirs as altered. In this research, we suggest a high clustered ESN called HCESN that its internal neurons are interconnected in form of clusters. Each of the clusters contains one backbone and a number of local nodes. We implemented a classical clustering algorithm, called K-means, and three optimization algorithms including genetic algorithm (GA), differential evolution (DE), and particle swarm optimization (PSO) to improve the clustering efficiency of the new reservoir and compared them with each other. For investigating the spectral radius and predictive power of the resulting reservoirs, we also applied them to the laser time series and the Mackey-Glass dynamical system. It is demonstrated that new clustered reservoirs have some specifications of biologic neural systems and complex networks like average short path length, high clustering coefficient, and power-law distribution. The empirical results illustrated that the ESN based on PSO could strikingly enhance echo state property (ESP) and obtains less chaotic time series prediction error compared with other works and the original version of ESN. Therefore, it can approximate nonlinear dynamical systems and predict the chaotic time series.

Controllability of linear and nonlinear systems governed by Stieltjes differential equations

In this manuscript, controllability of vector valued g-derivative Stieltjes differential system is considered. For linear system, explicit solution of homogeneous case is obtained by constructing a generalized matrix exponential function and nonhomogeneous solution is achieved with transferring the corresponding problem into a class of Stieltjes impulsive differential equations. Based on the analysis techniques of linear system theory, the Gramian criterion and the rank criterion for linear case are established, respectively. Existence of local and global solution for nonlinear system are presented by the well-knonw Banach contraction principle and the controllability result for nonlinear systems is shown by Krasnoselskii’s fixed point theorem. Simulations for linear and nonlinear systems are tackled respectively to verify the efficiency of theoretical results.

Bicycle dynamics and its circular solution on a revolution surface

In this paper, we study the dynamics of an idealized benchmark bicycle moving on a surface of revolution. We employ symbolic manipulations to derive the contact constraint equations from an ordered process, and apply the Lagrangian equations of the first type to establish the nonlinear differential algebraic equations (DAEs), leaving nine coupled differential equations, six contact equations, two holonomic constraint equations and four nonholonomic constraint equations. We then present a complete description of hands-free circular motions, in which the time-dependent variables are eliminated through a rotation transformation. We find that the circular motions, similar to those of the bicycle moving on a horizontal surface, nominally fall into four solution families, characterized by four curves varying with the angular speed of the front wheel. Then, we numerically investigate how the topological profiles of these curves change with the parameter of the revolution surface. Furthermore, we directly linearize the nonlinear DAEs, from which a reduced linearized system is obtained by removing the dependent coordinates and counting the symmetries arising from cyclic coordinates. The stability of the circular motion is then analyzed according to the eigenvalues of the Jacobian matrix of the reduced linearized system around the equilibrium position. We find that a stable circular motion exists only if the curvature of the revolution surface is very small and it is limited in small sections of solution families. Finally, based on the numerical simulation of the original nonlinear DAEs system, we show that the stable circular motion is not asymptotically stable.

The dynamical analysis of the modified rossler system

The synthesis of novel chaotic systems is a modern branch of nonlinear dynamics since deterministic chaos properties can be successfully applied in various engineering and scientific problems. In this paper we investigate changes in the dynamics of the modified Rossler system after applying coordinates transformation to the original model. We perform the bifurcation analysis of the obtained model and experimentally show that its behavior differs with the behavior of the prototype. We study the finite-difference schemes obtained for considered chaotic systems and find their similarity in simulation with different integration steps. We show that discretization effects are the source of the small differences between the two models. The obtained results can be used in theoretical nonlinear dynamics, nonlinear systems simulation, development of communication and control systems.

Numerical simulation of Kerr nonlinear systems; analyzing non-classical dynamics

We simulate coherent driven free dissipative Kerr nonlinear system numerically using Euler’s method by solving Heisenberg equation of motion and time evolving block decimation (TEBD) algorithm, and demonstrate how the numerical results are analogous to classical bistability. The comparison with analytics show that the TEBD numerics follow the quantum mechanical exact solution obtained by mapping the equation of motion of the density matrix of the system to a Fokker-Plank equation . Comparing between two different numerical techniques, we see that the semi-classical Euler’s method gives the dynamics of the system field of one among two coherent branches, whereas TEBD numerics generate the superposition of both of them. Therefore, the time dynamics determined by TEBD numerical method undergoes through a non-classical state which is also shown by determining second order correlation function.

Modeling and simulation of nonlinear dynamical system in the frame of nonlocal and non-singular derivatives

This paper considers mathematical analysis and numerical treatment for fractional reaction-diffusion system. In the model, the first-order time derivatives are modelled with the fractional cases of both the Atangana-Baleanu and Caputo-Fabrizio derivatives whose formulations are based on the notable Mittag-Leffler kernel. The main system is examined for stability to ensure the right choice of parameters when numerically simulating the full model. The novel Adam-Bashforth numerical scheme is employed for the approximation of these operators. Applicability and suitability of the techniques introduced in this work is justified via the evolution of the species in one and two dimensions. The results obtained show that modelling with fractional derivative can give rise to some Turing patterns.

VOLTERRA NEURAL NETWORK CONSTRUCTION IN THE NONLINEARDYNAMIC SYSTEMS MODELING PROBLEM

The features of using the theory of Volterraseries and neural networks in problems of nonlinear dynamic systems model-ing are considered.A comparative analysis of methods for constructing models of nonlinear dynamic systems based on the theory of Volterra series and neural networks is carried out;areas of effective application of each method are indicated. The problem statementis formulated, consisting in the creation of a mathematical apparatus for transforming models of nonlinear dynamic systems derived from the Volterra series apparatus into an artificial neural network of a certain structure.The three-layer structure of a direct signal propa-gation neural network has been substantiated and investigated forrepresent nonlinear dynamic systems. It is outlined a class of systems that can be efficiently approximated by this network.The dependence of the Volterra kernelscoefficients and the weighting coefficients of the hidden layer of the three-layer forward-propagation neural network is established.An algorithm for constructing an artificial neural network based on the Volterra series is given.The results of computer simulation of nonlinear dynamic systems using the Volterra neural network and direct signal propagation neural network are presented. The analysis of experimental data confirms theeffectiveness of using Volterraneural networks in problems of modeling nonlinear dynamic systems.Conclusions and recommendations on the effec-tive use of Volterra neural networks for modeling nonlinear dynamic systems are made.

Hypersonic flutter and flutter suppression system of a wind tunnel model

The aeroelastic behavior of a thin flat rudder model was numerically simulated and experimentally investigated in a hypersonic wind tunnel. In particular, a flutter suppression system taking advantage of collision within small gaps was proposed and a novel system for the flutter simulation of the whole nonlinear aeroelastic system including the flutter suppression system was developed. First, the critical flutter dynamic pressure of the rudder without the flutter suppression system was calculated with different methods. Then, the whole nonlinear aeroelastic system, including the flutter suppression system, was simulated to design the gap size. Finally, the flutter suppression system was experimentally validated in a hypersonic wind tunnel operating at Mach number 5. The typical phenomenon of Limit Cycle Oscillation (LCO) was observed, avoiding the structural failure of the model and the consistency between numerical and experimental results was demonstrated. The proposed suppression system can improve the design and reusability of test models of hypersonic flutter experiments.

Suboptimal Control of Nonlinear Dynamic System with Unlimited Transition Process Time

Abstract In this paper, we consider the issue of synthesizing optimal control of nonlinear dynamic systems in case of a problem with an unspecified time of the transition process. To solve this problem, an algorithmic method for constructing a suboptimal control is suggested. The basis of this method is the behavior of the Hamiltonian on the optimal trajectory, which corresponds to the required condition of the optimal system. For the synthesis of suboptimal control, the original nonlinear model is replaced by an equivalent model with a linear structure with state-dependent parameters. The use of the quadratic quality functional allows to make the transition from solving the Hamilton-Jacobi-Bellman equation to the Riccati-type equation with state-dependent parameters. A computer simulation of a simple nonlinear system, for which the Riccati equation with state-dependent parameters can be solved, has been carried out. Also, for this system a suboptimal control was created with the use of a new algorithmic parametric optimization method. The simulation results confirm the effectiveness of the method. In addition, the constructed suboptimal control is close to the control obtained by solving the Riccati equation.

Nonlinear mechanical system simulation based on modal coordinates and modal forces

In a project at Beihang University different methods to model, simulate and identify nonlinear mechanical systems are studied. One promising method to simulate forced response in nonlinear mechanical systems is based on modal coordinates and modal forces. The method allows many input forces and many nonlinearities to be handled simultaneously. The method is briefly described here together with a simple example.

Graphical interface as a teaching aid for nonlinear dynamical systems

Chaos theory is an important breakthrough in understanding nonlinear dynamical systems. It is generally accepted that chaotic systems involve transitivity, a high density of periodic points in metric space, and extreme sensitivity to changes in initial conditions. Many systems taught on undergraduate courses satisfy these conditions, such as electronic circuits or computer simulations of nonlinear dynamical systems. However, a good understanding of these concepts is broadly difficult to achieve on such courses, and especially so at the undergraduate level. In view of this, we have developed an interactive standalone graphical user interface, which is platform-independent. The interface didactically simulates six chaotic systems. It allows the user to study a particular system without prior knowledge of computer programming, demonstrating rich dynamical behaviour such as fixed points, periodic cycles and deterministic chaos. At the same time, the interface provides details of the simulation and the parameters used, as well as some of the leading references. We provide the complete code and a detailed installation guide, which should make it easy to use the interface reliably and with little overhead work. We also provide a comparison with other available software for reference, and as a guide to readers who would like to explore this fascinating subject in more depth or with alternative approaches. Lastly, we present the results of a survey of 30 students who used our graphical user interface, revealing that it is an easy-to-use tool in the process of learning about chaotic dynamical systems.

Numerical solution of Caputo fractional differential equations with infinity memory effect at initial condition

The simulation of linear and nonlinear fractional-order systems on digital computers is investigated. The Grünwald-Letnikov definition of the fractional-order derivative is analyzed in the light of the initial conditions and as a consequence a new modified scheme for the discretization and simulation of fractional order systems is proposed. For this new scheme, it will be shown a new result where Riemann-Liouville derivative with the lower limit at infinity is related with a Caputo derivative with the lower limit at a finite real value allowing the infinite memory effect of fractional calculus to be adequately dealt with. To illustrate the use of the proposed method, the numerical solution of a linear fractional-order system is compared to the available analytical solution and, in the case of nonlinear fractional systems, the solution is compared to one provided by using the Adams method proposed by Diethelm [1, 2, 3].

Order reduction of hierarchical interconnected dynamical systems

Abstract. The simulation of large scale nonlinear dynamical interconnected systems, as they arise in all modern engineering disciplines, is a usual task. Due to the high complexity of the considered systems, the principle of thinking in hierarchical structures is essential and common among engineers. Therefore, this contribution proposes an approach for the numerical simulation of large systems, which keeps the hierarchical system structure alive during the entire simulation process while simultaneously exploiting it for order reduction purposes. This is accomplished by embedding the trajectory piecewise linear order reduction scheme in a modified variant of the component connection modeling for building interconnected system structures. The application of this concept is demonstrated by means of a widely used benchmark example and a modified variant of it.

Pynamical: Model and visualize discrete nonlinear dynamical systems, chaos, and fractals

Author(s): Boeing, Geoff | Abstract: Pynamical is an educational Python package for introducing the modeling, simulation, and visualization of discrete nonlinear dynamical systems and chaos, focusing on one-dimensional maps (such as the logistic map and the cubic map). Pynamical facilitates defining discrete one-dimensional nonlinear models as Python functions with just-in-time compilation for fast simulation. It comes packaged with the logistic map, the Singer map, and the cubic map predefined. The models may be run with a range of parameter values over a set of time steps, and the resulting numerical output is returned as a pandas DataFrame. Pynamical can then visualize this output in various ways, including with bifurcation diagrams, two-dimensional phase diagrams, three-dimensional phase diagrams, and cobweb plots. These visualizations enable simple qualitative assessments of system behavior including phase transitions, bifurcation points, attractors and limit cycles, basins of attraction, and fractals.

Wave energy harvesting using nonlinear stiffness system

A nonlinear stiffness mechanism installed in a floating point absorber (FPA) in regular waves allowed for studying the influence of the nonlinear behavior on wave energy harvesting. Static analysis of nonlinear stiffness system and time domain numerical simulations based on Cummins’ equation evaluated the effects of power take-off (PTO) damping, system stiffness and geometry dimensions. The results may apply to the evaluation of the balance between energy harvesting performance and practical design limitations. The nonlinear stiffness system improved the efficiency of wave energy harvesting, increasing mean power, by both pushing up the natural period, and broadening resonance, therefore proving more competitiveness.

Validation of Simulation and Analytical Model of Nonlinear Passive Vehicle Suspension System for Quarter Car

In this paper the nonlinear behavior of passive suspension system for quarter car is studied. The mathematical model is developed with the help of equations of motions. In the actual practice, all system behaves nonlinearly but for simplification purpose it consider as linear. The quarter car two degree of freedom and MATLAB-SIMULINK model is developed for the nonlinear quarter car passive suspension system. The SIMULINK model is also validated with analytical model by State-Space approach. This paper is also emphasis the effect of nonlinear behavior of spring stiffness measured experimentally with the linear assumption of stiffness using validated SIMULINK model

Processing parameters influence on dynamics of vibratory drilling with adaptive control

Chip segmentation is one of necessary conditions of ensuring the deep hole drilling quality when processing hard-to-machine metals. It may be achieved through making drill bit harmonic oscillations in axial direction. Possible way to maintain such vibrations is to replace the standard drilling head with special vibratory head which includes elastic element giving the instrument a possibility to move in axial direction. Self-excitation of drill regenerative oscillations is possible if elastic element stiffness and processing parameters are chosen properly. It is advisable to complement this way of excitation by control action which is determined in feedback circuit and sustains required vibration process characteristics in wide range of processing parameters. In present paper the adaptive control algorithm of vibratory drilling process dynamics is proposed. Control action on oscillation system is proportional to drill vibrational velocity, the feedback gain is determined in adaptation circuit basing on comparison of actual peakto- peak vibrational displacement and its target value. Simulation of closedloop nonlinear system «elastic system – machining process – control system» dynamics has been performed for different values of processing parameters for cases with or without control. The simulation revealed efficiency of suggested algorithm in wide range of processing parameters. Joined influence of processing parameters and target peak-to-peak displacement values on chip breakage conditions is studied. Recommendations for choice of processing parameters and control parameters values are developed. Influence of control action magnitude limit on control system possibilities to achieve control target is analyzed.

An efficient Monte Carlo simulation strategy based on model order reduction and artificial neural networks

AbstractIn this paper, we present a new approach that enables the Monte Carlo simulation of complex nonlinear systems with considerably less computational effort compared to the classical Monte Carlo simulation. Hereby, we propose a combination of the proper orthogonal decomposition and neural networks. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Systematic Computation of Nonlinear Bilateral Dynamical Systems With a Novel Low-Power Log-Domain Circuit

Simulation of large-scale nonlinear dynamical systems on hardware with a high resemblance to their mathematical equivalents has been always a challenge in engineering. This paper presents a novel current-input current-output circuit supporting a systematic synthesis procedure of log-domain circuits capable of computing bilateral dynamical systems with considerably low power consumption and acceptable precision. Here, the application of the method is demonstrated by synthesizing four different case studies: 1) a relatively complex 2-D nonlinear neuron model; 2) a chaotic 3-D nonlinear dynamical system Lorenz attractor having arbitrary solutions for certain parameters; 3) a 2-D nonlinear Hopf oscillator, including bistability phenomenon sensitive to initial values; and 4) three small neurosynaptic networks comprising three FHN neuron models variously coupled with excitatory and inhibitory synapses. The validity of our approach is verified by nominal and Monte Carlo simulated results with realistic process parameters from the commercially available AMS 0.35- $\mu \text{m}$ technology. The resulting continuous-time, continuous-value, and low-power circuits exhibit various bifurcation phenomena, nominal time-domain responses in good agreement with their mathematical counterparts and fairly acceptable process variation results (less than 5% STD).

Nonlinear signal-based control with an error feedback action for nonlinear substructuring control

A nonlinear signal-based control (NSBC) method utilises the ‘nonlinear signal’ that is obtained from the outputs of a controlled system and its linear model under the same input signal. Although this method has been examined in numerical simulations of nonlinear systems, its application in physical experiments has not been studied. In this paper, we study an application of NSBC in physical experiments and incorporate an error feedback action into the method to minimise the error and enhance the feasibility in practice. Focusing on NSBC in substructure testing methods, we propose nonlinear substructuring control (NLSC), that is a more general form of linear substructuring control (LSC) developed for dynamical substructured systems.In this study, we experimentally and numerically verified the proposed NLSC via substructuring tests on a rubber bearing used in base-isolated structures. In the examinations, NLSC succeeded in gaining accurate results despite significant nonlinear hysteresis and unknown parameters in the substructures. The nonlinear signal feedback action in NLSC was found to be notably effective in minimising the error caused by nonlinearity or unknown properties in the controlled system. In addition, the error feedback action in NLSC was found to be essential for maintaining stability. A stability analysis based on the Nyquist criterion, which is used particularly for linear systems, was also found to be efficient for predicting the instability conditions of substructuring tests with NLSC and useful for the error feedback controller design.