Multidimensional Nonlinear Systems Research Articles

Chirikov and Nekhoroshev diffusion estimates: Bridging the two sides of the river

We present theoretical and numerical results pointing towards a strong connection between the estimates for the diffusion rate along simple resonances in multidimensional nonlinear Hamiltonian systems that can be obtained using the heuristic theory of Chirikov and a more formal one due to Nekhoroshev. We show that, despite a wide-spread impression, the two theories are complementary rather than antagonist. Indeed, although Chirikov’s 1979 review has thousands of citations, almost all of them refer to topics such as the resonance overlap criterion, fast diffusion, the Standard or Whisker Map, and not to the constructive theory providing a formula to measure diffusion along a single resonance. However, as will be demonstrated explicitly below, Chirikov’s formula provides values of the diffusion coefficient which are quite well comparable to the numerically computed ones, provided that it is implemented on the so-called optimal normal form derived as in the analytic part of Nekhoroshev’s theorem. On the other hand, Chirikov’s formula yields unrealistic values of the diffusion coefficient, in particular for very small values of the perturbation, when used in the original Hamiltonian instead of the optimal normal form. In the present paper, we take advantage of this complementarity in order to obtain accurate theoretical predictions for the local value of the diffusion coefficient along a resonance in a specific 3DoF nearly integrable Hamiltonian system. Besides, we compute numerically the diffusion coefficient and a full comparison of all estimates is made for ten values of the perturbation parameter, showing a very satisfactory agreement.

Lyapunov stability for continuous-time multidimensional nonlinear systems

This paper deals with the stability of continuous-time multidimensional nonlinear systems in the Roesser form. The concepts from 1D Lyapunov stability theory are first extended to 2D nonlinear systems and then to general continuous-time multidimensional nonlinear systems. To check the stability, a direct Lyapunov method is developed. While the direct Lyapunov method has been recently proposed for discrete-time 2D nonlinear systems, to the best of our knowledge what is proposed in this paper are the first results of this kind on stability of continuous-time multidimensional nonlinear systems. Analogous to 1D systems, a sufficient condition for the stability is the existence of a certain type of the Lyapunov function. A new technique for constructing Lyapunov functions for 2D nonlinear systems and general multidimensional systems is proposed. The proposed method is based on the sum of squares (SOS) decomposition, therefore, it formulates the Lyapunov function search algorithmically. In this way, polynomial nonlinearities can be handled exactly and a large class of other nonlinearities can be treated introducing some auxiliary variables and constrains.

Application of neural networks for synthesizing flight control algorithms. I neural network inverse dynamics method for aircraft flight control

The application of artificial neural networks for aircraft motion control, in particular, for creation of nonlinear algorithms of the aircraft remote control system (RCS) is considered. Aircraft as a control object is represented as a multidimensional nonlinear dynamic system and nonlinear control methods are used to operate this system. The control loop is constructed using the method of inverse dynamics based on the feedback linearization principle. The nonlinear control law is represented as a neural network being learned (adjusted) by recorded or incoming measurements of motion parameters. Synthesis and testing of neural network control algorithms is performed with the fully nonlinear mathematical model of a maneuverable aircraft for three control channels. Simulation results of the closed system are presented.

Multidimensional nonlinear systems of conservation laws

In this paper, we review the history and progress of the study in the field of multidimensional nonlinear systems of conservation laws. The course of the study can be described as local classical solutions, solutions with fan-shaped wave structure, solutions with flower-shaped wave structure, global solutions and mixed type equations. According to this line, we introduce main results obtained in recent decades, and present some open problems and difficulties.

一类高阶n维非线性伪双曲方程组的初边值问题

一类高阶n维非线性伪双曲方程组的初边值问题

Numerical methods for determine the equilibrium point's stability of multidimensional nonlinear dynamic system

SUMMARYThis work is concerned with the use of new nunumerical methods to establish some new criteria to determine the equilibrium point's stability of a multidimensional nonlinear dynamic system. Copyright © 2010 John Wiley & Sons, Ltd.

Vanishing viscosity with short wave–long wave interactions for multi-D scalar conservation laws

We consider a system coupling a multidimensional semilinear Schrödinger equation and a multidimensional nonlinear scalar conservation law with viscosity, which is motivated by a model of short wave–long wave interaction introduced by Benney (1977). We prove the global existence and uniqueness of the solution of the Cauchy problem for this system. We also prove the convergence of the whole sequence of solutions when the viscosity ε and the interaction parameter α approach zero so that α = o ( ε 1 / 2 ) . We also indicate how to extend these results to more general systems which couple multidimensional semilinear systems of Schrödinger equations with multidimensional nonlinear systems of scalar conservation laws mildly coupled.

On the Solution of High Dimensional Fokker Planck Equations using Orthogonal Polynomial Expansion

AbstractSolving the Fokker‐Planck‐Equation for multidimensional nonlinear systems is a great challenge in the field of stochastic dynamics. As for many mechanical systems a general idea about the shape of stationary solutions for the probability density function is known, it seems promising to use an approach that contains this knowledge. This is done using a Galerkin‐method which applies approximate solutions as weighting functions for the expansion of orthogonal polynomials, e.g. generalized Hermite polynomials [1]. As examples, nonlinear oscillators containing cubical restoring (Duffing oscillators) and cubical damping elements are considered. The method is applied to the two‐dimensional problem of a single‐degree‐of‐freedom oscillator and consecutively extended up to dimension ten. Results for probability density functions are presented and compared with results from Monte Carlo simulations. (© 2010 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Identification of multi-dimensional nonlinear systems based on differential evolution wavelet neural network

Identification of multi-dimensional nonlinear systems based on differential evolution wavelet neural network

Global stability analysis of computer networks with arbitrary topology and time-varying delays

In this paper, we determine the delay-dependent conditions of global asymptotic stability for a class of multi-dimensional nonlinear time-delay systems with application to computer communication networks. A nonlinear delayed model is considered for a rate-based congestion control system of a heterogeneous network with arbitrary topology and time-varying delays. We propose a Lyapunov-based method to obtain a sufficient condition under which global asymptotic stability of the equilibrium is guaranteed. The main contribution of the paper lies in considering time variations of delays in a heterogeneous network which may be applicable in actual networks. Moreover, we obtain conditions for Internet-style networks with multi-source multi-link topology. We first prove the stability for a class of nonlinear time-delay systems. Then, we apply the results to a Kelly’s rate-based approximation of the congestion control system.

Dynamical Bifurcation Study on Electromechanical Coupling Vibration in Rolling Mill’s Drive System

In order to solve the problem of low frequency vibration in rolling mill drive system driven by AC motor, the nonlinear dynamics equations of rolling mill drive system with coupling action of electromagnetic energy and mechanical energy are established on the basis of Lagrange-Maxwell theory, and the vibration characteristic of this kind of multidimensional nonlinear autonomous system is studied by using the theory of dynamical bifurcation. Considering the high dimensional bifurcation caused by the nonhyperbolic fixed points when the resistances in stator and rotor change, the nonlinear differential equations’ approximate analytical solutions are solved with the aid of multiple time scales and harmonic balance, and the third-order normal form is obtained directly without application of center manifold theory. According to the normal form, the double Hopf bifurcations, 2D tori and 3D tori bifurcations are studied, and the stability of periodic and quasi-periodic solutions is analyzed by using Hurwitz criterion, and the boundary conditions of stable regions are given for each solution. The numerical simulation verifies the correctness of the conclusions, thereby providing a theoretical basis for smooth running of the rolling mill drive system.

The theory of the structural identification of non-linear multidimensional systems

Abstract The qualitative properties of a posteriori dynamic processes, which allow of a continuous Kalman-Mesarovic realization in a class of non-linear time-invariant multidimensional differential systems of minimum dynamic order are investigated when there are no constantly acting preset controls. It is shown that, in the general case, there are structural obstacles on this route. A constructive procedure for obtaining such differential realizations is proposed which is illustrated taking the example of spatial rotational motion (with damping), described by Euler's equations.

Localization analysis of compact invariant sets of multi-dimensional nonlinear systems and symmetrical prolongations

In our paper we study the localization problem of compact invariant sets of nonlinear systems. Methods of a solution of this problem are discussed and a new method is proposed which is based on using symmetrical prolongations and the first-order extremum condition. Our approach is applied to the system modeling the Rayleigh–Bénard convection for which the symmetrical prolongation with the Lorenz system has been constructed.

Moment Vector Equation for Nonlinear Systems and Its Application to Optimal Control

A method was developed for deriving the control input for a multi-dimensional discrete-time nonlinear system so that a performance index is approximately minimized. First, a moment vector equation (MVE) is derived; it is a multi-dimensional linear equation that approximates a nonlinear system in the whole domain of the system state and control input. Next, the performance index is approximated by using a quadratic form with respect to the moment vector. On the basis of the MVE and the quadratic form, an approximate optimal controller is derived by solving the linear quadratic optimal control problem. A bilinear optimal control problem and a mountain-car problem were solved using this method, and the solutions were nearly optimal.

Characterisation of Model Error for Charpy Impact Energy of Heat Treated Steels Using Probabilistic Reasoning and a Gaussian Mixture Model

Data-driven modelling has gained much momentum recently, with modelling algorithms being evolved into more complex structures capable of dealing with highly non-linear multi-dimensional systems. However, it is widely accepted that data-driven models are typically obtained under the principle of error minimisation, with the assumption of normal error distribution. The latter assumption is often not valid in more complex modelling environments, leading to sub-optimal model predictions. In this paper, a new modelling strategy aimed at exploiting the rich information contained in the model error data using a Gaussian mixture model (GMM) is proposed. The GMM error model can provide a probability characterisation of the error distribution, which can then be used complementally with the original data model. This combination often produces improvements in prediction performances, as will be illustrated in the case study relating to the hybrid modelling of the Charpy impact energy of heat-treated steels.

Neuronal synchrony: peculiarity and generality.

Synchronization in neuronal systems is a new and intriguing application of dynamical systems theory. Why are neuronal systems different as a subject for synchronization? (1) Neurons in themselves are multidimensional nonlinear systems that are able to exhibit a wide variety of different activity patterns. Their "dynamical repertoire" includes regular or chaotic spiking, regular or chaotic bursting, multistability, and complex transient regimes. (2) Usually, neuronal oscillations are the result of the cooperative activity of many synaptically connected neurons (a neuronal circuit). Thus, it is necessary to consider synchronization between different neuronal circuits as well. (3) The synapses that implement the coupling between neurons are also dynamical elements and their intrinsic dynamics influences the process of synchronization or entrainment significantly. In this review we will focus on four new problems: (i) the synchronization in minimal neuronal networks with plastic synapses (synchronization with activity dependent coupling), (ii) synchronization of bursts that are generated by a group of nonsymmetrically coupled inhibitory neurons (heteroclinic synchronization), (iii) the coordination of activities of two coupled neuronal networks (partial synchronization of small composite structures), and (iv) coarse grained synchronization in larger systems (synchronization on a mesoscopic scale).

Homotopy perturbation method for multi-dimensional nonlinear coupled system of parabolic and hyperbolic equations

In this paper, the homotopy perturbation method (HPM) proposed by J. H. He is adopted for solving multi-dimensional nonlinear coupled system of parabolic and hyperbolic equations. The numerical results of the present method are compared with the exact solution of an artificial multi-dimensional nonlinear coupled system of parabolic and hyperbolic model to show the efficiency of the method. Moreover, comparison is made between the results obtained by the present method and that obtained by the Adomian decomposition method (ADM). It is found that the present method works extremely well, very efficient, simple and convenient.

Asymptotic behaviour of a finite-volume scheme for the transient drift-diffusion model

In this paper, we propose a finite-volume discretization for multidimensional nonlinear drift-diffusion system. Such a system arises in semiconductors modelling and is composed of two parabolic equations and an elliptic one. We prove that the numerical solution converges to a steady state when time goes to infinity. Several numerical tests show the efficiency of the method.

Generalized de Rham-Hodge complexes, the related characteristic Chern classes, and some applications to integrable multidimensional differential systems on Riemannian manifolds

We study the differential-geometric aspects of generalized de Rham-Hodge complexes naturally related to integrable multidimensional differential systems of the M. Gromov type, as well as the geometric structure of the Chern characteristic classes. Special differential invariants of the Chern type are constructed, their importance for the integrability of multidimensional nonlinear differential systems on Riemannian manifolds is discussed. An example of the three-dimensional Davey-Stewartson-type nonlinear integrable differential system is considered, its Cartan type connection mapping, and related Chern-type differential invariants are analyzed.

Development of data-based model-free representation of non-conservative dissipative systems

A general procedure is presented for developing data-based, non-parametric models of non-linear multi-degree-of-freedom, non-conservative, dissipative systems. Two broad classes of methods are discussed: one relying on the representation of the system restoring forces in a polynomial-basis format, and the other using artificial neural networks to map the complex transformations relating the system state variables to the needed system outputs. A non-linear two-degree-of-freedom system is used to formulate the approach under discussion and to generate synthetic data for calibrating the efficiency of the two methods in capturing complex non-linear phenomena (such as dry friction, hysteresis, dead-space non-linearities, and polynomial-type non-linearities) that are widely encountered in the applied mechanics field. Subsequently, a reconfigurable test apparatus was used to generate experimental measurements from a physical non-linear “joint” involving two-dimensional motion (translation and rotation) and complicated interaction forces between the different motion axes, among its internal elements. Both the polynomial-basis approach and the neural network method were used to develop high-fidelity, non-parametric models of the physical test article. The ability of the identified models to accurately “generalize” the essential features of the non-linear system was verified by comparing the predictions of the models with experimental measurements from data sets corresponding to different excitations than those used for identification purposes. It is shown that the identification techniques under discussion can be useful tools for developing accurate simulation models of complex multi-dimensional non-linear systems under broadband excitation.