Nonlinear Discrete Dynamical Systems Research Articles

Identification of Models of Geomagnetic Activity and Space Weather Prediction

The problem of prediction of geomagnetic activity by the example of the method of Dst-index is considered. For its solving we suggested to use a nonlinear discrete dynamical system, which is obtained by decomposition of perturbations into series relative to correlation functions. Identification of structure and model parameters is defined by solving the corresponding problems of mathematical programming on bounded, but large enough set of stable models. On identification of model structure we took into account the assumption that model describes the state of magnetic-spherical-plasma medium, which is to great extent caused by influence of velocities of solar wind and south magnetic component of interplanetary magnetic field. The principal result of the work consists in numerical substantiation of existence of common stable model of minimal complexity, which provides the best prediction.

Stability of Synchronous Oscillations in a Periodic Network

We derived a stability criterion for the totally synchronized state of a periodic network of coupled functional units. The periodicity constraint was the key assumption in deriving our circulant matrix-based stability criterion. The functional units were nonlinear discrete dynamical systems. We assumed exponentially decaying coupling strength versus distance in order to reduce the number of control parameters that would have been introduced due to the functional units’ coupling. As a concrete example, we determined the stability domain of the synchronous state for logistic type functional units.

Discrete optimal control model and bound error for microbial continuous fermentation

A discrete optimal control model is given according to the discrete nonlinear dynamic system of continuous fermentations of glycerol to 1,3-propanediol (1,3-PD). The property of some major functions and their bounds on approximation errors are studied in this paper. Then, the conclusion that the optimality function of discrete optimal control model is the consistent approximation to one of the continuous optimal control model is proved. The results presented in this work can be used as guidelines for the optimal algorithm and its convergence.

Reduction of slow–fast discrete models coupling migration and demography

This work deals with a general class of two-time scales discrete nonlinear dynamical systems which are susceptible of being studied by means of a reduced system that is obtained using the so-called aggregation of variables method. This reduction process is applied to several models of population dynamics driven by demographic and migratory processes which take place at two different time scales: slow and fast. An analysis of these models exchanging the role of the slow and fast dynamics is provided: when a Leslie type demography is faster than migrations, a multi-attractor scenario appears for the reduced dynamics; on the other hand, when the migratory process is faster than demography, the reduction process gives rise to new interpretations of well known discrete models, including some Allee effect scenarios.

Chaotic dynamics analysis and control of iterative procedure of capacity spectrum method

The capacity spectrum method (CSM), capable of predicting the demands of forces and deformations of the inelastic system, has been applied in the ATC and FEMA guidelines. The deformation of an inelastic system is solved iteratively by using the equivalent linearization for CSM, which actually forms a nonlinear map or discrete dynamical system. However, the iterative procedure of CSM did not converge for some inelastic systems, and the complicated dynamical phenomena for the solutions such as the periodic oscillation, period-doubling bifurcation and chaos may occur, which were shown in the bifurcation plots of iterative map of the simplified CSM in ATC40 and FEMA440. This paper presents a novel method to analyze and control the non-convergence of the iterative procedure of CSM from the perspective of chaotic dynamics. The Lyapunov exponent of the dynamical system is employed to identify the evolutional state and stability of solutions. Finally, the stability transformation method as a simple, versatile and effective chaos feedback control approach is applied to control the convergent failure of CSM in ATC40 and FEMA440. The numerical results illustrate that the stability transformation method can capture the desired fixed points of the dynamical system and obtain the stable convergent solutions of CSM.

Chaos control on the repeated game model in electric power duopoly

The chaos control method combining feedback control with parameter variation is successfully applied to the dynamic repeated Cournot model in electric power duopoly. Stability control of the period-doubling bifurcation and unstable periodic orbits in the chaotic attractor of the discrete non-linear dynamic system is achieved using this method, and the numerical simulation results show that this control method is effective. In practice, by utilizing the sensitivity of the model to disturbance, electric power producers could apply a small perturbation to the chaotic system and induce a large influence on electric output in order to obtain desirable behaviour.

Robust Algorithms of Ellipsoidal Estimate of One Kind of Nonlinear Dynamical Systems

Problems of ellipsoidal estimation of state vector of discrete, continuous and continuous-discrete nonlinear dynamical systems are solved by results of measurement of their outputs. We assume that nonlinearities satisfy Lipschitz condition.

Toward the Calculation of Higher-Dimensional Stable Manifolds and Stable Sets for Noninvertible and Piecewise-Smooth Maps

This paper presents a new numerical method for computing global stable manifolds and global stable sets of nonlinear discrete dynamical systems. For a given map f:ℝ d →ℝ d , the proposed method is capable of yielding large parts of stable manifolds and sets within a certain compact region M⊂ℝ d . The algorithm divides the region M in sets and uses an adaptive subdivision technique to approximate an outer covering of the manifolds. In contrast to similar approaches, the method requires neither the system’s inverse nor its Jacobian. Hence, it can also be applied to noninvertible and piecewise-smooth maps. The successful application of the method is illustrated by computation of one- and two-dimensional stable manifolds and global stable sets.

Quantitative Characterizations of Exponential Convergence Property of Nonlinear Discrete Dynamical Systems

In this technical note, we consider the exponential convergence property of the trajectories of nonlinear discrete dynamical systems with exponential stability. Based on an invariant quantity derived from Lyapunov functions of the systems, the essential and quantitative relationship between exponential convergence property of trajectories and the Lyapunov functions is revealed.

A state prediction scheme for discrete time nonlinear dynamic systems

A state prediction scheme is proposed for discrete time nonlinear dynamic systems with non-Gaussian disturbance and observation noises. This scheme is based upon quantization, multiple hypothesis testing, and dynamic programming. Dynamic models of the proposed scheme are as general as dynamic models of particle predictors, whereas the nonlinear models of the extended Kalman (EK) predictor are linear with respect to the disturbance and observation noises. The performance of the proposed scheme is compared with both the EK predictor and sampling importance resampling (SIR) particle predictor. Monte Carlo simulations have shown that the performances of the proposed scheme, EK predictor, and SIR particle predictor are all model-dependent, that is, one performs better than the others for a given example. Some examples, for which the proposed scheme performs better than the others do, are also given in the paper.

On Characterizations of Exponential Stability of Nonlinear Discrete Dynamical Systems on Bounded Regions

In this paper, we discuss the quantitative characterization problems of the exponential stability and trajectory convergence property of nonlinear discrete dynamical systems on a bounded set of the state space. Through introducing two new concepts-the Lip constant of a nonlinear operator and strongly equivalent metrics of a norm-we show that the nonlinear discrete dynamical systems are exponentially stable on the bounded set if and only if the corresponding nonlinear operator is contractive under some strongly equivalent metrics, or if and only if the Lip constant of the nonlinear operator is less than one. In the latter case, we further show that the infimum of the exponential bounds of trajectories of the system equals exactly to the Lip constant. Based on the obtained results, we clearly explain how trajectory convergence properties of the systems are determined quantitatively by the Lip constant and strongly equivalent metrics. The obtained results not only are of importance in understanding the essence of exponential stability and trajectory convergence properties of nonlinear discrete dynamical systems on bounded sets of the state space but also provide some new, useful criteria of testing exponential stability and estimating convergence speed of trajectories of the systems.

An Invariance Principle for Nonlinear Discrete Autonomous Dynamical Systems

This note proposes an extension of LaSalle's invariance principle for nonlinear discrete autonomous dynamical systems. The invariance principle is extended to allow the first difference of the auxiliar scalar function (usually a Lyapunov function) to be positive in some bounded regions. Moreover, a uniform version is proposed to deal with nonlinear discrete dynamical systems that vary with parameters. Both extensions have the original invariance principle as a particular case. As a consequence, a larger class of systems can be treated with this new theory. The extensions are very useful to obtain attractor estimates as well as their corresponding stability regions. The uniform version, in particular, is useful to obtain estimates that are uniform regarding parameters

SUDDEN OCCURRENCE OF CHAOS IN NONSMOOTH MAPS

This paper introduces a type of one-dimensional nonsmooth nonlinear discrete dynamic system. We find a direct route to chaos from stable period-two point, and this is called Sudden Occurrence of Chaos. It is completely different from the three routes from regular motion to chaos — period-doubling bifurcation chaos, intermittency and quasi-periodicity chaos. Furthermore, we present some examples of sudden occurrence of chaos from m-period directly to chaos.

Wavelet networks for nonlinear system modeling

This study presents a nonlinear systems and function learning by using wavelet network. Wavelet networks are as neural network for training and structural approach. But, training algorithms of wavelet networks is required a smaller number of iterations when the compared with neural networks. Gaussian-based mother wavelet function is used as an activation function. Wavelet networks have three main parameters; dilation, translation, and connection parameters (weights). Initial values of these parameters are randomly selected. They are optimized during training (learning) phase. Because of random selection of all initial values, it may not be suitable for process modeling. Because wavelet functions are rapidly vanishing functions. For this reason heuristic procedure has been used. In this study serial-parallel identification model has been applied to system modeling. This structure does not utilize feedback. Real system outputs have been exercised for prediction of the future system outputs. So that stability and approximation of the network is guaranteed. Gradient methods have been applied for parameters updating with momentum term. Quadratic cost function is used for error minimization. Three example problems have been examined in the simulation. They are static nonlinear functions and discrete dynamic nonlinear system.

On the New Method for the Control of Discrete Nonlinear Dynamic Systems Using Neural Networks

This correspondence points out an incorrect statement in Adetona et al, 2000, and Adetona et al., 2004, about the application of the proposed control law to nonminimum phase systems. A counterexample shows the limitations of the control law and, furthermore, its control capability to nonminimum phase systems is explained.

CONTROL OF TRAFFIC FLOW DYNAMICS IN URBAN ROAD NETWORKS

This paper studies a control of traffic flow dynamics in urban road networks. The signal control system of the congestion length is described by a nonlinear time-varying discrete dynamic system and synthesized by using the hierarchical control. A signal control algorithm in which the three signal control parameters consisting of the cycle length, green split and offset are searched systematically so as to minimize the sum of congestion lengths in the urban road network is presented. The dynamic route guidance system is consisted of both the evaluation for the mean OD travel time and the route search algorithm. The traffic flow control system, which is most effective to reduce the congestion lengths in urban road networks, is designed by connecting the signal control system with the dynamic route guidance system in real time.

The frequency spectrum of structured discrete time population models: its properties and their ecological implications

Much research effort has been devoted to the study of the interaction between environmental noise and discrete time nonlinear dynamical systems. A large part of this effort has involved numerical simulation of simple unstructured models for particular ranges of parameter values. While such research is important in encouraging discussion of important ecological issues it is often unclear how general are the conclusions reached. However, by restricting attention to weak noise it is possible to obtain analytical results that hold for essentially all discrete time models and still provide considerable insight into the properties of the noise‐dynamics interface. We follow this approach, focusing on the autocorrelation properties of the population fluctuations using the power (frequency) spectrum matrix as the analytic framework. We study the relationship between the spectral peak structure and the dynamical behaviour of the system and the modulation of this relationship by its internal structure, acting as an “intrinsic” filter and by colour in the noise acting as an “extrinsic” filter. These filters redistribute “power” between frequency components in the spectrum. The analysis emphasises the importance of eigenvalues in the identification of resonance, both in the system itself and in its subsystems, and the importance of noise configuration in defining which paths are followed on the network. The analysis highlights the complexity of the inverse problem (in finding, for example, the source of long term fluctuations) and the role of factors other than colour in the persistence of populations.

Control procedures using domains of attraction

A technique of gradual approximation of the domain of attraction in the case of nonlinear continuous and discrete analytical dynamical systems is presented. The technique is based on the gradual extension of the “embryo” of the optimal Lyapunov function and gives, at each step, a new, open and non-empty piece of the domain of attraction. For a nonlinear dynamical system with control, defined by an analytical autonomous system of differential equations or difference equations, using domains of attraction, it is shown that two steady states, which belong to an analytic path of asymptotically stable steady states, can be transferred gradually one in the other by successive changes of the values of the control parameters. Numerical examples are given for the ALFLEX reentry vehicle and for a discrete dynamical system considered recently in the literature.

A Novel Neural Approximate Inverse Control for Unknown Nonlinear Discrete Dynamical Systems

A novel neural approximate inverse control is proposed for general unknown single-input-single-output (SISO) and multi-input-multi-output (MIMO) nonlinear discrete dynamical systems. Based on an innovative input/output (I/O) approximation of neural network nonlinear models, the neural inverse control law can be derived directly and its implementation for an unknown process is straightforward. Only a general identification technique is involved in both model development and control design without extra training (online or offline) for the neural nonlinear inverse controller. With less approximation made on controller development, the control will be more robust to large variations in the operating region. The robustness of the stability and the performance of a closed-loop system can be rigorously established even if the nonlinear plant is of not well defined relative degree. Extensive simulations demonstrate the performance of the proposed neural inverse control.

Dynamics in numerics: on two different finite difference schemes for ODEs

We compare two finite difference schemes for Kolmogorov type of ordinary differential equations: Euler's scheme (a derivative approximation scheme) and an integral approximation (IA) scheme, from the view point of dynamical systems. Among the topics we investigate are equilibria and their stability, periodic orbits and their stability, and topological chaos of these two resulting nonlinear discrete dynamical systems.