Nonlinear Discrete Dynamical Systems Research Articles

Convergence of a Reactive Planning Algorithm

The reactive planning algorithms proposed by P. Maes in Artificial Intelligence for real-time decision making and other derived algorithms are empirically known to converge, but no mathematical proof has been proposed. The evolution equations of these incremental algorithms can be translated into discrete time non-linear dynamical systems. We prove the convergence of such dynamical systems, provided the necessary assumptions are fulfilled, and we give an expression for the limit.

A new approach for neural control of nonlinear discrete dynamic systems

This paper proposes a new approach to control nonlinear discrete dynamic systems, which relies on the identification of a discrete model of the system by a neural network. A locally equivalent optimal linear model is obtained from the neural network model at every operating point the system goes through during the control task. Based on the linear model, a linear state-space control design technique can be used to design a local control action to be applied at that particular operating point. The design procedure is repeated at every operating point of the system during the control task. The proposed approach was applied in three examples, which involved a linear and two nonlinear discrete single-input single-output (SISO) systems. In the first two examples, pole placement was chosen as the linear design technique. This method led to satisfactory tracking of the reference input but a steady-state error was present due to modeling inaccuracies. In the third example, the linear design technique involved an optimal control scheme with an integrator. This method led to satisfactory tracking of the reference input with zero steady-state error.

A Nonlinear Discrete Dynamical Model for Transcriptional Regulation: Construction and Properties

Transcriptional regulation is a fundamental mechanism of living cells, which allows them to determine their actions and properties, by selectively choosing which proteins to express and by dynamically controlling the amounts of those proteins. In this article, we revisit the problem of mathematically modeling transcriptional regulation. First, we adopt a biologically motivated continuous model for gene transcription and mRNA translation, based on first-order rate equations, coupled with a set of nonlinear equations that model cis-regulation. Then, we view the processes of transcription and translation as being discrete, which, together with the need to use computational techniques for large-scale analysis and simulation, motivates us to model transcriptional regulation by means of a nonlinear discrete dynamical system. Classical arguments from chemical kinetics allow us to specify the nonlinearities underlying cis-regulation and to include both activators and repressors as well as the notion of regulatory modules in our formulation. We show that the steady-state behavior of the proposed discrete dynamical system is identical to that of the continuous model. We discuss several aspects of our model, related to homeostatic and epigenetic regulation as well as to Boolean networks, and elaborate on their significance. Simulations of transcriptional regulation of a hypothetical metabolic pathway illustrate several properties of our model, and demonstrate that a nonlinear discrete dynamical system may be effectively used to model transcriptional regulation in a biologically relevant way.

Dynamics of the electron transport in a quantum wire coupled to a quantum-dot array

We study the transport properties of independent carriers through a quantum wire coupled to a quantum-dot array. The electrical conductance at zero temperature can be expressed through a non-linear discrete dynamical system as the number of quantum dots varies. The dynamical system shows a rich behavior, determining a non-trivial conductance dependence on the Fermi energy. The conductance depends smoothly on the Fermi energy far from the site-energy of the quantum dots. At the center of the band the conductance develops a complex pattern due to constructive and destructive interference in the ballistic channel.

Congestion Control System in Urban Road Networks

This paper studies a signal control system and a signal control algorithm according to the variation of the traffic demand in a two-way traffic network. The signal control system of the congestion length is described by a nonlinear time-varying discrete dynamic system and synthesized by using the feedback control based on the volume balance at each signalized intersection. A network 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 traffic network is presented. From the comparison of congestion lengths and signal control parameters between measurement values and simulation values, it is confirmed that the signal control system and the network control algorithm work effectively to minimize the congestion lengths in the traffic network.

Nonlinear dynamic system identification and control via constructivism inspired neural network

This paper deals with a novel idea of identification of nonlinear dynamic systems via a constructivism inspired neural network. The proposed network is known as growing multi-experts network (GMN). In GMN, the problem space is decomposed into overlapping regions by expertise domain and local expert models are graded according to their expertise level. The network output is computed by the smooth combination of local linear models. In order to avoid over-fitting problem, GMN deploys a redundant experts removal algorithm to remove the redundant local experts from the network. In addition, growing neural gas (GNG) algorithm is used to generate an induced Delaunay triangulation that is highly desired for optimal function approximation. A variety of examples are taken from literature to establish the efficacy of GMN. Discrete time nonlinear dynamic system modeling and water bath temperature control have been found to give excellent results via this novel neural network.

Hybrid control of period-doubling bifurcation and chaos in discrete nonlinear dynamical systems

It is a typical route to generate chaos via period-doubling bifurcations in some nonlinear systems. In this paper, we propose a new hybrid control strategy in which state feedback and parameter perturbation are used to control the period-doubling bifurcations and to stabilize unstable periodic orbits embedded in the chaotic attractor of a discrete chaotic dynamical system. Simulation shows that the higher stable 2 n -periodic orbit of the system can be controlled to lower stable 2 m -periodic orbits ( m< n) by this methods. Some other numerical simulations are also presented to verify the theoretical analysis.

Control of period-doubling bifurcation and chaos in a discrete nonlinear system by the feedback of states and parameter adjustment

In this paper, the control of delaying period-doubling bifurcations and unstable periodic orbits embedded in a chaotic attractor of a discrete nonlinear dynamical system is effectively realized by using the state variables feedback and parameter variation. Moreover, the 2n periodic orbits of the system can be controlled into the 2m (m&lt;n) periodic orbits by the methods proposed.

Moving horizon Nash strategies for a military air operation

Dynamic game theory has recently received considerable attention as a possible technology for formulating control actions for decision makers in an extended complex enterprise that involves an adversary. Examples of such enterprises are very common in military operations. Enterprises of this type are typically modeled by a highly nonlinear discrete time dynamic system whose state is controlled by two teams of decision makers each with a different objective function and possibly with a different hierarchy of decision making. Because of the complexity of such systems, the traditional solutions from dynamic game theory that involve optimizing objective functions over the entire time horizon of the system are computationally extremely difficult, if not impossible, to derive. We discuss a solution approach where at each step the controllers limit the computation of their actions to a short time horizon that may involve only the next few time steps. This moving horizon solution, although suboptimal in the global sense, is very useful in taking into account the possible near-term control actions of the adversary. To illustrate this solution methodology, we consider an example of an extended military enterprise that involves two opposing forces engaged in a battle.

Optimization of Signal Control Parameters for Two-Way Traffic Arterials

This paper studies a signal control method which controls congestion lengths on two-way traffic arterials systematically. The signal control system of the congestion length is described by a nonlinear time-varying discrete dynamic system and synthesized by using the feedback control. A balance 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 on the arterial is presented. From the comparison between measurement values and simulation values, it is confirmed that the balance control algorithm works effectively to reduce the congestion lengths on the arterial

Nonlinear adaptive control using neural networks and multiple models

In this paper, adaptive control of a class of nonlinear discrete time dynamical systems with boundedness of all signals is established by using a linear robust adaptive controller and a neural network based nonlinear adaptive controller, and switching between them by a suitably defined switching law. The linear controller, when used alone, assures boundedness of all the signals but not satisfactory performance. The nonlinear controller may result in improved response, but may also result in instability. By using a switching scheme, it is demonstrated that improved performance and stability can be simultaneously achieved.

Global Exponential Asymptotic Stability in Nonlinear Discrete Dynamical Systems

For the nonlinear discrete dynamical system xk+1=Txk on bounded, closed and convex set D⊂Rn, we present several sufficient and necessary conditions under which the unique equilibrium point is globally exponentially asymptotically stable. The infimum of exponential bounds of convergent trajectories is also derived.

Learning performance of a neurocomputer for nonlinear dynamical system identification

This paper investigates the learning performance of a RICOH neurocomputer RN-2000 for the identification problem of input and output map of a discrete nonlinear dynamical system. The results obtained show capability of on-chip learning, which is essential for many neural applications such as machine learning and control where real-time adaptation is required. In this paper, the method to use a neurocomputer is briefly presented for a nonlinear identification problem. The main significance of this research is to obtain a further guideline for designing a primitive artificial brain for robotics.

Observabilities and reachabilities of nonlinear DEDS and coloring graphs

From nonlinear discrete event dynamic systems with the applicable background of a large-scale digital integrated circuit, a new conception of coloring graphs on the system is advanced, the necessary and sufficient condition of upper-level observability is given, and the necessary and sufficient condition of respective reachability is simplified and improved.

Inversion of matrices and matrix functions as a nonlinear discrete system: Stability and sensitivity analysis

A technique for the recursive inversion of matrices and matrix functions is presented. The proposed method can be modelled as a discrete nonlinear dynamic system. Convergence, stability and robustness properties are discussed and eventually verified through various numerical experiments.

An Analysis of Recurrent Excitation Based on Non-Linear Discrete Dynamical System Theory and Its Relevance to Epileptogenesis

An Analysis of Recurrent Excitation Based on Non-Linear Discrete Dynamical System Theory and Its Relevance to Epileptogenesis

Nonlinear effects in a discrete-time dynamic model of a stock market

The time evolution of prices and savings in a stock market is modeled by a discrete time nonlinear dynamical system. The model proposed has a unique and unstable steady-state, so that the time evolution is determined by the nonlinear effects acting out of the equilibrium. The nonlinearities strongly influence the kind of long-run dynamics of the system. In particular, the global geometric properties of the noninvertible map of the plane, whose iteration gives the evolution of the system, are important to understand the global bifurcations which change the qualitative properties of the asymptotic dynamics. Such global bifurcations are studied by geometric and numerical methods based on the theory of critical curves, a powerful tool for the characterization of the global dynamical properties of noninvertible mappings of the plane. The model unfolds more complex chaotic and unpredictable trajectories as a consequence of increasing agents' “speculative” or “capital gain realizing” attitudes. The global analysis indicates that, for some ranges of the parameter values, the system has several coexisting attractors, and it may not be robust with respect to exogenous shocks due to the complexity of the basins of attraction.

A new method for the control of discrete nonlinear dynamic systems using neural networks

A new controller design method for nonaffine nonlinear dynamic systems is presented in this paper. An identified neural network model of the nonlinear plant is used in the proposed method. The method is based on a new control law that is developed for any discrete deterministic time-invariant nonlinear dynamic system in a subregion Phi(x) of an asymptotically stable equilibrium point of the plant. The performance of the control law is not necessarily dependent on the distance between the current state of the plant and the equilibrium state if the nonlinear dynamic system satisfies some mild requirements in Phi(x). The control law is simple to implement and is based on a novel linearization of the input-output model of the plant at each instant in time. It can be used to control both minimum phase and nonminimum phase nonaffine nonlinear plants. Extensive empirical studies have confirmed that the control law can be used to control a relatively general class of highly nonlinear multiinput-multioutput (MIMO) plants.

APPLICATION OF ORDER PARAMETER EQUATIONS FOR THE ANALYSIS AND THE CONTROL OF NONLINEAR TIME DISCRETE DYNAMICAL SYSTEMS

This work is based on the concept of order parameters of synergetics. The order parameter equations describe the behavior of a system in the vicinity of an instability and are used here not only for the analysis but also for the control of nonlinear time discrete dynamical systems. Usually, the dimensionality of the evolution equations of the order parameters is less than the dimensionality of the original evolution equations. It is, therefore, convenient to introduce control mechanisms, first in the order parameter equations, and then to use the obtained results for the control of the original system. The aim of the control in this case is to avoid chaotic behavior of the system. This is achieved by shifting appropriate bifurcation points of a period-doubling cascade. In this work we concentrate on the shifting of only the first bifurcation point. The used control mechanisms are delayed feedback schemes. As an example the well-known Hénon map is investigated. The order parameter equation is calculated using both the adiabatic elimination procedure and the center manifold theory. Using the order parameter concept two types of control mechanisms are constructed, analyzed and compared.

An elementary result in the stability theory of time-invariant nonlinear discrete dynamical systems

The stability of the equilibria of time-invariant nonlinear dynamical systems with discrete time scale is investigated. We present an elementary proof showing that in the case of a stable equilibrium and continuously differentiable state transition function, all eigenvalues of the Jacobian computed at the equilibrium must be inside or on the unit circle. We also demonstrate via numerical examples that if some eigenvalues are on the unit circle and all other eigenvalues are inside the unit circle, then the equilibrium maybe unstable, or marginally stable, or even asymptotically stable, which show that the necessary condition cannot be further restricted in general. In addition, the necessary condition is given in terms of spectral radius and matrix norms.