Flow invariance for competitive multi-modal neural networks

We present a new method of analyzing the dynamics of self-organizing neural networks with different time scales based on the theory of flow invariance. We are able to show the conditions under which the solutions of such a system are bounded being less restrictive than with the K-monotone theory, singular perturbation theory, or those based on supervised synaptic learning. We prove the existence and the uniqueness of the equilibrium. A strict Lyapunov function for the flow of a competitive neural system with different time scales is given, and based on it we are able to prove the global exponential stability of the equilibrium point.

We present new methods of analyzing the dynamics of a competitive neural system with different time scales: the K-monotone system theory developed by Kamke (1932) as a global analysis techniques, and the theory of singular perturbation as a local analysis method.

The dynamics of complex neural networks must include the aspects of long and short-term memory. The behaviour of the network is characterized by an equation of neural activity as a fast phenomenon and an equation of synaptic modification as a slow part of the neural system. We present a method of analyzing the dynamics of a system with different time scales based on the theory of flow invariance. We are able to show the conditions under which the solutions of such a system are bounded being less restrictive than with the K-monotone theory.

Global exponential stability of delayed competitive neural networks with different time scales

The dynamics of cortical cognitive maps developed by selforganization must include the aspects of long and short-term memory. The behavior of such a neural network is characterized by an equation of neural activity as a fast phenomenon and an equation of synaptic modification as a slow part of the neural system, besides, this model bases on unsupervised synaptic learning algorithm. In this paper, using theory of the topological degree and strict Liapunov functional methods, we prove existence and uniqueness of the equilibrium of competitive neural networks with continuously distributed delays and different time scales, and present some new criteria for its global robust stability.

The theory of singular perturbations and time scales has played an important role in aircraft performance analysis and in the development of feedback guidance laws for aerospace vehicles. This paper presents a survey of the applications of the theory of singular perturbations and time scales in guidance, navigation, and control of air and space transportation systems. In particular, emphasis will be placed on problem formulation and solution approaches that are useful in applying the theory for various types of problems arising in aerospace systems. Fhrther, the survey e m braces the related fields such as fluid dynamics, space structures and space robotics.

Global stability analysis of competitive neural networks with mixed time-varying delays and discontinuous neuron activations

A new approach based on sliding control is presented for modeling and simulation of thermo-fluid systems described by differential-algebraic equations (DAEs). The dynamics of thermo-fluid systems are often complicated by momentum interactions that occur on a time scale that is orders of magnitude faster than the time scale of interest. To address this problem the momentum equation is often modeled using algebraic steady state approximations. This will, in general, result in a model described by nonlinear DAEs for which few control methods are currently applicable. In this paper, the modeling problem is addressed using an approach that systematically constructs an explicit state space approximation of the DAEs. The state space model can in turn be used with existing control methods. This procedure, known as realization, is achieved by solving an associated nonlinear control problem by combining boundary layer sliding control with the singular perturbation method. The necessary criteria for key properties such as convergence are established. Further, the new approach is illustrated using a vapor compression cycle example. This demonstrates significant advantages over directly modeling momentum interactions.

Fine-Grained Parallel and Distributed Spatial Stochastic Simulation of Biological Reactions

Current challenges in industrial processes control include achieving optimum operation for systems with two-time-scale dynamics and unknown models. This paper presents, for the first time, the integration of singular perturbation theory and reinforcement learning to solve this problem. To this end, an optimal operational control (OOC) problem with two time scales is formulated to reach the desired operational indices. Then, a singularly perturbed dynamics for two-time-scale industrial operational processes is developed by introducing a perturbed scale, resulting in the separation of the original system dynamics. Thus, the original optimization problem is decomposed into a reduced slow subproblem and a boundary fast subproblem. The fact that the sum of the separate solutions of these subproblems is approximately equal to the solution of the OOC problem is proven. Then, two $Q$ -learning algorithms are proposed to obtain a composite feedback control. Finally, an industrial thickener example is employed to show the effectiveness of the proposed method.

We study time-scale separation and robust controller design for a class of singularly perturbed nonlinear systems under perfect state measurements. The system dynamics are taken to be jointly linear in the fast state variables, control and disturbance inputs, but nonlinear in the slow state variables. Since global timescale separation may not always be possible for nonlinear singularly perturbed systems, we restrict our attention here to some closed subset of the state space, on which a timescale separation holds for sufficiently small values of the singular perturbation parameter. We construct a slow controller and a composite controller based on the solutions of particular slow and fast games obtained using time-scale separation. For the class of systems for which the slow controller can be selected to be robust with respect to small regular structural perturbations on the slow subsystem, we show under some growth conditions that the composite controller can achieve any desired level of performance that is larger than the maximum of the performance levels for the slow and fast subsystems,. A slow controller, however, is not generally as robust as the composite controller; but, still under some conditions which are delineated in the paper, the fast dynamics can be totally ignored. The paper also presents a numerical example to illustrate the theoretical results.

The role of slow system dynamics in predicting the degeneracy of slow invariant manifolds: The case of vdP relaxation–oscillations

This paper investigates traveling wave solutions for the spatial diffusion predator-prey model with Allee effects. From the perspective of singular perturbations, we demonstrate the existence of the traveling fronts connecting periodic orbits and the positive equilibrium. On the one hand, we use geometric singular perturbation theory and exchange lemma to study the existence of periodic wave trains, including the degenerate case in which the periodic orbits pass through the small neighbourhoods of two folds on the critical manifold. We utilize the Fenichel normal form and central manifold theory to analyze the flows near the folds where normal hyperbolicity fails. Additionally, we derive the period functions of these periodic orbits. On the other hand, by invariant manifold theory, we prove the transversal intersection between the stable manifolds of positive equilibrium and the unstable manifolds of the periodic orbit. In particular, we identify two types of traveling fronts: pulled and pushed fronts connecting periodic orbits to the positive equilibrium with the steepest decay. Finally, numerical simulations illustrate these theoretical results.

This paper is concerned with the traveling wave solutions of a singularly perturbed system, which arises from the coupled arrays of Chua's circuit. By the geometric singular perturbation theory and invariant manifold theory, we prove that there exists a heteroclinic cycle consisting of the traveling front and back waves with the same wave speed. In particular, the expression of corresponding wave speed is also obtained. Furthermore, we show that the chaotic behavior induced by this heteroclinic cycle is hyperchaos.

In this paper, the regulation problem of a class of nonlinear singularly perturbed discrete-time systems is investigated. Using the theory of singular perturbations and time scales, the nonlinear system is decoupled into reduced-order slow and fast (boundary layer) subsystems. Then, a composite controller consisting of two sub-controllers for the slow and fast subsystems is developed using the discrete-time state-dependent Riccati equation (D-SDRE). It is proved that the equilibrium point of the original closed-loop system with a composite controller is locally asymptotically stable. Moreover, the region of attraction of the closed-loop system is estimated by using linear matrix inequality. One example is given to illustrate the effectiveness of the results obtained.