Limit Set Dichotomy Research Articles

Sharpened dynamics alternative and its C1-robustness for strongly monotone discrete dynamical systems

For strongly monotone dynamical systems, the dynamics alternative for smooth discrete-time systems turns out to be a perfect analogy of the celebrated Hirsch's limit-set dichotomy for continuous-time semiflows. In this paper, we first present a sharpened dynamics alternative for C1-smooth strongly monotone discrete-time dissipative system {F0n}n∈N (with an attractor A), which concludes that there is a positive integer m such that any orbit is either manifestly unstable; or asymptotic to a linearly stable cycle whose minimal period is bounded by m. Furthermore, we show the C1-robustness of the sharpened dynamics alternative, that is, for any C1-perturbed system {Fϵn}n∈N (Fϵ not necessarily monotone), any orbit initiated nearby A will admit the sharpened dynamics alternative with the same m. The improved generic convergence to cycles for the C1-system {F0n}n∈N, as well as for the perturbed system {Fϵn}n∈N, is thus obtained as by-products of the sharpened dynamics alternative and its C1-robustness. The results are applied to nonlocal C1-perturbations of a time-periodic parabolic equations and give typical convergence to periodic solutions whose minimal periods are uniformly bounded.

Convergent Dynamics of Nonreciprocal Differential Variational Inequalities Modeling Neural Networks

The paper addresses convergence of solutions for a class of differential inclusions termed differential variational inequalities (DVIs). Each DVI describes the dynamics of a neural network (NN) evolving in a closed hypercube of \BBR <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> and defined by a continuously differentiable, cooperative and (possibly) nonreciprocal vector field f. The main result in the paper is that under a new condition on f, which is called strong Kamke-Müller (SKM) condition, the solution semiflow generated by the DVI is strongly order preserving (SOP) and hence it satisfies a Limit Set Dichotomy and enjoys generic convergence properties. A characterization of the SKM condition is given in terms of the interconnection properties of the Jacobian matrix of f. In the case where f is an affine, or a linear, vector field the considered DVIs include two relevant classes of NNs, namely, the linear systems operating on a closed hypercube, also known as linear systems in saturated mode (LSSMs), and the full-range (FR) model of cellular neural networks (CNNs). By applying the results to LSSMs it is obtained that any cooperative LSSM with a (possibly) nonsymmetric and fully interconnected matrix is generically convergent. Analogous results hold for FRCNNs. All the obtained convergence results hold in the general case where the DVIs, and the LSSMs and FRCNNs, possess multiple equilibrium points.

Convergence on cooperative cascade systems with length one

Motivated by the open problems on the generic convergence of cooperative systems without the assumption of irreducibility independently proposed by Smith and Sontag, this paper investigates the generic convergence for the solutions of cooperative cascade systems with length one. First, by fixing a solution of a base system converging to an equilibrium, we establish both the Nonordering of Limit Sets and the Limit Set Dichotomy for the solutions of the cascade system. Combining these tools with the idea of limiting equation, we then prove the Sequential Limit Set Trichotomy and hence the quasiconvergence in generic meaning. The generic convergent result is finally obtained by improving the Limit Set Dichotomy.

Limit set dichotomy and multistability for a class of cooperative neural networks with delays.

Recent papers have pointed out the interest to study convergence in the presence of multiple equilibrium points (EPs) (multistability) for neural networks (NNs) with nonsymmetric cooperative (nonnegative) interconnections and neuron activations modeled by piecewise linear (PL) functions. One basic difficulty is that the semiflows generated by such NNs are monotone but, due to the horizontal segments in the PL functions, are not eventually strongly monotone (ESM). This notwithstanding, it has been shown that there are subclasses of irreducible interconnection matrices for which the semiflows, although they are not ESM, enjoy convergence properties similar to those of ESM semiflows. The results obtained so far concern the case of cooperative NNs without delays. The goal of this paper is to extend some of the existing results to the relevant case of NNs with delays. More specifically, this paper considers a class of NNs with PL neuron activations, concentrated delays, and a nonsymmetric cooperative interconnection matrix A and delay interconnection matrix A(τ). The main result is that when A+A(τ) satisfies a full interconnection condition, then the generated semiflows, which are monotone but not ESM, satisfy a limit set dichotomy analogous to that valid for ESM semiflows. It follows that there is an open and dense set of initial conditions, in the state space of continuous functions on a compact interval, for which the solutions converge toward an EP. The result holds in the general case where the NNs possess multiple EPs, i.e., is a result on multistability, and is valid for any constant value of the delays.

A study on semiflows generated by cooperative full‐range CNNs

ABSTRACTThe paper considers the full‐range (FR) model of cellular neural networks (CNNs) characterized by ideal hard‐limiter nonlinearities with two vertical segments in the current–voltage characteristic. It is shown that when the FRCNNs are cooperative, i.e., there are excitatory interconnections between distinct neurons, the generated solution semiflow is monotone and that monotonicity implies some fundamental restrictions on the geometry of omega‐limit sets. The result on monotonicity is a generalization to the class of differential inclusions describing the dynamics of FRCNNs of a classic result due to Kamke for cooperative ordinary differential equations. The paper also points out difficulties to use the standard theory of eventually strongly monotone (ESM) semiflows for addressing convergence of FRCNNs. By means of counterexamples, it is shown that, even assuming the irreducibility of the interconnections, the semiflow generated by a cooperative FRCNN is not ESM; furthermore, also the limit set dichotomy can be violated. Copyright © 2012 John Wiley &amp; Sons, Ltd.

Convergence of a Class of Cooperative Standard Cellular Neural Network Arrays

This paper considers a nonsymmetric standard (S) cellular neural network (CNN) array with cooperative (nonnegative) interconnections between neurons and a typical three-segment piecewise-linear (PL) neuron activation. The CNN is defined by a one-dimensional cell-linking (irreducible) cloning template with nearest-neighbor interconnections and has periodic boundary conditions. The flow generated by the SCNN is monotone but, due to the squashing effect of the horizontal segments in the PL activations, is not eventually strongly monotone (ESM). A new method for addressing convergence of the cooperative SCNN array is developed, which is based on the two main tools: (1) the concept of a frozen saddle, i.e., an unstable saddle-type equilibrium point (EP) enjoying certain dynamical properties that hold also for an asymptotically stable EP (a sink); (2) the analysis of the order relations satisfied by the sinks and saddle-type EPs. The analysis permits to show a fundamental result according to which any pair of ordered EPs of the SCNN contains at least a sink or a frozen saddle. On this basis it is shown that the flow generated by the SCNN enjoys a Limit Set Dichotomy and convergence properties analogous to those valid for ESM flows. Such results hold in the case where the SCNN displays either a local diffusion or a global propagation behavior.

Kalman Filtering With Intermittent Observations: Weak Convergence to a Stationary Distribution

The paper studies the asymptotic behavior of discrete time Random Riccati Equations (RRE) arising in Kalman filtering when the arrival of the observations is described by a Bernoulli independent and identically distributed (i.i.d.) process. We model the RRE as an order-preserving, strongly sublinear random dynamical system (RDS). Under a sufficient condition, stochastic boundedness, and using a limit-set dichotomy result for order-preserving, strongly sublinear RDS, we establish the asymptotic properties of the RRE: the sequence of random prediction error covariance matrices converges weakly to a unique invariant distribution, whose support exhibits fractal behavior. For stabilizable and detectable systems, stochastic boundedness (and hence weak convergence) holds for any nonzero observation packet arrival probability and, in particular, we can establish weak convergence at operating arrival rates well below the critical probability for mean stability (the resulting invariant measure in that situation does not possess a first moment). We apply the weak-Feller property of the Markov process governing the RRE to characterize the support of the limiting invariant distribution as the topological closure of a countable set of points, which, in general, is not dense in the set of positive semi-definite matrices. We use the explicit characterization of the support of the invariant distribution and the almost sure (a.s.) ergodicity of the sample paths to easily compute statistics of the invariant distribution. A one-dimensional example illustrates that the support is a fractured subset of the non-negative reals with self-similarity properties.

Limit Set Dichotomy and Convergence of Cooperative Piecewise Linear Neural Networks

This paper considers a class of nonsymmetric cooperative neural networks (NNs) where the neurons are fully interconnected and the neuron activations are modeled by piecewise linear (PL) functions. The solution semiflow generated by cooperative PLNNs is monotone but, due to the horizontal segments in the neuron activations, is not eventually strongly monotone (ESM). The main result in this paper is that it is possible to prove a peculiar form of the Limit Set Dichotomy for this class of cooperative PLNNs. Such a form is slightly weaker than the standard form valid for ESM semiflows, but this notwithstanding it permits to establish a result on convergence analogous to that valid for ESM semiflows. Namely, for almost every choice of the initial conditions, each solution of a fully interconnected cooperative PLNN converges toward an equilibrium point, depending on the initial conditions, as t → +∞. From a methodological viewpoint, this paper extends some basic techniques and tools valid for ESM semiflows, in order that they can be applied to the monotone semiflows generated by the considered class of cooperative PLNNs.

LIMIT SET DICHOTOMY AND CONVERGENCE OF SEMIFLOWS DEFINED BY COOPERATIVE STANDARD CNNs

The paper analyzes some fundamental properties of the solution semiflow of nonsymmetric cooperative standard (S) cellular neural networks (CNNs) with a typical three-segment piecewise-linear (pwl) neuron activation. Two relevant subclasses of SCNNs, corresponding to one-dimensional circular SCNNs with two-sided or single-sided positive interconnections between nearest neighboring neurons only, are considered. For these subclasses it is shown that the associated solution semiflow satisfies the fundamental properties of the CONVERGENCE CRITERION, the NONORDERING OF LIMIT SETS and the LIMIT SET DICHOTOMY, and that this is true although the semiflow is not eventually strongly monotone. As a consequence such CNNs are almost convergent, i.e., almost all solutions converge toward an equilibrium point as time tends to infinity. To the authors' knowledge the paper is the first rigorous investigation on the geometry of limit sets and convergence properties of cooperative SCNNs with a pwl neuron activation. All available convergence results in the literature indeed concern a modified cooperative CNN model where the original pwl activation of the SCNN model is replaced by a continuously differentiable strictly increasing sigmoid function. The main results in the paper are established by conducting a deep analysis of the properties of the omega-limit sets of the solution semiflow defined by the considered subclasses of SCNNs. In doing so the paper exploits and extends some mathematical tools for monotone systems in order that they can be applied to pwl vector fields that govern the dynamics of SCNNs. By using some transformations and referring to specific examples it is also shown that the treatment in the paper can be extended to other subclasses of SCNNs.

Unstable sets, heteroclinic orbits and generic quasi-convergence for essentially strongly order-preserving semiflows

AbstractWe present conditions guaranteeing the existence of non-trivial unstable sets for compact invariant sets in semiflows with certain compactness conditions, and then establish the existence of such unstable sets for an unstable equilibrium or a minimal compact invariant set, not containing equilibria, in an essentially strongly order-preserving semiflow. By appealing to the limit-set dichotomy for essentially strongly order-preserving semiflows, we prove the existence of an orbit connection from an equilibrium to a minimal compact invariant set, not consisting of equilibria. As an application, we establish a new generic convergence principle for essentially strongly order-preserving semiflows with certain compactness conditions.

Generic Quasi-Convergence for Essentially Strongly Order-Preserving Semiflows

AbstractBy employing the limit set dichotomy for essentially strongly order-preserving semiflows and the assumption that limit sets have infima and suprema in the state space, we prove a generic quasi-convergence principle implying the existence of an open and dense set of stable quasi-convergent points. We also apply this generic quasi-convergence principle to a model for biochemical feedback in protein synthesis and obtain some results about the model which are of theoretical and realistic significance.

Dynamics of smooth essentially strongly order-preserving semiflows with application to delay differential equations

In this paper, we introduce a class of smooth essentially strongly order-preserving semiflows and improve the limit set dichotomy for essentially strongly order-preserving semiflows. Generic convergence and stability properties of this class of smooth essentially strongly order-preserving semiflows are then developed. We also establish the generalized Krein–Rutman Theorem for a compact and eventually essentially strongly positive linear operator. By applying the main results of this paper to essentially cooperative and irreducible systems of delay differential equations, we obtain some results on generic convergence and stability, the linearized stability of an equilibrium and the existence of the most unstable manifold in these systems. The obtained results improve some corresponding ones already known.

A cooperative system which does not satisfy the limit set dichotomy

The fundamental property of strongly monotone systems, and strongly cooperative systems in particular, is the limit set dichotomy due to Hirsch: if x < y , then either ω ( x ) < ω ( y ) , or ω ( x ) = ω ( y ) and both sets consist of equilibria. We provide here a counterexample showing that this property need not hold for (non-strongly) cooperative systems.

?Laplacians? on finitely ramified, graph directed fractals

A finitely ramified graph directed fractal is approximated by an adapted sequence of increasingly refined graphs. The scaling problem for a corresponding sequence of “discrete Laplacians” is rephrased via a renormalization map comparing two subsequent graphs. A limit set dichotomy for this map is proved: The forward orbit always accumulates at periodic points, even if the corresponding models are disconnected. Thus these periodic points can be numerically approximated by an iteration scheme based on the Schur complement. This allows to turn numerical information via the “short-cut test” into theorems on the existence of Laplacians on fractals.

Convergence for Strongly Order-Preserving Semiflows

Convergence of almost all trajectories of strongly order-preserving semiflows is derived under suitable additional assumptions. These essentially consist in slightly sharpening the strongly order-preserving property and in the continuous differentiability of the flow with respect to the state variable. Required spectral properties of the linearizations of the flow around equilibria usually follow in the same way as the compactness and monotonicity assumptions for the flow itself. The proofs are based on sharpened versions of the limit set dichotomy and the sequential limit set trichotomy.

Quasi Convergence and Stability for Strongly Order-Preserving Semiflows

Hirsch’s results concerning quasi convergence of almost all trajectories of strongly monotone semiflows are derived under weaker assumptions adopted from Matano. The proofs are based on a sequential limit set trichotomy, which follows from the nonordering principle and the limit set dichotomy. The assumption excluding totally ordered arcs of equilibria, which is required for the set of asymptotically stable points to be dense, is verified for dynamical systems that are analytic on the state space.