Duccio Papini

Global exponential stability and global convergence in finite time of delayed neural networks with infinite gain

This paper introduces a general class of neural networks with arbitrary constant delays in the neuron interconnections, and neuron activations belonging to the set of discontinuous monotone increasing and (possibly) unbounded functions. The discontinuities in the activations are an ideal model of the situation where the gain of the neuron amplifiers is very high and tends to infinity, while the delay accounts for the finite switching speed of the neuron amplifiers, or the finite signal propagation speed. It is known that the delay in combination with high-gain nonlinearities is a particularly harmful source of potential instability. The goal of this paper is to single out a subclass of the considered discontinuous neural networks for which stability is instead insensitive to the presence of a delay. More precisely, conditions are given under which there is a unique equilibrium point of the neural network, which is globally exponentially stable for the states, with a known convergence rate. The conditions are easily testable and independent of the delay. Moreover, global convergence in finite time of the state and output is investigated. In doing so, new interesting dynamical phenomena are highlighted with respect to the case without delay, which make the study of convergence in finite time significantly more difficult. The obtained results extend previous work on global stability of delayed neural networks with Lipschitz continuous neuron activations, and neural networks with discontinuous neuron activations but without delays.

On the periodic boundary value problem and chaotic-like dynamics for nonlinear Hill's equations

Abstract We present some results which show the rich and complicated structure of the solutions of the second order differential equation ẍ + w(t)g(x) = 0 when the weight w(t) changes sign and g is sufficiently far from the linear case. New applications, motivated by recent studies on the superlinear Hill’s equation in [57, 58, 59], are then proposed for some asymptotically linear equations and for some sublinear equations with a sign-indefinite weight. Our results are based on a fixed point theorem for maps which satisfy a stretching condition along the paths on two-dimensional cells.

A topological approach to superlinear indefinite boundary value problems

We obtain the existence of infinitely many solutions with prescribed nodal properties for some boundary value problems associated to the second order scalar equation

Fixed points, periodic points, and coin-tossing sequences for mappings defined on two-dimensional cells

\ddot{x} + q(t) g(x) = 0

Invariant Measures of Tunable Chaotic Sources: Robustness Analysis and Efficient Estimation

, where

AN EFFICIENT AND ACCURATE METHOD FOR THE ESTIMATION OF ENTROPY AND OTHER DYNAMICAL INVARIANTS FOR PIECEWISE AFFINE CHAOTIC MAPS

g(x)

Analysis of a lagoon ecological model with anoxic crises and impulsive harvesting

has superlinear growth at infinity and

Global exponential stability of the periodic solution of a delayed neural network with discontinuous activations

q(t)

Common Asymptotic Behavior of Solutions and Almost Periodicity for Discontinuous, Delayed, and Impulsive Neural Networks

changes sign.

Superlinear Indefinite Equations on the Real Line and Chaotic Dynamics

We propose, in the general setting of topological spaces, a definition of two-dimensional oriented cell and consider maps which possess a property of stretching along the paths with respect to oriented cells. For these maps, we prove some theorems on the existence of fixed points, periodic points, and sequences of iterates which are chaotic in a suitable manner. Our results, motivated by the study of the Poincaré map associated to some nonlinear Hills equations, extend and improve some recent work. The proofs are elementary in the sense that only well known properties of planar sets and maps and a two-dimensional equivalent version of the Brouwer fixed point theorem are used.