Chaotic Itinerancy in Random Dynamical System Related to Associative Memory Models

Ricardo Bioni Liberalquino,Maurizio Monge,Luigi Marangio,Stefano Galatolo

doi:10.3390/math6030039

Ricardo Bioni Liberalquino, Maurizio Monge + Show 2 more

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https://doi.org/10.3390/math6030039

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Abstract

We consider a random dynamical system arising as a model of the behavior of a macrovariable related to a more complicated model of associative memory. This system can be seen as a small (stochastic and deterministic) perturbation of a determinstic system having two weak attractors which are destroyed after the perturbation. We show, with a computer aided proof, that the system has a kind of chaotic itineracy. Typical orbits are globally chaotic, while they spend a relatively long time visiting the attractor’s ruins.

Highlights

Chaotic itinerancy is a concept used to refer to a dynamical behavior in which typical orbits visit a sequence of regions of the phase space called “quasi attractors” or “attractor ruins” in some irregular way
Speaking, during this itinerancy, the orbits visit a neighborhood of a quasi attractor with a relatively regular and stable motion, for relatively long times and the trajectory jumps to another quasi attractor of the system after a relatively small chaotic transient
The simple model we study shares with the more complicated models from which it is derived the presence of quasi attractors, its undestanding can bring some light on the mathematical undestanding of chaotic itinearacy

Summary

Introduction

Chaotic itinerancy is a concept used to refer to a dynamical behavior in which typical orbits visit a sequence of regions of the phase space called “quasi attractors” or “attractor ruins” in some irregular way Speaking, during this itinerancy, the orbits visit a neighborhood of a quasi attractor (the attractor ruin) with a relatively regular and stable motion, for relatively long times and the trajectory jumps to another quasi attractor of the system after a relatively small chaotic transient. This behavior was observed in several models and experiments related to the dynamics of neural networks and related to neurosciences (see [1]) In this itinerancy, the visit near some attractors was associated to the appearance of some macroscopic aspect of the system, like the emergence of a perception or a memory, while the chaotic iterations from a quasi attractor to another are associated to an intelligent (history dependent with trial and errors) search for the thought or perception (see [2]).

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