A neuropysiological correlate of short-term memory has been found in the ability of some prefrontal neurons to hold task relevant information through ongoing neuronal activity. Such experimental findings, and the corresponding hypothesis of the mechanisms that enables the ongoing neuronal activity, have been captured successfully with continuous attractor neural network models. However, the bas...

A mathematical model for learning a nonlinear line of attractions is presented in this paper. This model encapsulates attractive fixed points scattered in the state space representing patterns with similar characteristics as an attractive line. The dynamics of this nonlinear line attractor network is designed to operate between stable and unstable states. These criteria can be used to circumven...

The attractor is a central object of an iterated function system (IFS), and fractal transformations are the natural maps from the attractor of one IFS to the attractor of another. This paper presents a global point of view, showing how to extend the domain of a fractal transformation from an attractor with non-empty interior to the ambient space. Intimitely related is the extension of addressin...

In this paper we consider a new type of bifurcation to chaotic motion. This bifurcation is characterized by the simultaneous appearance of a pair of unstable fixed points or periodic orbits, a, fractal (i.e., nondifferentiable) basin boundary, and a chaotic attractor (also commonly called a strange attractor). Just prior to this type of bifurcation, transient behavior with a chaotic character c...

In this work we derive lower bounds for the Hausdorff and fractal dimensions of the global attractor of the Sabra shell model of turbulence in different regimes of parameters. We show that for a particular choice of the forcing term and for sufficiently small viscosity term ν, the Sabra shell model has a global attractor of large Hausdorff and fractal dimensions proportional to logν−1 for all v...

An algorithm is introduced that trains a neural network to identify chaotic dynamics from a single measured timeseries. The algorithm has four special features: 1. The state of the system is extracted from the time-series using delays, followed by weighted Principal Component Analysis (PCA) data reduction. 2. The prediction model consists of both a linear model and a Multi-Layer-Perceptron (MLP...

in this paper, we consider a cahn-hillard/allen-cahn equation. by using the semigroup and the classical existence theorem of global attractors, we give the existence of the global attractor in h^k(0