Search results for "attractor"

Dynamic complexities in host-parasitoid interaction

1999

In the 1970s ecological research detected chaos and other forms of complex dynamics in simple population dynamics models, initiating a new research tradition in ecology. However, the investigations of complex population dynamics have mainly concentrated on single populations and not on higher dimensional ecological systems. Here we report a detailed study of the complicated dynamics occurring in a basic discrete-time model of host-parasitoid interaction. The complexities include (a) non-unique dynamics, meaning that several attractors coexist, (b) basins of attraction (defined as the set of the initial conditions leading to a certain type of an attractor) with fractal properties (pattern of…

Non-unique population dynamics: basic patterns

2000

We review the basic patterns of complex non-uniqueness in simple discrete-time population dynamics models. We begin by studying a population dynamics model of a single species with a two-stage, two-habitat life cycle. We then explore in greater detail two ecological models describing host‐macroparasite and host‐parasitoid interspecific interactions. In general, several types of attractors, e.g. point equilibria vs. chaotic, periodic vs. quasiperiodic and quasiperiodic vs. chaotic attractors, may coexist in the same mapping. This non-uniqueness also indicates that the bifurcation diagrams, or the routes to chaos, depend on initial conditions and are therefore non-unique. The basins of attrac…

Remarks on GRN-type systems

2020

Systems of ordinary differential equations that appear in gene regulatory networks theory are considered. We are focused on asymptotical behavior of solutions. There are stable critical points as well as attractive periodic solutions in two-dimensional and three-dimensional systems. Instead of considering multiple parameters (10 in a two-dimensional system) we focus on typical behaviors of nullclines. Conclusions about possible attractors are made.

Attraction in n ‐dimensional differential systems from network regulation theory

2018

On a Planar Dynamical System Arising in the Network Control Theory

2016

We study the structure of attractors in the two-dimensional dynamical system that appears in the network control theory. We provide description of the attracting set and follow changes this set suffers under the changes of positive parameters µ and Θ.

Retrieving infinite numbers of patterns in a spin-glass model of immune networks

2013

The similarity between neural and immune networks has been known for decades, but so far we did not understand the mechanism that allows the immune system, unlike associative neural networks, to recall and execute a large number of memorized defense strategies {\em in parallel}. The explanation turns out to lie in the network topology. Neurons interact typically with a large number of other neurons, whereas interactions among lymphocytes in immune networks are very specific, and described by graphs with finite connectivity. In this paper we use replica techniques to solve a statistical mechanical immune network model with `coordinator branches' (T-cells) and `effector branches' (B-cells), a…

Strange Attractors, Chaotic Behavior and Informational Aspects of Sleep EEG Data

1992

In order to perform a nonlinear dimensional analysis of the sleep EEG, we applied an algorithm proposed to calculate the correlation dimension D2 of different sleep stages. D2 characterizes the dynamics of the sleep EEG, estimates the degrees of freedom, and describes the complexity of the signal under study. An attempt is made to correlate dimensionality analysis and informational aspects of the sleep EEG. Information processing by the brain during different sleep stages of healthy subjects under the influence of lorazepam and in unmedicated acute schizophrenics is estimated.

Deterministic chaos and the first positive Lyapunov exponent: a nonlinear analysis of the human electroencephalogram during sleep

1993

Under selected conditions, nonlinear dynamical systems, which can be described by deterministic models, are able to generate so-called deterministic chaos. In this case the dynamics show a sensitive dependence on initial conditions, which means that different states of a system, being arbitrarily close initially, will become macroscopically separated for sufficiently long times. In this sense, the unpredictability of the EEG might be a basic phenomenon of its chaotic character. Recent investigations of the dimensionality of EEG attractors in phase space have led to the assumption that the EEG can be regarded as a deterministic process which should not be mistaken for simple noise. The calcu…

Attractors/Basin of Attraction

2020

It is a controversial issue to decide who first coined the term “attractor”. According to Peter Tsatsanis, the editor of the English version of Prédire n’est pas expliquer, it was René Thom who first introduced such a term. It is necessary, however, to remember that Thom thought that it was first introduced by the American mathe- matician Steven Smale, “although Smale says it was Thom that coined the neolo- gism “attractor”“(Tsatsanis 2010: 63–64 n. 20). From this point of view, Bob Williams expressed a more cautious opinion by saying that “the word “attractor” was invented by these guys, Thom and Smale” (Cucker and Wong 2000: 183). But other mathematicians are of the opinion that the term …

From Topology to Quasi-Topology. The Complexity of the Notional Domain

2016

This article examines a fundamental metalinguistic construction of the theory of enunciative operations: the notional domain. In particular, we try to explain some elementary topological concepts on which this construction is based and we try to show the key role they play in the description of some basic linguistic operations.