48 research outputs found
Complex patterns on the plane: different types of basin fractalization in a two-dimensional mapping
Basins generated by a noninvertible mapping formed by two symmetrically
coupled logistic maps are studied when the only parameter \lambda of the system
is modified. Complex patterns on the plane are visualised as a consequence of
basins' bifurcations. According to the already established nomenclature in the
literature, we present the relevant phenomenology organised in different
scenarios: fractal islands disaggregation, finite disaggregation, infinitely
disconnected basin, infinitely many converging sequences of lakes, countable
self-similar disaggregation and sharp fractal boundary. By use of critical
curves, we determine the influence of zones with different number of first rank
preimages in the mechanisms of basin fractalization.Comment: 19 pages, 11 figure
On the embedding of a (p-1)-dimensional non invertible map into a p-dimensional invertible map
This paper concerns the description of some properties of p-dimensional invertible real maps
Tb, turning into a (p - 1)-dimensional non invertible ones T0, p = 2, 3, when a parameter b of
the first map is equal to a critical value, say b=0. Then it is said that the noninvertible map is
embedded into the invertible one. More particularly properties of the stable, and the unstable
manifolds of a saddle fixed point are considered in relation with this embedding. This is made
by introducing the notion of folding as resulting from the crossing through a commutation curve
when p = 2, or a commutation surface when p = 3
Asset price dynamics in a financial market with fundamentalists and chartists
In this paper we consider a model of the dynamics of speculative markets involving the interaction of fundamentalists and chartists. The dynamics of the model are driven by a two-dimensional map that in the space of the parameters displays regions of invertibility and noninvertibility. The paper focuses on a study of local and global bifurcations which drastically change the qualitative structure of the basins of attraction of several, often coexistent, attracting sets. We make use of the theory of critical curves associated with noninvertible maps, as well as of homoclinic bifurcations and homoclinic orbits of saddles in regimes of invertibility
Fractal basins in an ecological model
Complex dynamics is detected in an ecological model of host-parasitoid interaction. It illustrates fractalization of basins with self-similarity and chaotic attractors. This paper describes these dynamic behaviors, bifurcations, and chaos. Fractals basins are displayed by numerical simulations
Using the basin entropy to explore bifurcations
Bifurcation theory is the usual analytic approach to study the parameter
space of a dynamical system. Despite the great power of prediction of these
techniques, fundamental limitations appear during the study of a given problem.
Nonlinear dynamical systems often hide their secrets and the ultimate resource
is the numerical simulations of the equations. This paper presents a method to
explore bifurcations by using the basin entropy. This measure of the
unpredictability can detect transformations of phase space structures as a
parameter evolves. We present several examples where the bifurcations in the
parameter space have a quantitative effect on the basin entropy. Moreover, some
transformations, such as the basin boundary metamorphoses, can be identified
with the basin entropy but are not reflected in the bifurcation diagram. The
correct interpretation of the basin entropy plotted as a parameter extends the
numerical exploration of dynamical systems