Narrow Results

In this paper we give an example of transition to fractal basin boundary in a two-dimensional map coming from the applicative context, in which the hard-fractal structure can be rigorously proved. That is, not only via numerical examples, although theoretically guided, as often occurs in maps coming from the applications, but also via analytical tools. The proposed example connects the two-dimensional maps of the real plane to the well-known complex map.

This paper concerns the description of some properties of p-dimensional invertible real maps T_b, turning into a (p-1)-dimensional noninvertible 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.

Many applications give rise to systems that can be described by maps that do not have a unique inverse. We consider here the case of a planar noninvertible map. Such a map folds the phase plane, so that there are regions with different numbers of preimages. The locus, where the number of preimages changes, is made up of so-called critical curves, that are defined as the images of the locus where the Jacobian is singular. A typical critical curve corresponds to a fold under the map, so that the number of preimages changes by two.

We consider the question of how the stable set of a hyperbolic saddle of a planar noninvertible map changes when a parameter is varied. The stable set is the generalization of the stable manifold for the case of an invertible map. Owing to the changing number of preimages, the stable set of a noninvertible map may consist of finitely or even infinitely many disjoint branches. It is now possible to compute stable sets with the Search Circle algorithm that we developed recently.

We take a bifurcation theory point of view and consider the two basic codimension-one interactions of the stable set with a critical curve, which we call the outer-fold and the inner-fold bifurcations. By taking into account how the stable set is organized globally, these two bifurcations allow one to classify the different possible changes to the structure of a basin of attraction that are reported in the literature. The fundamental difference between the stable set and the unstable manifold is discussed. The results are motivated and illustrated with a single example of a two-parameter family of planar noninvertible maps.

One of the most complex problems of nonlinear dynamics is the investigation of collective dynamics of ensembles consisting of a large number of coupled elements. A number of investigations were carried out for linear coupling of diffusion (difference) type. Recently, based on successful investigations of neurobiological objects [Abarbanel et al., 1996], an interest has grown in the study of dynamics of ensembles with nonlinear, nondiffusion type of coupling. In this paper we address pattern formation in a chain of bistable elements so coupled [Kuznetsov & Shalfeev, 2000].

Properties of the different configurations of Julia sets J, generated by the complex map T_Z: z′ = z² - c, are revisited when c is a real parameter, -1/4 < c < 2. This is done from a detailed knowledge of the fractal bifurcation organization "box-within-a-box", related to the real Myrberg's map T: x′ = x² - λ, first described in 1975. Part I of this paper constitutes a first step, leading to Part II dealing with an embedding of T_Z into the two-dimensional noninvertible map . For γ = 0, is semiconjugate to T_Z in the invariant half-plane (y ≤ 0). With a given value of c, and with γ decreasing, the identification of the global bifurcations sequence when γ → 0, permits to explain a route toward the Julia sets. With respect to other papers published on the basic Julia and Fatou sets, Part I consists in the identification of J singularities (the unstable cycles and their limit sets) with their localization on J. This identification is made from the symbolism associated with the "box-within-a-box" organization, symbolism associated with the unstable cycles of J for a given c-value. In this framework, Part I gives the structural properties of the Julia set of T_Z, which are useful to understand some bifurcation sequences in the more general case considered in Part II. Different types of Julia sets are identified.

Part I of this paper has been devoted to properties of the different Julia set configurations, generated by the complex map T_Z: z′ = z² - c, c being a real parameter, -1/4 < c < 2. These properties were revisited from a detailed knowledge of the fractal organization (called "box-within-a-box"), generated by the map x′ = x² - c with x a real variable. Here, the second part deals with an embedding of T_Z into the two-dimensional noninvertible map ; y′ = γ y + 4x²y, γ ≥ 0. For is semiconjugate to T_Z in the invariant half plane (y ≤ 0). With a given value of c, and with γ decreasing, the identification of the global bifurcations sequence when γ → 0, permits to explain a route toward the Julia sets, from a study of the basin boundary of the attractor located on y = 0.

In this work, we consider a family of Lotka–Volterra maps $(x^{\prime },y^{\prime })=(x(a-x-y),bxy)$ for $a>1$ and $b>0$ which unfold a map originally proposed by Sharkosky for $a=4$ and $b=1$. Multistability is observed, and attractors may exist not only in the positive quadrant of the plane, but also in the region $y<0$. Some properties and bifurcations are described. The $x$-axis is invariant, on which the map reduces to the logistic. For any $a>1$ an interval of values for $b$ exists for which all the cycles on the $x$-axis are transversely attracting. This invariant set is the source of several kinds of bifurcations. Riddling bifurcations lead to attractors in Milnor sense, not topological but with a stable set of positive measure, which may be the unique attracting set, or coexisting with other topological attractors. The riddling and blowout bifurcations are described related to chaotic intervals on the invariant set, and these global bifurcations have different dynamic results. Chaotic intervals which are not topological attractors may have all the cycles transversely attracting and as Milnor attractors. We show that Milnor attractors may also be related to attracting cycles on the $x$-axis at the bifurcation associated with the transverse and parallel eigenvalues. We show particular examples related to topological attractors with very narrow basins of attraction, when the majority of the trajectories are divergent.

Saddle fixed points are the centerpieces of complicated dynamics in a system. The one-dimensional stable and unstable manifolds of these saddle-points are crucial to understanding the dynamics of such systems. While the problem of sketching the unstable manifold is simple, plotting the stable manifold is not as easy. Several algorithms exist to compute the stable manifold of saddle-points, but they have their limitations, especially when the system is not invertible. In this paper, we present a new algorithm to compute the stable manifold of two-dimensional systems which can also be used for noninvertible systems. After outlining the logic of the algorithm, we demonstrate the output of the algorithm on several examples.

This text identifies some fractal sets generated by noninvertible maps. Two topics are presented: the fractal “box-within-a-box” bifurcations structure (described in 1975), produced by a dim 1 unimodal map, and the fractalization of basin boundaries generated by plane noninvertible maps. Some bifurcations of fractalization are described, depending on the basin boundary situation with respect to the critical curve (locus of points having two coincident preimages), and to the nature of strange repellers.

Many applications give rise to systems that can be described by maps that do not have a unique inverse. We consider here the case of a planar noninvertible map. Such a map folds the phase plane, so that there are regions with different numbers of preimages. The locus, where the number of preimages changes, is made up of so-called critical curves, that are defined as the images of the locus where the Jacobian is singular. A typical critical curve corresponds to a fold under the map, so that the number of preimages changes by two.

We consider the question of how the stable set of a hyperbolic saddle of a planar noninvertible map changes when a parameter is varied. The stable set is the generalization of the stable manifold for the case of an invertible map. Owing to the changing number of preimages, the stable set of a noninvertible map may consist of finitely or even infinitely many disjoint branches. It is now possible to compute stable sets with the Search Circle algorithm that we developed recently.

We take a bifurcation theory point of view and consider the two basic codimension-one interactions of the stable set with a critical curve, which we call the outer-fold and the inner-fold bifurcations. By taking into account how the stable set is organized globally, these two bifurcations allow one to classify the different possible changes to the structure of a basin of attraction that are reported in the literature. The fundamental difference between the stable set and the unstable manifold is discussed. The results are motivated and illustrated with a single example of a two-parameter family of planar noninvertible maps.