Two-dimensional Noninvertible Map Research Articles

Complex dynamics of the simplest neuron model: Singular chaotic Shilnikov attractor as specific oscillatory neuron activity

We study complex dynamics of the Chialvo model that is the simplest neuron-type model in form of a four-parameter family of two-dimensional noninvertible maps (endomorphisms). Main elements of bifurcation diagram in the plane of two parameters have been constructed in which regions corresponding to both quasi-periodic and chaotic oscillations are selected. We also indicate special regions corresponding to singular discrete chaotic Shilnikov attractors that we consider as a new type of the so-called snap-back repellers (over an unstable focus). The study of time series was carried out in which there were classified patterns of specific oscillatory activities in the cases when homoclinic orbits to the unstable focus exists and, when such orbits were not yet formed but a strange attractor already exists. New dynamical characteristics are proposed, with the help of which it is possible to assess the level of distinctness of atypical oscillatory activity.

Border collision bifurcations in a piecewise linear duopoly model

We consider a duopoly model characterized by a two-dimensional non-invertible continuous map T given by piecewise linear functions, with several partitions, defining a duopoly game. The structure of the game is such that it has separate second iterate so that its dynamics can be studied via a one-dimensional composite function, that is piecewise linear with multiple partitions in which the definition of the map changes. The number of partitions may change from 2 to 5, depending on the parameters. The dynamics are characterized by degenerate bifurcations and border collision bifurcations, which are typical in maps having kink points. Here the peculiarity is the multiplicity of the partitions, which leads to bifurcations different from those occurring in maps with only one kink point. We show several bifurcations, coexistence of cycles, attracting and superstable, as well chaotic attractors and chaotic repellors, related to the outcome of particular border collision bifurcations.

Codimension-one and -two bifurcation analysis of a two-dimensional coupled logistic map

This paper devotes to a detailed bifurcation analysis of a two-dimensional non-invertible map, obtained using a symmetric coupling between one-dimensional logistic maps. The critical normal form coefficients method is employed to detect bifurcations and to explore further critical conditions without explicit reduction to the center manifold. The results show that the two-dimensional map undergoes codimension-one (codim-1) bifurcations such as transcritical, pitchfork, period-doubling, Neimark–Sacker, and codim-2 bifurcations including transcritical-flip, pitchfork-flip, strong resonances 1:2, 1:3, 1:4. For each bifurcation, the critical normal form coefficients are calculated to check the non-degeneracy conditions and predict the bifurcation scenarios around the bifurcation points. To validate the theoretical results, all bifurcation curves of fixed points are plotted with the aid of the numerical continuation method. Weak resonances are also specified by the isoclines on the bi-parameter plane. The results will help in understanding the occurrence and the structure of bifurcation cascades observed in many coupled discrete systems.

The Lorenz model in discrete time

A two-dimensional noninvertible map proposed by Lorenz in 1989, depending on two parameters a and τ, is reconsidered. We show the two different bifurcation scenarios occurring for a>0 and a<0. Two particular degenerate cases are investigated, at and describing the related bifurcations as a function of the parameter a>0. For a straight line filled with 2-cycles and the occurrence of a resonant case with rotation number in the Neimark–Sacker bifurcation of two fixed points determine particular bifurcations, associated with the existence of four invariant sets of the map. For the critical curve degenerates into one point, the origin, which is also fixed for map T and the focal point for the inverse map, leading to chaotic attracting sets with infinitely many lobes issuing from the origin. The transition to chaos from attracting closed curves is also analysed, evidencing the occurrence of homoclinic tangles.

Dynamics of a minimal consumer network with bi-directional influence

We study the dynamics of a model of interdependent consumer behavior defined by a family of two-dimensional noninvertible maps. This family belongs to a class of coupled logistic maps with different nonlinearity parameters and coupling terms that depend on one variable only. In our companion paper we considered the case of independent consumers as well as the case of uni-directionally connected consumers. The present paper aims at describing the dynamics in the case of a bi-directional connection. In particular, we investigate the bifurcation structure of the parameter plane associated with the strength of coupling between the consumers, focusing on the mechanisms of qualitative transformations of coexisting attractors and their basins of attraction.

From wild Lorenz-like to wild Rovella-like dynamics

We consider a two-dimensional noninvertible map that was introduced by Bamón, Kiwi and Rivera-Letelier as a model of a wild Lorenz-like attractor in a vector field of dimension at least five; such an attractor contains an expanding equilibrium and a hyperbolic set with robust homoclinic tangencies. Advanced numerical techniques enable us to study how the stable, unstable and critical sets of the map change within the conjectured region of wild chaos in the transition from Lorenz-like to Rovella-like dynamics, that is, when the equilibrium of the vector field becomes contracting. We find numerical evidence for the existence of wild Rovella-like attractors, wild Rovella-like saddles and regions of multistability, where a Rovella-like attractor coexists with two fixed-point attractors. We identify bifurcations generating these different types of dynamics and compute them in two-parameter bifurcation diagrams.

Interactions of the Julia Set with Critical and (Un)Stable Sets in an Angle-Doubling Map on ℂ\{0}

We study a nonanalytic perturbation of the complex quadratic family z ↦ z2 + c in the form of a two-dimensional noninvertible map that has been introduced by Bamón et al. [2006]. The map acts on the plane by opening up the critical point to a disk and wrapping the plane twice around it; points inside the disk have no preimages. The bounding critical circle and its images, together with the critical point and its preimages, form the so-called critical set. For parameters away from the complex quadratic family we define a generalized notion of the Julia set as the basin boundary of infinity. We are interested in how the Julia set changes when saddle points along with their stable and unstable sets appear as the perturbation is switched on. Advanced numerical techniques enable us to study the interactions of the Julia set with the critical set and the (un)stable sets of saddle points. We find the appearance and disappearance of chaotic attractors and dramatic changes in the topology of the Julia set; these bifurcations lead to three complicated types of Julia sets that are given by the closure of stable sets of saddle points of the map, namely, a Cantor bouquet and what we call a Cantor tangle and a Cantor cheese. We are able to illustrate how bifurcations of the nonanalytic map connect to those of the complex quadratic family by computing two-parameter bifurcation diagrams that reveal a self-similar bifurcation structure near the period-doubling route to chaos in the complex quadratic family.

FROM THE BOX-WITHIN-A-BOX BIFURCATION STRUCTURE TO THE JULIA SET PART II: BIFURCATION ROUTES TO DIFFERENT JULIA SETS FROM AN INDIRECT EMBEDDING OF A QUADRATIC COMPLEX MAP

Part I of this paper has been devoted to properties of the different Julia set configurations, generated by the complex map TZ: z′ = z2 - c, c being a real parameter, -1/4 &lt; c &lt; 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 TZ into the two-dimensional noninvertible map [Formula: see text]; y′ = γ y + 4x2y, γ ≥ 0. For [Formula: see text] is semiconjugate to TZ 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.

Novel routes to chaos through torus breakdown in non-invertible maps

The paper describes a number of new scenarios for the transition to chaos through the formation and destruction of multilayered tori in non-invertible maps. By means of detailed, numerically calculated phase portraits we first describe how three- and five-layered tori arise through period-doubling and/or pitchfork bifurcations of the saddle cycle on an ordinary resonance torus. We then describe several different mechanisms for the destruction of five-layered tori in a system of two linearly coupled logistic maps. One of these scenarios involves the destruction of the two intermediate layers of the five-layered torus through the transformation of two unstable node cycles into unstable focus cycles, followed by a saddle-node bifurcation that destroys the middle layer and a pair of simultaneous homoclinic bifurcations that produce two invariant closed curves with quasiperiodic dynamics along the sides of the chaotic set. Other scenarios involve different combinations of local and global bifurcations, including bifurcations that lead to various forms of homoclinic and heteroclinic tangles. We finally demonstrate that essentially the same scenarios can be observed both for a system of nonlinearly coupled logistic maps and for a couple of two-dimensional non-invertible maps that have previously been used to study the properties of invariant sets.

FROM THE BOX-WITHIN-A-BOX BIFURCATION ORGANIZATION TO THE JULIA SET PART I: REVISITED PROPERTIES OF THE SETS GENERATED BY A QUADRATIC COMPLEX MAP WITH A REAL PARAMETER

Properties of the different configurations of Julia sets J, generated by the complex map TZ: z′ = z2 - c, are revisited when c is a real parameter, -1/4 &lt; c &lt; 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 TZ into the two-dimensional noninvertible map [Formula: see text]. For γ = 0, [Formula: see text] is semiconjugate to TZ 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 TZ, which are useful to understand some bifurcation sequences in the more general case considered in Part II. Different types of Julia sets are identified.

Complex dynamics in a two-dimensional noninvertible map

A two-dimensional noninvertible map is investigated. The conditions of existence for pitchfork bifurcation, flip bifurcation and Naimark–Sacker bifurcation are derived by using center manifold theorem and bifurcation theory. Chaotic behavior in the sense of Marotto’s definition of chaos is proven. And numerical simulations not only show the consistence with the theoretical analysis but also exhibit the complex dynamical behaviors, including period-34, period-5 orbits, quasi-period orbits, intermittency, boundary crisis as well as chaotic transient. The computation of Lyapunov exponents conforms the dynamical behaviors.

A method to construct irregular fractal curves

In this paper, we present a method of constructing irregular fractal curves. This method is based on a sequence ( Φ [ n ] ) n ∈ N of piecewise differentiable maps defined succinctly in Narayaninsamy [T. Narayaninsamy, On basin boundaries, Appl. Math. Comput. 99 (1999) 261–274]. The experimental results concerning the basins of attraction of attractors which are given in Gardini, Mira et al. and Narayaninsamy [L. Gardini, Some global bifurcations of two-dimensional endomorphisms by use of critical lines, Nonlinear Anal. Theory Meth. Appl. 18(4) (1991) 361–399; C, Mira, D. Fournier-Prunaret, L. Gardini, H. Kawakami, J.-C. Cathala, Basin bifurcations of two-dimensional noninvertible maps: fractalization of basins, J. Bifurcat. Chaos 2 (1994) 343–381; T. Narayaninsamy, Contribution à l’Etude de l’Itération Fractionaire et à Celle des Endomorphismes Bi-Dimensionnels, Doctorat Mathématiques Appliquées de l’Université Paul Sabatier (Toulouse III)] constitute the basic tool of this sequence.

Crisis induced by an escape from a fat strange set

This article reports a characteristic crisis observed in a two-dimensional discontinuous and noninvertible map. The discontinuity border in the definition range of the mapping oscillates as the discrete time progresses so that the forward images of the border form a fat fractal. By choosing particular parameters the iterations on the fat fractal display chaotic motion, and the transient iterations from the initial values in a certain region of the phase space are attracted to the fat fractal. At a threshold of a control parameter an elliptic periodic orbit and the elliptic islands around it suddenly appear inside the fat strange set so that the iterations on the set escape to the islands. The fat chaotic attractor thus suddenly transfers to a fat transient set. The effect of the feature of the crisis on the rule of the lifetime in the transient set is discussed. It shows that the dependence of the lifetime on the control parameter follows a universal scaling law suggested by Grebogy, Ott, and Yorke [Phys. Rev. Lett. 57, 1284 (1986)], and the scaling exponent can be approximated according to the variation rules of the elliptic islands and the measure of the fat fractal. The strange repeller, which appears after the crisis, is also a fat fractal.

A Simple Hyperchaos Generator Based on Impulsive Switching

This paper studies a simple hyperchaos generator. The circuit consists of three capacitors, three voltage-controlled current sources, and one state-dependent impulsive switch. The circuit dynamics are described by a normal form equation and a two-dimensional noninvertible return map is derived. The return map is calculated precisely using the exact piecewise solution and hyperchaos generation is guaranteed by two positive Lyapunov exponents of the return map. Using a simple test circuit, a typical hyperchaotic attractor can be verified in the laboratory.

On the Fractal Structure of Basin Boundaries in Two-Dimensional Noninvertible Maps

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.

RIDDLED BASINS AND UNSTABLE DIMENSION VARIABILITY IN CHAOTIC SYSTEMS WITH AND WITHOUT SYMMETRY

Riddling occurs in dissipative dynamical systems with more than one attractor, when the basin of one attractor is punctured with holes belonging to the basins of the other attractors. The basin of a chaotic attractor is riddled if (i) it has a positive Lebesgue measure; (ii) in the vicinity of every point belonging to the basin of the attractor, there is a positive Lebesgue measure set of points that asymptote to another attractor. We investigate the presence of riddled basins in a two-dimensional noninvertible map with a symmetry-breaking term. In the symmetric case the onset of riddling is characterized by an unstable–unstable pair bifurcation, which also leads to unstable dimension variability in the invariant chaotic set. The nonsymmetric case exhibits a chaotic attractor, but a riddled basin occurs only at the bifurcation point, since after that the attractor becomes a chaotic saddle. We analyze the presence of unstable dimension variability in the symmetric case by computing the finite-time transverse Lyapunov exponents. We point out some consequences of those facts to the synchronization properties of coupled chaotic systems.

Universality and scaling for the breakup of phase synchronization at the onset of chaos in a periodically driven Rössler oscillator.

Universal behavior discovered earlier in two-dimensional noninvertible maps is found numerically in a periodically driven Rössler system. The critical behavior is associated with the limit of a period-doubling cascade at the edge of the Arnold tongue, and may be reached by variation of two control parameters. The corresponding scaling regularities, distinct from those of the Feigenbaum cascade, are demonstrated. Presence of a critical quasiattractor, an infinite set of stable periodic orbits of quadrupled periods, is outlined. As argued, this type of critical behavior may occur in a wide class of periodically driven period-doubling systems.

Global dynamics in a non-linear model of the equity ratio

A model for firms' financial conditions is proposed, which ultimately reduces to a two-dimensional non-invertible map in the variables mean and variance of the equity ratio. The possible dynamics of the model and the global behaviour are investigated. We describe the mechanism of bifurcations leading to fractalization of the basins and/or fractalization of their boundaries, showing how a locally stable attractor may be almost globally unstable. Multistability is also investigated. Two, three or four co-existing attractors have been found and we describe the mechanism of bifurcations leading their basins to become chaotically intermingled, and thus to unpredictability of the asymptotic state in a wide region. The knowledge of such regimes, besides those associated with simple dynamics, may be of help for the operators. While the use of the technical tools we propose to study the global dynamics and bifurcations may be of help for further investigations.

Global properties of symmetric competition models with riddling and blowout phenomena

In this paper the problem of chaos synchronization, and the related phenomena of riddling, blowout and on–off intermittency, are considered for discrete time competition models with identical competitors. The global properties which determine the different effects of riddling and blowout bifurcations are studied by the method of critical curves, a tool for the study of the global dynamical properties of two‐dimensional noninvertible maps. These techniques are applied to the study of a dynamic market‐share competition model.

Chaos and fractal properties induced by noninvertibility of models in the form of maps

Many classes of discrete dynamical systems give rise to models in the form of noninvertible maps. With respect to invertible maps, noninvertible maps introduce a singularity of a different nature: the critical set of rank-one, as the geometrical locus of points having at least two coincident preimages. Such new singularities play a fundamental role in the definition of attractors, basins and their bifurcations. The purpose of this paper is a survey of some fundamental results related to two-dimensional noninvertible maps leading to specific chaotic behaviors, as fractal sets, characterizing irreversibility properties of a class of discrete systems.