Two-dimensional Piecewise Linear Map Research Articles

Bifurcation Sequences and Multistability in a Two-Dimensional Piecewise Linear Map

Bifurcation mechanisms in piecewise linear or piecewise smooth maps are quite different with respect to those occurring in smooth maps, due to the role played by the borders. In this work, we describe bifurcation mechanisms associated with the appearance/disappearance of cycles, which may be related to several cases: (A) fold border collision bifurcations, (B) degenerate flip bifurcations, supercritical and subcritical, (C) degenerate transcritical bifurcations and (D) supercritical center bifurcations. Each of these is characterized by a particular dynamic behavior, and may be related to attracting or repelling cycles. We consider different bifurcation routes, showing the interplay between all these kinds of bifurcations, and their role in the phase plane in determining attracting sets and basins of attraction.

Dynamics in the transition case invertible/non-invertible in a 2D Piecewise Linear Map

• We investigate the dynamics of a two-dimensional piecewise linear map at the transition invertible/noninvertible. • We show how complex the dynamics may be also in such degenerate cases (with multistability and chaotic dynamics). • We show how the use of the first return map in a suitable segment may help in some cases. • We prove a regime in which a 3-cycle attracts almost all the points: the whole plane except for three curves. We consider the dynamics of a family of two-dimensional piecewise linear maps at the transition between the invertible and non-invertible cases. This leads to a degeneracy consisting of a half plane which is mapped onto a straight line, the critical line LC. In these regimes the ω -limit set of the trajectories must be on the images of some segment of LC. Thus, the first return map along this line can help in defining the global dynamic behavior of the two-dimensional map. In other cases, the first return map helps in determining some attracting sets, although not the unique attractors of the two-dimensional map.

Renormalization of two-dimensional piecewise linear maps: Abundance of 2-D strange attractors

For a two parameter family of two-dimensional piecewise linear maps and for every natural number \begin{document} $n$ \end{document} , we prove not only the existence of intervals of parameters for which the respective maps are n times renormalizable but also we show the existence of intervals of parameters where the coexistence of at least \begin{document} $2^n$ \end{document} strange attractors takes place. This family of maps contains the two-dimensional extension of the classical one-dimensional family of tent maps.

Hierarchical structure in sharply divided phase space for the piecewise linear map.

We have studied a two-dimensional piecewise linear map to examine how the hierarchical structure of stable regions affects the slow dynamics in Hamiltonian systems. In the phase space there are infinitely many stable regions, each of which is polygonal-shaped, and the rest is occupied by chaotic orbits. By using symbolic representation of stable regions, a procedure to compute the edges of the polygons is presented. The stable regions are hierarchically distributed in phase space and the edges of the stable regions show the marginal instability. The cumulative distribution of the recurrence time obeys a power law as ∼t^{-2}, the same as the one for the system with phase space, which is composed of a single stable region and chaotic components. By studying the symbol sequence of recurrence trajectories, we show that the hierarchical structure of stable regions has no significant effect on the power-law exponent and that only the marginal instability on the boundary of stable regions is responsible for determining the exponent. We also discuss the relevance of the hierarchical structure to those in more generic chaotic systems.

Border collision bifurcations and chaotic sets in a two-dimensional piecewise linear map

A two-dimensional piecewise linear continuous model is analyzed. It reflects the dynamics occurring in a circuit proposed as chaos generator, in a simplified case. The parameter space is investigated in order to classify completely regions of existence of stable cycles, and regions associated with chaotic behaviors. The border collision bifurcation curves are analytically detected, as well as the degenerate flip bifurcations of k-cycles and the homoclinic bifurcations occurring in cyclic chaotic regions leading to chaos in one-piece.

Snap-back repellers and chaotic attractors

When homoclinic orbits to an expanding periodic point exist, the point is called a snap-back repeller. Here, we consider the two-dimensional piecewise-linear map in canonical form, continuous and discontinuous, showing how snap-back repellers may be associated with robust chaotic attracting sets (not only with chaotic repellers). Examples are given both for the continuous and discontinuous maps.

Accumulation of unstable periodic orbits and the stickiness in the two-dimensional piecewise linear map

We investigate the stickiness of the two-dimensional piecewise linear map with a family of marginal unstable periodic orbits (FMUPOs), and show that a series of unstable periodic orbits accumulating to FMUPOs plays a significant role to give rise to the power law correlation of trajectories. We can explicitly specify the sticky zone in which unstable periodic orbits whose stability increases algebraically exist, and find that there exists a hierarchy in accumulating periodic orbits. In particular, the periodic orbits with linearly increasing stability play the role of fundamental cycles as in the hyperbolic systems, which allows us to apply the method of cycle expansion. We also study the recurrence time distribution, especially discussing the position and size of the recurrence region. Following the definition adopted in one-dimensional maps, we show that the recurrence time distribution has an exponential part in the short time regime and an asymptotic power law part. The analysis on the crossover time T(c)(*) between these two regimes implies T(c)(*) approximately -log[micro(R)] where micro(R) denotes the area of the recurrence region.

CENTER BIFURCATION FOR TWO-DIMENSIONAL BORDER-COLLISION NORMAL FORM

In this work we study some properties associated with the border-collision bifurcations in a two-dimensional piecewise-linear map in canonical form, related to the case where a fixed point of one of the linear maps has complex eigenvalues and undergoes a center bifurcation when its eigenvalues pass through the unit circle. This problem is faced in several applied piecewise-smooth models, such as switching electrical circuits, impacting mechanical systems, business cycle models in economics, etc. We prove the existence of an invariant region in the phase space for parameter values related to the center bifurcation and explain the origin of a closed invariant attracting curve after the bifurcation. This problem is related also to particular border-collision bifurcations leading to such curves which may coexist with other attractors. We show how periodicity regions in the parameter space differ from Arnold tongues occurring in smooth models in case of the Neimark–Sacker bifurcation, how so-called dangerous border-collision bifurcations may occur, as well as multistability. We give also an example of a subcritical center bifurcation which may be considered as a piecewise-linear analogue of the subcritical Neimark–Sacker bifurcation.

Stability of fixed points placed on the border in the piecewise linear systems

In this paper we consider two-dimensional piecewise linear maps characterized by nondifferentiability on a curve in the phase space. According to the stability of the fixed point without having its Jacobian information, recently found dangerous border-collision bifurcations could happen. It is thus important to determine the stability of the nondifferential fixed point. We investigate the global behavior of trajectories near the fixed point, which can be characterized by the dynamics of a map defined on the unit circle with the assigned dilation ratios, and then introduce a novel method to determine the stability of nondifferential fixed points of piecewise linear systems. We also present a special bifurcation phenomenon exhibiting the unbounded behavior of orbits before and after the critical bifurcation value, but the stable fixed point at the critical bifurcation value, which is one of unexpected phenomena in smooth bifurcation theory.

Singularities in the fluctuation of on-off intermittency

For a two-dimensional piecewise linear map exhibiting on-off intermittency, the scaling property of fluctuation, i.e., the large deviation property is investigated. It is shown that there are three phases of fluctuation and the q-weighted average of an observed quantity has singularities such as jumps or a plateau due to transitions between the phases. At the onset of on-off intermittency, the width of the plateau vanishes due to the disappearance of one of the three phases and the singularity becomes weaker but more probable. The singularity at the onset of on-off intermittency is also examined on the coupled logistic map.

Multifractal structure of a riddled basin.

For a two-dimensional piecewise linear map with a riddled basin, a multifractal spectrum f(gamma), which characterizes the "skeletons" of the riddled basin, is introduced. With f(gamma), the uncertainty exponent is obtained by a variational principle, which enables us to introduce a concept of a "boundary" for the riddled basin. A conjecture on the relation between f(gamma) and the "stable sets" of various ergodic measures, which coexist with the natural invariant measure on the chaotic attractor, is also proposed. (c) 2001 American Institute of Physics.

Large deviation properties of on-off intermittency.

The large deviation property of on-off intermittency is investigated by introducing a two-dimensional piecewise linear map, which can be mapped to an infinite Markov chain. It is shown that nonanalyticity, in the q-weighted average of the portion of time spent in burst state, appears as a second-order phase transition for an interval of control parameter with the bifurcation point of on-off intermittency as its end point.

About Two Mechanisms of Reunion of Chaotic Attractors

Considering a family of two-dimensional piecewise linear maps, we discuss two different mechanisms of reunion of two (or more) pieces of cyclic chaotic attractors into a one-piece attracting set, observed in several models. It is shown that, in the case of so-called contact bifurcation of the 2 nd kind, the reunion occurs immediately due to homoclinic bifurcation of some saddle cycle belonging to the basin boundary of the attractor. In the case of so-called contact bifurcation of the 1 st kind, the reunion is a result of a contact of the attractor with its basin boundary which is fractal, including the stable set of a chaotic invariant hyperbolic set appeared after the homoclinic bifurcation of a saddle cycle on the basin boundary

Some global bifurcations of two-dimensional endomorphisms by use of critical lines

MANY STUDIES have been devoted to discrete dynamical systems represented by homeomorphisms and diffeomorphisms since the works by Smale [l] and Gavrilov and Shilnikov [2, 31, but comparatively few works have been devoted to endomorphisms, that is to maps with a nonunique inverse. However, the mathematical models derived from the description of evolutive phenomena in several applicative fields are often represented by discrete dynamical systems, or maps, with a nonunique inverse. This is particularly true in the socio-economic field [4-61, but examples can also be found in ecology [7, 81, biology [9-l l] and control systems [ 121. This work concerns the study of the global dynamics of recurrences represented by twodimensional endomorphisms of applicative interest in the economic context. Useful notions for the local and global analysis of nonlinear endomorphisms have been introduced by Gumowski and Mira [ll, 13, 141. For two-dimensional endomorphisms, in particular, this is the notion of critical lines (or critical curves), representing the twodimensional extension of the critical points in one-dimensional maps, and the notions of absorbing area and chaotic area (bounded by elements of critical lines), which determine phase plane domains where chaotic solutions are located. These notions, which will be recalled in the next section, together with some bifurcations involving the chaotic areas, are the basic concepts and properties that we shall use in the analysis of the subsequent sections. The aim of this paper is to illustrate by an example how powerful the critical lines may be in studying the global dynamics and bifurcations of plane nonlinear endomorphisms, particularly when they are used in combination with more traditional techniques. It is known that in diffeomorphisms, global bifurcations which qualitatively change the structure of a chaotic attractor, called interior crises or catastrophes by some authors [ 15-181, are characterized by the stable manifold of a saddle, approaching and touching the chaotic attractor. We shall find that this phenomenology also occurs in the endomorphisms of our model, with a particular role being played by the critical lines. In Section 3 we shall describe our mathematical model, a two-dimensional piecewise-linear map which we call map F, together with some of its elementary properties and the definition of three classes of endomorphisms into which the analysis can be conveniently subdivided. Endomorphisms of the first class have been the object of a previous work [ 191. Here we shall show, in Section 4, how several bifurcations involving chaotic areas can be explained by use of critical lines. The main result is the analytical determination of the bifurcation which discriminates between the global attractivity of an absorbing area and the nonglobal attractivity

Correlation functions and generalized Lyapunov exponents.

Correlation functions of one- and two-dimensional piecewise linear maps are analytically investigated. The asymptotic time behavior is shown to be given by the average inverse multiplier 〈${\ensuremath{\mu}}_{1}^{\mathrm{\ensuremath{-}}1}$(\ensuremath{\tau})〉, for one-dimensional maps with absolutely continuous invariant measure. The decay rate \ensuremath{\gamma} coincides with the generalized Lyapunov exponent \ensuremath{\Lambda}(scrq) at scrq=2, if the sign of the multiplier does not change during the time evolution, while, in general, it is larger than \ensuremath{\Lambda}(2). The analysis of two-dimensional maps reveals the importance of the average second multiplier 〈${\ensuremath{\mu}}_{2}$(\ensuremath{\tau})〉 and of the average ratio 〈${\ensuremath{\mu}}_{2}$(\ensuremath{\tau})/${\ensuremath{\mu}}_{1}$(\ensuremath{\tau})〉 which, in some cases, can provide the leading long-time contribution.