Iryna Sushko

DEGENERATE BIFURCATIONS AND BORDER COLLISIONS IN PIECEWISE SMOOTH 1D AND 2D MAPS

We recall three well-known theorems related to the simplest codimension-one bifurcations occurring in discrete time dynamical systems, such as the fold, flip and Neimark–Sacker bifurcations, and analyze these bifurcations in presence of certain degeneracy conditions, when the above mentioned theorems are not applied. The occurrence of such degenerate bifurcations is particularly important in piecewise smooth maps, for which it is not possible to specify in general the result of the bifurcation, as it strongly depends on the global properties of the map. In fact, the degenerate bifurcations mainly occur in piecewise smooth maps defined in some subspace of the phase space by a linear or linear-fractional function, although not necessarily only by such functions. We also discuss the relation between degenerate bifurcations and border-collision bifurcations.

Growing through chaotic intervals

We consider a growth model proposed by Matsuyama [K. Matsuyama, Growing through cycles, Econometrica 67 (2) (1999) 335-347] in which two sources of economic growth are present: the mechanism of capital accumulation (Solow regime) and the process of technical change and innovations (Romer regime). We will shown that no stable cycle can exist, except for a fixed point and a cycle of period two. The Necessary and Sufficient conditions for regular or chaotic regimes are formulated. The bifurcation structure of the two-dimensional parameter plane is completely explained. It is shown how the border-collision bifurcation leads from the stable fixed point to pure chaotic regime (which consists either in 4-cyclical chaotic intervals, 2-cyclical chaotic intervals or in one chaotic interval).

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.

Bifurcations of Chaotic Attractors in One-Dimensional Piecewise Smooth Maps

In this work, we classify the bifurcations of chaotic attractors in 1D piecewise smooth maps from the point of view of underlying homoclinic bifurcations of repelling cycles which are located before the bifurcation at the boundary of the immediate basin of the chaotic attractor.

BIFURCATION STRUCTURES IN A BIMODAL PIECEWISE LINEAR MAP: REGULAR DYNAMICS

In this work, we investigate the bifurcation structure of the parameter space of a generic 1D continuous piecewise linear bimodal map focusing on the regions associated with chaotic attractors (cyclic chaotic intervals). The boundaries of these regions corresponding to chaotic attractors with different number of intervals are identified. The results are obtained analytically using the skew tent map and the map replacement technique.

Typical bifurcation scenario in a three region identical New Economic Geography model

We study global dynamics of the New Economic Geography model which describes spatial distribution of industrial activity in the long run across three identical regions depending on the balancing of agglomeration and dispersion forces. It is defined by a two-dimensional piecewise smooth map depending on four parameters. Based on the numerical evidence we discuss typical bifurcation scenarios observed in the model: starting from the symmetric fixed point (related to equal distribution of the industrial activity in all the three regions) two different scenarios are realized depending on whether the transportation cost parameter is increased or decreased. Emergence of the Wada basins of coexisting attractors leading to the so-called final state sensitivity is discussed, as well as final bifurcation of the chaotic attractor.

Border collision bifurcations of superstable cycles in a one-dimensional piecewise smooth map

We study the dynamics of a one-dimensional piecewise smooth map defined by constant and logistic functions. This map has qualitatively the same dynamics as the one defined by constant and unimodal functions, coming from an economic application. Namely, it contributes to the investigation of a model of the evolution of corruption in public procurement proposed by Brianzoni et al. [4]. Bifurcation structure of the economically meaningful part of the parameter space is described, in particular, the fold and flip border-collision bifurcation curves of the superstable cycles are obtained. We show also how these bifurcations are related to the well-known saddle-node and period-doubling bifurcations.

A Gallery of Bifurcation Scenarios in Piecewise Smooth 1D Maps

We give a brief overview of several bifurcation scenarios occurring in 1D piecewise monotone maps defined on two partitions, continuous or discontinuous. A collection of some basic blocks is proposed, which may be observed in particular bifurcation sequences of a system of interest both in regular and chaotic parameter domains.

Original article: Border collision bifurcations in one-dimensional linear-hyperbolic maps

Abstract: In this paper we consider a continuous one-dimensional map, which is linear on one side of a generic kink point and hyperbolic on the other side. This kind of map is widely used in the applied context. Due to the simple expression of the two functions involved, in particular cases it is possible to determine analytically the border collision bifurcation curves that characterize the dynamic behaviors of the model. In the more general model we show that the steps to be performed are the same, although the analytical expressions are not given in explicit form.

Center Bifurcation for a Two-Dimensional Piecewise Linear Map

It is already well known that the main bifurcation scenario which can be realized considering a business cycle model in dynamic context, is related to a fixed point losing stability with a pair of complex-conjugate eigenvalues. In the case in which such a model is discrete and defined by some smooth nonlinear fianctions, the Neimark-Sacker bifiarcation theorem can be used, described in the previous chapter. While for piecewise linear, or piecewise smooth, functions which are also quite often used for business cycle modeling, the bifurcation theory is much less developed. The purpose of this chapter is to describe a so-called center bifurcation occurring in a family of two-dimensional piecewise linear maps whose dynamic properties are, to our knowledge, not well known. Namely, we shall see that in some similarity to the Neimark-Sacker bifurcation occurring for smooth maps, for piecewise linear maps the bifurcation of stability loss of a fixed point with a pair of complex-conjugate eigenvalues on the unit circle can also result in the appearance of a closed invariant attracting curve homeomorphic to a circle. However, differently from what occurs in the smooth case, the closed invariant curve is not a smooth, but a piecewise linear set, appearing not in a neighborhood of the fixed point, as it may be very far from it. In fact, we shall see that the position of the closed invariant curve depends on the distance of the fixed point from the boundary of the region in which the linear map is defined (i.e., from what we shall call critical line LC-i),