Border Collision Bifurcation Curves Research Articles

Bifurcation structures of a two-dimensional piecewise linear discontinuous map: analysis of a cobweb model with regime-switching expectations

AbstractWe consider the bifurcations occurring in a two-dimensional piecewise-linear discontinuous map that describes the dynamics of a cobweb model in which firms rely on a regime-switching expectation rule. In three different partitions of the phase plane, separated by two discontinuity lines, the map is defined by linear functions with the same Jacobian matrix, having two real eigenvalues, one of which is negative and one equal to 0. This leads to asymptotic dynamics that can belong to two or three critical lines. We show that when the basic fixed point is attracting, it may coexist with at most three attracting cycles. We have determined their existence regions, in the two-dimensional parameter plane, bounded by border collision bifurcation curves. At parameter values for which the basic fixed point is repelling, chaotic attractors may exist - either one that is symmetric with respect to the basic fixed point, or, if not symmetric, the symmetric one also exists. The homoclinic bifurcations of repelling cycles leading to the merging of chaotic attractors are commented by using the first return map on a suitable line. Moreover, four different kinds of homoclinic bifurcations of a saddle 2-cycle, leading to divergence of the generic trajectory, are determined.

1d piecewise smooth map: exploring a model of investment dynamics under financial frictions with three types of investment projects

UDC 517.9 We consider a 1D continuous piecewise smooth map, which depends on seven parameters. Depending on the values of parameters, it may have up to six branches. This map was proposed by Matsuyama [Theor. Econ., <strong>8</strong>, 623–651 (2013); Section 5]. It describes the macroeconomic dynamics of investment and credit fluctuations in which three types of investment projects compete in the financial market. We introduce a partitioning of the parameter space according to different branch configurations of the map and illustrate this partitioning for a specific parameter setting. Then we present an example of the bifurcation structure in a parameter plane, which includes periodicity regions related to superstable cycles. Several bifurcation curves are obtained analytically, in particular, the border-collision bifurcation curves of fixed points. We show that the intersection point of two curves of this kind is an organizing center from which infinitely many other bifurcation curves are originated.

On the Price Dynamics of a Two-Dimensional Financial Market Model with Entry Levels

This paper aims to extend the model developed by Tramontana et al. By adding trend followers who pay attention to the most recent observed price trend, we formulate a financial market model driven by a new two-dimensional discontinuous piecewise linear (PWL) map with three branches. The dynamic behavior of the mapping system is studied in two cases according to different trend followers’ expectation of the stock price. The existence and stability conditions of periodic attractors and other bounded attractors are derived by using qualitative and quantitative methods, theoretical analysis, and numerical simulation. When trend followers are neutral on the stock market, we present that the basin of locally attracting fixed points can be determined by the preimages of two borderlines. We also prove that one of its surprising features is that model simulations may appear to be chaotic, although only regular dynamics can emerge. When trend followers are bullish or bearish on the stock market, we present the existence conditions of attracting coexistence fixed point, globally and locally attracting fixed point, and periodic and other bounded attractors. The transversal homoclinic theory of flip saddle periodic point is applied to prove the existence of chaotic attractor. We also give the calculation methods of border collision bifurcation (BCB) curves. This paper advances our knowledge of discontinuous PWL systems and reveals the endogenous evolution of bubbles and crashes and excessive volatility in financial markets from a new perspective with new methods.

Chaos, border collisions and stylized empirical facts in an asset pricing model with heterogeneous agents

An asset pricing model with chartists, fundamentalists and trend followers is considered. A market maker adjusts the asset price in the direction of the excess demand at the end of each trading session. An exogenously given fundamental price discriminates between a bull market and a bear market. The buying and selling orders of traders change moving from a bull market to a bear market. Their asymmetric propensity to trade leads to a discontinuity in the model, with its deterministic skeleton given by a two-dimensional piecewise linear dynamical system in discrete time. Multiple attractors, such as a stable fixed point and one or more attracting cycles or cycles and chaotic attractors, appear through border collision bifurcations. The multi-stability regions are underlined by means of two-dimensional bifurcation diagrams, where the border collision bifurcation curves are detected in analytic form at least for basic cycles with symbolic sequences $${\hbox {LR}}^{n}$$ and $${\hbox {RL}}^{n}$$ . A statistical analysis of the simulated time series of the asset returns, generated by perturbing the deterministic dynamics with a random walk process, indicates that this is one of the simplest asset pricing models which are able to replicate stylized empirical facts, such as excess volatility, fat tails and volatility clustering.

BCB Curves and Contact Bifurcations in Piecewise Linear Discontinuous Map Arising in a Financial Market

In this paper, we further study a financial market model established in our earlier paper. The model dynamics is driven by a two-dimensional piecewise linear discontinuous map, which is investigated analytically and numerically for one-sided fixed points being flip saddle and two-sided fixed points being attractors. The existence of chaotic orbit is explained by using the theory of homoclinic intersection between stable and unstable manifolds of the flip saddle invariant set. The structure of chaotic attractor is disclosed. It consists of finite segments rooted on both sides of the [Formula: see text]-axis which are unstable manifolds of flip saddle invariant set. The basins and their structural changes of bounded attractors and coexisting attractors are presented by contact bifurcation theory and numerical simulations. The border collision bifurcation (BCB for short) curves are calculated and coexisting multiattractors are disclosed by overlapping periodicity regions. The results can deepen our understanding of financial markets and dynamical systems.

Period-adding bifurcations in a discontinuous system with a variable gap

The period-adding bifurcations in a discontinuous system with a variable gap are observed for two control parameters. Various period-adding bifurcations are found by simulations. The bifurcation diagram can be divided into two different zones: chaos and period. The period attractor takes up a considerable part of the parameter space, and all of them show stable period attractors. The periodic zone can also be divided into three different zones: stable period-5 attractor, period-adding bifurcations on the right side of period-5 attractor, and period-adding bifurcations on the right side of period-5 attractor. We choose various control parameters to plot the cobweb of period attractor, and find that it will exhibit a border-collision bifurcation and the period orbit loses its stability, once the position of iteration reaches discontinuous boundary. The discontinuous system has two kinds of border-collision bifurcations: one comes from the gap on the right side, and the other from the gap on the left side. The results show that the period-adding phenomena are due to the border-collision bifurcation at two boundaries of the forbidden area. In order to determine the condition of the period orbit existence, we also choose various control parameters to plot the cobweb of period attractor. The results show that the iteration sequence of period trajectory has a certain sequence with different iteration units. The period trajectory of period-adding bifurcation on the left side of period-5 attractor consists of period-4 and period-5 iteration units, forming period-9, period-13 and period-14 attractor. The period trajectory of period-adding bifurcation on the right side of period-5 attractor consists of period-6 and period-5 iteration units, forming period-11, period-16 and period-21 attractor. All attractors can be easily shown analytically, owing to the piecewise linear characteristics of the map. We analyze its underlying mechanisms from the viewpoint of border-collision bifurcations. The result shows that the period attractor can be determined by two border-collision bifurcations and the condition of stability. Based on the theoretical and iteration unit, the border-collision bifurcations, two border collision bifurcation curves are obtained analytically. The result shows that the theoretical and numerical results are in excellent agreement.

Bifurcation and dynamics of a normal form map

This paper investigates the dynamics and stability properties of a so-called planar truncated normal form map. This kind of map is widely used in the applied context, especially in normal form coefficients of n-dimensional maps. We determine analytically the border collision bifurcation curves that characterize the dynamic behaviors of the system. We first analyze stability of the fixed points and the existence of local bifurcations. Our analysis shows the presence of a rich variety of local bifurcations, namely stable fixed points, periodic cycles, quasiperiodic cycles that are constraints to stable attractors called invariant closed curves, and chaos, where dynamics of the system change erratically. Our study is based on the numerical continuation method under variation of 1 and 2 parameters and computation of different bifurcation curves of the system and its iterations. For the all codimension 1 and 2 bifurcation points, we compute the corresponding normal form coefficients to reveal the criticality of the corresponding bifurcations as well as to identify different bifurcation curves that emerge around the corresponding bifurcation point. We further perform numerical simulations to characterize qualitatively different dynamical behaviors within each regime of parameter space.

Codimension-2 Border Collision, Bifurcations in One-Dimensional, Discontinuous Piecewise Smooth Maps

We consider a two-parametric family of one-dimensional piecewise smooth maps with one discontinuity point. The bifurcation structures in a parameter plane of the map are investigated, related to codimension-2 bifurcation points defined by the intersections of two border collision bifurcation curves. We describe the case of the collision of two stable cycles of any period and any symbolic sequences. For this case, we prove that the local monotonicity of the functions constituting the first return map defined in a neighborhood of the border point at the parameter values related to the codimension-2 bifurcation point determines, under suitable conditions, the kind of bifurcation structure originating from this point; this can be either a period adding structure, or a period incrementing structure, or simply associated with the coupling of colliding cycles.

Analysis of stable periodic orbits in the one dimensional linear piecewise-smooth discontinuous map

In this paper, we consider one dimensional linear piecewise-smooth discontinuous maps. It is well known that stable periodic orbits exist for such maps, in some parameter region. It is also known that the corresponding bifurcation phenomena (termed as period adding bifurcation) exhibit a special structure. In the last couple of years, several authors have analyzed this structure using border collision bifurcation curves and given the characterization for various parameter regions. In this paper, we have analyzed a specific parameter range employing a different approach. We show that this approach enables one to pose some interesting questions like: what is the number of distinct periodic orbits of any given cardinality? We prove that there are precisely ϕ(n) distinct orbits of period n, where ϕ is the Euler’s totient function. We propose an algorithm which calculates the location of fixed points of all these ϕ(n) distinct orbits and gives the precise range of existence of these orbits with respect to the parameters. Further, we show how the amount of computations required to find these ranges of existence can be optimized.

Organizing centers in parameter space of discontinuous 1D maps. The case of increasing/decreasing branches

This work contributes to classify the dynamic behaviors of piecewise smooth systems in which border collision bifurcations characterize the qualitative changes in the dynamics. A central point of our investigation is the intersection of two border collision bifurcation curves in a parameter plane. This problem is also associated with the continuity breaking in a

Sufficient conditions for a period incrementing big bang bifurcation in one-dimensional maps

Typically, big bang bifurcation occurs for one (or higher)-dimensional piecewise-defined discontinuous systems whenever two border collision bifurcation curves collide transversely in the parameter space. At that point, two (feasible) fixed points collide with one boundary in state space and become virtual, and, in the one-dimensional case, the map becomes continuous. Depending on the properties of the map near the codimension-two bifurcation point, there exist different scenarios regarding how the infinite number of periodic orbits are born, mainly the so-called period adding and period incrementing. In our work we prove that, in order to undergo a big bang bifurcation of the period incrementing type, it is sufficient for a piecewise-defined one-dimensional map that the colliding fixed points are attractive and with associated eigenvalues of different signs.

Border collision bifurcations in a two-dimensional piecewise smooth map from a simple switching circuit

In recent years, the study of chaotic and complex phenomena in electronic circuits has been widely developed due to the increasing number of applications. In these studies, associated with the use of chaotic sequences, chaos is required to be robust (not occurring only in a set of zero measure and persistent to perturbations of the system). These properties are not easy to be proved, and numerical simulations are often used. In this work, we consider a simple electronic switching circuit, proposed as chaos generator. The object of our study is to determine the ranges of the parameters at which the dynamics are chaotic, rigorously proving that chaos is robust. This is obtained showing that the model can be studied via a two-dimensional piecewise smooth map in triangular form and associated with a one-dimensional piecewise linear map. The bifurcations in the parameter space are determined analytically. These are the border collision bifurcation curves, the degenerate flip bifurcations, which only are allowed to occur to destabilize the stable cycles, and the homoclinic bifurcations occurring in cyclical chaotic regions leading to chaos in 1-piece.

Border collision bifurcation curves and their classification in a family of 1D discontinuous maps

In this paper we consider a one-dimensional piecewise linear discontinuous map in canonical form, which may be used in several physical and engineering applications as well as to model some simple financial markets. We classify three different kinds of possible dynamic behaviors associated with the stable cycles. One regime (i) is the same existing in the continuous case and it is characterized by periodicity regions following the period increment by 1 rule. The second one (ii) is the regime characterized by periodicity regions of period increment higher than 1 (we shall see examples with 2 and 3), and by bistability. The third one (iii) is characterized by infinitely many periodicity regions of stable cycles, which follow the period adding structure, and multistability cannot exist. The analytical equations of the border collision bifurcation curves bounding the regions of existence of stable cycles are determined by using a new approach.

Endogenous cycles in discontinuous growth models

In this paper we consider a discontinuous one-dimensional piecewise linear model describing a neoclassical growth model. These kind of maps are widely used in the applied context. We determine the analytical expressions of border collision bifurcation curves, responsible for the observed dynamics, which consists of attracting cycles of any period and of quasiperiodic trajectories in exceptional cases.

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.

CALCULATION OF BIFURCATION CURVES BY MAP REPLACEMENT

The complex bifurcation structure in the parameter space of the general piecewise-linear scalar map with a single discontinuity — nowadays known as nested period adding structure — was completely studied analytically by N. N. Leonov already 50 years ago. He used an elegant and very efficient recursive technique, which allows the analytical calculation of the border-collision bifurcation curves, causing the nested period adding structure to occur. In this work, we have demonstrated that the application of Leonov's technique is not resticted to that particular bifurcation structure. On the contrary, the presented map replacement approach, which is an extension of Leonov's technique, allows the analytical calculation of border-collision bifurcation curves for periodic orbits with high periods and complex symbolic sequences using appropriate composite maps and the bifurcation curves for periodic orbits with much lower periods.

BORDER-COLLISION BIFURCATIONS IN 1D PIECEWISE-LINEAR MAPS AND LEONOV'S APPROACH

50 years ago (1959) in a series of publications by Leonov, a detailed analytical study of the nested period adding bifurcation structure occurring in piecewise-linear discontinuous 1D maps was presented. The results obtained by Leonov are barely known, although they allow the analytical calculation of border-collision bifurcation subspaces in an elegant and much more efficient way than it is usually done. In this work we recall Leonov's approach and explain why it works. Furthermore, we slightly improve the approach by avoiding an unnecessary coordinate transformation, and also demonstrate that the approach can be used not only for the calculation of border-collision bifurcation curves.

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.

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.

The dynamics of the NAIRU model with two switching regimes

We consider a model of inflation and unemployment proposed in Ferri et al. (JEBO, 2001), in which the dynamics are described by a discontinuous piecewise linear map, made up of two branches. We shall show that the bounded dynamics may be classified in two cases: we may have either regular dynamics with stable cycles of any period or quasiperiodic trajectories, or only chaotic dynamics ( pure chaos in which a unique absolutely continuous invariant ergodic measure exists, and structurally stable), in a rich variety of cyclical chaotic intervals. The main results are the analytical formulation of the border collision bifurcation curves, through which we give a complete picture of the possible outcomes of the model.