Piecewise Linear Discontinuous Map Research Articles

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

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

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

We 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.

Bifurcation, chaos and multi-stability regions in an asset pricing model with three subsystems

An asset pricing model with two types of chartists and fundamentalists and trend followers is considered, it is driven by a two-dimensional piecewise linear discontinuous map with three subsystems. There are great differences in the dynamic behaviour between expected offset (the expectations of trend followers offset the difference between the expectations of Type 1 traders in bull and bear markets) and expected non-offset. It is proven that there is no chaos in the dynamic of system with expected offset. However, chaos may exist in the dynamic of system with expected non-offset. We present a systematic approach to the problem of analysing the bifurcation phenomena associated with the appearance/disappearance of cycles, which may be related to several bifurcations. The multi-stability regions in parameter plane and related basins of multi-attractors in phase space are investigated. This paper aims to uncover the endogenous law of the unpredictability and excessive volatility in financial markets.

Multistability Regions and Related Basins of Multiattractors in Piecewise Linear Discontinuous Map from Financial Markets

By adding trend followers, we extend the model given by Tramontana et al. from one-dimensional ([Formula: see text]D) piecewise linear discontinuous (PWLD) map to a new 2D PWLD map. Using this map in financial markets, we describe the bifurcation mechanisms associated with the appearance/disappearance of cycles, which may be related to several cases: border collision bifurcations; Poincaré equator collision bifurcations; degenerate flip bifurcations in both supercritical and subcritical cases. We investigate the multistability regions in the parameter plane and related basins of multiattractors to uncover the reason for the unpredictability of the internal law of price fluctuations in financial market.

Memory effects on binary choices with impulsive agents: Bistability and a new BCB structure.

After the seminal works by Schelling, several authors have considered models representing binary choices by different kinds of agents or groups of people. The role of the memory in these models is still an open research argument, on which scholars are investigating. The dynamics of binary choices with impulsive agents has been represented, in the recent literature, by a one-dimensional piecewise smooth map. Following a similar way of modeling, we assume a memory effect which leads the next output to depend on the present and the last state. This results in a two-dimensional piecewise smooth map with a limiting case given by a piecewise linear discontinuous map, whose dynamics and bifurcations are investigated. The map has a particular structure, leading to trajectories belonging only to a pair of straight lines. The system can have, in general, only attracting cycles, but the related periods and periodicity regions are organized in a complex structure of the parameter space. We show that the period adding structure, characteristic for the one-dimensional case, also persists in the two-dimensional one. The considered cycles have a symbolic sequence which is obtained by the concatenation of the symbolic sequences of cycles, which play the role of basic cycles in the bifurcation structure. Moreover, differently from the one-dimensional case, the coexistence of two attracting cycles is now possible. The bistability regions in the parameter space are investigated, evidencing the role of different kinds of codimension-two bifurcation points, as well as in the phase space and the related basins of attraction are described.

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.

Phase order in one-dimensional piecewise linear discontinuous map**Project supported by the National Natural Science Foundation of China (Grant No. 11645005) and the Interdisciplinary Incubation Project of Shaanxi Normal University (Grant No. 5).

The phase order in a one-dimensional (1D) piecewise linear discontinuous map is investigated. The striking feature is that the phase order may be ordered or disordered in multi-band chaotic regimes, in contrast to the ordered phase in continuous systems. We carried out an analysis to illuminate the underlying mechanism for the emergence of the disordered phase in multi-band chaotic regimes, and proved that the phase order is sensitive to the density distribution of the trajectories of the attractors. The scaling behavior of the net direction phase at a transition point is observed. The analytical proof of this scaling relation is obtained. Both the numerical and analytical results show that the exponent is 1, which is controlled by the feature of the map independent on whether the system is continuous or discontinuous. It extends the universality of the scaling behavior to systems with discontinuity. The result in this work is important to understanding the property of chaotic motion in discontinuous systems.

2D discontinuous piecewise linear map: Emergence of fashion cycles.

We consider a discrete-time version of the continuous-time fashion cycle model introduced in Matsuyama, 1992. Its dynamics are defined by a 2D discontinuous piecewise linear map depending on three parameters. In the parameter space of the map periodicity, regions associated with attracting cycles of different periods are organized in the period adding and period incrementing bifurcation structures. The boundaries of all the periodicity regions related to border collision bifurcations are obtained analytically in explicit form. We show the existence of several partially overlapping period incrementing structures, that is, a novelty for the considered class of maps. Moreover, we show that if the time-delay in the discrete time formulation of the model shrinks to zero, the number of period incrementing structures tends to infinity and the dynamics of the discrete time fashion cycle model converges to those of continuous-time fashion cycle model.

Bifurcations and Chaos for 2D Discontinuous Dynamical Model of Financial Markets

We develop a financial market model with interacting chartists and fundamentalists and chase sellers, the model dynamics is driven by a two-dimensional discontinuous piecewise linear map. Assume that the fixed point on the left side of border is restricted to regular saddle, we provide a more or less complete analytical treatment of the model dynamics by characterizing its possible outcomes in parameter space. The interpretation of structure for basin boundary and chaotic attractor is given by using contact bifurcation resulting from the contact between invariant set and the border. The critical value of occurring boundary crisis is given. In addition, we show that quite different scenarios can trigger real world phenomena such as bull and bear market dynamics and excess volatility.

A Simple Financial Market Model with Heterogeneous Speculators: 2-dimensional Discontinuous System

This paper aims at a financial market model with interacting chartists and fundamentalists and chase sell, the model dynamics is driven by a two-dimensional discontinuous piecewise linear map. The parameter space is partitioned into regions depending on the types of the fixed points at the two sides of the border. For the fixed points of both sides are attractors (regular/ spiral/flip), the existence and coexistence of periodic points are studied analytically and numerically. The existence of chaotic orbit is explained by using the theory of homoclinic intersection for stable and unstable manifold of periodic orbit. The basins of coexisting multi-attractors are presented. The results can deep our knowledge of both financial market and dynamical system.

Wild oscillations in a nonlinear neuron model with resets: (Ⅰ) Bursting, spike-adding and chaos

In a series of two papers, we investigate the mechanisms by which complex oscillations are generated in a class of nonlinear dynamical systems with resets modeling the voltage and adaptation of neurons. This first paper presents mathematical analysis showing that the system can support bursts of any period as a function of model parameters, and that these are organized in a period-incrementing structure. In continuous dynamical systems with resets, such structures are complex to analyze. In the present context, we use the fact that bursting patterns correspond to periodic orbits of the adaptation map that governs the sequence of values of the adaptation variable at the resets. Using a slow-fast approach, we show that this map converges towards a piecewise linear discontinuous map whose orbits are exactly characterized. That map shows a period-incrementing structure with instantaneous transitions. We show that the period-incrementing structure persists for the full system with non-constant adaptation, but the transitions are more complex. We investigate the presence of chaos at the transitions.

Symbolic dynamics of Belykh-type maps

The symbolic dynamics of a Belykh-type map (a two-dimensional discontinuous piecewise linear map) is investigated. The admissibility condition for symbol sequences named the pruning front conjecture is proved under a hyperbolicity condition. Using this result, a symbolic dynamics model of the map is constructed according to its pruning front and primary pruned region. Moreover, the boundary of the parameter region in which the map is chaotic of a horseshoe type is given.

A simple financial market model with chartists and fundamentalists: Market entry levels and discontinuities

We present a simple financial market model with interacting chartists and fundamentalists. Since some speculators only become active when a certain misalignment level has been crossed, the model dynamics is driven by a discontinuous piecewise linear map. Recent mathematical techniques allow a comprehensive study of the model's dynamical system. One of its surprising features is that model simulations may appear to be chaotic, although only regular dynamics can emerge. While our deterministic model is able to produce stylized bubbles and crashes we also show that a stochastic version of our model is able to match the finer details of financial market dynamics.

The bull and bear market model of Huang and Day: Some extensions and new results

We develop a financial market model with interacting chartists and fundamentalists that embeds the famous bull and bear market model of Huang and Day as a special case. Their model is given by a one-dimensional continuous piecewise-linear map. Our model, on the other hand, is more flexible and is represented by a one-dimensional discontinuous piecewise-linear map. Nevertheless, we are able to provide a more or less complete analytical treatment of the model dynamics by characterizing its possible outcomes in parameter space. In addition, we show that quite different scenarios can trigger real-world phenomena such as bull and bear market dynamics and excess volatility.

One-dimensional maps with two discontinuity points and three linear branches: mathematical lessons for understanding the dynamics of financial markets

We develop a simple financial market model with heterogeneous interacting speculators. The dynamics of our model is driven by a one-dimensional discontinuous piecewise linear map, having two discontinuity points and three linear branches. On the one hand, we study this map analytically and numerically to advance our knowledge about such dynamical systems. In particular, not much is known about discontinuous maps involving three branches. On the other hand, we seek to improve our understanding of the functioning of financial markets. We find, for instance, that such maps can generate complex bull and bear market dynamics.

Breaking the continuity of a piecewise linear map

Knowledge about the behavior of discontinuous piecewise-linear maps is important for a wide range of applications. An efficient way to investigate the bifurcation structure in 2D parameter spaces of such maps is to detect specific codimension-2 bifurcation points, called organizing centers, and to describe the bifurcation structure in their neighborhood. In this work, we present the organizing centers in the 1D discontinuous piecewise-linear map in the generic form, which can be used as a normal form for these bifurcations in other 1D discontinuous maps with one discontinuity. These organizing centers appear when the continuity of the system function is broken in a fixed point. The type of an organizing center depends on the slopes of the piecewise-linear map. The organizing centers that occur if the slopes have an absolute value smaller than one were already described in previous works, so we concentrate on presenting the organizing centers that occur if one or both slopes have absolute values larger than one. By doing this, we also show that the behavior for each organizing center can be explained using four basic bifurcation scenarios: the period incrementing and the period adding scenarios in the periodic domain, as well as the bandcount incrementing and the bandcount adding scenarios in the chaotic domain.

Heterogeneous Speculators and Asset Price Dynamics: Further Results from a One-Dimensional Discontinuous Piecewise-Linear Map

In this paper we continue exploring a recently introduced financial market model in which boundedly rational agents follow technical and fundamental trading rules to determine their orders. Amongst other things, our model reveals that interactions between heterogeneous speculators can generate interesting boom-bust cycles. In addition, we provide an extensive analytical treatment of the model's underlying dynamical system, which is given by a one-dimensional discontinuous piecewise-linear map. One result is that we detect a period-adding bifurcation sequence, implying the existence of infinitely many stable cycles. Moreover, we analytically determine the parameter space that yields stable, cyclical and chaotic asset price fluctuations.

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.

Snap-back repellers in non-smooth functions

In this work we consider the homoclinic bifurcations of expanding periodic points. After Marotto, when homoclinic orbits to expanding periodic points exist, the points are called snap-back-repellers. Several proofs of the existence of chaotic sets associated with such homoclinic orbits have been given in the last three decades. Here we propose a more general formulation of Marotto’s theorem, relaxing the assumption of smoothness, considering a generic piecewise smooth function, continuous or discontinuous. An example with a two-dimensional smooth map is given and one with a two-dimensional piecewise linear discontinuous map.

Influence of Control Valve Delay and Dead Zone on the Stability of a Simple Hydraulic Positioning System

This paper deals with the PI control of a highly simplified dynamic model of a hydraulic cylinder. It is assumed that the hydraulic fluid is incompressible and that the pump provides constant flow rates, which results in the possibility of velocity control. Two types of anomalies are taken into account: (a) the time delay due to the controller computations and the internal pressure dynamics and (b) the dead zone of the controller valve. This results in a nonlinear system are described by a piecewise linear discontinuous map. Nonlinear behavior of the system is explored and the practically globally stable parameter domains are identified.