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.

Codimension-two border collision bifurcation in a two-class growth model with optimal saving and switch in behavior

We consider a two-class growth model with optimal saving and switch in behavior. The dynamics of this model is described by a two-dimensional (2D) discontinuous map. We obtain stability conditions of the border and interior fixed points (known as Solow and Pasinetti equilibria, respectively) and investigate bifurcation structures observed in the parameter space of this map, associated with its attracting cycles and chaotic attractors. In particular, we show that on the x-axis, which is invariant, the map is reduced to a 1D piecewise increasing discontinuous map, and prove the existence of a corresponding period adding bifurcation structure issuing from a codimension-two border collision bifurcation point. Then, we describe how this structure evolves when the related attracting cycles on the x-axis lose their transverse stability via a transcritical bifurcation and the corresponding interior cycles appear. In particular, we show that the observed bifurcation structure, being associated with the 2D discontinuous map, is characterized by multistability, that is impossible in the case of a standard period adding bifurcation structure.

Research of Periodic Orbits and Chaos Produced in 1-D Liner Piecewise-Smooth Maps with Single Discontinuity, Positive and Negative Slopes

Last four decades have seen a major development in the theory of piece-wise smooth discontinuous maps for the analysis of typical bifurcation phenomena for systems that can be modelled as such. The major focus of this paper is the analysis of 1-D linear piecewise smooth maps with a discontinuity, one positive and another negative slope. Interestingly, this type of map analysis has been carried out by various authors and the results have been reported in literature. For example, the existence of period adding cascade in particular parameter regions specified by the parameters ‘a’ and ‘b’ was proven. The new range of parameters this work presents are a ϵ (0,1), b ϵ (-1, 0) and a ϵ (0,1), b ϵ (- , -1). Elementary algebraic and geometric tools have been used to analyze the periodicities in the 1-D linear piecewise smooth discontinuous map with respect to parameters a, b, µ and l. Various examples have been illustrated along with the plotted bifurcation curves. The analysis of the behaviour of the system with varied parameter ranges indicates that non-trivial cases are present for negative values of one or more parameters. A sample basin of attraction plot is illustrated as well. Further, an analytic proof for the existence of LnR orbits for the region a ϵ (0,1), b ϵ (-1, 0) was successfully produced, which is unpublished till date. The research concentrates on the theoretical results.

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.

Bifurcation Analysis of Five-Level Cascaded H-Bridge Inverter Using Proportional-Resonant Plus Time-Delayed Feedback

In this paper, a traditional five-level cascaded H-bridge inverter is studied and regulated by a proportional-resonant (PR) controller. In order to extend the range of the gain of PR controller, for the purpose of achieving a fast response, a time-delayed feedback controller (TDFC) is used. Similar to the pulse width modulation (PWM) current-mode single phase H-bridge inverter that exhibits bifurcation and chaos when parameters vary, we demonstrate for the first time that the cascaded H-bridge inverter also shows similar features. From the perspective of a discontinuous map, the cascaded H-bridge inverter generally displays extraordinary complexity. Moreover, a new virtual ergodic method (VEM) is proposed to establish the mathematical model of the whole system, which helps to understand the observed bifurcation phenomena. Simulation results are given to verify the analysis.

Dynamics of a 2D Piecewise Linear Braess Paradox Model: Effect of the Third Partition

In this work, we investigate the dynamics of a piecewise linear 2D discontinuous map modeling a simple network showing the Braess paradox. This paradox represents an example in which adding a new route to a specific congested transportation network makes all the travelers worse off in terms of their individual travel time. In the particular case in which the modeled network corresponds to a binary choice situation, the map is defined on two partitions and its dynamics has already been described. In the general case corresponding to a ternary choice, a third partition appears leading to significantly more complex bifurcation structures formed by border collision bifurcations of stable cycles with points located in all three partitions. Considering a map taking a constant value on one of the partitions, we provide a first systematic description of possible dynamics for this case.