Two-dimensional Discontinuous Map Research Articles

Regular and chaotic dynamics in a 2D discontinuous financial market model with heterogeneous traders

We develop a financial market model where three types of traders operate simultaneously: fundamentalists and chartists of two types, namely, trend followers and contrarians. The dynamics of this model is described by a two-dimensional discontinuous map defined by two linear functions, where one acts in the partition between two (parallel) discontinuity lines and the other one acts outside this partition. Our analysis shows that despite the linearity of the map components, its dynamics can be quite complex, with various, possibly coexisting attracting cycles and chaotic attractors. As a first step towards the understanding how the overall bifurcation structure observed in the parameter space of the map is organized, we obtain analytically the boundaries of periodicity regions related to the simplest attracting period-n cycles, n≥3, with one point in the middle partition and n−1 points outside it. These boundaries can be related to border-collision bifurcations (when a point of the cycle collides with a discontinuity line) as well as to degenerate bifurcations (associated with eigenvalues on the unit circle). We show also some elements of period-adding and period-incrementing bifurcation structures for which the cycles mentioned above are basic. From an economic point of view, our study confirms that a fairly simple financial market model with heterogeneous agents is able to produce complicated boom-bust dynamics typical of real financial markets.

On the dynamics of two-dimensional dissipative discontinuous maps

• In this work we consider a dissipative two-dimensional discontinuous mapping. • We show that from a large initial action, the dynamics converges exponentially to chaotic attractors. • The maximum of the chaotic attractors as a function of the control parameters provides power laws. • We also investigate the positive Lyapunov exponents as a function of the control parameters. Some dynamical properties for a dissipative two-dimensional discontinuous standard mapping are considered. The mapping, in action-angle variables, is parameterized by two control parameters; namely, k ≥ 0 controlling the intensity of the nonlinearity and γ ∈ [0, 1] representing the dissipation. The case of γ = 0 recovers the non-dissipative model while any γ ≠ 0 yields to the breaking of area preservation; hence leading to the existence of attractors, including chaotic ones. We show that when starting from a large initial action, the dynamics converges to chaotic attractors through an exponential decay in time, while the speed of the decay depends on the dissipation intensity. We also investigate the positive Lyapunov exponents and describe their behavior as a function of the control parameters.

Period adding structure in a 2D discontinuous model of economic growth

We study the dynamics of a growth model formulated in the tradition of Kaldor and Pasinetti where the accumulation of the ratio capital/workers is regulated by a two-dimensional discontinuous map with triangular structure. We determine analytically the border collision bifurcation boundaries of periodicity regions related to attracting cycles, showing that in a two-dimensional parameter plane these regions are organized in the period adding structure. We show that the cascade of flip bifurcations in the base one-dimensional map corresponds for the two-dimensional map to a sequence of pitchfork and flip bifurcations for cycles of even and odd periods, respectively.

Synchronization in time-discrete model of two electrically coupled spike-bursting neurons

Dynamics of the ensemble of two model neurons interacting through electrical synapse is investigated. Both neurons are described by two-dimensional discontinuous map. It is shown that in four-dimensional phase space a chaotic attractor of relaxation type exists corresponding to spike-bursting chaotic oscillations. A new effect of recurrent transitory chaotic oscillations underlies a dynamical mechanism of chaotic bursts formation. It is shown that, under coupling, the transient from chaotic bursts generation into rest state occurs with a time delay. A new characteristic estimating the degree of spike-bursting synchronization is introduced. Dependence of the synchronism degree on the coupling strength is shown for some coupling interval where only activity synchronization occurs. A probabilistic study provides a dynamical explanation of these phenomena.

Spike-burst synchronization in an ensemble of electrically coupled discrete model neurons

In this paper, we study the dynamics of a system of two model neurons interacting via the electrical synapse. Each neuron is described by a two-dimensional discontinuous map. A chaotic relaxational-type attractor, which corresponds to the spiking-bursting chaotic oscillations of neurons is shown to exist in a four-dimensional phase space. It is found that the dynamical mechanism of formation of chaotic bursts is based on a new phenomenon of generation of transient chaotic oscillations. It is demonstrated that transition from the chaotic-burst generation to the state of relative rest occurs with a certain time delay. A new characteristic which estimates the degree of synchronization of the spiking-bursting oscillations is introduced. The dependence of the synchronization degree on the strength of coupling of the ensemble elements is studied.

Bifurcation phenomena in two-dimensional piecewise smooth discontinuous maps

In recent years the theory of border collision bifurcations has been developed for piecewise smooth maps that are continuous across the border and has been successfully applied to explain nonsmooth bifurcation phenomena in physical systems. However, there exist a large number of switching dynamical systems that have been found to yield two-dimensional piecewise smooth maps that are discontinuous across the border. In this paper we present a systematic approach to the problem of analyzing the bifurcation phenomena in two-dimensional discontinuous maps, based on a piecewise linear approximation in the neighborhood of the border. We first motivate the analysis by considering the bifurcations occurring in a familiar physical system—the static VAR compensator used in electrical power systems—and then proceed to formulate the theory needed to explain the bifurcation behavior of such systems. We then integrate the observed bifurcation phenomenology of the compensator with the theory developed in this paper. This theory may be applied similarly to other systems that yield two-dimensional discontinuous maps.

On the existence of low-period orbits in n-dimensional piecewise linear discontinuous maps

Discontinuous maps occur in many practical systems, and yet bifurcation phenomena in such maps is quite poorly understood. In this paper, we report some important results that help in analyzing the border collision bifurcations that occur in n-dimensional discontinuous maps. For this purpose, we use the piecewise linear approximation in the neighborhood of the plane of discontinuity. Earlier, Feigin had made a similar analysis for general n-dimensional piecewise smooth continuous maps. In this paper, we extend that line of work for maps with discontinuity to obtain the general conditions of existence of period-1 and period-2 fixed points before and after a border collision bifurcation. The application of the method is then illustrated using a specific example of a two-dimensional discontinuous map.

Tongues of periodicity in a family of two-dimensional discontinuous maps of real Möbius type

In this paper we consider a two-dimensional piecewise-smooth discontinuous map representing the so-called “relative dynamics” of an Hicksian business cycle model. The main features of the dynamics occur in the parameter region in which no fixed points at finite distance exist, but we may have attracting cycles of any periods. The bifurcations associated with the periodicity tongues of the map are studied making use of the first-return map on a suitable segment of the phase plane. The bifurcation curves bounding the periodicity tongues in the parameter plane are related with saddle-node and border-collision bifurcations of the first-return map. Moreover, the particular “sausages structure” of the bifurcation tongues is also explained.

A Prelude Staircase to a Type V Intermittency in Two-Dimensional Discontinuous Maps

A sequence of periodic attractors has been observed in a two-dimensional discontinuous map, which can be considered as a model of impact oscillator. The so-called “transfer number”, which is defined as the mean number of transfer from non-impact state to impact state per iteration, is locked onto a lot of rational values to form a curve consisting of many steps. Our numerical investigation confirms that every step is confined by conditions created by the collision between the periodic orbit and the discontinuous boundary of the system. After the last collision the system shows a chaotic motion with intermittent characteristics. Therefore the staircase can be addressed as a “prelude staircase to type V intermittency”. The similar phenomenon has also been observed in a model of electric circuit. These results of our study suggest that this kind of staircases is common in two (or even higher) dimensional discontinuous maps.