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We develop a three-dimensional nonlinear dynamic model in which the stock
markets of two countries are linked through the foreign exchange market. Connections are due to the trading activity of heterogeneous speculators. Using
analytical and numerical tools, we seek to explore how the coupling of the markets may affect the emergence of

Financial market models with heterogeneous interacting agents have proven to be quite successful in the recent past. For instance, these nonlinear dynamical systems have the potential to replicate some important stylized facts of financial markets—such as the emergence of bubbles and crashes—quite well and thereby help us to understand what is going on in these markets. For pioneering contributions and related further developments see Day and Huang [

The seminal model of Day and Huang [

In this paper we develop and explore a nonlinear model in which the stock markets of two countries, say H(ome) and A(broad), are linked via and with the foreign exchange market. So far, most of these models focus on one speculative market and not much is known about the implications of market interactions. A few exceptions include Westerhoff [

To make matters as simple as possible, we assume that stock market traders only rely on a (linear) fundamental trading rule. If we allow stock market traders from country A to become active in country H, then the stock market H and the foreign exchange market are linked and coevolve in a two-dimensional nonlinear dynamical system. Our model turns into a three-dimensional dynamical system if stock market traders from country H also invest in country A. The expansion of the trading activity of stock market speculators, via the introduction of international connections, therefore results in a gradual increase of the dimension of the dynamical system. As it turns out, the

A related model of interacting markets with a similar nonlinear structure was recently investigated by Dieci and Westerhoff [

The two-dimensional and the full three-dimensional cases of the present model can thus be regarded as generalizations of the one-dimensional model by Day and Huang [

Let us describe in greater detail the key dynamic features of the model. As is well known, the typical

Finally, the three-dimensional case, obtained by removing any restriction on trading activities across different countries, can be understood, via numerical experiments, due to the knowledge of the dynamics occurring in the one- and two-dimensional cases. We will see that the global bifurcations due to contacts between different invariant sets are still present, leading to dynamics which are the natural extension to a three-dimensional space of those occurring also in the two-dimensional one.

The structure of the paper is as follows. In Section

This section is devoted to the description of the three-dimensional discrete-time dynamic model of internationally connected markets, which will then be analyzed in the lower dimensional subcases before exploring some of its properties in the full three-dimensional model.

We consider two stock markets which are linked

In the following subsections we describe each market in detail.

Let us start with a description of the stock market in country

The orders placed by fundamental traders from country

Fundamental traders from abroad may benefit from a price correction in the stock market and in the foreign exchange market. Denote the fundamental value of the exchange rate by

Let us now turn to the stock market in country

Let us now consider the dynamics of the exchange rate (

The orders submitted by technical and fundamental speculators in the foreign exchange market are denoted by

Fundamentalists seek to exploit misalignments using a nonlinear trading rule

The complete dynamic model is given by (

Map (

Stable non-fundamental steady states. (a) and its enlargement (b) are obtained using the following set of parameters:

Basins of attraction. In (a) the immediate basin of the steady state

For

Periodic and chaotic attractors. In (a) a stable 2-cycle is obtained using the same set of parameters of Figure

Homoclinic bifurcation of

Figures

Homoclinic bifurcation of

Put differently, the two disjoint symmetric attractors exist as long as each unimodal part of the map behaves as the standard logistic map,

Bifurcation diagram versus parameter

This kind of dynamics persists as long as the chaotic interval is inside the repelling two cycle; that is,

In Appendix

The bimodal map in (

From an economic point of view it is interesting to note that already the one-dimensional nonlinear map for the foreign exchange market is able to generate endogenous dynamics ( i.e., excess volatility) and bubbles and crashes. For a more detailed economic interpretation of this scenario see the related setup of Day and Huang [

In this section we analyze the case in which stock market traders from

With regard to system (

Three steady states therefore coexist when the reaction parameter

In order to understand better which kind of bifurcations occur, Appendix

The two dimensional map in (

As cannot have the explicit expressions of the new pair of equilibria, cannot perform analytically their local stability analysis. Thus in the following we describe the results via numerical simulations. Note that we keep parameters

Of the two new equilibria, the stable one, which we call

Change of stability in the two-dimensional case. Parameters are

At

As parameter

Bifurcation diagrams (b.d. for short). In blue the b.d. corresponding to an initial condition close to

Although the two attractors

Basins of attraction. Basin

The previous subsection has shown how, under increasing values of parameter

Bifurcation diagrams. The b.d. in (a) corresponds to an initial condition close to

As stated above, for a wide interval of values of

First homoclinic bifurcation of

For

By increasing parameter

One-side homoclinic bifurcation of

Homoclinic bifurcation of

The asymmetry of the map implies that the contacts between the chaotic attractors and the stable manifold of

We remark that in the interval of values of

From an economic perspective our findings imply that the famous

So far, we have observed several homoclinic bifurcations involving the chaotic area. It is worth noting that the homoclinic bifurcations occurring at

Homoclinic bifurcation of

It follows that almost all initial conditions inside the previous basin

In this section we deal with the complete three-dimensional model. Our analysis (mainly via numerical simulations) will show that the dynamic phenomena highlighted in the previous lower dimensional models also persist in the full model. In particular, we shall see that (as for the model in the previous section), the origin is always an equilibrium, and two more equilibria appear as parameter

In the full model, stock market traders from countries

By imposing the fixed point condition to (

By substituting (

We remark that the analytical investigation of the local stability properties of the fundamental fixed point

We will now study the local and global bifurcations via numerical investigation, supported by our knowledge of the model behavior in the simplified, two-dimensional case. Our base parameter selection is the following:

Bifurcation diagram of the dynamic behavior of

Note that the emergence of endogenous

We have considered a three-dimensional discrete-time dynamic model of internationally connected financial markets, where two stock markets, populated by national and foreign fundamental traders, interact with each other via the foreign exchange market. In the latter, heterogeneous speculators are active, and their nonlinear trading rules are at the origin of complicated endogenous fluctuations across all three markets, similar to the well-known

The possibility to reduce the dimension of the dynamical system, via restrictions imposed on the activity of foreign traders, results in simplified one- and two-dimensional setups, whose analysis is simpler and helped in the understanding of those dynamic phenomena occurring in the complete three-dimensional model. While the one-dimensional case has the same qualitative dynamics of the Day and Huang [

In this appendix we compare the behavior, as the parameter

From (

In this appendix we provide an analytical study of the eigenvalues of the Jacobian matrix evaluated at the fundamental steady state.

The Jacobian matrix of system (

lie between

In this appendix we provide the equation of the critical curve

Given the parameters selection used in this work (that is,

In our case the condition (i) is satisfied if