The Emergence of Bull and Bear Dynamics in a Nonlinear Model of Interacting Markets

Abstract

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 bull and bear market dynamics. The dimension of the model can be reduced by restricting investors' trading activity, which enables the dynamic analysis to be performed stepwise, from low-dimensional cases up to the full three-dimensional model. In our paper we focus mainly on the dynamics of the one-and twodimensional cases, with numerical experiments and some analytical results, and also show that the main features persist in the three-dimensional model.

Introducing a price variation limiter mechanism into a behavioral financial market model

“…On the other hand, for some intermediate values for the asymptotes and particular configurations of the other parameters, in addition to two coexisting attractors similar to those detected in Ref. 33, we find (at least) one more attractor, which may be periodic or chaotic, independently of the nature of the other two attractors. According to the considered parameter set, it may happen that the third attractor persists, while the other two disappear, or vice versa, that the third attractor disappears when the other two are still present.…”

“…33, and we show that our system admits no divergent trajectories; in Section IV, we investigate the occurrence of the first flip bifurcation for our steady states and we prove the existence of complex dynamics when the asymptotes are sufficiently distant; in Section V, we present some global scenarios with multistability phenomena, characterized by the presence of two or (at least) three coexisting attractors; finally, in Section VI, we draw some conclusions and discuss our results.…”

“…Second, from a normative viewpoint, we aim to propose a method in view of reducing volatility and controlling chaos, that is, of diminishing turbulence, as well as of decreasing the number, the size, and the complexity of the attractor in the phase space, in order to achieve the convergence to a fixed point. To show the effectiveness and the functioning of our mechanism, we use as benchmark framework a special case of the model in Tramontana et al 33 …”

“…Moreover, when dynamics are chaotic in Ref. 33, we can stabilize the system through a suitable choice of the asymptotes. Finally, when in Ref.…”

“…33, i.e., a fundamental steady state, which is always unstable, and two steady states symmetric with respect to it, for which we derive the corresponding stability conditions. In Proposition 3.2, we prove that under suitable conditions on the asymptotes the map generating our dynamical system is increasing and we describe the basins of attraction in this simple framework, employing then similar arguments to exclude the presence of negative trajectories, as well as of divergent trajectories under any condition on the parameters.…”