The complexity of an investment competition dynamical model with imperfect information in a security market

We present a nonlinear discrete dynamical model of investment competition with imperfect information for N heterogeneous oligopolists in a security market. In this paper, our focus is on a given three-dimensional model which exhibits highly rich dynamical behaviors. Based on Wen’s Hopf bifurcation criterion [Wen GL. Criterion to identify Hopf bifurcations in maps of arbitrary dimension. Phys Rev E 2005;72:026201–3; Wen GL, Xu DL, Han X. On creation of Hopf bifurcations in discrete-time nonlinear systems. Chaos 2002;12(2):350–5] and Kuznetsov’s normal form theory [Kuznetsov YA. Elements of applied bifurcation theory. New York: Springer-Verlag; 1998. p. 125–37], we study the model’s stability, criterion and direction of Neimark–Sacker bifurcation. Moreover, we numerically simulate a complexity evolution route: fixed point, closed invariant curve, double closed invariant curves, fourfold closed invariant curves, strange attractor, period- 3 closed invariant curve, period-3 2-tours, period-4 closed invariant curve, period-4 2-tours.