x^{f}_{t} = exp(α(p^{∗} − p_{t})) − 1, α > 0 —– (1)

where α is the parameter that denotes the strength of the non-linearity of the fundamentalist excess demand function (1). The fundamentalist’s excess demand function (1) is derived in a one-period utility optimizing framework. The technical details of the derivation of (1) within a utility maximizing framework are given in * Kaizoji*. We can see that Equation (1) has captured the distinctive features of the fundamentalist’s strategy. While fundamentalists calculate the fundamental value, chartists estimate a trend in the price change. Chartists can be assumed to form their expectation of the price of the risky asset according to the simple adaptive scheme:

p^{e}_{t+1} = p^{e}_{t} + µ(p_{t} − p^{e}_{t}), 0 < µ ≥ 1 —– (2)

x^{c}_{t} = exp(β(p^{e}_{t+1} − p_{t})) − 1, β > 0 —– (3)

p_{t+1} − p_{t} = θN[(1 − κ)x^{f}_{t} + κx^{c}_{t}] —– (4)

where θ denotes the speed of the adjustment of the price, and N the total number of traders…..

[…] Substituting (1), (2) and (3) to (4) from here, the dynamical system can be obtained as […]