if the two complex conjugate eigenvalues are near the unit circle. Solving p(ξ)=0 under the conditions in (6.46) yields

d
c
=
4-(1-µ+
v
R
+K)2
(1-µ+
v
R
+K)2
, with K=
2 a γ σ2
λ R
qma
qfund
.
For 0<µ<1 this is an increasing function of µ and hence θ is an increasing function of µ. Thus the period of the wave of the price fluctuation close to the unstable steady state is a decreasing function of µ. Recall that in the computation of the exponential moving average, the larger µ, the more weight is placed on current prices, the more closely the moving average follows the price series, and the more earlier a change in direction of the price series is detected. Thus, a larger µ causes price to return to the fundamental value with a higher frequency than a smaller µ.

6.6  Numerical analysis

In the last section we studied the local stability of the steady state analytically. We determined what kind of bifurcations can occur if the value of one of the model parameters is varied. In this section we study the global dynamical behavior numerically, especially when the steady state is unstable, with the aid of time series plots, phase diagrams, delay plots, bifurcation diagrams and the computation of Lyapunov exponents.

6.6.1  Lyapunov characteristic exponents

The Lyapunov characteristic exponents (LCEs) measure the average rate of divergence (or convergence) of nearby initial states, along an attractor in several directions. Consider the dynamical model zt+1=Φ(zt), where Φ is a k-dimensional map. After n periods the distance between two nearby initial state vectors z0 and z0+v0 has grown approximately to
||Φn(z0+v0)-Φn(z0)|| ≈ ||DΦn(z0) v0 ||,
where v0 is the initial perturbation vector, DΦn(z0) is the Jacobian matrix of the n-th iterate of Φ evaluated at z0 and ||.|| denotes the Euclidean distance. The exponent λ(z0, v0) measuring the exponential rate of divergence has to satisfy
||Φn(z0+v0)-Φn(z0)|| ≈ ||DΦn(z0) v0 || = en λ(z0, v0) ||v0||.
For a k dimensional system there exist k distinct LCEs, ordered as λ1 ≥ λ2 ≥ ... ≥ λk, each measuring the average expansion or contraction along an orbit in the different directions.
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