When solving for the gross risk-free interest rate parameter R a Hopf bifurcation occurs given the parameter set {Θ} \ R if RH satisfies
| RH= |
| 2 a γ σ2 |
|
| λ |
⎛
⎜
⎜
⎝ |
1-(1-µ) |
|
⎞
⎟
⎟
⎠ |
|
|
|
. |
Because
R>1, there are possibly cases for which no Hopf bifurcation occurs, when varying
R and keeping the other parameters constant.
When solving the first condition for the fundamental belief parameter v a Hopf bifurcation occurs given the parameter set {Θ} \ v if vH satisfies
Again however, because 0 ≤
v ≤ 1, there are cases for which no Hopf bifurcation occurs, when varying
v and keeping the other parameters constant.
When solving the first condition for the exponential moving average parameter µ a Hopf bifurcation occurs given the parameter set {Θ} \ µ and v>0 if µH satisfies
and where
v should additionally satisfy 0<
v≤ 1. Now also, because 0 < µ < 1, there are cases for which no Hopf bifurcation occurs, when varying µ and keeping the other parameters constant.
When solving for the moving average belief parameter γ a Hopf bifurcation occurs given the parameter set {Θ} \ γ if γH satisfies
| γH= |
⎛
⎜
⎜
⎝ |
1-(1-µ) |
|
⎞
⎟
⎟
⎠ |
|
|
|
>0 . |
When solving for the moving average belief parameter λ a Hopf bifurcation occurs given the parameter set {Θ} \ λ if λ
H satisfies
| λH= |
| 2 a γ σ2 |
|
| R |
⎛
⎜
⎜
⎝ |
1-(1-µ) |
|
⎞
⎟
⎟
⎠ |
|
|
|
>0. |
Hence when one of the parameters
a, σ
2, γ or λ is varied while keeping the other parameters constant, a Hopf bifurcation always arises for some parameter value.
If the Jacobian matrix of Φ at the steady state
z has two complex conjugate eigenvalues, ξ
1=
c+
di and ξ
2=
c-
di, then the price series, and therefore also the exponential moving average series, follows a wavelike pattern. For fluctuation close to the steady state the period of the wave is approximately equal to
|
, where tan(θ)= |
|
, with c>0, |
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