complex conjugate of modulus 1 if ξ1 ξ2=1 and |ξ1+ξ2|<2. Using (6.43) this leads to the conditions

(1-µ)
v
R
+
2 a γ σ2
λ R
qma
qfund
= 1 and |1-µ +
v
R
+
2 a γ σ2
λ R
qma
qfund
|<2.     (46)
Substituting the first condition in the second yields
|2-µ ⎛
⎜
⎜
⎝		 1-
v
R
⎞
⎟
⎟
⎠		 |<2.
For 0<µ<1 and 0≤ v≤ 1<R this condition is always satisfied. Hence for parameters satisfying the first condition a Hopf bifurcation should occur. However, when solving the first condition for β, η, v or µ it turns out that not always a Hopf bifurcation occurs when varying one of these four parameters while keeping the other parameters constant, because for these parameters additional restrictions apply.

When solving the first condition for the risk aversion parameter a a Hopf bifurcation occurs given the parameter set {Θ} \ a if aH satisfies

aH= ⎛
⎜
⎜
⎝		 1-(1-µ)
v
R
⎞
⎟
⎟
⎠
λ R
2 γ σ2
qfund
qma
>0 .

When solving the first condition for the intensity of choice parameter β a Hopf bifurcation occurs given the parameter set {Θ} \ β and Cfund>Cma if βH satisfies
βH=
1-η
Cfund-Cma
ln(c),
where c is defined as
c=
b (1-mma)-mma
(1-mfund)-b mfund
,with b= ⎛
⎜
⎜
⎝		 ⎛
⎜
⎜
⎝		 1-(1-µ)
v
R
⎞
⎟
⎟
⎠
λ R
2 a γ σ2
⎞
⎟
⎟
⎠		 >0.
However, because β ≥ 0, there are possibly cases for which no Hopf bifurcation occurs, when varying β and keeping the other parameters constant if the logarithm ln(c) is taken over a value smaller than one.

When solving the first condition for the memory parameter η a Hopf bifurcation occurs given the parameter set {Θ} \ η if ηH satisfies

ηH=1-
β (Cfund-Cma)
ln(c)
.
However, because 0 ≤ η < 1, there are cases for which no Hopf bifurcation occurs, when varying η and keeping the other parameters constant.

When solving for the expected dispersion in return parameter σ2 a Hopf bifurcation occurs given the parameter set {Θ} \ σ2 if σH2 satisfies

σH2= ⎛
⎜
⎜
⎝		 1-(1-µ)
v
R
⎞
⎟
⎟
⎠
λ R
2 a γ
qfund
qma
>0 .
262