**4.2. Stability of the steady state and bifurcations**

The local stability of a steady state is again determined by the eigenvalues of the Jacobian matrix. At the fundamental steady state *Xeq* = (0, 0, 0, 0, 0) the Jacobian matrix exhibits the characteristic equation that is in our case given by:

$$\mathbf{g}(\boldsymbol{\lambda}) = -\left(\boldsymbol{\lambda}^2 - \left(\boldsymbol{w} - \frac{2}{3R}k\rho\theta b^2(\boldsymbol{w} - 1)\right)\boldsymbol{\lambda} - \frac{2}{3}k\rho\theta b^2(\boldsymbol{w} - 1)\right)\boldsymbol{\lambda}\left(\boldsymbol{w} - \boldsymbol{\lambda}\right)^2 = \boldsymbol{0} \,\tag{62}$$

which has the following three solutions, two of them being double: 1 0 , 2,3 *w* and <sup>2</sup> <sup>2</sup> <sup>2</sup> 2 2 4,5 <sup>1</sup> 2 (1 ) 3 2 ( 1) 3 24 (1 ) <sup>6</sup> *b k w Rw b k w Rw b k w R R* .

The fundamental steady state is stable for 1 , which in our case is limited to the product of eigenvalues 4,5 being smaller than one, i.e. <sup>2</sup> <sup>2</sup> ( 1) 1 <sup>3</sup> *kb w* . In terms of the intensity of choice this happens for 2 3 2 ( 1) *kb w* , while in terms of the memory strength this is guaranteed for 2 <sup>3</sup> <sup>1</sup> 2 *w k b* .

Thus we can state the following lemma.

**Lemma 5:** *The fundamental steady state in case fundamentalists versus opposite biased beliefs is globally stable for* <sup>2</sup> 3 2 ( 1) *kb w . Memory affects the stability of this steady state by restricting it to the given interval of the parameter value.* 

### **Proof of Lemma 5:**

334 Nonlinearity, Bifurcation and Chaos – Theory and Applications

exp (1 )

*expression (56) we obtain a new expression for the fractions:* 

*dynamic system defined by (55) and (57) is thus of the form:* 

exp (1 )

exp (1 )

exp (1 )

,

*where Uh t*,

*h t H*

,

*is increasing and* \* ( ) *V rx*

*h t H*

*where the right-hand side function is defined as:* 

*and (57) or expression (58) are determined by:* 

*(1998, p. 1271), a straightforward computation shows that:* 

*After subtracting off identical terms from the exponents of both numerator and denominator in* 

1 2 ( ), *Rx V x Rx t kt t* 

*wU y w kb y V y bn b wU y w kb y*

,2 1

exp ( ) (1 )

*it t i t i*

exp ( ) (1 )

<sup>1</sup> <sup>1</sup> d d exp exp ( ) exp d d exp exp

> 1 1 1 *H H H*

*y y kb y kb y*

 *kn b kn b n b*

*<sup>i</sup> <sup>i</sup> <sup>i</sup> <sup>i</sup>*

1 ,2 1

*Since it follows from (52) and (53) that*  \* \* \* <sup>1</sup> *U kx R b Rx h h , steady states of expressions (55)* 

\* \*\* \* ( ) () *Rx V x Rx V rx*

*where r = R – 1. Since a steady state has to satisfy expression (60), following Brock and Hommes* 

*H H h h h k h i H i h H*

*kb kb y kb y V y kb y b*

*k t H h hh h*

( ) .

 

*ht t h t H*

 

1 1 <sup>2</sup>

*<sup>k</sup> decreasing in x\*. It then follows from expression (60) that the steady* 

 

 

2 2

<sup>2</sup> <sup>2</sup> 0, *h h*

*h hh hh hh h h hh hh h*

*kn b kn b b*

*where the inequality follows from the fact that the term between square brackets can be interpreted as the variance of the stochastic process, where each bh is drawn with probability nh. Therefore,* ( ) *V y*

*wU w k x Rx b n h H wU w k x Rx b*

1 , 2 1 2

*i t t ti i*

 , 2 1 2

 *is the fitness of trader type h, adjusted by subtracting off identical terms as above. The* 

*h t t th*

*h t t tht*

1 , 2 1 2 2

*wU w k x Rx b Rx n h H wU w k x Rx b Rx*

*i t t tit i*

 , 2 1 2 2

;1 .

;1 ,

(57)

(58)

1

*k*

(59)

*k k* (60)

 

*kb b* (61)

(56)

From the characteristic equation (62) we can observe five eigenvalues. The first three eigenvalues always assure stability, while the last two eigenvalues limit stability. Given k > 0, b > 0, 0, R > 1 and 0 w 1, the condition for stability in terms of implies 2 3 2 ( 1) *kb w* . Similarly, the condition for stability in terms of w indicates 2 <sup>3</sup> <sup>1</sup> 2 *w k b* .

Memory therefore affects the stability of the steady state as shown.

If we now take a look at the eigenvalues 4,5 of the characteristic equation (62), which are of interest in our case, we can observe that a saddle-node bifurcation would occur for:

$$
\beta = \frac{3R}{2b^2k(1-R)}.\tag{63}
$$

Memory and Asset Pricing Models with Heterogeneous Beliefs 337

*. This happens either when* 

a bifurcation occurs at

we arrive at more

*\*, memory affects the* 

. The same

 = \*,

( 1) 1 <sup>3</sup> *kb w* 

\* *the eigenvalues have to be complex and thus a Hopf bifurcation occurs.* 

\* complex dynamical behaviour appears, which is

*product of eigenvalues* 4,5

<sup>3</sup> \* 2 ( 1) *kb w*

**4.3. Numerical analysis** 

equilibrium or into an attractor.

bifurcation occurs later. For

 =   

*Since* <sup>2</sup>

*conclude that for* 

four analytical tools will be used than in Section 3.4.

clearly demonstrated by the simulations. For increasing

 > 

observed period doubling route to chaos (rational route to randomness). By examining largest Lyapunov characteristic exponent (LCE) plots of

 *has to be equal one, i.e.* <sup>2</sup> <sup>2</sup>

*Since the memory strength parameter is present in the expression for* 

*we have two real eigenvalues with product equal to one or a complex conjugate pair of eigenvalues.* 

*emergence of this bifurcation; the higher the value of this parameter, the higher the bifurcation value.* 

Our numerical analysis in the case of fundamentalists and opposite biased beliefs will be conducted for fixed values of parameters *R* = 1.1, *k* = 1.0, *b*2 = 0.2 and *b*3 = –0.2. We shall thus

Dynamic behaviour of the system can again first and foremost be determined by investigating bifurcation diagrams. From Figure 5 we can observe that for low values of

we have a stable steady state, i.e. the fundamental steady state. As has been proven in Lemma 4, the position of this steady state, i.e. *xeq* = 0, is independent of the memory, which is

which is a Hopf bifurcation; the steady state becomes unstable and an attractor appears, consisting of an invariant circle around the (unstable) steady state. It is again a supercritical Hopf bifurcation, where the steady state gradually changes either into an unstable

The bifurcation value varies with changing memory strength parameter, as given by expression in Lemma 6. As can also be seen from Figure 5 at higher memory strength the

interspersed with stable cycles. As we have already discovered in Section 4.2, irrespective of the amount of additional memory that is taken into account such a (bifurcation) route to complicated dynamics is different from that in the presence of contrarians, where we

precise conclusions about the dynamic behaviour of the system. It can be seen from Figure 5 that the largest LCE is smaller than 0 and the system is thus stable until the bifurcation. At the bifurcation value a qualitative change in dynamics occurs, i.e. a Hopf bifurcation. The dynamics is somewhat more complicated. Namely, we can observe that the largest LCE after

\* is non-positive, but mainly close to 0, which implies periodic and quasi-periodic dynamics, i.e. for high values of the intensity of choice only regular (quasi-)periodic fluctuations around the unstable fundamental steady state occur. An important finding is that the predominating quasi-periodic dynamics does not seem to evolve to chaotic dynamics and the route to complex dynamics is indeed different from the routes examined so far.

vary the memory strength parameter *w* and the intensity of choice parameter

 *falls into the interval (65) for any given finite memory strength, we can* 

This can never hold, since 0 and the left-hand side is always non-negative, while *R* 1 and the right-hand side is always negative. On the other hand, a period doubling bifurcation would occur for:

$$\beta = \frac{3R(w+1)}{2b^2k(R+1)(w-1)} \,. \tag{64}$$

This can never hold either, since 0 and the left-hand side is again always non-negative, while 0 1 *w* and the right-hand side is either negative or not defined.

The remaining qualitative change of the three discussed in Section 4.3 is the Hopf bifurcation. For this to occur, a complex conjugate pair of eigenvalues has to cross the unit circle. Eigenvalues 4,5 are complex for <sup>2</sup> <sup>2</sup> 2 2 2 ( 1) 3 24 (1 ) 0 *b k w Rw b k w R* , which produces the following interval of values:

$$\frac{R\left(3w - 6R - 2\sqrt{R(R - w)}\right)}{2b^2k(w - 1)} < \beta < \frac{R\left(3w - 6R + 2\sqrt{R(R - w)}\right)}{2b^2k(w - 1)}.\tag{65}$$

We therefore state the following lemma.

**Lemma 6:** There exists an intensity of choice value \* such that the fundamental steady state, which is stable for 0 \* , becomes unstable and remains such for \* . For 2 <sup>3</sup> \* 2 ( 1) *kb w* the system exhibits a Hopf bifurcation. Memory affects the emergence of

this bifurcation, viz. with more memory the bifurcation occurs later.

As we have just established, in the case of fundamentalists versus opposite biased beliefs increasing intensity of choice to switch predictors destabilizes the fundamental steady state. This happens through a Hopf bifurcation. We can thus conclude, as did Brock and Hommes (1998) for the simpler version of the model, that in the presence of biased agents the first step towards complicated price fluctuations is different from that in the presence of contrarians. This fact does not change when we take memory into account.

### **Proof of Lemma 6:**

*When increases, terms with in the expressions for the eigenvalues 4,5 increase as well, and one of the eigenvalues has to cross the unit circle at some critical*  \* *. The fundamental steady state thus becomes unstable. Since it is obvious from the characteristic equation (62) that for all*  0 *we have g*(1) 0  *and g*( 1) 0 *, a bifurcation has to occur. At the moment of the bifurcation the*  *product of eigenvalues* 4,5  *has to be equal one, i.e.* <sup>2</sup> <sup>2</sup> ( 1) 1 <sup>3</sup> *kb w . This happens either when we have two real eigenvalues with product equal to one or a complex conjugate pair of eigenvalues. Since* <sup>2</sup> <sup>3</sup> \* 2 ( 1) *kb w falls into the interval (65) for any given finite memory strength, we can conclude that for*  \* *the eigenvalues have to be complex and thus a Hopf bifurcation occurs. Since the memory strength parameter is present in the expression for \*, memory affects the emergence of this bifurcation; the higher the value of this parameter, the higher the bifurcation value.* 
