**3.4. Numerical analysis**

Our numerical analysis in the case of fundamentalists and contrarians will be conducted for fixed values of parameters *R* = 1.1, *k* = 1.0, *C* = 1.0 and *g* = –1.5. We shall thus vary the intensity of choice parameter and of course the memory strength parameter *w*. Four analytical tools will be used2; bifurcation diagrams, largest Lyapunov characteristic exponent (LCE) plots, phase plots, and time series plots.

The dynamic behaviour of the system can first and foremost be determined by investigating bifurcation diagrams. In Figure 1 the bifurcation diagrams for two different values of the memory strength parameter are presented. 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 1, the position of this steady state, i.e. *xeq* = 0, is independent of the memory, which is clearly demonstrated by the simulations. For increasing a (primary) period doubling bifurcation occurs at = \*; the steady state becomes unstable and a stable period two-cycle appears, as

<sup>2</sup> However, we will not discuss these tools here in more detail, since they are fairly well-known; instead we will direct the interested reader to more detailed discussions in Arrowsmith and Place (1990), Shone (1997), and Brock and Hommes (1998).

proven in Lemma 3. As can be seen from the simulations, this bifurcation value is also independent of the memory. The stability of the steady state is thus unaffected by the memory, as proven in Lemma 2.

326 Nonlinearity, Bifurcation and Chaos – Theory and Applications

2

of the attractor numerically in the next section.

**3.4. Numerical analysis** 

intensity of choice parameter

occurs at

Hommes (1998).

 =   

*. If expression (38) is satisfied, it then follows from expressions* \* <sup>2</sup> <sup>1</sup> *<sup>R</sup>*

*. Since the memory strength parameter does not affect the difference in* 

 = 

and of course the memory strength parameter *w*. Four

*(40) that*  \*\* \*\* ( , ),( , ) *xm xm is a period two-cycle. Finally, for –2R < g < –R, the fundamental* 

*steady state is unstable and expression (40) has solutions* \* *x if and only if* 

As in the paper of Brock and Hommes (1998), very strong contrarians with *g* < –2*R* may lead to the existence of a period two-cycle, even when there are no costs for fundamentalists (*C* = 0). When the fundamentalists' costs are positive (*C* > 0), strong contrarians with –2*R* < *g* < –*R* may

bifurcation occurs in which the fundamental steady state becomes unstable and a (stable) period two-cycle is created, with one point above and the other one below the fundamental.

the period two-cycle becomes unstable and a Hopf bifurcation of this period two-cycle occurs, as in Brock and Hommes (1998). The model would then get an attractor consisting of two invariant circles around each of the two (unstable) period two-points, one lying above and the other one below the fundamental. Immediately after such a Hopf bifurcation, the price dynamics is either periodic or quasi-periodic, jumping back and forth between the two circles. The proof of this phenomenon is not straightforward due to the non-zero period points, although the 5-dimensional system (31) – (35) is still symmetric with respect to the origin. We shall thus demonstrate the occurrence of the Hopf bifurcation and the emergence

Our numerical analysis in the case of fundamentalists and contrarians will be conducted for fixed values of parameters *R* = 1.1, *k* = 1.0, *C* = 1.0 and *g* = –1.5. We shall thus vary the

analytical tools will be used2; bifurcation diagrams, largest Lyapunov characteristic

The dynamic behaviour of the system can first and foremost be determined by investigating bifurcation diagrams. In Figure 1 the bifurcation diagrams for two different values of the

a stable steady state, i.e. the fundamental steady state. As has been proven in Lemma 1, the position of this steady state, i.e. *xeq* = 0, is independent of the memory, which is clearly

2 However, we will not discuss these tools here in more detail, since they are fairly well-known; instead we will direct the interested reader to more detailed discussions in Arrowsmith and Place (1990), Shone (1997), and Brock and

\*; the steady state becomes unstable and a stable period two-cycle appears, as

memory strength parameter are presented. We can observe that for low values of

*fractions of belief types, memory does not affect the position of the period two-cycle.* 

When the intensity of choice further increases, we are likely to find a value

lead to a period two-cycle. As the intensity of choice increases to

exponent (LCE) plots, phase plots, and time series plots.

demonstrated by the simulations. For increasing

*m*

*g and* 

\*, a period doubling

a (primary) period doubling bifurcation

we have

\*\*, for which

 = 

*solutions, x\**

*m m*

\* tanh

 *and –x\**

*eq C*

If increases further, indeed a (secondary) Hopf bifurcation occurs at = \*\*, as has been claimed in Section 3.3; the period two-cycle becomes unstable and an attractor appears consisting of two invariant circles around each of the two (unstable) period two-points, one lying above and the other one below the fundamental. It is a supercritical Hopf bifurcation, where the steady state gradually changes either into an unstable equilibrium or into an attractor (*cf*. Guckenheimer and Holmes, 1983; Frøyland, 1992; Kuznetsov, 1995). The position of the period two-cycle is independent of the memory, but it is not independent of the intensity of choice, as can be seen from expression (40). Numerical simulations suggest that the secondary bifurcation value also does not vary with changing memory strength parameter *w*. For > \*\* chaotic dynamic behaviour appears, which is interspersed with many (mostly higher order) stable cycles. Such a bifurcation route to chaos was also called the rational route to randomness (Brock and Hommes, 1997a), while the last part of it has been referred to as the breaking of an invariant circle.

**Notes:** Horizontal axis represents the intensity of choice (). Vertical axis represents deviations of the price from the fundamental value (*x*) in the upper two diagrams and the value of the largest LCE in the lower two diagrams, respectively. The diagrams differ with respect to the memory strength parameter *w*; the left one corresponds to *w* = 0.3, while the right one corresponds to *w* = 0.9.

**Figure 1.** Bifurcation diagrams and Largest LCE plots of in case of fundamentalists versus contrarians

By examining largest Lyapunov characteristic exponent (LCE) plots of we arrive at the same conclusions about the dynamic behaviour of the system. It can be seen from Figure 1 that the largest LCE is smaller than 0 and the system is thus stable until the primary bifurcation, which is independent of memory. At the bifurcation value, a qualitative change in dynamics occurs, i.e. a period doubling bifurcation and we obtain a stable period twocycle. Largest LCE is again smaller than 0 and the system is thus stable until the secondary bifurcation. At this bifurcation value, again a qualitative change in dynamics occurs, i.e. a Hopf bifurcation, but the dynamics is more complicated.

Memory and Asset Pricing Models with Heterogeneous Beliefs 329

= 9

= 25 (with small noise)

= 9

= 25 (with small noise)

**Notes:** Horizontal axis represents deviations of the price from the fundamental value (*xt*). Vertical axis represents lagged deviations of the price from the fundamental value (*xt*–1). The groups of four diagrams differ with respect to the memory strength parameter *w*; the left group corresponds to *w* = 0.3, while the right group corresponds to *w* = 0.9.

= 4

= 25 (without noise)

= 4

= 25 (without noise)

**Notes:** Horizontal axis represents deviations of the price from the fundamental value (*xt*). Vertical axis represents the fraction of fundamentalists (*n*1,*<sup>t</sup>*). The groups of four diagrams differ with respect to the memory strength parameter *w*;

the left group corresponds to *w* = 0.3, while the right group corresponds to *w* = 0.9. **Figure 3.** Phase plots of (*xt*, *n*1,*<sup>t</sup>*) in case of fundamentalists versus contrarians

= 25 (with small noise)

**Figure 2.** Phase plots of (*xt*, *xt*–1) in case of fundamentalists versus contrarians

= 9

= 25 (with small noise)

= 9

= 4

= 25 (without noise)

= 4

= 25 (without noise)

For lower values of *w* the largest LCE after \*\* is non-positive, but close to 0, which implies quasi-periodic dynamics. After some transient period the largest LCE becomes mainly positive with exceptions, which implies chaotic dynamics, interspersed with stable cycles. In fact, the largest LCE plot has a fractal structure (*cf*. Brock and Hommes, 1998, p. 1258). In the case of *w* = 0.9 the global dynamics after \*\* immediately becomes chaotic. Memory thus certainly affects the dynamics after the secondary bifurcation. Since the latter is a period doubling bifurcation, we are talking about period doubling routes to chaos.

Next, we shall examine plots of the attractors in the (*xt*, *xt*–1) plane and in the (*xt*, *n*1,*<sup>t</sup>*) plane3 without noise and with IID noise added to the supply of risky shares. In the upper left plot of each of the four parts of Figures 2 and 3 we can first observe the appearance of an attractor for the intensity of choice beyond the secondary bifurcation value. The orbits converge on such an attractor consisting of two invariant 'circles' around each of the two (unstable) period two-points4, one lying above and the other one below the fundamental value. As the intensity of choice increases, the circles 'move' closer to each other. In the upper right and lower left plot of each of the four parts of Figures 2 and 3 we can observe that the system seems already to be close to having a homoclinic orbit. The stable manifold of the fundamental steady state, (0, ) *<sup>s</sup> eq W m* , contains the vertical segment, *xeq* = 0, whereas the unstable manifold, (0, ) *<sup>u</sup> eq W m* , has two branches, one moving to the right and one to the left. Both of them are then 'folding back' close to the stable manifold.

For as Brock and Hommes (1998, p. 1254) have proven for the asset pricing model without additional memory, at infinite intensity of choice and strong contrarians, *g* < –*R*, that unstable manifold (0, 1) *W<sup>u</sup>* is bounded and all orbits converge on the saddle point (0, –1). In particular, all points of the unstable manifold converge on (0, –1) and are thus also on the stable manifold. Consequently, the system has homoclinic orbits for infinite intensity of choice. In the case of strong contrarians and high intensity of choice it is therefore reasonable to expect that we will obtain a system close to having a homoclinic intersection between the stable and unstable manifolds of the fundamental steady state. This is indeed what can be observed from the lower left plot of each of the two parts of Figures 2 and 3 and it suggests the occurrence of chaos for high intensity of choice. As can be seen from the lower right plot of each of the two parts of Figures 2 and 3, the addition of small dynamic noise to the system does not alter our findings.

<sup>3</sup> Attractors in the (*xt*, *n*2,*<sup>t</sup>*) plane are just flipped (rotated by 180 degrees) images of attractors in the (*xt*, *n*1,*<sup>t</sup>*) plane and will thus not be separately examined.

<sup>4</sup> Though we are topologically speaking about circles, the actual shape of such an attractor can be quite diverse, as seen from the figures.

**Notes:** Horizontal axis represents deviations of the price from the fundamental value (*xt*). Vertical axis represents lagged deviations of the price from the fundamental value (*xt*–1). The groups of four diagrams differ with respect to the memory strength parameter *w*; the left group corresponds to *w* = 0.3, while the right group corresponds to *w* = 0.9.

**Figure 2.** Phase plots of (*xt*, *xt*–1) in case of fundamentalists versus contrarians

328 Nonlinearity, Bifurcation and Chaos – Theory and Applications

Hopf bifurcation, but the dynamics is more complicated.

For lower values of *w* the largest LCE after

case of *w* = 0.9 the global dynamics after

of the fundamental steady state, (0, )

noise to the system does not alter our findings.

will thus not be separately examined.

from the figures.

By examining largest Lyapunov characteristic exponent (LCE) plots of

same conclusions about the dynamic behaviour of the system. It can be seen from Figure 1 that the largest LCE is smaller than 0 and the system is thus stable until the primary bifurcation, which is independent of memory. At the bifurcation value, a qualitative change in dynamics occurs, i.e. a period doubling bifurcation and we obtain a stable period twocycle. Largest LCE is again smaller than 0 and the system is thus stable until the secondary bifurcation. At this bifurcation value, again a qualitative change in dynamics occurs, i.e. a

doubling bifurcation, we are talking about period doubling routes to chaos.

left. Both of them are then 'folding back' close to the stable manifold.

quasi-periodic dynamics. After some transient period the largest LCE becomes mainly positive with exceptions, which implies chaotic dynamics, interspersed with stable cycles. In fact, the largest LCE plot has a fractal structure (*cf*. Brock and Hommes, 1998, p. 1258). In the

certainly affects the dynamics after the secondary bifurcation. Since the latter is a period

Next, we shall examine plots of the attractors in the (*xt*, *xt*–1) plane and in the (*xt*, *n*1,*<sup>t</sup>*) plane3 without noise and with IID noise added to the supply of risky shares. In the upper left plot of each of the four parts of Figures 2 and 3 we can first observe the appearance of an attractor for the intensity of choice beyond the secondary bifurcation value. The orbits converge on such an attractor consisting of two invariant 'circles' around each of the two (unstable) period two-points4, one lying above and the other one below the fundamental value. As the intensity of choice increases, the circles 'move' closer to each other. In the upper right and lower left plot of each of the four parts of Figures 2 and 3 we can observe that the system seems already to be close to having a homoclinic orbit. The stable manifold

the unstable manifold, (0, ) *<sup>u</sup> eq W m* , has two branches, one moving to the right and one to the

For as Brock and Hommes (1998, p. 1254) have proven for the asset pricing model without additional memory, at infinite intensity of choice and strong contrarians, *g* < –*R*, that unstable manifold (0, 1) *W<sup>u</sup>* is bounded and all orbits converge on the saddle point (0, –1). In particular, all points of the unstable manifold converge on (0, –1) and are thus also on the stable manifold. Consequently, the system has homoclinic orbits for infinite intensity of choice. In the case of strong contrarians and high intensity of choice it is therefore reasonable to expect that we will obtain a system close to having a homoclinic intersection between the stable and unstable manifolds of the fundamental steady state. This is indeed what can be observed from the lower left plot of each of the two parts of Figures 2 and 3 and it suggests the occurrence of chaos for high intensity of choice. As can be seen from the lower right plot of each of the two parts of Figures 2 and 3, the addition of small dynamic

3 Attractors in the (*xt*, *n*2,*<sup>t</sup>*) plane are just flipped (rotated by 180 degrees) images of attractors in the (*xt*, *n*1,*<sup>t</sup>*) plane and

4 Though we are topologically speaking about circles, the actual shape of such an attractor can be quite diverse, as seen

\*\* is non-positive, but close to 0, which implies

\*\* immediately becomes chaotic. Memory thus

*<sup>s</sup> eq W m* , contains the vertical segment, *xeq* = 0, whereas

we arrive at the

**Notes:** Horizontal axis represents deviations of the price from the fundamental value (*xt*). Vertical axis represents the fraction of fundamentalists (*n*1,*<sup>t</sup>*). The groups of four diagrams differ with respect to the memory strength parameter *w*; the left group corresponds to *w* = 0.3, while the right group corresponds to *w* = 0.9.

**Figure 3.** Phase plots of (*xt*, *n*1,*<sup>t</sup>*) in case of fundamentalists versus contrarians

Again, we can observe that memory has an impact on the global dynamics of the system. That is, both the convergence of the system on an attractor consisting of two invariant 'circles' around each of the two unstable period two-points and the 'moving' of the circles closer to each other seem to be happening faster (at lower intensity of choice) when more memory is present in the model. Moreover, at the same intensity of choice we seem to be closer to obtaining a system that has a homoclinic intersection between the stable and unstable manifolds of the fundamental steady state when the memory strength is higher.

Memory and Asset Pricing Models with Heterogeneous Beliefs 331

**Notes:** Horizontal axis represents the time (*t*). Vertical axis in each pair of time series plots first represents deviations of the price from the fundamental value (*xt*), and then the fraction of fundamentalists (*n*1,*<sup>t</sup>*). The plots on the left-hand side and the right-hand side of the figure differ with respect to the memory strength parameter *w*; the ones on the left

**Figure 4.** Time series of prices and fractions in case of fundamentalists versus contrarians

correspond to *w* = 0.3, while the ones on the right to *w* = 0.9.

Finally, we shall examine time series plots of deviations of the price from the fundamental value and of the fraction of fundamentalists5. Figure 4 shows some time series corresponding to the attractors in Figures 2 and 3, with and without noise added to the supply of risky shares. Similarly to the findings of Brock and Hommes (1998), we can observe that the asset prices are characterized by an irregular switching between a stable phase with prices close to their (unstable) fundamental value and an unstable phase of up and down price fluctuations with increasing amplitude.

This irregular switching is of course reflected in the fractions of fundamentalists and contrarians in the market. Namely, when the oscillations of the price around the unstable steady state gain sufficient momentum, it becomes profitable for the trader to follow efficient market hypothesis fundamental value despite the costs that are involved in this strategy. The fraction of fundamentalists approaches unity and the asset price stabilizes. But then the nonzero costs of fundamentalists bring them into position where they are unable to compete in the market; the fraction of fundamentalists rapidly decreases to zero, while the fraction of contrarians with no costs approaches unity with equal speed. The higher the intensity of choice, *ceteris paribus*, the faster this transition is complete; when approaches the neoclassical limit, the entire mass of traders tends to use the best predictor with respect to costs, i.e. the strategy with the highest fitness.\

Additional memory does not change the pattern of asset prices *per se*, but it does affect its period. Namely, at the same intensity of choice and higher memory strength the period of this irregular cycle appears to be elongated on average, in such a way that the stable phase with prices close to their fundamental value lasts longer, while the duration of the unstable phase of up and down price fluctuations does not change significantly. The effect of including more memory thus mainly appears to be stabilizing with regard to asset prices. With regard to fractions of different trader types we could say that including additional memory affects the transition from the short period of fundamentalists' dominance to the longer period of contrarians' dominance in the market. This transition takes more time to complete at the same intensity of choice. More memory thus causes the traders to stick longer to the strategy that has been profitable in the past, but might not be so profitable in the recent periods.

<sup>5</sup> Since the fraction of contrarians is just the unity complement of the fraction of fundamentalists, i.e. *n*1,*<sup>t</sup>* + *n*2,*<sup>t</sup>* = 1, the former will thus not be separately graphically examined.

and down price fluctuations with increasing amplitude.

to costs, i.e. the strategy with the highest fitness.\

former will thus not be separately graphically examined.

the recent periods.

Again, we can observe that memory has an impact on the global dynamics of the system. That is, both the convergence of the system on an attractor consisting of two invariant 'circles' around each of the two unstable period two-points and the 'moving' of the circles closer to each other seem to be happening faster (at lower intensity of choice) when more memory is present in the model. Moreover, at the same intensity of choice we seem to be closer to obtaining a system that has a homoclinic intersection between the stable and unstable manifolds of the fundamental steady state when the memory strength is higher.

Finally, we shall examine time series plots of deviations of the price from the fundamental value and of the fraction of fundamentalists5. Figure 4 shows some time series corresponding to the attractors in Figures 2 and 3, with and without noise added to the supply of risky shares. Similarly to the findings of Brock and Hommes (1998), we can observe that the asset prices are characterized by an irregular switching between a stable phase with prices close to their (unstable) fundamental value and an unstable phase of up

This irregular switching is of course reflected in the fractions of fundamentalists and contrarians in the market. Namely, when the oscillations of the price around the unstable steady state gain sufficient momentum, it becomes profitable for the trader to follow efficient market hypothesis fundamental value despite the costs that are involved in this strategy. The fraction of fundamentalists approaches unity and the asset price stabilizes. But then the nonzero costs of fundamentalists bring them into position where they are unable to compete in the market; the fraction of fundamentalists rapidly decreases to zero, while the fraction of contrarians with no costs approaches unity with equal speed. The higher the

the neoclassical limit, the entire mass of traders tends to use the best predictor with respect

Additional memory does not change the pattern of asset prices *per se*, but it does affect its period. Namely, at the same intensity of choice and higher memory strength the period of this irregular cycle appears to be elongated on average, in such a way that the stable phase with prices close to their fundamental value lasts longer, while the duration of the unstable phase of up and down price fluctuations does not change significantly. The effect of including more memory thus mainly appears to be stabilizing with regard to asset prices. With regard to fractions of different trader types we could say that including additional memory affects the transition from the short period of fundamentalists' dominance to the longer period of contrarians' dominance in the market. This transition takes more time to complete at the same intensity of choice. More memory thus causes the traders to stick longer to the strategy that has been profitable in the past, but might not be so profitable in

5 Since the fraction of contrarians is just the unity complement of the fraction of fundamentalists, i.e. *n*1,*<sup>t</sup>* + *n*2,*<sup>t</sup>* = 1, the

approaches

intensity of choice, *ceteris paribus*, the faster this transition is complete; when

**Notes:** Horizontal axis represents the time (*t*). Vertical axis in each pair of time series plots first represents deviations of the price from the fundamental value (*xt*), and then the fraction of fundamentalists (*n*1,*<sup>t</sup>*). The plots on the left-hand side and the right-hand side of the figure differ with respect to the memory strength parameter *w*; the ones on the left correspond to *w* = 0.3, while the ones on the right to *w* = 0.9.

**Figure 4.** Time series of prices and fractions in case of fundamentalists versus contrarians
