**6.1 Chaotic financial model**

We consider a dynamic finance model composed of three coupled first order differential equation. This model describes the temporal evolution of the state variables viz. the interest rate *X*, the investment demand *Y* and the price index *Z*. The model is described as [39]:

$$\begin{aligned} \frac{dX}{dt} &= Z + (Y - a)X, \\ \frac{dY}{dt} &= \mathbf{1} - bY - X^2, \\ \frac{dZ}{dt} &= -X - cZ. \end{aligned} \tag{23}$$

Here *a*, *b* and *c* are positive constants and represent the saving amount, cost per investment and elasticity of demand of the commercial markets. First equation appears, representing the changes in *X*, as a result of contradiction in the investment market and structural adjustment from the goods prices. Second equation representing the changes in *Y* appears due to proportionality to the rate of investment and also to an inversion of the cost of investment and interest rate. The third equation emerges due to contradiction between supply and demand in commercial markets which is influenced by interest rates.

Eq. (23) has been numerically integrated using fourth-order Runge-Kutta method to obtain the time series of the dynamic variables *X*, *Y* and *Z*, shown in **Figure 21** with *a* ¼ 3*:*0, *b* ¼ 0*:*1, *c* ¼ 1*:*0 and initial condition ð Þ¼ *X*0, *Y*0, *Z*<sup>0</sup> ð Þ 2, 3, 2 . Similar choice of the parameter were made in [45].

*y* ð Þ*s <sup>j</sup>* <sup>¼</sup> <sup>1</sup> *s*

*Chaotic Dynamics and Complexity in Real and Physical Systems*

*DOI: http://dx.doi.org/10.5772/intechopen.96573*

the original time series.

<sup>1</sup>þ<sup>1</sup> <sup>≤</sup>*yn*<sup>þ</sup> *<sup>j</sup>*

<sup>2</sup>þ<sup>1</sup> ≤⋯≤*yn*<sup>þ</sup> *<sup>j</sup>*

*<sup>n</sup>*þ1

*yn*<sup>þ</sup> *<sup>j</sup>*

**Figure 23.**

**53**

X *js*

*Xi* (24)

*p*ð Þ Π *ln* ð Þ Π *:* (25)

� �,

*i*¼ð Þ *j*�1 *τ*þ1

where 1≤*i* ≤ *N=s* and *s* defines the scale factor. Each time series length is of size that is an integral multiple of *N=s*. For *s* ¼ 1, the coarse-grained time series is just

The second step involves the computation of permutation entropy [47] for each of the coarse-grained time series. For a coarse-grained time series *y <sup>j</sup>* we first consider the series of vector of length *<sup>m</sup>*, and obtain *Sm*ð Þ¼ *<sup>n</sup> yn*, *yn*þ1, <sup>⋯</sup>, *yn*þ*m*�<sup>1</sup>

<sup>1</sup>≤*<sup>n</sup>* <sup>≤</sup>ð Þ� *<sup>N</sup>=<sup>s</sup> <sup>m</sup>* <sup>þ</sup> 1. Subsequently, *Sm*ð Þ *<sup>n</sup>* is arranged in an increasing order viz.,

h i. For *<sup>m</sup>* different numbers, there will be *<sup>m</sup>*! possible

order patterns/structures Π which are termed as permutations. If *f*ð Þ Π denotes the frequency of order pattern Π, then the relative frequency and hence the probability *p* ¼ *f*ð Þ Π *=*ð Þ *N=s* � *m* þ 1 . The permutation entropy *H m*ð Þ therefore is given by

Π¼1

The maximum value of *H m*ð Þ is *log m*ð Þ! thus showing all permutations to have equal probability. Also, the time series is termed as regular if minimum value of *H m*ð Þ is zero. Therefore *H m*ð Þ the permutation entropy provide a quantitative measure of dynamical complexity of a time series as it refers to its local structures. It may be noted that the permutation entropy depends on the chosen value of *m*. For *m* < 3 , there will be very few distinct states and the foregoing scheme does not work satisfactorily. In the present analysis we have considered sufficiently large time

For the financial model, Eq. (23), we have simulated the permutation entropy as

a function of the scale *s* for *m* ¼ 6. The simulated results have been shown in **Figure 23a** where we observe a saturation behavior with increased value of the scale factor *s*. It is also observed that the permutation entropy at any scale *s* for the interest rate *X* time series is higher than the investment demand time series *Y* which

*H m*ð Þ¼�X*<sup>m</sup>*!

series and chosen *m* ¼ 6 to estimate the complexity measure *MPE*.

*Multiscale permutation of (a) Financial model time series, (b) Rossler model.*

**Figure 21.** *Temporal evolution of Finance model.*

**Figure 22.** *(a) Phase portrait of Finance model, (b) 3-D attractor of the Finance model.*

The representation of two dimensional phase portrait ð Þ *X*, *Y* and the attractor are shown in **Figure 22**. Obviously they represent chaotic dynamics of the temporal behavior shown in **Figure 22**.
