**6. Chaotic dynamics in finance model**

threshold. Models of observed tonic firing and spike bursting were simulated by considering both periodic and noisy form of the threshold variation. The effect of a mixed mode threshold on the spiking FHN system was also investigated. Here we present results of bifurcation analysis of different states of neuronal firing of FHN neuron by considering a discrete form of the system [34, 35]. Following [34], the

*Advances in Dynamical Systems Theory, Models, Algorithms and Applications*

where Δ refers to the integral step size and is treated here as a bifurcation

complex behavior as shown in bifurcation diagram (**Figure 19a**). It is readily observed that the temporal behavior is chaotic in the region 0*:*42≤ Δ ≤0*:*68. Thereafter windows of regular and chaotic regimes are observe till Δ ¼ 0*:*8 for *I* ¼ 0. A slight increase in *I* changes the dynamics to a more complex behavior as shown in

*Phase portrait showing quasiperiodic behavior for (a) I = 0 and chaotic behavior for (b) I = 1.*

In case of mixed mode threshold variation, the membrane potential *v* exhibits a

*vn*þ<sup>1</sup> ¼ *vn* þ Δ*α*½ � �*vn*ð Þ *vn* � 1 ð Þ� *vn* � *bn w* þ *I* (21) *wn*þ<sup>1</sup> ¼ *wn* þ Δð Þ *vn* � *δwn* (22)

discrete form of the FHN system may be written as:

parameter.

**Figure 19.**

**Figure 20.**

**50**

*Bifurcation of membrane potential. (a) I = 0; (b) I = 1.*

In mainstream economics, economic dynamics has assume great importance in recent years in view of the availability of market and other data. Economic dynamics has therefore influenced both micro- and macroeconomics. Therefore lot of research output has poured in explaining irregular micro-economic fluctuation, erratic business cycles, irregular growth and aperiodic behavior of economic data etc. Nonlinear systems provides an alternative simple and deterministic framework that easily can explain aperiodic or chaotic behaviors of various financial systems. One of the important features of nonlinear system is that the irregular/chaotic behavior supports an endogenous mechanism for the observed complexity in economic time series. As a result nonlinear dynamic framework has been applied to economic modeling and several examples are available in [36–46].

In the present work we revisit the synthetic chaotic financial model discussed in [45, 46] which is based on interest rate, investment demand and price index as dynamical variables. We numerically explored and analyze the complexity of the model using the multiscale entropy (MPE) frame work. In this section, we briefly describe the chaotic financial model and its basic characteristics. We also outlines the procedure of MPE for analyzing the complexity of the finance model.
