**8. References**

[1] Abramov, A. A., Ul'yanova, V. I. & Yukhno, L. F. (1998). The Argument Principle in a Spectral Problem for Systems of Ordinary Differential Equations with Singularities, *Comput. Math. Math. Phys,* Vol. 38: 57-63.

[2] Andriychuk, M. I., Voitovich, N. N., Savenko, P. O. & Tkachuk, V. P. (1993). *The antenna synthesis according to prescribed amplitude radiation pattern: numerical methods and algorithms*, Naukova Dumka, Kiev. [in Russian].

310 Nonlinearity, Bifurcation and Chaos – Theory and Applications

2 2

2 2

12 1 2 *F*( , ) 1 ( )/2

the function 1 2 *F*(,)

 

 

> 

12 1 2 *F*( , ) 1 ( )/2

for Cauchy problem for each curve.

*Comput. Math. Math. Phys,* Vol. 38: 57-63.

**Author details** 

B. M. Podlevskyi

**8. References** 

For the problems in which the function 1 2 *F*(,)

diagrams, where the variables are separated.

electrodynamic characteristics of the radiating system.

 

and 2 2

and 2 2

 

*F*

no known results. The results have been obtained for the first time.

there numerical results obtained for some directivity patterns 1 2 *F*(,)

is. In such approach it is necessary to solve the Cauchy problem

*F*

 

we have found the solutions to the problem (19) (for example, for

shown in Fig. 12 and Fig. 13, respectively), which are supposed to exist only for the

1 2 1 2 ( , ) cos <sup>2</sup>

We have calculated the bifurcation point of eigenvalue curves for the problems in which

Since the spectral parameters are the geometric and electromagnetic characteristics of radiating systems, the solution of this problem makes it possible to obtain the necessary information at the design stage, choosing the optimal ones with respect to the size and

Note that such two-dimensional problem was studied also in the works [10, 17, 19], but

To complete we shall mark, that the offered algorithm of calculation of derivatives of matrix determinant can be used and in the approach in which basis the implicit function theorem

det ( , )

 

 

*n*

det ( , )

*n*

,

for which the right part of equation (51) can be calculated by the algorithm of calculation of derivatives of matrix determinant. Besides by the algorithm, given in this paper, it is possible numerically to define a number of eigenvalues, and, therefore, the eigenvalue curves, which are in the given range of spectral parameters and to calculate the initial value

[1] Abramov, A. A., Ul'yanova, V. I. & Yukhno, L. F. (1998). The Argument Principle in a Spectral Problem for Systems of Ordinary Differential Equations with Singularities,

,

1 1

 

*T d d T*

*Institute of Applied Problems of Mechanics and Mathematics of NASU, Ukraine* 

1 2 1 2 ( , ) cos <sup>2</sup>

 are 1

does not allow separation of variables (eg, 2 2

 

does not allow separation of variables,

 ( ) 

). For these diagrams there are

   and 2 ( ) ,

12 1 2 *F*( , ) 1 ( ) / 2,

are not reliable.

(51)

 

	- [19] Savenko, Petro & Tkach, Myroslava (2010). Numerical Approximation of Real Finite Nonnegative Function by Modulus of Discrete Fourier Transform, *Applied Math.*, Vol. 1 (1): 65-75.

**Chapter 12** 

© 2012 Verbič, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

distribution, and reproduction in any medium, provided the original work is properly cited.

and reproduction in any medium, provided the original work is properly cited.

Heterogeneous agent models are present in various fields of economic analysis, such as market maker models, exchange rate models, monetary policy models, overlapping generations models and models of socio-economic behaviour. Yet the field with the most systematic and perhaps most promising nonlinear dynamic approach seems to be asset price modelling. Contributions by Brock and Hommes (1998), LeBaron (2000), Hommes *et al*. (2002), Chiarella and He (2002), Chiarella *et al*. (2003), Gaunersdorfer *et al*. (2003), Brock *et al*. (2005), Hommes *et al*. (2005), and Hommes (2006) thoroughly demonstrate how a simple standard pricing model is able to lead to complex dynamics that makes it extremely hard to predict the evolution of prices in asset markets. The main framework of analysis of such asset pricing models constitutes a financial market application for the evolutionary selection of expectation rules, introduced by Brock and Hommes (1997a) and is called the adaptive

As a model in which different agents have the ability to switch beliefs, the adaptive belief system in a standard discounted value asset pricing set-up is derived from mean-variance maximization and extended to the case of heterogeneous beliefs (Hommes, 2006, p. 47). It can be formulated in terms of deviations from a benchmark fundamental and therefore used in experimental and empirical testing of deviations from the rational expectations benchmark. Agents are boundedly rational, act independently of each other and select a forecasting or investment strategy based upon its recent relative performance. The key feature of such systems, which often incorporate active learning and adaptation, is endogenous heterogeneity (*cf*. LeBaron, 2002), which means that markets can move through periods that support a diverse population of beliefs, and others in which these beliefs and

**Memory and Asset Pricing Models** 

**with Heterogeneous Beliefs** 

Additional information is available at the end of the chapter

strategies might collapse down to a very small set.

Miroslav Verbič

**1. Introduction** 

belief system.

http://dx.doi.org/10.5772/48070

[20] Vainberg, M. M. & Trenogin, V. A. (1969). *Branching theory of the solution of nonlinear equations,* Nauka, Moscow. [in Russian].

**Chapter 12** 
