**7. Concluding remarks**

308 Nonlinearity, Bifurcation and Chaos – Theory and Applications

**Figure 12.** Eigenvalue curves for 2 2 ( , ) 1 ( )/2 12 1 2 *<sup>F</sup>*

**Figure 13.** Eigenvalue curves for 2 2 ( , ) cos 12 1 2 <sup>2</sup> *F*

 

  Numerical experiments with the calculation of eigenvalues and eigenvectors, realized for certain specified types of directivity pattern by the proposed algorithms, and comparison them with existing results obtained by other methods shows their efficiency (in terms of bilateral approximations and convergence rate). Developed and implemented algorithms for numerical finding of the branching lines of nonlinear integral equations, the kernels of which nonlinearly depend on two spectral parameters and their bifurcation points, yielded the new results, namely:


12 1 2 ( , ) cos |sin | <sup>2</sup> *F* this is 3 ( ) , what is shown in Fig. 10 and Fig. 11, respectively), which corresponds to the synthesized directivity patterns in which the variables are not separated.

 For the problems in which the function 1 2 *F*(,) does not allow separation of variables, we have found the solutions to the problem (19) (for example, for 2 2 12 1 2 *F*( , ) 1 ( )/2 and 2 2 1 2 1 2 ( , ) cos <sup>2</sup> *F* are 1 ( ) and 2 ( ) , shown in Fig. 12 and Fig. 13, respectively), which are supposed to exist only for the

diagrams, where the variables are separated. We have calculated the bifurcation point of eigenvalue curves for the problems in which

$$\text{the function } F(\xi\_1, \xi\_2) \text{ does not allow separation of variables (e.g. } F(\xi\_1, \xi\_2) = 1 - (\xi\_1^2 + \xi\_2^2)/2,$$

$$\pi / \boxed{1.3}$$

2 2 12 1 2 *F*( , ) 1 ( )/2 and 2 2 1 2 1 2 ( , ) cos <sup>2</sup> *F* ). For these diagrams there are

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

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 electrodynamic characteristics of the radiating system.

Note that such two-dimensional problem was studied also in the works [10, 17, 19], but there numerical results obtained for some directivity patterns 1 2 *F*(,) are not reliable.

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

$$\begin{split} \frac{d\boldsymbol{\lambda}}{d\mu} &= -\frac{\left[\det T\_n(\boldsymbol{\lambda}, \boldsymbol{\mu})\right]' \boldsymbol{\mu}}{\left[\det T\_n(\boldsymbol{\lambda}, \boldsymbol{\mu})\right]' \boldsymbol{\lambda}}, \\ \boldsymbol{\lambda}(\boldsymbol{\mu}\_1) &= \boldsymbol{\lambda}\_1. \end{split} \tag{51}$$

Numerical Algorithms of Finding the Branching Lines

and Bifurcation Points of Solutions for One Class of Nonlinear Integral Equations 311

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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 for Cauchy problem for each curve.
