**2.2. Problem of finding the branching lines**

The points of possible branching of solutions of integral equation (7) are such values of real physical parameters <sup>2</sup> 1 2 *cc R* , , in which homogeneous integral equation [18]

$$\mathfrak{u}(\tilde{\xi}\_{1},\tilde{\xi}\_{2},c\_{1},c\_{2}) = \tilde{\mathcal{T}}(c\_{1},c\_{2})\mathfrak{u}(\tilde{\xi}\_{1},\tilde{\xi}\_{2},c\_{1},c\_{2}) = \prod\_{\Omega} \mathcal{F}(\tilde{\xi}\_{1}^{\prime},\tilde{\xi}\_{2}^{\prime}) \mathcal{K}(\tilde{\xi}\_{1}^{\prime},\tilde{\xi}\_{2}^{\prime},\tilde{\xi}\_{1}^{\prime},c\_{1},c\_{2}) \frac{\mathfrak{u}(\tilde{\xi}\_{1}^{\prime},\tilde{\xi}\_{2}^{\prime},c\_{1},c\_{2})}{f\_{0}(\tilde{\xi}\_{1}^{\prime},\tilde{\xi}\_{2}^{\prime},c\_{1},c\_{2})} d\tilde{\xi}\_{1}^{\prime} d\tilde{\xi}\_{2}^{\prime}, \quad \text{(12)}$$

obtained by linearization of equation (7), has solutions distinct from identical zero [20]. Thus, we have obtained the nonlinear (with respect to parameters 1*c* and 2*c* ) twoparameter eigenvalue problem

$$\mu\left(\tilde{T}(\mathcal{A}\_1, \mathcal{A}\_2) - I\right)\mu(\tilde{\xi}\_1, \tilde{\xi}\_2, \mathcal{A}\_1, \mathcal{A}\_2) = 0,\ \mathcal{A}\_1 = c\_1,\ \mathcal{A}\_2 = c\_2.\tag{13}$$

It is easy to be convinced, that at arbitrary finite values 1*c* 0 , 2*c* 0 , the function <sup>01212</sup> *f* (, ,,) *c c* is the eigenfunction of equation (12). From this it follows, that the operator 1 2 *Tc c* (,) has a spectrum, which coincides with the first quadrant of the plane <sup>2</sup> *<sup>R</sup>* .

The problem consists in finding such range of real parameters 1 1 *c* and 2 2 *c* of the problem (13), for which there appear the solutions different from 01212 *f* (, ,,) *c c* .

It should be noted that in a special case, when it is possible to separate variables in the function 1 2 *F*( , ) , i.e. 1 2 *F*( , ) to present as 12 11 2 *F FF* ( , ) ( ) ( ) , the equation (12), provided that function 1212 *u cc* (, ,,) can also be presented as 1 212 11 22 *u cc u c u c* (, ,,) (,) (,) , decomposes in two independent one-parameter equations, i.e.

$$
\mu\_j(\xi\_{j'}c\_j) = T(c\_j)\mu\_j(\xi\_{j'}c\_j), \quad j = 1,2,3
$$

with operators

290 Nonlinearity, Bifurcation and Chaos – Theory and Applications

**Figure 8.** Amplitude directivity pattern of solution branched off from 01212

**Figure 9.** Optimal distribution of current on the radiators, which creates the branching solution

*f cc* (, ,,) 

$$T\_j(c\_j)u(\xi\_j, c\_j) = \int\_{-1}^{1} \frac{F\_j(\xi\_j)}{f\_0(\xi\_j, c\_j)} \frac{\sin N\_j \frac{c\_j}{\mathfrak{Q}}(\xi\_j - \xi\_j')}{\sin \frac{c\_j}{\mathfrak{Q}}(\xi\_j - \xi\_j')} u(\xi\_j', c\_j) d\xi\_j' \quad j = 1, 2... \times 1$$

The study of such equations is carried out in [2, 18], and it is possible to apply , for example, the algorithms of the work [11, 13, 15] to solution of such equations.

In the given work the numerical algorithms to solve more complicated problem when the variables are not separated, are proposed.
