**6. Analysis of numerical results**

In conducting a series of numerical experiments on the synthesis of antenna arrays, Algorithms 1 and 2 were used to find the curves of eigenvalues for two-parameter eigenvalue problem, which are the branching lines of solutions of nonlinear synthesis equation (7) and their bifurcation points. Numerical calculations were carried out as for the problems in which in the function 1 2 *F*(,) that describes the given directivity pattern of the array, the variables are separated and are not separated.

In Fig. 10 - Fig. 13 are shown the curves of eigenvalues for four problems, in which the given

directivity patterns are defined by the formulas 1 2 *F*(,)1 , 12 1 2 ( , ) cos |sin | <sup>2</sup> *F* ,

2 2 12 1 2 *F*( , ) 1 ( )/2 and 2 2 1 2 1 2 ( , ) cos <sup>2</sup> *F* , respectively.

In conducting numerical experiments the interval of changing of parameter is divided into sequence intervals, each of which is puted in a circle of corresponding radius with respective center. Number of points partition of boundary of each circle was constant and equal 512 *N* . On the step 2 of algorithm the values of was selected with the interval [0,7 1,5] with a step 0,05 as well 0 .

Table. 1 presents the bifurcation points for four directivity patterns 1 2 *F*(,) , when the variables are separated and three directivity patterns when the variables are not separated. For the first four diagrams a bifurcation point is shown also, which can be obtained by other methods, provided that in the function 1 2 *f*(,) the variables are separated.

The Table shows that when the variables in the functions 1 2 *F*(,) and 1 2 *f*(,) are separated, the results obtained by different approaches (reduction of one-parameter problems to the transcendental equations and solving them [2], by methods of descent [18] and also by bilateral methods proposed in [11, 13, 15]) are the same.

Note that the bifurcation points (at least their rough estimates) may be obtained graphically from Fig. 10 - Fig. 13, and may be clarified by the Algorithm 2.

**Figure 10.** Eigenvalue curves for (, )1 1 2 *F* 

306 Nonlinearity, Bifurcation and Chaos – Theory and Applications

matrix of the second derivatives of (48). **Step 6.** Compute the next approximation to

> 

 , (,) *m m f*

 **then go to** *Step 10*.

 

 

 , ( , ) *m m f* 

**6. Analysis of numerical results** 

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

2 2

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

 

[0,7 1,5] with a step 0,05

array, the variables are separated and are not separated.

directivity patterns are defined by the formulas 1 2 *F*(,)1

equal 512 *N* . On the step 2 of algorithm the values of

as well

methods, provided that in the function 1 2 *f*(,)

and 2 2

 

In conducting numerical experiments the interval of changing of parameter

0 .

 

Table. 1 presents the bifurcation points for four directivity patterns 1 2 *F*(,)

The Table shows that when the variables in the functions 1 2 *F*(,)

and also by bilateral methods proposed in [11, 13, 15]) are the same.

from Fig. 10 - Fig. 13, and may be clarified by the Algorithm 2.

*F*

 

(,) *m m f* 

**Step 7. end for** *m*

*Step 3*. **Step 10. The end**

**Step 8. if** (,) *mm f f* 

 

**Step 5.** Using the decomposition (49) and relations (50) we calculate ( , ) *m m f*

 , (,) *m m f* 

> and

**Step 9. else** Initialize a different initial approximation to the bifurcation point and **go to**

In conducting a series of numerical experiments on the synthesis of antenna arrays, Algorithms 1 and 2 were used to find the curves of eigenvalues for two-parameter eigenvalue problem, which are the branching lines of solutions of nonlinear synthesis equation (7) and their bifurcation points. Numerical calculations were carried out as for the

In Fig. 10 - Fig. 13 are shown the curves of eigenvalues for four problems, in which the given

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

into sequence intervals, each of which is puted in a circle of corresponding radius with respective center. Number of points partition of boundary of each circle was constant and

variables are separated and three directivity patterns when the variables are not separated. For the first four diagrams a bifurcation point is shown also, which can be obtained by other

separated, the results obtained by different approaches (reduction of one-parameter problems to the transcendental equations and solving them [2], by methods of descent [18]

Note that the bifurcation points (at least their rough estimates) may be obtained graphically

 , respectively.

 

the variables are separated.

 

 and (,) *m m f* 

by the formula (47)

that describes the given directivity pattern of the

*F*

 

, 12 1 2 ( , ) cos |sin | <sup>2</sup>

 

 and 1 2 *f*(,) 

was selected with the interval

 

,

 

is divided

, when the

are

 

  and construct the

 ,

**Figure 11.** Eigenvalue curves for ( , ) cos |sin | 12 1 2 <sup>2</sup> *F* 

Numerical Algorithms of Finding the Branching Lines

Bifurcation point, obtained by other methods



, that we are interested in.

, what is shown in Fig. 10 and Fig. 11,

admits separation of variables, another

 

and

and Bifurcation Points of Solutions for One Class of Nonlinear Integral Equations 309

 

 

 

 

<sup>1</sup> ( , ) (2.832715, 2.832715) *b b*

<sup>1</sup> ( , ) (4.207065, 4.207065) *b b*

<sup>1</sup> ( , ) (2.855425, 2.855425) *b b*

<sup>1</sup> ( , ) (4.207065, 2.855425) *b b*


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

We have found all valid solutions (the curves of eigenvalues) of the problem (19), which

 and 

 

respectively), which corresponds to the synthesized directivity patterns in which the

solution to the problem (19) has been found (for example, for 1 2 *F*(,)1

 ( ) 

1 2 *F*(,) 

1 1 cos cos 2 2

1 2 sin sin 

1 <sup>2</sup> cos sin 2 

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

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

 

 

2 2 <sup>2</sup> 1 2 cos 

**7. Concluding remarks** 

the new results, namely:

*F*

 

Bifurcation point

*b b* 

*b b* 

*b b* 

 1 2 3 (,) (,)

*b b* 

> *b b b b b b*

> *b b b b b b*

 

1 2

 

1 2

 

 

 

 

 

 

 

 

 

fall in the interval of modified parameters

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

 this is 3

12 1 2 ( , ) cos |sin | <sup>2</sup>

variables are not separated.

 

3 (,) (,)

3 (,) (,)

3 (,) (,)

> 1 2 3

> 1 2 3

 

<sup>1</sup> ( , ) (4.503957, 4.503957) *b b*

**Table 1.** Bifurcation points for different types of the given directivity pattern

*bb bb*

*bb bb*

*bb bb*

*bb bb*

   

 

 

 

const 1 1 2

(,) *b bi* 

( , ) (2.832715, 2.832715)

( , ) (4.207065, 4.207065)

( , ) (2.855425, 2.855425)

( , ) (4.207065, 2.855425)

( , ) (3.064250, 3.064250) ( , ) (3.064250, 3.186696) ( , ) (3.186696, 3.064250)

( , ) (3.302395, 3.302395) ( , ) (3.302395, 3.565660) ( , ) (3.565660, 3.302395)

**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* 


**Table 1.** Bifurcation points for different types of the given directivity pattern
