**2. Preliminary. Nonlinear synthesis problem**

We consider the radiating system, which consists of identical and identically oriented radiators of the same for all radiators directivity pattern (DP), in which the phase centers are located on the plane *XOY* (grid plane) of Cartesian coordinate system. We believe that the coordinates of radiators ( , ) *n m x y* form a rectangular equidistant grid, focused on the axes and symmetric with respect to these axes. Then the function that describes the DP (plane array factor) of equidistant plane system of radiators (plane array) has the form [2].

$$f(\mathcal{G}, \mathcal{op}) = \sum\_{n=-M\_1}^{M\_1} \sum\_{m=-M\_2(n)}^{M\_2(n)} I\_{nm} e^{ik\{x\_n \sin \mathcal{G} \cos \varphi + y\_m \sin \mathcal{G} \sin \varphi\}}\,\tag{1}$$

where *nm I* are the complex currents on the radiators, , are the angular coordinates of a spherical coordinate system (,,) *R* whose center coincides with the center of the Cartesian coordinate system *XOY* , 2 *M* ( ) *n* is the integer function that sets the number of elements 2 2 *Nn Mn* () 2 () 1 in the *n* th row of the array. Thus, the number of elements *N* in this array is equal to <sup>1</sup> <sup>2</sup> (2 ( ) 1) *M n M M n* .

1 We introduce the generalized variables

$$\tilde{\xi}\_1 = \sin \mathcal{G} \cos \rho \prime \,\, \tilde{\xi}\_2 = \sin \mathcal{G} \sin \rho \prime$$

and denote by 1 *d* and 2 *d* , respectively, the distance between adjacent radiators along the axes *Ox* and *Oy* . Then the coordinates of the radiators are calculated as

$$\propto\_n = d\_1 n \,, \ y\_m = d\_2 m \,\,\mu$$

and the plane array factor (1) can be represented as

$$f(\tilde{\xi}\_1, \tilde{\xi}\_2) = \sum\_{n=-M\_1}^{M\_1} \sum\_{m=-M\_2(n)}^{M\_2(n)} I\_{nm} e^{i(\tilde{c}\_1 n \tilde{\xi}\_1 + \tilde{c}\_2 m \tilde{\xi}\_2)}\,,\tag{2}$$

where

282 Nonlinearity, Bifurcation and Chaos – Theory and Applications

to the nonlinear two-parameter eigenvalue problem

for algebraic, differential and integral equations.

**2. Preliminary. Nonlinear synthesis problem** 

 

spherical coordinate system (,,) *R*

*M*

*n M*

We introduce the generalized variables

1

array is equal to <sup>1</sup>

where *nm I* are the complex currents on the radiators,

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

*M n*

.

 

(see, eg, [5, 6] and references therein)

with an integral operator *T*(,)

.

Some of these problems is reflected in the monographs [3, 4, 20], as well as in several articles

Application of the cited above approach to the nonlinear integral operator arising at synthesis of the antenna systems according to the given amplitude directivity pattern, brings

> *T ff* (,)

The essential difference of the two-parameter problems from the one-parameter ones is that the two-parameter problem can not have at all the solutions or, on the contrary, to have them as a continuum set, which in the case of real parameters are the curves of eigenvalues. Such problems are still not investigated because there are still many open questions connected with this problem such as, for example, the existence of solutions and their number, and also the development of numerical methods of solving such spectral problems

In the given work an algorithm of finding the branching lines of the integral equation arising in the variational statement of the synthesis problem of antenna array according to

We consider the radiating system, which consists of identical and identically oriented radiators of the same for all radiators directivity pattern (DP), in which the phase centers are located on the plane *XOY* (grid plane) of Cartesian coordinate system. We believe that the coordinates of radiators ( , ) *n m x y* form a rectangular equidistant grid, focused on the axes and symmetric with respect to these axes. Then the function that describes the DP (plane

array factor) of equidistant plane system of radiators (plane array) has the form [2].

( )

( ) (,) , *n m*

*nm*

coordinate system *XOY* , 2 *M* ( ) *n* is the integer function that sets the number of elements 2 2 *Nn Mn* () 2 () 1 in the *n* th row of the array. Thus, the number of elements *N* in this

1 2

*n Mm M n f I e*

 

1 

*M Mn*

1 2

 sin cos , 2 

the given amplitude directivity pattern as, for example, in [2] is proposed.

analytically depending on two spectral parameters

( sin cos sin sin )

(1)

 

whose center coincides with the center of the Cartesian

are the angular coordinates of a

*ik x y*

 

> ,

 sin sin  and

$$
\tilde{c}\_1 = kd\_{1\prime} \quad \tilde{c}\_2 = kd\_2 \dots
$$

Note that the function 1 2 *f*(,) is periodic with a period 1 2 / *c* for the variable 1 and with a period 2 2 / *c* for the variable 2 . Denote by *R*2 the region that corresponds to one period *R cc* 2 1 12 2 : /, / and assume that the required amplitude directivity pattern 1 2 *F*(,) is given in some region *<sup>R</sup>*2 and is described by the function that is continuous and nonnegative in and is equal to zero outside.

We must find such currents *nm I* on radiators that created by them directivity pattern will approach by the amplitude to the given directivity pattern 1 2 *F*(,) in the best way. To this end, we consider the variational statement of the problem as, for example, in [2] or [18].

## **2.1. Variational statement of the synthesis problem**

Thus, the synthesis problem we formulate as a problem of minimizing the functional [18]

$$
\sigma(I) = \iint \left[ F(\tilde{\xi}\_1, \tilde{\xi}\_2) - \left| f(\tilde{\xi}\_1, \tilde{\xi}\_2) \right| \right]^2 d\tilde{\xi}\_1 d\tilde{\xi}\_2 + \iint\_{\mathbb{R}\_2 \backslash \Omega} \left| f(\tilde{\xi}\_1, \tilde{\xi}\_2) \right|^2 d\tilde{\xi}\_1 d\tilde{\xi}\_2 \tag{3}
$$

on the space *<sup>N</sup> H C <sup>I</sup>* , i.e.

$$
\sigma(I) \to \min\_{I \in H\_I} \quad I\_{nm} \in H\_{I'}
$$

which characterizes the magnitude of mean-square deviation of modules of the given directivity pattern and the synthesized one in the region .

From the necessary condition of the functional ( )*I* minimum, we obtain a nonlinear system of equations for the optimum currents on radiators

$$I\_{nm} = \frac{\tilde{c}\_1 \tilde{c}\_2}{\left(2\pi\right)^2} \iint\_{\Omega} F(\tilde{\xi}\_1, \tilde{\xi}\_2) e^{-i(\tilde{c}\_1 n \tilde{\xi}\_1 + \tilde{c}\_2 m \tilde{\xi}\_2)} \times \exp\left\{i \arg \sum\_{n=-M\_1}^{M\_1} \sum\_{m=-M\_2(n)}^{M\_2(n)} I\_{nm} e^{i(\tilde{c}\_1 n \tilde{\xi}\_1 + \tilde{c}\_2 m \tilde{\xi}\_2)}\right\} d\tilde{\xi}\_1 d\tilde{\xi}\_2.$$

$$\mathbf{M} \cdot (\mathbf{n} = -\mathbf{M}\_1 \mathbf{\dot{\cdot}} \mathbf{\dot{\cdot}} M\_{1'} \ \mathbf{m} = -\mathbf{M}\_2 \mathbf{\dot{\cdot}} \mathbf{\dot{\cdot}} M\_2 \ \mathbf{\dot{\cdot}} \tag{4}$$

Numerical Algorithms of Finding the Branching Lines

and Bifurcation Points of Solutions for One Class of Nonlinear Integral Equations 285

11 2 2

(9)

 

> 

the values of functional (3), which it

( )*I* 0.719989 - 1 2 *c c* 0.60,

 

( )*I* 0.493709 - 1 2 *c c*

0 

*c c* at the

( )*I* 0.734128,

(11)

*c c* for different values of the main parameters 1*c*

( )*<sup>I</sup>* 0.741211, 3

2 

*c c* , is shown in Fig. 3, and the optimal current on the

( )

*icn cm*

 

 

(10)

a solution of equation (8), the corresponding to it solution of equation (7) is determined by

1 2

*M M*

 

 . It is easy to see that one of possible solutions of equation (7) (call it trivial) is

*n Mm M f I e*

1 2

12 11 2 2 arg ( , ) ( ) 1 2 <sup>2</sup> 1 2 1 2 (,) . (2 ) *i f cn cm*

Since equations (7) and (8) are nonlinear equations (Hammerstein type), they may have nonunique solutions, the number and properties of which depend on the number of elements in the antenna array and their placement, and also on the properties of the given

01212 12 121212 1 2 *f cc F K ccdd* ( , , , ) ( , )( , , , , , ) .

Experimental results of numerical synthesis of the directivity pattern for different values of parameters 1*c* and 2*c* show that with growth of parameters 1*c* and 2*c* there are other solutions that branch off from a trivial solution and they are more effective in terms of the

> 

( )*I* 0.559552 - 1 2 *c c* 0.70, 5

( )*I* 0.739769, <sup>0</sup>

Numerical examples of the trivial and branching solutions for the given directivity pattern

In Fig. 1 shows the trivial solution, which creates a symmetrical inphase current distribution on the radiators of array (Fig. 2). The amplitude of the synthesized directivity pattern, which

radiators that it creates, is asymmetric and is shifted to the first quadrant relatively of the

and the basic parameters of 1 2 *c c* 0.75 are shown in Fig. 1 -- Fig. 4.

( )*I* 0.661929), respectively by 0.03%, 2.86%, 12.24%, 20.95% and 25.41%.

0.75) is smaller than the values of functional (3) for the trivial solution 01212 *f* (, ,,)

0 

*c c I Fe d d*

 

 

1 2 (,) ,

*nm*

is a solution of equation (7), the corresponding to it solution of equation (8)

the formula

and if 1 2 *f*(,) 

and 2*c* ( <sup>1</sup>

1 2 *F*(,)1 

3 

4 0 

( )*<sup>I</sup>* 0.707903, 5

center of array (Fig. 4).

is determined by the relation

*nm*

 

values of functional (3), from 0% to 75 %

takes on the optimal solution 1212 *f*(, ,,)

same parameter values 1*c* and 2*c* ( <sup>1</sup>

0 

branches off from 01212 *f* (, ,,)

 

( )*I* 0.644291 - 1 2 *c c* 0.65, 4

In particular, for the given directivity pattern 1 2 *F*(,)1

 

( )*I* 0.739543, correspond, to 1 2 *c c* 0.57, 2

amplitude directivity pattern 1 2 *F*(,)

or the equation for the optimum directivity pattern, which is equivalent to (4)

$$f(\tilde{\xi}\_1, \tilde{\xi}\_2) = \frac{\overline{c}\_1 \overline{c}\_2}{\left(2\pi\right)^2} \iint \limits\_{\Omega} F(\tilde{\xi}\_1', \tilde{\xi}\_2') \text{K}(\tilde{\xi}\_1, \tilde{\xi}\_2, \tilde{\xi}\_1', \tilde{\xi}\_2', \tilde{c}\_1, \tilde{c}\_2) e^{i \arg f(\tilde{\xi}\_1', \tilde{\xi}\_2')} d\tilde{\xi}\_1' d\tilde{\xi}\_2' \tag{5}$$

where

$$K(\tilde{\xi}\_1, \tilde{\xi}\_2, \tilde{\xi}\_1', \tilde{\xi}\_2', \tilde{c}\_1, \tilde{c}\_2) = \sum\_{n=-M\_1}^{M\_1} \sum\_{m=-M\_2(n)}^{M\_2(n)} e^{i[\tilde{\varepsilon}\_1 n(\tilde{\xi}\_1 - \tilde{\xi}\_1') + \tilde{c}\_2 m(\tilde{\xi}\_2 - \tilde{\xi}\_2')]}$$

is the kernel, which essentially depends on the coordinates of antenna array.

Next, consider the rectangular grid with geometric center at the origin, which consists of 12 1 2 *NN N M M* (2 1)(2 1) elements. Here 2 2 *M Mn* ( ) const . We believe also that the amplitude directivity pattern 1 2 *F*(,) is given in the region 1 12 2 :{| | ,| | } *b b* . Denote by 1 2 and 2 2 the intervals of change of the angle in the region at 0 and / 2 , respectively, and introduce new variables

$$
\xi\_1 = \tilde{\xi}\_1 / \sin a\_1, \,\xi\_2 = \tilde{\xi}\_2 / \sin a\_2.
$$

Then 1 2 :{| | 1,| | 1} , and the kernel in equation (5) is real and takes the form [2]

$$\mathbf{K}(\boldsymbol{\xi}\_{1},\boldsymbol{\xi}\_{2},\boldsymbol{\xi}\_{1}^{\prime},\boldsymbol{\xi}\_{2}^{\prime},\boldsymbol{c}\_{1},\boldsymbol{c}\_{2}) = \sum\_{n=-M\_{1}}^{M\_{1}} \sum\_{m=-M\_{2}}^{M\_{2}} e^{i[\boldsymbol{\xi}\_{1}n(\boldsymbol{\xi}\_{1}-\boldsymbol{\xi}\_{1}^{\prime}) + \boldsymbol{c}\_{2}m(\boldsymbol{\xi}\_{2}-\boldsymbol{\xi}\_{2}^{\prime})]} = \frac{\sin N\_{1}\frac{\boldsymbol{\mathcal{C}\_{1}}}{2}(\boldsymbol{\xi}\_{1}-\boldsymbol{\xi}\_{1}^{\prime})}{2} \cdot \frac{\sin N\_{2}\frac{\boldsymbol{\mathcal{C}\_{2}}}{2}(\boldsymbol{\xi}\_{2}-\boldsymbol{\xi}\_{2}^{\prime})}{\sin \frac{\boldsymbol{\mathcal{C}\_{2}}}{2}(\boldsymbol{\xi}\_{2}-\boldsymbol{\xi}\_{2}^{\prime})},\tag{6}$$

where 1 1 12 2 2 1 *c kd c kd N* sin , sin , and *N*2 are the main parameters of the problem. Thus, equation (5) for optimal DP takes the form

$$f(\boldsymbol{\xi}\_1', \boldsymbol{\xi}\_2') = \frac{c\_1 c\_2}{(2\pi)^2} \iint\_{\Omega} \mathbf{F}(\boldsymbol{\xi}\_1', \boldsymbol{\xi}\_2') \mathbf{K}(\boldsymbol{\xi}\_1', \boldsymbol{\xi}\_2', \boldsymbol{\xi}\_1', \boldsymbol{\xi}\_2', c\_1, c\_2) e^{i \arg f(\boldsymbol{\xi}\_1', \boldsymbol{\xi}\_2')} d\boldsymbol{\xi}\_1' d\boldsymbol{\xi}\_2' \tag{7}$$

and equation (4) for optimal currents takes the form

$$I\_{nm} = \frac{c\_1 c\_2}{\left(2\pi\right)^2 \Omega} \iint \left(\tilde{\xi}\_1, \tilde{\xi}\_2\right) e^{-i\left(c\_1 n \tilde{\varepsilon}\_1 + c\_2 m \tilde{\varepsilon}\_2\right)} \times \exp\left\{i \arg \sum\_{n=-M\_1}^{M\_1} \sum\_{m=-M\_2}^{M\_2} I\_{nm} e^{i\left(c\_1 n \tilde{\varepsilon}\_1 + c\_2 m \tilde{\varepsilon}\_2\right)}\right\} d\xi\_1 d\xi\_2,\tag{8}$$
 
$$\{n = -M\_1 \div M\_1, \ m = -M\_2 \div M\_2\}.\tag{8}$$

Equivalence of equations (7) and (8) means that between the solutions of these equations one-to-one correspondence exists, i.e., to each solution of equation (7) corresponds the solution of equation (8) and vice versa. This means that if 11 22 , ,, ,, *nm I n MM m MM* is a solution of equation (8), the corresponding to it solution of equation (7) is determined by the formula

284 Nonlinearity, Bifurcation and Chaos – Theory and Applications

121212

(, ,, ,,)

/ 2 , respectively, and introduce new variables

1 2

*M M*

*n Mm M*

Thus, equation (5) for optimal DP takes the form

1 2

and equation (4) for optimal currents takes the form

11 2 2 1 2 ( )

 

*icn cm*

1 2

 

 

the amplitude directivity pattern 1 2 *F*(,)

 and 2 2

> 

*K cc e*

where 1 1 12 2 2 1 *c kd c kd N* sin , sin , 

> 

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

 

*c c I Fe*

 

where

Denote by 1 2

  Then 1 2 :{| | 1,| | 1} 

<sup>121212</sup> (, ,, ,,)

*nm*

or the equation for the optimum directivity pattern, which is equivalent to (4)

*K cc e*

is the kernel, which essentially depends on the coordinates of antenna array.

 

11 1

the intervals of change of the angle

 

1 11 2 22

[ ( ) ( )]

 

1 2 1 2 arg ( , ) 1 2 <sup>2</sup> 12 121212 1 2 ( , ) ( , )( , , , , , ) , (2 ) *c c i f <sup>f</sup> F K cce dd*

Equivalence of equations (7) and (8) means that between the solutions of these equations one-to-one correspondence exists, i.e., to each solution of equation (7) corresponds the solution of equation (8) and vice versa. This means that if 11 22 , ,, ,, *nm I n MM m MM* is

*icn cm*

1 2 1 2 arg ( , ) 1 2 <sup>2</sup> 12 121212 1 2 ( , ) ( , )( , , , , , ) , (2 ) *c c i f <sup>f</sup> F K cce dd*

(5)

1 2

*n Mm M n*

Next, consider the rectangular grid with geometric center at the origin, which consists of 12 1 2 *NN N M M* (2 1)(2 1) elements. Here 2 2 *M Mn* ( ) const . We believe also that

> / sin , 22 2

, and the kernel in equation (5) is real and takes the form [2]

 

*M Mn*

1 2

( )

( )

11 22 (, ) *n M Mm M M* (4)

 

1 11 2 22

is given in the region 1 12 2 :{| | ,| | }

 / sin .

and *N*2 are the main parameters of the problem.

(7)

1 2

*M M*

*n Mm M*

1 2 exp arg ,

 

11 22 (*n M Mm M M* , ). (8)

*nm*

*i I e dd*

in the region at

1 2 1 11 2 22

 

(6)

 

1 2 1 1 2 2 sin ( ) sin ( ) 2 2 ,

> 

11 2 2

 

( )

*icn cm*

 

sin ( ) sin ( ) 2 2

*c c N N*

 

*c c*

 

 *b b* .

> 0 and

[ ( ) ( )]

 

*icn cm*

 

$$f(\boldsymbol{\xi}\_1, \boldsymbol{\xi}\_2) = \sum\_{n=-M\_1}^{M\_1} \sum\_{m=-M\_2}^{M\_2} I\_{nm} e^{i(c\_1 n \boldsymbol{\xi}\_1 + c\_2 m \boldsymbol{\xi}\_2)}\,,\tag{9}$$

and if 1 2 *f*(,) is a solution of equation (7), the corresponding to it solution of equation (8) is determined by the relation

$$I\_{nm} = \frac{c\_1 c\_2}{\left(2\pi\right)^2} \iint\limits\_{\Omega} F(\xi\_1, \xi\_2) e^{i\left[\arg f(\xi\_1, \xi\_2) - \left(c\_1 n \xi\_1 + c\_2 m \xi\_2\right)\right]} d\xi\_1 d\xi\_2. \tag{10}$$

Since equations (7) and (8) are nonlinear equations (Hammerstein type), they may have nonunique solutions, the number and properties of which depend on the number of elements in the antenna array and their placement, and also on the properties of the given amplitude directivity pattern 1 2 *F*(,) .

It is easy to see that one of possible solutions of equation (7) (call it trivial) is

$$f\_0(\xi\_1, \xi\_2, c\_1, c\_2) = \iint\_{\Omega} F(\xi\_1', \xi\_2') K(\xi\_1, \xi\_2, \xi\_1', \xi\_2', c\_1, c\_2) d\xi\_1' d\xi\_2'.\tag{11}$$

Experimental results of numerical synthesis of the directivity pattern for different values of parameters 1*c* and 2*c* show that with growth of parameters 1*c* and 2*c* there are other solutions that branch off from a trivial solution and they are more effective in terms of the values of functional (3), from 0% to 75 %

In particular, for the given directivity pattern 1 2 *F*(,)1 the values of functional (3), which it takes on the optimal solution 1212 *f*(, ,,) *c c* for different values of the main parameters 1*c* and 2*c* ( <sup>1</sup> ( )*I* 0.739543, correspond, to 1 2 *c c* 0.57, 2 ( )*I* 0.719989 - 1 2 *c c* 0.60, 3 ( )*I* 0.644291 - 1 2 *c c* 0.65, 4 ( )*I* 0.559552 - 1 2 *c c* 0.70, 5 ( )*I* 0.493709 - 1 2 *c c* 0.75) is smaller than the values of functional (3) for the trivial solution 01212 *f* (, ,,) *c c* at the same parameter values 1*c* and 2*c* ( <sup>1</sup> 0 ( )*I* 0.739769, <sup>0</sup> 2 ( )*<sup>I</sup>* 0.741211, 3 0 ( )*I* 0.734128, 4 0 ( )*<sup>I</sup>* 0.707903, 5 0 ( )*I* 0.661929), respectively by 0.03%, 2.86%, 12.24%, 20.95% and 25.41%.

Numerical examples of the trivial and branching solutions for the given directivity pattern 1 2 *F*(,)1 and the basic parameters of 1 2 *c c* 0.75 are shown in Fig. 1 -- Fig. 4.

In Fig. 1 shows the trivial solution, which creates a symmetrical inphase current distribution on the radiators of array (Fig. 2). The amplitude of the synthesized directivity pattern, which branches off from 01212 *f* (, ,,) *c c* , is shown in Fig. 3, and the optimal current on the radiators that it creates, is asymmetric and is shifted to the first quadrant relatively of the center of array (Fig. 4).

Numerical Algorithms of Finding the Branching Lines

 *c c*

and Bifurcation Points of Solutions for One Class of Nonlinear Integral Equations 287

**Figure 3.** Amplitude directivity pattern of the solution branched from 01212 *f* (, ,,)

**Figure 4.** Optimal current distribution on radiators, which creates the branched solution

**Figure 1.** Amplitude directivity pattern of a trivial solution 01212 *f cc* (, ,,) 

**Figure 2.** Current cophasal distribution on the radiators, which creates the diagram 01212 *f cc* (, ,,) 

Numerical Algorithms of Finding the Branching Lines and Bifurcation Points of Solutions for One Class of Nonlinear Integral Equations 287

**Figure 3.** Amplitude directivity pattern of the solution branched from 01212 *f* (, ,,) *c c*

286 Nonlinearity, Bifurcation and Chaos – Theory and Applications

**Figure 1.** Amplitude directivity pattern of a trivial solution 01212

**Figure 2.** Current cophasal distribution on the radiators, which creates the diagram 01212


<sup>4</sup> <sup>0</sup>


1

*f cc* (, ,,) 


*f cc* (, ,,) 

0.1

5

0.2

0.3

**Figure 4.** Optimal current distribution on radiators, which creates the branched solution

Numerical Algorithms of Finding the Branching Lines

and Bifurcation Points of Solutions for One Class of Nonlinear Integral Equations 289

 

**Figure 6.** Amplitude directivity pattern of trivial solution 01212 *f cc* (, ,,)

**Figure 7.** Inphase current distribution on the radiators, which creates the diagram 01212 *f cc* (, ,,)

 

**Figure 5.** Given amplitude directivity pattern <sup>1</sup> <sup>2</sup> ( ) cos |sin | <sup>2</sup> *F* 

The branching solutions are still more effective for directivity patterns that do not have central symmetry. For example, for the given directivity pattern <sup>1</sup> <sup>2</sup> ( ) cos |sin | <sup>2</sup> *F* and the main parameters 1 2 *c c* 1.0, 0.85 , which is shown in Fig. 5, numerical examples of the trivial and the branching solutions are shown in Fig. 6 - Fig. 9. From these figures we see that the branching solution (the amplitude directivity pattern of which is shown in Fig. 8, and the optimal distribution of the current on the radiators that it generates is shown in Fig. 9) more accurately than the trivial solution (11) (amplitude directivity pattern is shown in Fig. 6, which is created by symmetric inphase current (Fig. 7)) approximates the given directivity pattern not only in the mean square approximation (in terms of the values of functional (3) - 0.050109 in comparison with 0.185960) to about 73%, but also with respect to the form.

Thus, in most cases from the practical point of view the nontrivial solution, that branches off from 01212 *f* (, ,,) *c c* with growth of parameters 1*c* and 2*c* is interesting.

Numerical Algorithms of Finding the Branching Lines and Bifurcation Points of Solutions for One Class of Nonlinear Integral Equations 289

**Figure 6.** Amplitude directivity pattern of trivial solution 01212 *f cc* (, ,,) 

288 Nonlinearity, Bifurcation and Chaos – Theory and Applications

**Figure 5.** Given amplitude directivity pattern <sup>1</sup>

the form.

from 01212 *f* (, ,,) 

<sup>2</sup> ( ) cos |sin | <sup>2</sup>

*F*

<sup>2</sup> ( ) cos |sin | <sup>2</sup>

and

*F*

central symmetry. For example, for the given directivity pattern <sup>1</sup>

The branching solutions are still more effective for directivity patterns that do not have

the main parameters 1 2 *c c* 1.0, 0.85 , which is shown in Fig. 5, numerical examples of the trivial and the branching solutions are shown in Fig. 6 - Fig. 9. From these figures we see that the branching solution (the amplitude directivity pattern of which is shown in Fig. 8, and the optimal distribution of the current on the radiators that it generates is shown in Fig. 9) more accurately than the trivial solution (11) (amplitude directivity pattern is shown in Fig. 6, which is created by symmetric inphase current (Fig. 7)) approximates the given directivity pattern not only in the mean square approximation (in terms of the values of functional (3) - 0.050109 in comparison with 0.185960) to about 73%, but also with respect to

Thus, in most cases from the practical point of view the nontrivial solution, that branches off

*c c* with growth of parameters 1*c* and 2*c* is interesting.

**Figure 7.** Inphase current distribution on the radiators, which creates the diagram 01212 *f cc* (, ,,) 

**Figure 8.** Amplitude directivity pattern of solution branched off from 01212 *f cc* (, ,,) 

Numerical Algorithms of Finding the Branching Lines

1212

*c c* , . (13)

 *c* and 2 2 

> 

*c* of the

 ,

 

01212

*f cc* 

 

and Bifurcation Points of Solutions for One Class of Nonlinear Integral Equations 291

The points of possible branching of solutions of integral equation (7) are such values of real

1212 12 1212 12 121212 1 2

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* ) two-

It is easy to be convinced, that at arbitrary finite values 1*c* 0 , 2*c* 0 , the function

It should be noted that in a special case, when it is possible to separate variables in the function

( , ) ( ) ( , ) , 1,2, *jjj jjjj u c Tc u c j*

sin ( ) ( ) <sup>2</sup> ( )( , ) (,) , (,) sin ( ) <sup>2</sup>

*F N Tcu c u cd f c c*

 *j j jj j j jj jj jj j j j j*

 

 

*c*

The study of such equations is carried out in [2, 18], and it is possible to apply , for example,

In the given work the numerical algorithms to solve more complicated problem when the

*j j*

 *<sup>j</sup>* 1, 2. .

 

can also be presented as 1 212 11 22 *u cc u c u c* (, ,,) (,) (,)

 

(as in one-dimensional case [18]) in

for arbitrary 1*c* 0 and 2*c* 0 is a positive kernel of the

*c c* is the eigenfunction of equation (12). From this it follows, that the operator

(, ,,) ( , , , ) ( , )( , , , ) ( , )( , , , , , ) , (, ,,) *u cc u c c Tc c u c c F K c c d d*

*T Iu* ( , ) ( , , , ) 0,

1 2 *Tc c* (,) has a spectrum, which coincides with the first quadrant of the plane <sup>2</sup> *<sup>R</sup>* .

problem (13), for which there appear the solutions different from 01212 *f* (, ,,)

 to present as 12 11 2 *F FF* ( , ) ( ) ( ) 

 12 1212 1 12 2

The problem consists in finding such range of real parameters 1 1

1 2 *cc R* , , in which homogeneous integral equation [18]

 

 (12)

 

 *c c* .

, the equation (12), provided that

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

 

decomposes in two independent one-parameter equations, i.e.

1

variables are not separated, are proposed.

First we will show that the kernel 121212 *K cc* (, ,, ,,)

  1 0

the algorithms of the work [11, 13, 15] to solution of such equations.

physical parameters <sup>2</sup>

parameter eigenvalue problem

 , i.e. 1 2 *F*( , ) 

function 1212 *u cc* (, ,,) 

**3. Basic equations** 

integral operator

the region 1 2 :{| | 1,| | 1} 

 

<sup>01212</sup> *f* (, ,,) 

1 2 *F*( , ) 

with operators

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