**5. Algorithm of finding the bifurcation points of eigenvalue curves**

Note that if two eigenvalue curves intersect at some point, this point is called the point of bifurcation (or branch the point). Sufficient criterion for the existence of such points have been known long ago (see, for example, [8]) and consists in that the point ( , ) *b b* is a bifurcation point of equation

$$f(\mathcal{A}, \mu) \equiv \det D\_n(\mathcal{A}, \mu) = 0,\tag{45}$$

if conditions (,) <sup>0</sup> *<sup>f</sup>* , (,) <sup>0</sup> *<sup>f</sup>* are satisfied, and the second order partial derivatives

are different from zero. But this criterion was not often used in practical calculations, because it requires calculation of derivatives of the determinant of matrix.

Using the algorithm of computing derivatives of the determinant of the matrix proposed in section 2 this criterion can be effectively used to calculate the bifurcation points of equation (45).

Thus, the problem consists in determining such parameters and which are the solutions of two nonlinear algebraic equations

$$\begin{aligned} \frac{\left\| \widehat{f}(\lambda,\mu) \right\|}{\left\| \lambda \right\|} & \equiv \left[ \det D\_n(\lambda,\mu) \right]^\prime \_\lambda = 0 \,, \\\frac{\left\| \widehat{f}(\lambda,\mu) \right\|}{\left\| \mu \right\|} & \equiv \left[ \det D\_n(\lambda,\mu) \right]^\prime \_\mu = 0 . \end{aligned} \tag{46}$$

Numerical Algorithms of Finding the Branching Lines

, ' 1,1 <sup>1</sup> (,) (,) *w v ii ii*

 , 1,2 *W* (,) 

> 

 

> 

> 

 

 

 ,

 .

 , ( , ) *m m f* 

 

 and *<sup>m</sup>* 

> , ,

 , (,) *m m f*

 ,

to calculate

(49)

 

 

 

 ,

 and 2,1 *W* (,) 

> 

> 

> >

> >

 

 

are the diagonal

and Bifurcation Points of Solutions for One Class of Nonlinear Integral Equations 303

 

 

> 

 

> 

> 

 

 

1,2 1 2

2,1 2 1

, ' <sup>2</sup>

 

*f w uv v u*

 

 

 , <sup>2</sup> *V* (,) 

> 

 

> 

> 

 

 

> 

 

 

2,2 <sup>2</sup> (,) (,) *w v ii ii*

in decompositions

 

 

''

''

 

 

Here ' <sup>1</sup> *M L* (,) (,)

2,2 <sup>2</sup> *N M* (,) (,)

(,) *m m f* 

   

 

'

 

 

> 

> 

 

 

'

'

 

 

'

From this it follows that to calculate ( , ) *m m f*

, ' <sup>2</sup> *M L* (,) (,)

, ' 1,2 <sup>1</sup> *N M* (,) (,)

11 1 22 2

,

*B M U LV B M U LV*

 

*D LU*

, ' 1,2 <sup>1</sup> (,) (,) *w v ii ii*

where ' <sup>1</sup> (,) (,) *ii ii v u*

 

elements of matrices <sup>1</sup> *V* (,)

''

''

> 

> 

which are obtained from decomposition

 

 and (,) *m m f* 

 

 

*f w uv v u*

1 1, 1 1, 1, , ( , ) ( , ) ( , ) ( , ) ( , ) ( , ), *n n n n n*

1 1, 1 1, 1, , (, ) (, ) (, ) (, ) (, ) (, ), *n n n n n*

1 1 <sup>1</sup> *D B M U LV* ( , ) ( , ) ( , ) ( , ) ( , ) ( , ),

2 2 <sup>2</sup> *D B M U LV* ( , ) ( , ) ( , ) ( , ) ( , ) ( , ),

1,1 1,1 1 1 1,1 *D C N U M V LW* ( , ) ( , ) ( , ) ( , ) 2 ( , ) ( , ) ( , ) ( , ),

1,1 1,1 1 1 1,1 *D C N U M V LW* ( , ) ( , ) ( , ) ( , ) 2 ( , ) ( , ) ( , ) ( , ),

1,2 1,2 1 2 2 1 1,2 *D C N U M V M V LW* ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ),

2,1 2,1 2 1 1 2 2,1 *D C N U M V M V LW* ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ),

**D LU** ( , ) ( , ) ( , ).

 

 

> 

1,1 1,1 1 1 1,1 2,2 2,2 2 2 2,2

*C N U M V LW C N U M V LW*

 

, ,

1,2 1,2 1 2 2 1 1,2 2,1 2,1 2 1 1 2 2,1

*C N U M V M V LW C N U M V M V LW*

 

2 , 2 ,

it is necessary for fixed *<sup>m</sup>*

 

 

 

 

 

 

 

 , 1,1 *W* (,) 

 

 

*kk ii kk jj ii k i ik k j jk i i ki j*

 

*kk ii kk jj ii k i ik k j jk i i ki j*

(,) (,) *ii ii v u*

 

, ' 2,1 <sup>2</sup> (,) (,) *w v ii ii*

 

 

 

 

> 

, ' 1,1 <sup>1</sup> *N M* (,) (,)

, ' 2,1 <sup>2</sup> *N M* (,) (,)

 , (,) *m m f*

 

 

 

 

 , 2,2 *W* (,) 

 

Note that, with some approximation to solution of (46), for its solution the iterative process of Newton's method can be applied as in [12]

$$
\begin{bmatrix}
\boldsymbol{\lambda}\_{m+1} \\
\boldsymbol{\mu}\_{m+1}
\end{bmatrix} = \begin{bmatrix}
\boldsymbol{\lambda}\_{m} \\
\boldsymbol{\mu}\_{m}
\end{bmatrix} - \begin{bmatrix}
\boldsymbol{J}(\boldsymbol{\lambda}\_{m'}, \boldsymbol{\mu}\_{m})
\end{bmatrix}^{-1} \begin{bmatrix}
\left\lceil \boldsymbol{f}(\boldsymbol{\lambda}\_{m'}, \boldsymbol{\mu}\_{m}) \right\rceil\_{\boldsymbol{\lambda}} \\
\left\lceil \int \boldsymbol{f}(\boldsymbol{\lambda}\_{m'}, \boldsymbol{\mu}\_{m}) \right\rceil\_{\boldsymbol{\mu}}'
\end{bmatrix}, m = 0, 1, \ldots \tag{47}
$$

where

$$f(\boldsymbol{\lambda}\_{m'}\boldsymbol{\mu}\_m) = \begin{bmatrix} \left[ f(\boldsymbol{\lambda}\_{m'}\boldsymbol{\mu}\_m) \right]\_{\boldsymbol{\lambda}}^{\ast} & \left[ f(\boldsymbol{\lambda}\_{m'}\boldsymbol{\mu}\_m) \right]\_{\boldsymbol{\lambda}}^{\ast} \\\\ \left[ f(\boldsymbol{\lambda}\_{m'}\boldsymbol{\mu}\_m) \right]\_{\boldsymbol{\mu}}^{\ast} & \left[ f(\boldsymbol{\lambda}\_{m'}\boldsymbol{\mu}\_m) \right]\_{\boldsymbol{\mu}}^{\ast} \end{bmatrix}^{\ast} \tag{48}$$

Further we assume that the determinant of matrix of the second derivatives (48) whose elements are calculated at point different ( , ) *m m* from zero.

Thus, at each step iterative process to compute the function *f*( , ) det ( , ) **Τ** and its partial derivatives (first and second) only for fixed values of the parameters and . This can be realized in a numerical procedure using the *LU*-decomposition of matrix **T**(,) , namely:

$$f'\_{\lambda}(\lambda,\mu) = \sum\_{k=1}^{n} v^1\_{kk}(\lambda,\mu) \prod\_{i=1, i \neq k}^{n} u\_{ii}(\lambda,\mu),$$

$$f'\_{\mu}(\lambda,\mu) = \sum\_{k=1}^{n} v^2\_{kk}(\lambda,\mu) \prod\_{i=1, i \neq k}^{n} u\_{ii}(\lambda,\mu),$$

$$f''\_{\lambda\lambda}(\lambda,\mu) = \sum\_{k=1}^{n} w^1\_{kk}(\lambda,\mu) \prod\_{i=1, i \neq k}^{n} u\_{ii}(\lambda,\mu) + \sum\_{k=1}^{n} v^1\_{kk}(\lambda,\mu) \left(\sum\_{j=1, j \neq k}^{n} v^1\_{jj}(\lambda,\mu) \prod\_{i=1, i \neq k, i \neq j}^{n} u\_{ii}(\lambda,\mu)\right).$$

$$f''\_{\mu\mu}(\lambda,\mu) = \sum\_{k=1}^{n} w^2\_{kk}(\lambda,\mu) \prod\_{i=1, i \neq k}^{n} u\_{ii}(\lambda,\mu) + \sum\_{k=1}^{n} v^2\_{kk}(\lambda,\mu) \left(\sum\_{j=1, j \neq k}^{n} v^2\_{jj}(\lambda,\mu) \prod\_{i=1, i \neq k, i \neq j}^{n} u\_{ii}(\lambda,\mu)\right).$$

Numerical Algorithms of Finding the Branching Lines and Bifurcation Points of Solutions for One Class of Nonlinear Integral Equations 303

> 

 ,

$$\begin{split} &f^{\pi}\_{\lambda\mu}(\lambda,\mu) = \sum\_{k=1}^{n} w^{1}\_{kk}^{12}(\lambda,\mu) \prod\_{i=1, i\neq k}^{n} u\_{ii}(\lambda,\mu) + \sum\_{k=1}^{n} v^{1}\_{kk}(\lambda,\mu) \Big( \sum\_{j=1, j\neq k}^{n} v^{2}\_{jj}(\lambda,\mu) \prod\_{i=1, i\neq k, i\neq j}^{n} u\_{ii}(\lambda,\mu) \Big), \\ &f^{\pi}\_{\mu\lambda}(\lambda,\mu) = \sum\_{k=1}^{n} w^{2}\_{kk}^{12}(\lambda,\mu) \prod\_{i=1, i\neq k}^{n} u\_{ii}(\lambda,\mu) + \sum\_{k=1}^{n} v^{2}\_{kk}(\lambda,\mu) \Big( \sum\_{j=1, j\neq k}^{n} v^{1}\_{jj}(\lambda,\mu) \prod\_{i=1, i\neq k, i\neq j}^{n} u\_{ii}(\lambda,\mu) \Big), \\ &\text{re} \qquad v^{1}\_{ii}(\lambda,\mu) = \Big[ u\_{ii}(\lambda,\mu) \Big]\_{\lambda'} \qquad v^{2}\_{ii}(\lambda,\mu) = \Big[ u\_{ii}(\lambda,\mu) \Big]\_{\mu'} \qquad \qquad w^{11}\_{ii}(\lambda,\mu) = \Big[ v^{1}\_{ii}(\lambda,\mu) \Big]^{\dagger}, \end{split}$$

(,) (,) *ii ii v u*

 

where ' <sup>1</sup> (,) (,) *ii ii v u*

302 Nonlinearity, Bifurcation and Chaos – Theory and Applications

solutions of two nonlinear algebraic equations

of Newton's method can be applied as in [12]

1

 

 

*m m*

*J*

elements are calculated at point different ( , ) *m m*

 

Thus, the problem consists in determining such parameters

 

 

1 1

*J*

*m m m m m m m m m m*

 

(45).

where

 

 

Using the algorithm of computing derivatives of the determinant of the matrix proposed in section 2 this criterion can be effectively used to calculate the bifurcation points of equation

(,) det ( , ) 0,

 

 

(, ) ,

*f*

(,) (,)

 

(,) (,) *m m m m*

*m m m m*

(,) ,

*f f* 

Further we assume that the determinant of matrix of the second derivatives (48) whose

*f f*

 

1

2

*fv u*

1,1 1 1

2,2 2 2

*f w uv v u*

 

 

*f w uv v u*

*fv u*

1 1, ( , ) ( , ) ( , ), *n n*

1 1, ( , ) ( , ) ( , ), *n n*

1 1, 1 1, 1, , ( , ) ( , ) ( , ) ( , ) ( , ) ( , ), *n n n n n*

1 1, 1 1, 1, , ( , ) ( , ) ( , ) ( , ) ( , ) ( , ), *n n n n n*

 

 

 

 

*kk ii k i ik*

*kk ii k i ik*

*kk ii kk jj ii k i ik k j jk i i ki j*

*kk ii kk jj ii k i ik k j jk i i ki j*

 

 

Thus, at each step iterative process to compute the function *f*( , ) det ( , )

realized in a numerical procedure using the *LU*-decomposition of matrix **T**(,)

derivatives (first and second) only for fixed values of the parameters

 

 

 

  *f*

*<sup>f</sup> <sup>D</sup>*

*<sup>f</sup> <sup>D</sup>*

(,) det ( , ) 0. *n*

Note that, with some approximation to solution of (46), for its solution the iterative process

*n*

(,)

 

   

(,)

" "

 

" "

 

from zero.

 

 

> 

 

and

(46)

*m* 0,1, ... (47)

(48)

. This can be

, namely:

 

 

 **Τ** and its partial

> 

 and 

which are the

 ' 2,2 <sup>2</sup> (,) (,) *w v ii ii* , ' 1,2 <sup>1</sup> (,) (,) *w v ii ii* , ' 2,1 <sup>2</sup> (,) (,) *w v ii ii* are the diagonal

, ' <sup>2</sup>

 

elements of matrices <sup>1</sup> *V* (,) , <sup>2</sup> *V* (,) , 1,1 *W* (,) , 2,2 *W* (,) , 1,2 *W* (,) and 2,1 *W* (,) in decompositions

' 1 1 <sup>1</sup> *D B M U LV* ( , ) ( , ) ( , ) ( , ) ( , ) ( , ), ' 2 2 <sup>2</sup> *D B M U LV* ( , ) ( , ) ( , ) ( , ) ( , ) ( , ), '' 1,1 1,1 1 1 1,1 *D C N U M V LW* ( , ) ( , ) ( , ) ( , ) 2 ( , ) ( , ) ( , ) ( , ), '' 1,1 1,1 1 1 1,1 *D C N U M V LW* ( , ) ( , ) ( , ) ( , ) 2 ( , ) ( , ) ( , ) ( , ), '' 1,2 1,2 1 2 2 1 1,2 *D C N U M V M V LW* ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ), '' 2,1 2,1 2 1 1 2 2,1 *D C N U M V M V LW* ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ), 

which are obtained from decomposition

$$\mathbf{D}(\mathcal{X}, \mu) = \mathbf{L}(\mathcal{X}, \mu)\mathbf{U}(\mathcal{X}, \mu).$$

$$\begin{split} \text{Here } \ M^{1}(\lambda,\mu) = \left[L(\lambda,\mu)\right]\_{\lambda'}^{\cdot}, \text{ } M^{2}(\lambda,\mu) = \left[L(\lambda,\mu)\right]\_{\mu'}^{\cdot}, \text{ } N^{1,1}(\lambda,\mu) = \left[M^{1}(\lambda,\mu)\right]\_{\lambda'} \\ \text{ } \text{ } N^{2,2}(\lambda,\mu) = \left[M^{2}(\lambda,\mu)\right]\_{\mu'}^{\cdot}, \text{ } N^{1,2}(\lambda,\mu) = \left[M^{1}(\lambda,\mu)\right]\_{\mu'}^{\cdot}, \text{ } N^{2,1}(\lambda,\mu) = \left[M^{2}(\lambda,\mu)\right]\_{\lambda'}^{\cdot}. \end{split}$$

From this it follows that to calculate ( , ) *m m f* , (,) *m m f* , ( , ) *m m f* , (,) *m m f* , (,) *m m f* and (,) *m m f* it is necessary for fixed *<sup>m</sup>* and *<sup>m</sup>* to calculate

$$\begin{aligned} D &= L\mathcal{U}\_{\prime} \\ B^1 &= M^1 \mathcal{U} + L\mathcal{V}^1 \\ B^2 &= M^2 \mathcal{U} + L\mathcal{V}^2 \\ \mathcal{C}^{1,1} &= N^{1,1} \mathcal{U} + 2M^1 \mathcal{V}^1 + L\mathcal{W}^{1,1} \\ \mathcal{C}^{2,2} &= N^{2,2} \mathcal{U} + 2M^2 \mathcal{V}^2 + L\mathcal{W}^{2,2} \\ \mathcal{C}^{1,2} &= N^{1,2} \mathcal{U} + M^1 \mathcal{V}^2 + M^2 \mathcal{V}^1 + L\mathcal{W}^{1,2} \\ \mathcal{C}^{2,1} &= N^{2,1} \mathcal{U} + M^2 \mathcal{V}^1 + M^1 \mathcal{V}^2 + L\mathcal{W}^{2,1} \end{aligned} \tag{49}$$

from which

$$f'\_{\lambda}(\lambda\_m, \mu\_m) = \sum\_{k=1}^n v\_{kk}^1 \prod\_{i=1, i \neq k}^n u\_{ii} \quad f'\_{\mu}(\lambda\_m, \mu\_m) = \sum\_{k=1}^n v\_{kk}^2 \prod\_{i=1, i \neq k}^n u\_{ii},$$

$$f''\_{\lambda \lambda}(\lambda\_m) = \sum\_{k=1}^n w\_{kk}^{1,1} \prod\_{i=1, i \neq k}^n u\_{ii} + \sum\_{k=1}^n v\_{kk}^1 \left( \sum\_{j=1, j \neq k}^n v\_{jj}^1 \prod\_{i=1, i \neq k, i \neq j}^n u\_{ii} \right).$$

$$f''\_{\mu \mu}(\lambda\_m) = \sum\_{k=1}^n w\_{kk}^{2,2} \prod\_{i=1, i \neq k}^n u\_{ii} + \sum\_{k=1}^n v\_{kk}^2 \left( \sum\_{j=1, j \neq k}^n v\_{jj}^2 \prod\_{i=1, i \neq k, i \neq j}^n u\_{ii} \right). \tag{50}$$

$$f''\_{\lambda \mu}(\lambda\_m) = \sum\_{k=1}^n w\_{kk}^{1,2} \prod\_{i=1, i \neq k}^n u\_{ii} + \sum\_{k=1}^n v\_{kk}^1 \left( \sum\_{j=1, j \neq k}^n v\_{jj}^2 \prod\_{i=1, i \neq k, i \neq j}^n u\_{ii} \right).$$

$$f''\_{\mu k}(\lambda\_m) = \sum\_{k=1}^n w\_{kk}^{2,1} \prod\_{i=1, i \neq k}^n u\_{ii} + \sum\_{k=1}^n v\_{kk}^2 \left( \sum\_{j=1, j \neq k}^n v\_{jj}^1 \prod\_{i=1, i \neq k, i \neq j}^n u\_{ii} \right).$$

Numerical Algorithms of Finding the Branching Lines

(,) (,) *<sup>n</sup> B D*

 ,

 ,

and with

and Bifurcation Points of Solutions for One Class of Nonlinear Integral Equations 305

( 2 ) , , ... , ,

*ir n* 1, ... , ,

( 2 ) , , ... , ,

*ir n* 1, ... , ,

( ) , , ... , ,

*ir n* 1, ... , ,

( ) , , ... , ,

*ir n* 1, ... , , ,

**do** 

, <sup>1</sup>

 for , *m m* 

, 2,2(,) (,) *C Dn*

 .

 

> 

1

*r*

*j*

1

*r*

*j*

1

1

*r*

*j*

1

*r*

*j*

1

1

*r*

*j*

**Algorithm 2.** 

2

1

respect of the function - *<sup>f</sup>*

 , 

> 

> >

**Step 2.** Initialize 0 0

(,) (,) *<sup>n</sup> B D*

1,2(,) (,) *C Dn*

1

1

1

*r*

*j*

1

*r*

*j*

1

1

*r*

*j*

1

which are generalization of recurent relations of Section 3.2.

parametric spectral problem consists of the following steps.

**Step 4.** Calculate the matrix ( , ) ( , ) *DTI n n*

and 1,1(,) (,) *C Dn*

, 2,1(,) (,) *C Dn*

**Step 3. for** *m* =1,2, … up to achievement of accuracy *<sup>p</sup>*

> 

1

2,2 2,2 2,2 2 2 2,2 2 2 2,2

1,2 1,2 1,2 1 2 2 1 1,2 1 2 2 1 1,2

2,1 2,1 2,1 2 1 1 2 2,1 2 1 1 2 2,1

*ir ir ij jr ij jr ij jr ij jr ir rr ir rr ir rr rr*

**Step 1.** To set the accuracy of calculations: with respect of the parameters - *<sup>p</sup>*

 

*n c n u mv mv lw mv mv lw u*

*ir ir ij jr ij jr ij jr ij jr ir rr ir rr ir rr rr*

2,1 2,1 2,1 2 1 1 2 2,1

*rk rk rj jk rj jk rj jk rj jk*

 

*n c n u mv mv lw mv mv lw u*

*ir ir ij jr ij jr ij jr ir rr ir rr rr*

1,2 1,2 1,2 1 2 2 1 1,2

*rk rk rj jk rj jk rj jk rj jk*

 

*n c n u mv lw mv lw u*

1,1 1,1 1,1 1 1 1,1 1 1 1,1

*ir ir ij jr ij jr ij jr ir rr ir rr rr*

2,2 2,2 2,2 2 2 2,2

*rk rk rj jk rj jk rj jk*

 

*n c n u mv lw mv lw u*

1,1 1,1 1,1 1 1 1,1

*w c n u mv lw k r n*

( 2 )2 / ,

*w c n u mv lw k r n*

( 2 )2 / ,

*w c n u mv mv lw k r n*

( ) / ,

*w c n u mv mv lw k r n*

( ) / ,

Thus, the algorithm of finding of the bifurcation points of eigenvalue curves of two-

 

*rk rk rj jk rj jk rj jk*

The elements of matrix from decomposition (49) can be calculated directly using the recurrent relations

$$\begin{aligned} r &= 1, 2, \dots, n\_r \\\\ u\_{rk} &= d\_{rk} - \sum\_{j=1}^{r-1} l\_{rj} u\_{jk} \quad , \quad k = r, \dots, n\_r \\\\ l\_{ir} &= \left( d\_{ir} - \sum\_{j=1}^{r-1} l\_{ij} u\_{jr} \right) / \left( u\_{rr} \quad , \quad i = r+1, \dots, n\_r \\\\ \upsilon\_{rk}^1 &= b\_{rk}^1 - \sum\_{j=1}^{r-1} \left( m\_{rj}^1 u\_{jk} + l\_{rj} \upsilon\_{jk}^1 \right) \quad , \quad k = r, \dots, n\_r \\\\ m\_{ir}^1 &= \left[ b\_{ir}^1 - \sum\_{j=1}^{r-1} \left( m\_{jr}^1 u\_{jr} + l\_{rj} \upsilon\_{jr}^1 \right) - l\_{ir} \upsilon\_{ir}^1 \right] / u\_{rr} \quad , \quad i = r+1, \dots, n\_r \\\\ \upsilon\_{rk}^2 &= b\_{rk}^2 - \sum\_{j=1}^{r-1} \left( m\_{rj}^2 u\_{jk} + l\_{rj} \upsilon\_{jk}^2 \right) \quad , \quad k = r, \dots, n\_r \\\\ m\_{ir}^2 &= \left[ b\_{ir}^2 - \sum\_{j=1}^{r-1} \left( m\_{jr}^2 u\_{jr} + l\_{rj} \upsilon\_{jr}^2 \right) - l\_{ir} \upsilon\_{rr}^2 \right] / u\_{rr} \quad , \quad i = r+1, \dots, n\_r \end{aligned}$$

Numerical Algorithms of Finding the Branching Lines and Bifurcation Points of Solutions for One Class of Nonlinear Integral Equations 305

1 1,1 1,1 1,1 1 1 1,1 1 ( 2 ) , , ... , , *r rk rk rj jk rj jk rj jk j w c n u mv lw k r n* 1 1,1 1,1 1,1 1 1 1,1 1 1 1,1 1 ( 2 )2 / , *r ir ir ij jr ij jr ij jr ir rr ir rr rr j n c n u mv lw mv lw u ir n* 1, ... , , 1 2,2 2,2 2,2 2 2 2,2 1 ( 2 ) , , ... , , *r rk rk rj jk rj jk rj jk j w c n u mv lw k r n* 1 2,2 2,2 2,2 2 2 2,2 2 2 2,2 1 ( 2 )2 / , *r ir ir ij jr ij jr ij jr ir rr ir rr rr j n c n u mv lw mv lw u ir n* 1, ... , , 1 1,2 1,2 1,2 1 2 2 1 1,2 1 ( ) , , ... , , *r rk rk rj jk rj jk rj jk rj jk j w c n u mv mv lw k r n* 1 1,2 1,2 1,2 1 2 2 1 1,2 1 2 2 1 1,2 1 ( ) / , *r ir ir ij jr ij jr ij jr ij jr ir rr ir rr ir rr rr j n c n u mv mv lw mv mv lw u ir n* 1, ... , , 1 2,1 2,1 2,1 2 1 1 2 2,1 1 ( ) , , ... , , *r rk rk rj jk rj jk rj jk rj jk j w c n u mv mv lw k r n* 1 2,1 2,1 2,1 2 1 1 2 2,1 2 1 1 2 2,1 1 ( ) / , *r ir ir ij jr ij jr ij jr ij jr ir rr ir rr ir rr rr j n c n u mv mv lw mv mv lw u ir n* 1, ... , , ,

which are generalization of recurent relations of Section 3.2.

Thus, the algorithm of finding of the bifurcation points of eigenvalue curves of twoparametric spectral problem consists of the following steps.

### **Algorithm 2.**

304 Nonlinearity, Bifurcation and Chaos – Theory and Applications

 

 

 

 

 

recurrent relations

1 1 1, (,) , *n n m m kk ii k i ik*

<sup>2</sup>

 

1,1 1 1 1 1, 1 1, 1, , ( ) , *n nn n n m kk ii kk jj ii k i ik k j jk i i ki j f w uv v u*

2,2 2 2 1 1, 1 1, 1, , ( ) , *n n n n n m kk ii kk jj ii k i ik k j jk i i ki j f w uv v u*

1,2 1 2 1 1, 1 1, 1, , ( ) , *n n n n n m kk ii kk jj ii k i ik k j jk i i ki j f w uv v u*

2,1 2 1 1 1, 1 1, 1, , ( ) . *n n n n n m kk ii kk jj ii k i ik k j jk i i ki j f w uv v u*

The elements of matrix from decomposition (49) can be calculated directly using the

*r n* 1,2, ... , ,

*u d lu k r n* 

*l d lu u i r n*

*v b mu lv k r n*

*m b mu lv lv u i r n*

*v b mu lv k r n*

*m b mu lv lv u i r n*

1

*r rk rk rj jk j*

1

 

1

*r ir ir ij jr rr j*

1

1 11 1 1 1

*ir ir ij jr ij jr ir rr rr*

1 22 2 2 1

*ir ir ij jr ij jr ir rr rr*

*r rk rk rj jk rj jk j*

1 11 1 1 1

*r*

*j*

1

1 22 2 2 2

*r*

*j*

1

*r rk rk rj jk rj jk j*

 

 

 

 

 

, , ... , ,

/ , 1, ... , ,

( ) , , ... , ,

( ) / , 1, ... , ,

( ) , , ... , ,

( ) / , 1, ... , ,

1 1, (,) , *n n m m kk ii k i ik*

*f v u*

(50)

*f v u*

from which


$$\textbf{Step 4.}\quad \textbf{Calculate}\quad \textbf{ the}\qquad \textbf{matrix}\qquad \textbf{matrix}\qquad D\_{\pi}(\lambda,\mu) = \mathrm{T}\_{n}(\lambda,\mu) - \mathrm{I},\qquad\qquad \mathsf{B}^{1}(\lambda,\mu) = \left[\mathrm{D}\_{n}(\lambda,\mu)\right]\_{\lambda^{\prime},\mu}^{\top},$$

$$\textbf{B}^{2}(\lambda,\mu) = \left[\mathrm{D}\_{n}(\lambda,\mu)\right]\_{\mu}^{\top}\quad\text{and}\qquad\mathsf{C}^{1,1}(\lambda,\mu) = \left[\mathrm{D}\_{n}(\lambda,\mu)\right]\_{\lambda\lambda^{\prime}}^{\top},\qquad\mathsf{C}^{2,2}(\lambda,\mu) = \left[\mathrm{D}\_{n}(\lambda,\mu)\right]\_{\mu\mu^{\prime}}^{\top},$$

$$\textbf{C}^{1,2}(\lambda,\mu) = \left[\mathrm{D}\_{n}(\lambda,\mu)\right]\_{\lambda^{\prime}}^{\top},\ \mathsf{C}^{2,1}(\lambda,\mu) = \left[\mathrm{D}\_{n}(\lambda,\mu)\right]\_{\mu\lambda}^{\top}\text{ for}\quad\lambda = \lambda\_{\mathrm{m}},\ \ \mu = \mu\_{\mathrm{m}}.$$

**Step 5.** Using the decomposition (49) and relations (50) we calculate ( , ) *m m f* , (,) *m m f* , ( , ) *m m f* , (,) *m m f* , (,) *m m f* and (,) *m m f* and construct the matrix of the second derivatives of (48).

Numerical Algorithms of Finding the Branching Lines

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

 

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

   

 

and Bifurcation Points of Solutions for One Class of Nonlinear Integral Equations 307


**Step 10. The end**
