**4. Algorithm of finding the eigenvalue curves**

294 Nonlinearity, Bifurcation and Chaos – Theory and Applications

*nm*

*nm*

Then equation (16) takes the form

 

linear algebraic equations

*b*

 

 

 

*a cc K cc*

1 2

*M M*

*n Mm M*

1 2

where

where

where

1 2 1 2

*M M M M*

 

1212

1212

*n Mm M n Mm M*

1 2 1 2

*c c <sup>F</sup> K cc <sup>e</sup>*

*c c <sup>F</sup> K cc <sup>e</sup>*

 

> 

 

2

 

( )

 

*kl kl nm nm*

( ) 2 1

*nm kl nm*

*cc b*

 

1 11 2 2

2 11 2 2

*nm q F e dd*

1 2

*n Mm M*

 

1 2

1212 1212 1 2 ( , , , )( , , , ) , *nm nm b K cc ccdd*

1 1 1 1 2 1212 1212 1212 <sup>1</sup> (, ,,) (, ,,) (, ,,) ,

1 2

 *q q* 

*M M*

*n Mm M*

1 2 (,) ,

*nm nm st st nm*

1 2

and the unknown coefficients *nm b* are determined as solutions of a homogeneous system of

11 22 *k MM l MM* , , ,,

12 1212 1212 1 2 (,) (, ,,) (, ,,) , *kl*

 *cc a cca ccdd* 

 

*c c a cc b*

*nm q F e dd*

11 22 <sup>1212</sup> <sup>1212</sup> (, ,,) (, ,,) ,

01212

*f cc* 

01212 (,) (, ,,) , 2 (, ,,)

*f cc* 

1 2 1 2 (,) , *icn cm*

1 2 1 2 (,) . *icn cm*

1 <sup>1212</sup> <sup>1212</sup> (, ,,) (, ,,) , *M M*

 

1 2

*M M*

*s Mt M*

1 2 ,

*nm nm*

 

 

1 2

*K cc q q*

 

 

 

*nm nm*

 

> 

> >

 

*K cc q K cc q*

1 1 2 1 2 11 2 2

2 1 2 1 2 11 2 2

 

(,) (, ,,) , 2 (, ,,)

 

*nm nm nm nm*

 

*icn cm*

*icn cm*

 

 

> The main calculational part of algorithm proposed is the implementation method proposed in [14, 15] to compute all eigenvalues of the nonlinear matrix spectral problem

$$\mathbf{T}\_n(\mathcal{A}, \mu)\mu\_n = 0,\tag{21}$$

belonging to some given range of the spectral parameter at the given value of parameter . In the problem (21) *<sup>n</sup> <sup>n</sup> u* , and ( , ) *<sup>n</sup>* **T** is the real ( ) *n n* matrix whose elements depend nonlinearly on the parameters and . In order to detail how the method [15] is applied to the problem under consideration in this paper, we present the necessary results from [15].

Thus, we replace in the problem (21), for example, the parameter by the expression and consider the appropriate one-parameter problem

$$\mathbf{T\_n}(\mathcal{\lambda})\boldsymbol{\mu}\_n \equiv \mathbf{T\_n}(\mathcal{\lambda}, \boldsymbol{\alpha}, \boldsymbol{\beta})\boldsymbol{\mu}\_n = \mathbf{0},\tag{22}$$

at the given fixed values and . Then, obviously, the eigenvalues of problem (21) are zeros of function

$$f(\mathcal{A}) \equiv \det \mathbf{T}\_n(\mathcal{A}) = 0\_{\prime\prime}$$

where ( ) *<sup>n</sup>* **T** is a real ( ) *n n* matrix whose elements depend nonlinearly on the parameter .

One should determine how many zeros of the function *f*( ) , and, therefore, the eigenvalues of the problem are in some given range of change of parameter , *k k c d* **[ ]** and calculate each of them.
