# [

###### Abstract

We consider two-point non-self-adjoint boundary eigenvalue problems for linear matrix differential operators. The coefficient matrices in the differential expressions and the matrix boundary conditions are assumed to depend analytically on the complex spectral parameter and on the vector of real physical parameters . We study perturbations of semi-simple multiple eigenvalues as well as perturbations of non-derogatory eigenvalues under small variations of . Explicit formulae describing the bifurcation of the eigenvalues are derived. Application to the problem of excitation of unstable modes in rotating continua such as spherically symmetric MHD -dynamo and circular string demonstrates the efficiency and applicability of the theory.

Multiparameter perturbation theory for non-self-adjoint operator matrices]
Perturbation of multiparameter

non-self-adjoint boundary eigenvalue

problems for
operator matrices
O.N. Kirillov]O.N. Kirillov

This work was completed with the support of the Alexander von Humboldt Foundation and the grant DFG HA 1060/43-1.\par\@mkboth\shortauthors\shorttitle \par [\par

Mathematics Subject Classification (2000). Primary 34B08; Secondary 34D10.

Keywords. operator matrix, non-self-adjoint boundary eigenvalue problem, Keldysh chain, multiple eigenvalue, diabolical point, exceptional point, perturbation, bifurcation, stability, veering, spectral mesh, rotating continua.

Non-self-adjoint boundary eigenvalue problems for matrix differential operators describe distributed non-conservative systems with the coupled modes and appear in structural mechanics, fluid dynamics, magnetohydrodynamics, to name a few. \parPractical needs for optimization and rational experiment planning in modern applications allow both the differential expression and the boundary conditions to depend analytically on the spectral parameter and smoothly on several physical parameters (which can be scalar or distributed). According to the ideas going back to von Neumann and Wigner [1], in the multiparameter operator families, eigenvalues with various algebraic and geometric multiplicities can be generic [12]. In some applications additional symmetries yield the existence of spectral meshes [33] in the plane ‘eigenvalue versus parameter’ containing infinite number of nodes with the multiple eigenvalues [10, 15, 30, 38]. As it has been pointed out already by Rellich [2] sensitivity analysis of multiple eigenvalues is complicated by their non-differentiability as functions of several parameters. Singularities corresponding to the multiple eigenvalues [12] are related to such important effects as destabilization paradox in near-Hamiltonian and near-reversible systems [5, 18, 25, 26, 27, 35], geometric phase [31], reversals of the orientation of the magnetic field in MHD dynamo models [32], emission of sound by rotating continua interacting with the friction pads [38] and other phenomena [23]. \parAn increasing number of multiparameter non-self-adjoint boundary eigenvalue problems and the need for simple constructive estimates of critical parameters and eigenvalues as well as for verification of numerical codes, require development of applicable methods, allowing one to track relatively easily and conveniently the changes in simple and multiple eigenvalues and the corresponding eigenvectors due to variation of the differential expression and especially due to transition from one type of boundary conditions to another one without discretization of the original distributed problem, see e.g. [11, 15, 18, 20, 36, 37, 38]. \parA systematical study of bifurcation of eigenvalues of a non-self-adjoint linear operator due to perturbation , where is a small parameter, dates back to 1950s. Apparently, Krein was the first who derived a formula for the splitting of a double eigenvalue with the Jordan block at the Hamiltonian resonance, which was expressed through the generalized eigenvectors of the double eigenvalue [3]. In 1960 Vishik and Lyusternik and in 1965 Lidskii created a perturbation theory for nonsymmetric matrices and non-self-adjoint differential operators allowing one to find the perturbation coefficients of eigenvalues and eigenfunctions in an explicit form by means of the eigenelements of the unperturbed operator [4, 6]. Classical monographs by Rellich [9], Kato [7], and Baumgärtel [13], mostly focusing on the self-adjoint case, contain a detailed treatment of eigenvalue problems linearly or quadratically dependent on the spectral parameter. \parRecently Kirillov and Seyranian proposed a perturbation theory of multiple eigenvalues and eigenvectors for two-point non-self-adjoint boundary eigenvalue problems with scalar differential expression and boundary conditions, which depend analytically on the spectral parameter and smoothly on a vector of physical parameters, and applied it to the sensitivity analysis of distributed non-conservative problems prone to dissipation-induced instabilities [16, 17, 19, 21, 20, 24, 26, 27, 38]. An extension to the case of intermediate boundary conditions with an application to the problem of the onset of friction-induced oscillations in the moving beam was considered in [34]. In [33] this approach was applied to the study of MHD -dynamo model with idealistic boundary conditions. \parIn the following we develop this theory further and consider boundary eigenvalue problems for linear non-self-adjoint -th order matrix differential operators on the interval . The coefficient matrices in the differential expression and the matrix boundary conditions are assumed to depend analytically on the spectral parameter and smoothly on a vector of real physical parameters . The matrix formulation of the boundary conditions is chosen for the convenience of its implementation in computer algebra systems for an automatic derivation of the adjoint eigenvalue problem and perturbed eigenvalues and eigenvectors, which is especially helpful when the order of the derivatives in the differential expression is high. Based on the eigenelements of the unperturbed problem explicit formulae are derived describing bifurcation of the semi-simple multiple eigenvalues (diabolical points) as well as non-derogatory eigenvalues (branch points, exceptional points) under small variation of the parameters in the differential expression and in the boundary conditions. Finally, the general technique is applied to the investigation of the onset of oscillatory instability in rotating continua. \par\par\@xsect \parFollowing [8, 19, 22, 24, 26, 27] we consider the boundary eigenvalue problem

(\theequation) |

where . The differential expression of the operator is

(\theequation) |

and the boundary forms are

(\theequation) |

Introducing the block matrix and the vector

(\theequation) |

the boundary conditions can be compactly rewritten as [26, 27]

(\theequation) |

where and . It is assumed that the matrices , , and are analytic functions of the complex spectral parameter and smooth functions of the real vector of physical parameters . For some fixed vector the eigenvalue , to which the eigenvector corresponds, is a root of the characteristic equation obtained after substitution of the general solution to equation into the boundary conditions (\@setrefs7) [8]. \parLet us introduce a scalar product [8]

(\theequation) |

where the asterisk denotes complex-conjugate transpose (). Taking the scalar product of and a vector-function and integrating it by parts yields the Lagrange formula for the case of operator matrices (cf. [8, 22, 26, 27])

(\theequation) |

with the adjoint differential expression [8, 22]

(\theequation) |

the vector

(\theequation) |

and the block matrix

(\theequation) |

where the matrices are

(\theequation) |

Extend the original matrix (cf. (\@setrefs7)) to a square matrix , which is made non-degenerate in a neighborhood of the point and the eigenvalue by an appropriate choice of the auxiliary matrices and

(\theequation) |

Similar transformation for the adjoint boundary conditions yields

(\theequation) |

Then, the form in (\@setrefs13) can be represented as [8]

(\theequation) |

so that without loss in generality we can assume [26, 27]

(\theequation) |

Differentiating the equation (\@setrefs28) we find

(\theequation) |

Hence, we obtain the formula for calculation of the matrix of the adjoint boundary conditions and the auxiliary matrix

(\theequation) |

which exactly reproduces and extends the corresponding result of [26, 27]. \par\par\@xsect \parAssume that in the neighborhood of the point the spectrum of the boundary eigenvalue problem (\@setrefs1) is discrete. Denote and . Let us consider a smooth perturbation of parameters in the form where and is a small real number. Then, as in the case of analytic matrix functions [25], the Taylor decomposition of the differential operator matrix and the matrix of the boundary conditions are [19, 24, 26, 27]

(\theequation) |

with

(\theequation) |

where dot denotes differentiation with respect to at and partial derivatives are evaluated at , . Our aim is to derive explicit expressions for the leading terms in the expansions for multiple-semisimple and non-derogatory eigenvalues and for the corresponding eigenvectors. \par\par\@xsect \parLet at the point the spectrum contain a semi-simple -fold eigenvalue with linearly-independent eigenvectors , , , . Then, the perturbed eigenvalue and the eigenvector are represented as Taylor series in [4, 24, 28, 29, 33]

(\theequation) |

Substituting expansions (\@setrefp2) and (\@setrefp6) into (\@setrefs1) and collecting the terms with the same powers of we derive the boundary value problems

(\theequation) |

(\theequation) |

Scalar product of (\@setrefp8) with the eigenvectors , of the adjoint boundary eigenvalue problem

(\theequation) |

yields equations

(\theequation) |

With the use of the Lagrange formula (\@setrefs13), (\@setrefs28) and the boundary conditions (\@setrefp8) the left hand side of (\@setrefp11) takes the form

(\theequation) |

Together (\@setrefp11) and (\@setrefp13) result in the equations

(\theequation) |

Assuming in the equations (\@setrefp14) the vector as a linear combination

(\theequation) |

and taking into account that

(\theequation) |

where , we arrive at the matrix eigenvalue problem (cf. [LMZ03])

(\theequation) |

The entries of the matrices and are defined by the expressions

(\theequation) |

Therefore, in the first approximation the splitting of the semi-simple eigenvalue due to variation of parameters is , where the coefficients are generically distinct roots of the -th order polynomial

(\theequation) |

For the formulas (\@setrefp18) and (\@setrefp19) describe perturbation of a simple eigenvalue

(\theequation) |

The formulas (\@setrefp18),(\@setrefp19), and (\@setrefp20) generalize the corresponding results of the works [24] to the case of the multiparameter non-self-adjoint boundary eigenvalue problems for operator matrices. \par\par\@xsect \parLet at the point the spectrum contain a -fold eigenvalue with the Keldysh chain of length , consisting of the eigenvector and the associated vectors . The vectors of the Keldysh chain solve the following boundary value problems [8, 22]

(\theequation) |

(\theequation) |

Consider vector-functions , , , . Let us take scalar product of the differential equation (\@setrefp21) and the vector-function . For each we take the scalar product of the equation (\@setrefp22) and the vector-function . Summation of the results yields the expression

(\theequation) |

Applying the Lagrange identity (\@setrefs13), (\@setrefs28) and taking into account relation (\@setrefs29), we transform (\@setrefp23) to the form

(\theequation) |

Equation (\@setrefp24) is satisfied in case when the vector-functions , , , originate the Keldysh chain of the adjoint boundary value problem, corresponding to the -fold eigenvalue [26, 27]

(\theequation) |

(\theequation) |

Taking the scalar product of equation (\@setrefp22) and the vector and employing the expressions (\@setrefs13), (\@setrefs28) we arrive at the orthogonality conditions

(\theequation) |

Substituting into equations (\@setrefs1) the Newton-Puiseux series for the perturbed eigenvalue and eigenvector [7, 13, 26, 27]

(\theequation) |

where , taking into account expansions (\@setrefp2) and (\@setrefp26) and collecting terms with the same powers of , yields boundary value problems serving for determining the functions ,

(\theequation) |

(\theequation) |

where and , , are positive integers. The vector-function is a solution of the following boundary value problem

(\theequation) |

(\theequation) |

Comparing equations (\@setrefp31) and (\@setrefp32) with the expressions (\@setrefp22) we find the first functions in the expansions (\@setrefp26)

(\theequation) |

With the vectors (\@setrefp33) we transform the equations (\@setrefp31) and (\@setrefp32) into

(\theequation) |

(\theequation) |

Applying the expression following from the Lagrange formula

(\theequation) | |||||

and taking into account the equations for the adjoint Keldysh chain (\@setrefp24a) and (\@setrefp25a) yields the coefficient in (\@setrefp26). Hence, the splitting of the -fold non-derogatory eigenvalue due to perturbation of the parameters is described by the following expression, generalizing the results of the works [19, 24, 26, 27]

(\theequation) |

For equation (\@setrefp37) is reduced to the equation (\@setrefp20) for a simple eigenvalue. \par\par\@xsect \parConsider a circular string of displacement , radius , and mass per unit length that rotates with the speed and passes at through a massless eyelet generating a constant frictional follower force on the string, as shown in Fig. \@setreffig1. The circumferential tension in the string is assumed to be constant; the stiffness of the spring supporting the eyelet is and the damping coefficient of the viscous damper is . Introducing the non-dimensional variables and parameters

(\theequation) |

and assuming we arrive at the non-self-adjoint boundary eigenvalue problem for a scalar differential operator [15]

(\theequation) |

(\theequation) |

where prime denotes differentiation with respect to . Parameters , , , and express the speed of rotation, and damping, stiffness, and friction coefficients. \par

For the unconstrained rotating string with , , and the eigenfunctions and of the adjoint problems, corresponding to purely imaginary eigenvalue and , coincide. With assumed as a solution of (\@setrefst4), the characteristic equation follows from (\@setrefst5)

(\theequation) |

The eigenvalues with the eigenfunctions , , found from equation (\@setrefst8) form the spectral mesh in the plane , Fig. \@setreffig1. The lines and , where , intersect each other at the node with

(\theequation) |

where the double eigenvalue has two linearly independent eigenfunctions

(\theequation) |

Using the perturbation formulas for semi-simple eigenvalues (\@setrefp18) and (\@setrefp19) with the eigenelements (\@setrefst11) and (\@setrefst13) we find an asymptotic expression for the eigenvalues originated after the splitting of the double eigenvalues at the nodes of the spectral mesh due to interaction of the rotating string with the external loading system

(\theequation) |

with and

(\theequation) | |||||

Due to action of gyroscopic forces and an external spring double eigenvalues split in the subcritical region (, and ) as

(\theequation) |

while in the supercritical region (, and , )

(\theequation) |

Therefore, for the spectral mesh collapses into separated curves demonstrating avoided crossings; for the eigenvalue branches overlap forming the bubbles of instability with eigenvalues having positive real parts, see Fig. \@setreffig2. From (\@setrefst22) a linear approximation follows to the boundary of the domains of supercritical flutter instability in the plane (gray resonance tongues in Fig. \@setreffig2)

(\theequation) |

At the nodes with the external damper creates a circle of complex eigenvalues being a latent source of subcritical flutter instability responsible for the emission of sound in the squealing brake and the singing wine glass [38]

(\theequation) |

(\theequation) |

as shown in Fig. \@setreffig3. Non-conservative perturbation yields eigenvalues with

(\theequation) |

(\theequation) |

so that both the real and imaginary parts of the eigenvalue branches show a degenerate crossing, touching at the node , Fig. \@setreffig3. Deformation patterns of the spectral mesh and first-order approximations of the instability tongues obtained by the perturbation theory and shown in Fig. \@setreffig2 and Fig. \@setreffig3 are in a good qualitative and quantitative agreement with the results of numerical calculations of [15]. \par\par\@xsect \parConsider a non-self-adjoint boundary eigenvalue problem appearing in the theory of MHD -dynamo [14, 32, 33, 37] \par

(\theequation) |

with the matrices of the differential expression

(\theequation) |

and of the boundary conditions

(\theequation) |

where it is assumed that with . For the fixed the differential expression depends on the parameters and , while interpolates between the idealistic () and physically realistic () boundary conditions [14, 32, 33, 36]. \parThe matrix of the boundary conditions and auxiliary matrix for the adjoint differential expression follow from the formula (\@setrefs31) where the matrices , and are chosen as

(\theequation) |

In our subsequent consideration we assume that and interpret and as perturbing parameters. It is known [33] that for and the spectrum of the unperturbed eigenvalue problem (\@setrefdm1) forms the spectral mesh in the plane , as shown in Fig. \@setreffig4. The eigenelements of the spectral mesh are

(\theequation) |

(\theequation) |

The branches (\@setrefdm6) intersect and originate a double semi-simple eigenvalue with two linearly independent eigenvectors (\@setrefdm7) at the node , where [33]

(\theequation) |

Taking into account that the components of the eigenfunctions of the adjoint problems are related as and , we find from equations (\@setrefp18) and (\@setrefp19) the asymptotic formula for the perturbed eigenvalues, originating after the splitting of the double semi-simple eigenvalues at the nodes of the spectral mesh

where

(\theequation) |

When and , one of the two simple eigenvalues (\@setrefdm10) remains unshifted in all orders of the perturbation theory with respect to the parameter : . The sign of the first-order increment to another eigenvalue depends on the sign of , which is directly determined by the Krein signature of the modes involved in the crossing [33]. This is in the qualitative and quantitative agreement with the results of numerical calculations of [36] shown in Fig. \@setreffig4. \par