AP07_2-3.vp 1 Preliminaries The magnetic fields of planets, stars and galaxies are maintained by dynamo effects in conducting fluids or plasmas [1, 2, 3]. These dynamo effects are caused by a topologically nontrivial interplay of fluid (plasma) motions and a bal- anced self-amplification of the magnetic fields – and can be described within the framework of magnetohydrodynamics (MHD) [1, 2]. For physically realistic dynamos the coupled system of Maxwell and Navier-Stokes equations has, in general, to be solved numerically. For a qualitative understanding of the oc- curring effects, semi-analytically solvable toy models play an important role. One of the simplest dynamo models is the so called �2-dynamo with a spherically symmetric �-pro- file(1) (see, e.g. [2]). For such a dynamo the magnetic field can be decomposed into poloidal and toroidal components, expanded over spherical harmonics [2, 4] and unitarily re- -scaled [5]. As a result, one arrives at a set of mode decoupled matrix differential eigenvalue problems [2, 4, 5] A� � � � � � � � � � � Q Q Q [ ] [ ] [ ] 1 1 , Q r r l l r r r[ ]: ( ) ( ) ( ) � � � � �� � 1 2 , (1) with boundary conditions (BCs) which have to be imposed in dependence on the concrete physical setup and which will be discussed below. The �-profile describes the net effect of small scale helical turbulence on the magnetic field [2]. It can be assumed real-valued �( )r � � and sufficiently smooth. We note that the reality of the differential expression (1), in- dependently from the concrete BCs, implies an operator spectrum which is symmetric with regard to the real axis, i.e. which consists of purely real eigenvalues and of complex conjugate eigenvalue pairs. In [4] it was shown that the differential expression (1) of this operator has the fundamental (canonical) symmetry [6, 7] A A� �� J J † , J I I � � � � � � 0 0 . (2) In case of BCs compatible with this fundamental sym- metry the operator turns out self-adjoint in a Krein space(2) © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 75 Acta Polytechnica Vol. 47 No. 2–3/2007 The Spherically Symmetric �2–dynamo and Some of its Spectral Peculiarities U. Günther, O. N. Kirillov, B. F. Samsonov, F. Stefani A brief overview is given of recent results on the spectral properties of spherically symmetric MHD �2-dynamos. In particular, the spectra of sphere-confined fluid or plasma configurations with physically realistic boundary conditions (BCs) (surrounding vacuum) and with idealized BCs (super-conducting surrounding) are discussed. The subjects comprise third-order branch points of the spectrum, self-adjointness of the dynamo operator in a Krein space as well as the resonant unfolding of diabolical points. It is sketched how certain classes of dynamos with a strongly localized �-profile embedded in a conducting surrounding can be mode decoupled by a diagonalization of the dynamo operator matrix. A mapping of the dynamo eigenvalue problem to that of a quantum mechanical Hamiltonian with energy dependent potential is used to obtain qualitative information about the spectral behavior. Links to supersymmetric Quantum Mechanics and to the Dirac equation are indicated. Keywords: MHD dynamo, operator spectrum, Krein space, boundary conditions, supersymmetric Quantum Mechanics, diabolical points, resonance, KdV soliton potential. Fig. 1: Real and imaginary components of the �2-dynamo spec- trum as functions of the scale factor C of an �-profile �( ) ( . . . )r C r r r� � � � � � �1 26 09 5364 28 222 3 4 in the case of angular mode number l � 1 and physically re- alistic boundary conditions (3). The concrete coefficients in the quartic polynomial �( )r have their origin in nu- merical simulations of field reversal dynamics (see Ref. [20, 21]). Only the imaginary components with �� 0 are shown. The symmetrically located complex conjugate �� 0 components are omitted for the sake of brevity. (�J, [.,.]J) [4, 5] and in this way it behaves similar like Hamil- tonians of ��-symmetric Quantum Mechanics (PTSQM) [9–14]. Subsequently, we first present a sketchy overview of some recent results on the spectral behavior of �2-dynamos ob- tained in [5, 15–19], which we extend by a discussion of the transition from �2-dynamo configurations confined in a box to dynamos living in an unconfined conducting surrounding. 2 Physically realistic BCs and spectral triple points For roughly spherically symmetric dynamical systems like the Earth, the conducting fluid is necessarily confined with- in the core of the Earth so that the �-effect resulting from the fluid motion has to be confined to this core. Setting the surface of the outer core at a radius r �1, one can assume �( )r � �1 0 and a behavior of the magnetic field at r �1 like in a vacuum. A multi-pole-like decay of the magnetic field at r � � then leads to mixed effective BCs at r �1 (see, e.g. [2]) and a corresponding operator domain of the type �� �( ) ~ ( , ) ( , ) (A u u� � � � �L L r2 201 01 �0 0) ,� Bu r � �1 0� , u:� � � � � � u u 1 2 , (3) B:� � � � � � � �r l r 0 0 1 . From the domain �( )†A� of the adjoint operator �� �( ) � � ( , ) ( , ) �(†A u u� � � � �L L r2 201 01 �0 0) , � �� Bu r � �1 0� , �: � � u � � � � � � u u 1 2 , (4) � : ( ) B � �� � � � � � � � �r r l r r 0 1 , one reads off that � �( ) ( )†A A� �� and, hence, the dynamo operator itself is not self-adjoint even in a Krein space. In the case of constant �-profiles and arbitrary l � �, the spectrum is implicitly given by a characteristic equation built from spherical Bessel functions [2]. In all other cases numeri- cal studies are required. A typical spectral branch graph is depicted in Fig. 1. Obviously, for the specific �-profile it con- tains a large number of spectral phase transitions from real spectral branches to complex ones and back. There are strong indications that phase transition points (second order branch points/exceptional points) of the spectrum close to the � � 0 line play an important role in polarity reversals of the mag- netic field (see [20–23] for numerical studies and [24] for recent experiments). Apart from the second-order branch points visible in Fig. 1 there may occur third- and higher-order branch points. They are located on hyper-surfaces of higher co-dimension in parameter space and they therefore require the tuning of more parameters to pin them down(3). Corresponding re- sults have been obtained in [16] and are illustrated in Fig. 2. The triple points result from coalescing second-order branch points, correspond to 3×3 Jordan blocks in the spectral decomposition of the operator and are accompanied by a merging or disconnecting of two complex spectral sectors 76 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 47 No. 2–3/2007 Fig. 2: �2-dynamo with a r C r( ) [ ( . . ) ( . . ) ( .� � �21 465 2 467 426 412 167 928 8062� � 729 436 289 392 276 2729913 4 . ) ( . . ) ]� �r r and a spectral triple point at ( . , . )� � �0 45 0 86C . Highlighted (fat) lines correspond to purely real branches of the spectrum. The cusp in the imaginary component (lower right graphics) indicates the closely located triple point. over the parameter space. An implicit indication of a closely located triple point is the presence of cusps in the imaginary components as shown in Figs.1, 2. 3 Idealized BCs and Krein-space related perturbation theory In order to gain some deeper insight into possible dynamo-related processes, semi-analytical toy model consid- erations play a crucial role. A certain simplification of the eigenvalue problem was achieved in [17, 18] by considering a reduced and idealized (auxiliary) problem(4) with Dirichlet BCs imposed at r �1, i.e. by setting u( )r � �1 0. In this case it holds � �( ) ( )†A A� �� and the operator A� is self-adjoint in a Krein space ( , [.,. ] )� J J [5]. For con- stant �-profiles � �( )r � �0 const the eigenvalue problem ( )A u� �0 0� � becomes exactly solvable in terms of ortho- normalized Riccati-Bessel functions � � � � u r N r J r N J u u u n n l n n l n m n mn n ( ) , : , ( , ) , � � � � 1 2 1 2 3 2 2 1 � � � (5) with �n � 0 the squares of Bessel function roots � �Jl n �1 2 0� . The solutions of the eigenvalue problem have the form un n nu L � � � � � � � � � 1 0 12 � � ( , ), (6) are Krein space orthonormalized [u u [u u u u u m n n mn m n n n n � � � � � � � � � � � � � , ] , , ] , , : , 2 0� � �� � � � � (7) and correspond to eigenvalue branches � � �� ��n n n� � 0 which scale linearly with �0. In the ( , )� �0 � -plane the branches � n and � n � of states un , un � of positive and negative Krein space type form a spectral mesh (see Fig. 3). The inter- section points (nodes of the mesh) are semisimple double eigenvalues, i.e. eigenvalues of geometric and algebraic mul- tiplicity two – so-called diabolical points (DPs) [26]. Two given branches � ��n ( )0 and � � � m( )0 intersect at the single point � � �� � � � � � � � � � � � � 0 0 0: , :n m n m, and we obtain that branches from states of opposite Krein space type � �� � intersect for � 0 0� , whereas states of the same type (� �� ) intersect at � 0 0� . Under small inho- mogeneous perturbations � � � � ( ) ( ) ( )r r r� � 0 0� � the diabolical points split � � � 0 0 1� � � into two real or complex points (see also [27] for similar considerations) with leading contribution �1 resulting from the quadratic equation © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 77 Acta Polytechnica Vol. 47 No. 2–3/2007 Fig. 3: The spectral mesh of the operator matrix A� for l � 0 (a); its resonant deformation due to harmonic perturbations of a con- stant �-profile (b), (c); the formation of overcritical oscillatory dynamo regimes for � increasing from � � �0 0( )� to � � � � �35 0 0( ,� � for some branches) (d); and the resonant unfolding of DPs in the complex plane (e), (f). � � � � � � �� � � � � 1 2 1 2 2 � � � � � � � [ , ] [ , ] [ Bu u Bu u Bu n n n m m m n � � � � � � � � , ][ , ] [ , ] , u Bu u Bu un m m n m n m � � 4 0 (8) where [ , ] ( ) Bu um n n m m n m n l l r u u u u� � � �� � �� � � � � � � � � ! " # $ 1 2 dr 0 1 % . (9) The unfolding of the DPs follows the typical Krein space rule. When they result from branches of the same type (� �� , � 0 0� ) then the corresponding DPs unfold purely real-val- ued, whereas DPs from branches of opposite type (� �� � , � 0 0� ) may unfold into complex conjugate eigenvalue pairs. This behavior is clearly shown in Fig. 3b,c. Direct inspection reveals that the spectral meshes of unperturbed operators A� 0 for l � 0 and 0 � �l � show strong qualitative similari- ties so that results obtained for the quasi-exactly solvable l � 0 – model will also qualitatively hold for models with 0 � �l � . Via Fourier expansion of �( )r a very pronounced resonance has been found along parabolas in the ( , )� �0 � – plane indicated by white and colored dots in Fig. 3a – leav- ing regions away from these parabolas almost unaffected. An especially pronounced resonance is induced by cosine pertur- bations which in linear approximation affect only the single parabola j k� 2 , Fig. 3b,e. Sine perturbations act strongest on parabolas j k� �2 1 with decreasing effect on j k m� �2 for increasing m (see Fig. 3c,f). Physically important is the fact that higher mode numbers k (shorter wave lengths of the � �(r) perturbations) affect more negative ��. Due to a mag- netic field behavior & e t� this is the mathematical formulation of the physically plausible fact that small-scale perturbations decay faster than large-scale perturbations. Numerical in- dications for the importance of this behavior in the subtle interplay of polarity reversals and so-called excursions (“aborted” reversals) of the magnetic field have recently been given in [23]. 4 Diagonalizable �2-dynamo operators, SUSYQM and the Dirac equation Another approach to obtain quasi-exact solution classes of the eigenvalue problem ( )A u� �� � 0 consists in a �-de- pendent diagonalization of the operator matrix (1). The basic feature of this technique, as demonstrated in [19], is a two-step procedure consisting of a gauge transformation which diagonalizes the kinetic term and a subsequent global (coordinate-independent) diagonalization of the potential term. Such an operator diagonalization is possible for �-pro- files satisfying the constraint �� � �� � �( ) ( ) ( )r r a r 1 2 03 2 (10) with a � �const � a free parameter. Solutions �(r) of this autonomous differential equation (DE) can be expressed in terms of elliptic integrals. In order to maximally explore the similarities to known QM type models(5) a strongly localized �-profile has been assumed which smoothly vanishes toward r � �. Physically, such a setup can be imagined as a strongly localized dynamo-maintaining turbulent fluid/plasma motion embedded in an unbounded conducting surrounding (plasma) with fixed homogeneous conductivity. The only �-profile with �( )r � � � 0 satisfying (10) has the form of a Korteweg-de Vries(KdV)-type one-soliton potential �( ) cosh[ ( )] r a a r r � � 2 0 . (11) This amazing finding points to deep structural links to KdV and supersymmetric quantum mechanics (SUSYQM) and opens up a completely new exploration approach to �2-dynamos (6). In [19] we restricted our consideration to the most elementary solution properties of such models. The decoupled equation set after a parameter and coordinate rescaling has been found in terms of two quadratic pencils � � �� ! " #$ � � � �� � � � x l l x F x x 2 2 2 2 0 1 1 2 1 2 0 2 ( ) , cosh( � � � ) (12) in the new variable x ar:� and with new auxiliary spectral pa- rameter � �� � �12 1 2� . The equivalence transformation from ( )A u� �� � 0 to (12) is regular for � � 0 and becomes singular at � � 0 where (12) has to be replaced by a Jordan type equa- tion system � � x x V V V 2 0 1 2 0 1 00 0 � � � � � � � � � � � � � � � (13) with potentials V l l x0 2 1 2 21 1� � ��( ) ( )� , V1 � � �. In terms of the original spectral parameter � the eigenvalue problems (12) read � � � �� � � � � � ! ! ! " # $ $ $ � �� �� � � � �x l l x F F2 2 2 1 21 1 2 1 2 ( ) (14) and can be related to the spectral problem of a QM Hamil- tonian with energy E � �� and energy-dependent potential component � � � �� �12 12 1 2 1 2� � � �� � �� �x E x . For physical reasons asymptotically vanishing field configurations with F x� � � �( ) 0, 0 1 0, ( )x � � � are of interest. These Di- richlet BCs at infinity imply the self-adjointness of the opera- tor A� in a Krein space �J – with (12), (13) as a special 78 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 47 No. 2–3/2007 representation of the eigenvalue problem ( )A u� �� � 0. From the structure of (12), (14) it follows that the only free parameter apart from the angular mode number l � � is the maximum position x0 of the �-profile �(x) (the minimum po- sition of the potential component � �2 2( )x ) so that the solu- tion branches will be functions �( )x0 . With the help of SUSY techniques it has been shown in [19] that (14) has a single bound state (BS) type solution which via E � � �� 0 corresponds to an overcritical dynamo mode � � 0. It has been found that the BS solutions of (14) be- have differently for x x J0 � and x x J0 � , where for x x J0 � the description in terms of (12) breaks down and has to be replaced by the singular Jordan type representation (13). By a SUSY inspired factorization ansatz � � �� �x l l x L L2 2 21 1 2 1 2 ( ) † , (15) L wx� � � , L wx † � � , w u u � � , (16) an equivalence relation between (12) and a system of two Dirac equations H� � ��� �� , H Vx� �� �� (17) �� � � � � � � � � �� � � � � � � � � � � � � � � � � � � 1 2 1 0 1 1 0 , , : , , F LF� V w w� � � � � � � �� 0 (18) has been established for models with x x J0 � . General results on Dirac equations then allowed for the conclusion that in the case of x x J0 � the bound state related spectrum has to be real. A perturbation theory with the distance � � �x x J0 from the Jordan configuration as a small parameter sup- plemented by a bootstrap analysis showed that the Dirichlet BCs F x� � � �( ) 0 render only the solution F x ( ) non-trivial and with a real eigenvalue, whereas F x�( ) has to vanish identi- cally F x� '( ) 0. The single spectral branch in terms of �( )x0 and �( )x0 is depicted in Fig. 4 for angular mode numbers l � 0 1 2 3, , , . © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 79 Acta Polytechnica Vol. 47 No. 2–3/2007 Fig. 4: Spectra � ( )x0 (a) and �( )x0 (b) in the case of angular mode numbers l � 0 1 2 3, , , . For numerical reasons the Dirichlet BC has been imposed at the large distance x � 100. Fig. 5: Cutoff (X-)dependence of the spectral branches with radial mode numbers n � 1 2, and angular mode numbers l � 0 (a) and l � 2 (b) for cutoffs (box-lengths) X � 10 20 40, , . Clearly visible are the X-independence of the overcritical BS type modes (n � 1) and the tendency � & �1 2X for the undercritical (box-type) mode. The modes with n � 3 show the same qualitative � & �1 2X behavior as the n � 2 mode and are note depicted here. Assuming the dynamo model with strongly localized �-profile (11), (12) confined in a large box, i.e. with Dirichlet BCs imposed at large x X� �0, one can study the dynamo spectrum in the infinite box limit. Figs. 5a,b show the corresponding behavior. Due to its lo- calization the BS-related overcritical dynamo mode is almost insensitive to the X � � limit. This is in contrast to the un- der-critical (decaying) modes which behave as expected for a sign inverted box spectrum of QM. For fixed mode number n � 2 and X � � the energies En decrease like E Xn &1 2 � 0 and the corresponding part of the spectrum becomes quasi-continuous and related to the continuous (essential) spectrum of QM scattering states of a particle moving in the energy dependent potential l l x x E x ( ) ( ) ( ) � � � � � � 1 1 2 1 22 2 1 2� �� . For the associated dynamo eigenvalues this implies � & �1 2X �0 – as is clearly shown in Figs. 5a,b. 5 Concluding remarks A brief overview of some recent results on the spectra of dynamo operators has been given. The obtained structural features like the resonance effects in the unfolding of diaboli- cal points as well as the unexpected link to KdV soliton poten- tials, elliptic integrals, SUSYQM and the Dirac equation ap- pear capable to open new semi-analytical approaches to the study of �2-dynamos. Acknowledgment The work reviewed here has been supported by the Ger- man Research Foundation DFG, grant GE 682/12-3 (U.G.), by the CRDF-BRHE program and the Alexander von Humboldt Foundation (O.N.K.), as well as by RFBR-06-02-16719 and SS-5103.2006.2 (B.F.S.). Remarks (1) The �-profile �(x) plays the role of an effective potential for the �2-dynamo. (2) For comprehensive discussions of operators in Krein spaces see, e.g., [6–8]. (3) An explicit hyper-surface parametrization of second- -order branch point configurations embedded in a ��- symmetric 3 × 3 matrix model with corresponding 2 × 2 Jordan-block preserving modes can be found e.g. in the recent work [25]. (4) From a physical point of view such �2-dynamos can be re- garded as embedded in a superconducting surrounding. (5) For early comments on structural links between MHD dynamo models and QM-related eigenvalue problems, see e.g. [28]. (6) The question of whether this new class of quasi-exactly solvable �2-dynamo models might be structurally re- lated (via dynamical embedding) to the recently studied ��-symmetrically extended KdV solitons [29, 30] remains to be clarified. References [1] Moffatt, H. K.: Magnetic Field Generation in Electrically Conducting Fluids. Cambridge University Press, Cam- bridge, 1978. [2] Krause, F., Rädler, K.-H.: Mean-Field Magnetohydro- dynamics and Dynamo Theory. Akademie-Verlag, Berlin and Pergamon Press, Oxford, 1980, Chapter 14. [3] Zeldovich, Ya. B., Ruzmaikin, A. A., Sokoloff, D. D.: Magnetic Fields in Astrophysics. Gordon & Breach Science Publishers, New York, 1983. [4] Günther, U., Stefani, F.: Isospectrality of Spherical MHD Dynamo Operators: Pseudo-Hermiticity and a No-Go Theorem. Journal Math. Phys. Vol. 44 (2003), p. 3097–3111, math-ph/0208012. [5] Günther, U., Stefani, F., Znojil, M.: MHD �2-dynamo, Squire Equation and ��-symmetric Interpolation Be- tween Square Well and Harmonic Oscillator. J. Math. Phys. Vol. 46 (2005), 063504, math-ph/0501069. [6] Azizov, T. Ya., Iokhvidov, I. S.: Linear Operators in Spaces with an Indefinite Metric. Wiley-Interscience, New York, 1989. [7] Dijksma, A., Langer, H.: Operator Theory and Ordinary Differential Operators. In: Böttcher A. (ed.) et al.: Lec- tures on Operator Theory and its Applications. Fields Institute Monographs, Vol. 3 (1996), p. 75, Am. Math. Soc., Provi- dence, RI. [8] Bognár, J.: Indefinite Inner Product Spaces. Springer, New York, 1974. [9] Bender, C. M., Boettcher, S.: Real Spectra in Non- -Hermitian Hamiltonians Having PT Symmetry. Phys. Rev. Lett. Vol. 80 (1998), p. 5243–5246, physics/9712001 [10] Bender, C. M., Boettcher, S., Meisinger, P. N.: ��-sym- metric Quantum Mechanics. J. Math. Phys. Vol. 40 (1999), p. 2201–2229, quant-ph/9809072. [11] Znojil, M.: ��-symmetric Harmonic Oscillators. Phys. Lett. Vol. A259, (1999), p. 220–223, quant-ph/9905020; Znojil, M.: Quasi-Exactly Solvable Quartic Potentials with Centrifugal and Coulombic Terms. J. Phys. A: Math. Gen. Vol. 33 (2000), p. 4203–4211, math-ph/0002036; Znojil, M., Cannata, F., Bagchi, B., Roychoudhury, R.: Supersymmetry without Hermiticity. Phys. Lett. Vol. B483 (2000), p. 284–289, hep-th/0003277. [12] Mostafazadeh, A.: Pseudo-Hermiticity versus �� Sym- metry: The Necessary Condition for the Reality of the Spectrum of a Non-Hermitian Hamiltonian. J. Math. Phys. Vol. 43 (2002), p. 205–214, math-ph/0107001; Pseudo-Hermiticity versus ��-symmetry II: A Complete Characterization of Non-Hermitian Hamiltonians with a Real Spectrum. ibid. Vol. 43 (2002), p. 2814–2816, math-ph/0110016; Pseudo-Hermiticity versus ��-Sym- metry III: Equivalence of Pseudo-Hermiticity and the Presence of Antilinear Symmetries. ibid. Vol. 43 (2002), p. 3944–3951, math-ph/0203005. [13] Bender, C. M., Brody, D. C., Jones, H. F.: Complex Ex- tension of Quantum Mechanics. Phys. Rev. Lett. Vol. 89 (2002), 270401, quant-ph/0208076. [14] Bender, C. M.: Making Sense of Non-Hermitian Hamil- tonians. hep-th/0703096. 80 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 47 No. 2–3/2007 [15] Günther, U., Stefani, F., Gerbeth, G.: The MHD �2-dy- namo, �2- Graded Pseudo-Hermiticity, Level Crossings, and Exceptional Points of Branching Type. Czech. J. Phys. Vol. 54 (2004), p. 1075–1090, math-ph/0407015. [16] Günther, U., Stefani, F.: Third Order Spectral Branch Points in Krein Space Related Setups: ��-symmetric Ma- trix Toy Model, MHD �2-dynamo, and Extended Squire Equation. Czech. J. Phys. Vol. 55 (2005), p. 1099–1106, math-ph/0506021. [17] Günther, U., Kirillov, O. N.: Krein Space Related Per- turbation Theory for MHD �2-dynamos and Resonant Unfolding of Diabolical Points. J. Phys. A: Math. Gen. Vol. 39 (2006), p. 10057–10076, math-ph/0602013. [18] Kirillov, O. N., Günther, U.: On Krein Space Related Perturbation Theory for MHD �2-dynamos. Proc. Appl. Math. Mech. (PAMM) Vol. 6 (2006), p. 637–638. [19] Günther, U., Samsonov, B. F., Stefani, F.: A Globally Diagonalizable �2-dynamo Operator, SUSY QM and the Dirac Equation. J. Phys. A: Math. Theor. Vol. 40 (2007), p. F169–F176, math-ph/0611036. [20] Stefani, F., Gerbeth, G.: Asymmetric Polarity Reversals, Bimodal Field Distribution, and Coherence Resonance in a Spherically Symmetric Mean-Field Dynamo Model. Phys. Rev. Lett. Vol. 94 (2005), 184506, physics/0411050. [21] Stefani, F., Gerbeth, G., Günther, U., Xu, M.: Why Dy- namos are Prone to Reversals. Earth Planet. Sci. Lett. Vol. 243 (2006), p. 828–840, physics/0509118. [22] Stefani, F., Gerbeth, G., Günther, U.: A Paradigmatic Model of Earth’s Magnetic Field Reversals. Magneto- hydrodynamics, Vol. 42 (2006), p. 123–130, phys- ics/0601011. [23] Stefani, F., Xu, M., Sorriso-Valvo, L., Gerbeth, G., Günther, U.: Oscillation or Rotation: a Comparison of Two Simple Reversal Models. Geophys. Astrophys. Fluid Dyn. Vol. 101 (2007), p. 227–248. [24] Berhanu, M. et al.: Magnetic Field Reversals in an Ex- perimental Turbulent Dynamo. Eur. Phys. Lett. Vol. 77 (2007), 59001, physics/0701076. [25] Znojil, M.: A Return to Observability Near Exceptional Points in a Schematic ��-symmetric Model, Phys. Lett. Vol. B647 (2007), p. 225–230, quant-ph/0701232. [26] Berry, M. V., Wilkinson, M.: Diabolical Points in the Spectra of Triangles. Proc. R. Soc. Lond. Vol. A392 (1984), p. 15–43. [27] Kirillov, O. N., Seyranian, A. P.: Collapse of the Keldysh Chains and Stability of Continuous Nonconservative Systems. SIAM Journal Appl. Math. Vol. 64 (2004), p. 1383–1407. [28] Meinel, R.: Generation of Localized Magnetic Fields by Dynamos with Conducting Surroundings. Astron. Nachr. Vol. 310 (1989), p. 1–6. [29] Bender, C. M., Brody, D. C., Chen, J., Furlan, E.: ��-symmetric Extension of the Korteweg-de Vries Equa- tion. J. Phys. A: Math. Theor. Vol. 40 (2007), p. F153–F160, math-ph/0610003. [30] Fring, A.: ��-symmetric Deformations of the Korte- weg-de Vries Equation. J. Phys. A: Math. Theor. Vol. 40 (2007), p. 4215–4224, math-ph/0701036. Dr. Uwe Günther phone: +49 351 260 2415 email: u.guenther@fzd.de Research Center Dresden-Rossendorf POB 510119 D-01314 Dresden, Germany Dr. Oleg N. Kirillov phone: +7 7095 939 2039 email: kirillov@imec.msu.ru Institute of Mechanics Moscow State Lomonosov University Michurinskii pr. 1 119192 Moscow, Russia Prof. Dr. Boris F. Samsonov phone: +7 3822 913 019 email: samsonov@phys.tsu.ru Physics Department Tomsk State University 36 Lenin Avenue 634050 Tomsk, Russia Dr. Frank Stefani phone: +490 351 260 3069 email: f.stefani@fz-rossendorf.de Research Center Dresden-Rossendorf POB 510119 D-01314 Dresden, Germany © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 81 Acta Polytechnica Vol. 47 No. 2–3/2007