O N Kirillov

Campbell diagrams of weakly anisotropic flexible rotors

Keywords: Campbell diagram, flexible rotor, dissipation-induced instabilities, subcritical flutter, symplectic (Krein) signature, non-Hermitian degeneracies

We consider an axi-symmetric flexible rotor perturbed by dissipative, conservative and non-conservative positional forces originated at the contact with the anisotropic stator. The Campbell diagram of the unperturbed system is a mesh-like structure in the frequency-speed plane with double eigenfrequencies at the nodes. The diagram is convenient for the analysis of the travelling waves in the rotating elastic continuum. Computing sensitivities of the doublets, we find that at every particular node the unfolding of the mesh into the branches of complex eigenvalues in the first approximation is generically determined by only four 2 × 2 sub-blocks of the perturbing matrix. Selection of the unstable modes that cause self-excited vibrations in the subcritical speed range is governed by the exceptional points at the corners of the singular eigenvalue surfaces-'double coffee filter' and 'viaduct'-which are sharply associated with the crossings of the unperturbed Campbell diagram with the definite symplectic (Krein) signature. The singularities connect the problems of wave propagation in the rotating continua with that of electromagnetic and acoustic wave propagation in non-rotating anisotropic chiral media. As mechanical examples a model of a rotating shaft with two degrees of freedom and a continuous model of a rotating circular string passing through the eyelet are studied in detail.

Introduction

Bending waves propagate in the circumferential direction of an elastic body of revolution rotating about its axis of symmetry [START_REF] Bryan | On the beats in the vibrations of a revolving cylinder or bell[END_REF][START_REF] Southwell | On the free transverse vibrations of a uniform circular disc clamped at its centre; and on the effect of rotation[END_REF][START_REF] Srinivasan | Traveling waves in rotating cylindrical shells[END_REF][START_REF] Ouyang | Unstable travelling waves in the friction-induced vibration of discs[END_REF][START_REF] Genta | Dynamics of rotating systems[END_REF]. The frequencies of the waves plotted against the rotational speed are referred to as the Campbell diagram [START_REF] Campbell | The protection of steam-turbine disk wheels from axial vibration[END_REF][START_REF] Genta | Dynamics of rotating systems[END_REF]. The spectrum of a perfect rotationally symmetric rotor at standstill has infinitely many double semisimple eigenvalues-the doublet modes. Indeed, for R 2×2 A = diag(ω 2 1 , ω 2 2 ) and R = cos θ sin θ sin θ cos θ the restriction R T AR = A imposed by equivariance of the equations of motion with respect to the action of the circle group implies ω 2 1 = ω 2 2 [START_REF] Dellnitz | Generic bifurcation of Hamiltonian vector fields with symmetry[END_REF][START_REF] Nagata | Bifurcations in gyroscopic systems with an application to rotating shafts[END_REF]. By this reason, the Campbell diagram contains the eigenvalue branches originated after splitting of the doublets by gyroscopic forces [START_REF] Bryan | On the beats in the vibrations of a revolving cylinder or bell[END_REF]. The branches correspond to simple pure imaginary eigenvalues and intersect each other forming a spectral mesh [START_REF] Günther | A Krein space related perturbation theory for MHD α 2dynamos and resonant unfolding of diabolical points[END_REF] in the frequency-speed plane with the doublets at the nodes (figure 1a). Perturbations of the axially symmetric rotor by dissipative, conservative and non-conservative positional forces, caused by its contact with the anisotropic stator, generically unfold the spectral mesh of pure imaginary eigenvalues of the Campbell diagram into separate branches of complex eigenvalues in the (Ω, Im λ, Re λ)-space (figure 1d). Nevertheless, the eigenvalue branches in the perturbed Campbell diagram can both avoid crossings and cross each other (figure 1e). Moreover, the real parts of the perturbed eigenvalues plotted against the rotational speed-decay rate plots [START_REF] Genta | Dynamics of rotating systems[END_REF])-can also intersect each other and inflate into 'bubbles' (figure 1f ). This complicated behaviour is difficult to predict and even to interpret according to the studies of numerous mechanical systems [START_REF] Ono | Stability analysis for the head-disk interface in a flexible disk drive[END_REF][START_REF] Chen | Mathematical structure of modal interactions in a spinning diskstationary load system[END_REF][START_REF] Mottershead | Flutter instability of circular discs with frictional follower loads[END_REF][START_REF] Yang | Interactions between an idealized rotating string and stationary constraints[END_REF][START_REF] Tian | Self-excited vibration in flexible rotating discs subjected to various transverse interactive forces: a general approach[END_REF][START_REF] Xiong | Instability mechanism of a rotating disc subjected to various transverse interactive forces[END_REF][START_REF] Canchi | Parametric instability of a circular ring subjected to moving springs[END_REF][START_REF] Young | Stability of a spinning disk under a stationary oscillating unit[END_REF][START_REF] Genta | Dynamics of rotating systems[END_REF][START_REF] Lesaffre | Contact analysis of a flexible bladed-rotor[END_REF][START_REF] Spelsberg-Korspeter | In-and out-of-plane vibrations of a rotating plate with frictional contact: investigations on squeal phenomena[END_REF]. The present work reveals that the unfolding of the Campbell diagrams is determined by a limited number of local scenarios for eigenvalues as a function of parameters, which form stratified manifolds.

A model of a weakly anisotropic rotor system

In general, the imperfections in the rotor and stator complicate the linearized equations of motion, making them non-self-adjoint with time-dependent coefficients. Nevertheless, an axially symmetric rotor with an anisotropic stator as well as an asymmetric rotor with an isotropic stator are autonomous nonconservative gyroscopic systems [START_REF] Genta | Dynamics of rotating systems[END_REF]. Neglecting the centrifugal stiffness without loss of generality, we consider the finite-dimensional anisotropic rotor system

ẍ + (2ΩG + δD)ẋ + (P + Ω 2 G 2 + κK + νN)x = 0, (2.1)
which is a perturbation of the isotropic one [START_REF] Nagata | Bifurcations in gyroscopic systems with an application to rotating shafts[END_REF])

ẍ + 2ΩGẋ + (P + Ω 2 G 2 )x = 0, (2.2) where x = R 2n , P = diag(ω 2 1 , ω 2 1 , ω 2 2 , ω 2 2 , . . . , ω 2 n , ω 2 n )
is the stiffness matrix and G = -G T is the matrix of gyroscopic forces defined as

G = blockdiag(J, 2J, . . . , nJ) and J = 0 -1 1 0 . (2.
3)

The matrices of non-Hamiltonian perturbation corresponding to velocitydependent dissipative forces, D = D T , and non-conservative positional forces, N = -N T , as well as the matrix K = K T of the Hamiltonian perturbation that breaks the rotational symmetry, can depend on the rotational speed Ω. The intensity of the perturbation is controlled by the parameters δ, κ and ν. Putting κ = 0 and ν = 0 in equation (2.1) yields the model considered in [START_REF] Nagata | Bifurcations in gyroscopic systems with an application to rotating shafts[END_REF].

At Ω = 0, the eigenvalues ±iω s , ω s > 0, of the isotropic rotor equation (2.2) are double semisimple with two linearly independent eigenvectors. The sequence of the frequencies ω s , where s is an integer index, is usually different for various bodies of revolution. For example, ω s = s corresponds to the natural frequency f s = s/2πr √ P/ρ of a circular string of radius r, circumferential tension P and mass density ρ per unit length [START_REF] Yang | Interactions between an idealized rotating string and stationary constraints[END_REF].

Substituting x = u exp(λt) into equation (2.2), we arrive at the eigenvalue problem

L 0 (Ω)u := (Iλ 2 + 2ΩGλ + P + Ω 2 G 2 )u = 0.
(2.4)

The eigenvalues of the operator L 0 are found in the explicit form

λ + s = iω s + isΩ, λ - s = -iω s + isΩ, λ - s = iω s -isΩ and λ + s = -iω s -isΩ, (2.5) s
where the overbar denotes complex conjugate. The eigenvectors of λ s + and λ -are u + 1 = (-i, 1, 0, 0, . . . , 0, 0) T , . . . , u + n = (0, 0, . . . , 0, 0, -i, 1) T , (2.6)

where the imaginary unit holds the (2s -1)st position in the vector u + s . The eigenvectors, corresponding to the eigenvalues λ - s and λ + s , are simply u - s = u + s . For Ω > 0, simple eigenvalues λ + s and λ - s correspond to the forward and backward travelling waves, respectively, that propagate in the circumferential direction of the rotor. At the angular velocity Ω cr s = ω s /s the frequency of the sth backward travelling wave vanishes to zero, so that the wave remains stationary in the non-rotating frame. We assume further in the text that the sequence of the doublets iω s has the property ω s+1ω s ≥ Ω cr s , which implies the existence of the minimal critical speed Ω cr = Ω cr 1 = ω 1 . When the speed of rotation exceeds the critical speed, some backward waves, corresponding to the eigenvalues λ - s , travel slower than the disc rotation speed and appear to be travelling forward (reflected waves).

In figure 1a, the spectral mesh (2.5) is shown for the six d.f. system (2.2) with the frequencies ω 1 = 1, ω 2 = 3 and ω 3 = 6 that imitate the distribution of the doublets of a circular ring [START_REF] Canchi | Parametric instability of a circular ring subjected to moving springs[END_REF]. To illustrate typical unfolding of the Campbell diagram, we plot in figure 1d-f the eigenvalues of the six d.f. system (2.1) with κ = 0.2, δ = 0.1, ν = 0.2, ω 1 = 1, ω 2 = 3 and ω 3 = 6 for the specific symmetry-breaking matrix

K = K 1 , (K 1 = K T 1 ), whose non-zero entries are k 11 = 1, k 12 = 2, k 13 = 1, k 14 = 2, k 22 = 1, k 23 = 3, k 24 = 4, k 33 = -3, k 44 = -2.
D 1 = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ -1 2 1 7 2 -2 2 3 -2 -4 3 1 1 -2 1 8 2 1 7 -4 8 3 -2 3 2 3 2 -2 5 5 -2 1 1 3 5 6 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ and N 1 = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 -1 1 -1 -3 8 1 0 2 3 2 4 -1 -2 0 7 1 3 1 -3 -7 0 8 2 3 -2 -1 -8 0 2 -8 -4 -3 -2 -2 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ .
In the following, we classify and interpret the typical behaviour of the eigenvalues of weakly anisotropic rotor system (2.1) with the use of the perturbation formula for the doublets of the spectral mesh (2.5), which we derive in the next section.

Perturbation of the doublets

Introducing the indices α, β, ε, σ = ±1, we find that the eigenvalue branches λ ε s = iαω s + iεsΩ and λ σ t = iβω t + iσ tΩ cross each other at Ω = Ω 0 with the origination of the double eigenvalue λ 0 = iω 0 with two linearly independent eigenvectors u ε s and u σ t , where

Ω 0 = αω s -βω t σ t -εs and ω 0 = ασ ω s t -βεω t s σ t -εs . (3.1)
Let M be one of the matrices D, K or N. In the following, we decompose the matrix M ∈ R 2n×2n into n 2 blocks M st ∈ R 2×2 , where s, t = 1, 2, . . . , n

M = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ * * * * * * M ss • • • M st * * . . . . . . . . . * * M ts • • • M tt * * * * * * ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ and M st = m 2s-1,2t-1 m 2s-1,2t m 2s,2t-1 m 2s,2t . (3.2)
Note that D st = D T ts , K st = K T ts and N st = -N T ts . We consider a general perturbation of the matrix operator of the isotropic rotor L 0 (Ω) + L(Ω). The size of the perturbation L(Ω) = δλD + κK + νN ∼ is small, where = L(Ω 0 ) is the Frobenius norm of the perturbation at Ω = Ω 0 . For small Ω = |Ω -Ω 0 | and , the increment to the doublet λ 0 = iω 0 with the eigenvectors u ε s and u σ t is given by the formula det(R + (λλ 0 )Q) = 0 [START_REF] Kirillov | Unfolding of eigenvalue surfaces near a diabolic point due to a complex perturbation[END_REF][START_REF] Kirillov | Subcritical flutter in the acoustics of friction[END_REF], where the entries of the 2 × 2 matrices Q and R are

Q εσ st = 2iω 0 ( ūε s ) T u σ t + 2Ω 0 ( ūε s ) T Gu σ t , R εσ st = (2iω 0 ( ūε s ) T Gu σ t + 2Ω 0 ( ūε s ) T G 2 u σ t )(Ω -Ω 0 ) +iω 0 ( ūε s ) T Du σ t δ + ( ūε s ) T Ku σ t κ + ( ūε s ) T Nu σ t ν. ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ (3.3)
Calculating the coefficients (3.3) with the eigenvectors (2.6), we find the real and imaginary parts of the sensitivity of the doublet λ 0 = iω 0 at the crossing (3.1)

Re λ = - 1 8 Im A 1 αω s + Im B 1 βω t ± |c| -Re c 2 , Im λ = ω 0 + Ω 2 (sε + tσ ) + κ 8 tr K ss αω s + tr K tt βω t ± |c| + Re c 2 , ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ (3.4) where c = Re c + i Im c with Im c = αω t Im A 1 -βω s Im B 1 8ω s ω t (sε -tσ ) Ω +κ (αω s tr K tt -βω t tr K ss )(αω s Im B 1 -βω t Im A 1 ) 32ω 2 s ω 2 t -αβκ Re A 2 tr K st J εσ -Re B 2 tr K st I εσ 8ω s ω t , Re c = tσ -s 2 Ω + κ βω s tr K tt -αω t tr K ss 8ω s ω t 2 +αβ (tr K st J εσ ) 2 + (tr K st I εσ ) 2 16ω s ω t κ 2 - (αω s Im B 1 -βω t Im A 1 ) 2 + 4αβω s ω t ((Re A 2 ) 2 + (Re B 2 ) 2 ) 64ω 2 s ω 2 t . ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ (3.5)
The coefficients A 1 , A 2 and B 1 , B 2 depend only on those entries of the matrices D, K and N that belong to the four 2 × 2 blocks (3.2) with the indices s and t A 1 = δλ 0 tr D ss +κ tr K ss +ε2iνn 2s-1,2s ,

A 2 = σ ν tr N st I εσ +i(δλ 0 tr D st J εσ +κ tr K st J εσ ),

B 1 = δλ 0 tr D tt +κ tr K tt +σ 2iνn 2t-1,2t , B 2 = σ ν tr N st J εσ -i(δλ 0 tr D st I εσ +κ tr K st I εσ ), ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ (3.6)
where

I εσ = ε 0 0 σ and J εσ = 0 -σ ε 0 . (3.7)
Therefore, we have identified the elements of the perturbing matrices that control practically important eigenvalue assignment [START_REF] Ouyang | Prediction and assignment of latent roots of damped asymmetric systems by structural modifications[END_REF]Kirillov 2009a,b) near every particular node (Ω 0 , ω 0 ) of the spectral mesh.

MacKay's eigenvalue cones and instability bubbles

Modification of the stiffness matrix induced by the elastic support interacting with the rotating continua is typical in the models of rotating shafts [START_REF] Shieh | Some general principles of dynamic instability of solid bodies[END_REF]), computer disk drives [START_REF] Ono | Stability analysis for the head-disk interface in a flexible disk drive[END_REF][START_REF] Chen | Mathematical structure of modal interactions in a spinning diskstationary load system[END_REF], circular saws [START_REF] Yang | Interactions between an idealized rotating string and stationary constraints[END_REF][START_REF] Tian | Self-excited vibration in flexible rotating discs subjected to various transverse interactive forces: a general approach[END_REF][START_REF] Xiong | Instability mechanism of a rotating disc subjected to various transverse interactive forces[END_REF], car brakes [START_REF] Mottershead | Flutter instability of circular discs with frictional follower loads[END_REF][START_REF] Hervé | Analysis of squeal noise and mode coupling instabilities including damping and gyroscopic effects[END_REF][START_REF] Spelsberg-Korspeter | In-and out-of-plane vibrations of a rotating plate with frictional contact: investigations on squeal phenomena[END_REF]) and turbine wheels [START_REF] Genta | Dynamics of rotating systems[END_REF][START_REF] Lesaffre | Contact analysis of a flexible bladed-rotor[END_REF].

Assuming δ = 0 and ν = 0 in equation (3.4), we find that the eigenvalues of the system (2.5) with the stiffness modification κK either are pure imaginary (Re λ = 0) and form a conical surface in the (Ω, κ, Im λ)-space with the apex at the point (Ω 0 , 0, ω 0 )

Im λ -ω 0 - κ 8 tr K ss αω s + tr K tt βω t - Ω -Ω 0 2 (sε + tσ ) 2 = Re c, (4.1)
see figure 2a, or they are complex and in the (Ω, κ, Re λ)-space their real parts originate a cone (Re λ) 2 = -Re c with the apex at the point (Ω 0 , 0, 0) figure 2c.

In the (Ω, κ, Im λ)-space, the corresponding imaginary parts belong to the plane

Im λ = ω 0 + κ 8 tr K ss αω s + tr K tt βω t + Ω -Ω 0 2 (sε + tσ ), (4.2)
which is attached to the cone (4.1) as shown in figure 2b.

The existence of eigenvalues with Re λ = 0 depends on the sign of αβ. It is negative only if the crossing in the Campbell diagram is formed by the eigenvalue branch of the reflected wave and by that of either the forward or backward travelling wave. Otherwise, αβ > 0. Owing to the property ω s+1ω s ≥ Ω cr s , the crossings of the reflected wave with the forward and backward travelling waves occur only in the supercritical speed range |Ω| ≥ Ω cr . The crossings with αβ > 0 are situated in both the super-and subcritical (|Ω| < Ω cr ) ranges. Therefore, the eigenvalues with Re λ = 0 originate only near the supercritical crossings of the eigenvalue branches λ ε s and λ σ t with αβ < 0, when the parameters in the (Ω, κ)-plane are in the sector Re c < 0 bounded by the straight lines Re c = 0

Im λ Im λ Re λ Ω Ω Ω κ κ κ (a) ( b) ( c)
κ = 4(sε -tσ )(Ω -Ω 0 ) (k 2t-1,2t-1 +k 2t,2t )/βω t -(k 2s-1,2s-1 +k 2s,2s )/αω s ±2 ((εk 2s-1,2t-1 + σ k 2s,2t ) 2 + (εk 2s-1,2t -σ k 2s,2t-1 ) 2 )/ -αβω s ω t . (4.3)
As for αβ < 0 the cones of the real parts (Re λ) 2 = -Re c are near-horizontally oriented and extended along the κ-axis in the (Ω, κ, Re λ)-space, their crosssections by the planes κ = const. are ellipses, as shown in figures 1c and 2c. As a part of the ellipse corresponds to the eigenvalues with positive real parts, the ellipse is called the bubble of instability [START_REF] Mackay | Stability of water waves[END_REF]. Equation (4.3) is, therefore, a linear approximation to the boundary of the domain of instability, which is divergence (parametric resonance) for Ω 0 = Ω cr s and flutter (combination resonance) otherwise. The near-horizontal orientation of the corresponding cones of imaginary parts (4.1) in the (Ω, κ, Im λ)-space explains deformation in the presence of the perturbation κK of the crossings with αβ < 0 into the branches of a hyperbola connected by a straight line in the Campbell diagram (figures 1b and 2b).

Near the crossings with αβ > 0, the perturbed eigenvalues are pure imaginary (stability). The corresponding cones of imaginary parts (4.1) are near-vertically oriented in the (Ω, κ, Im λ)-space, figure 2a. In the plane κ = const., this yields the avoided crossing (MacKay 1986; MacKay & Saffman 1986), which is approximated by a hyperbola shown by the bold lines in figure 2a (cf. figure 1b).

The conical singularities of the eigenvalue surfaces in the Hamiltonian systems are traced back to the works of Hamilton himself, who predicted the effect of conical refraction of light in birefringent crystals [START_REF] Hamilton | On a general method of expressing the paths of light, and of the planets, by the coefficients of a characteristic function[END_REF][START_REF] Berry | Chapter 2 Conical diffraction: Hamilton's diabolical point at the heart of crystal optics[END_REF]. Later on, the conical singularities of eigenvalue surfaces were found in atomic, nuclear and molecular physics [START_REF] Von Neumann | Über das Verhalten von Eigenwerten bei adiabatischen Prozessen[END_REF][START_REF] Teller | The crossing of potential surfaces[END_REF][START_REF] Mondragon | Degeneracy and crossing of resonance energy surfaces[END_REF]. Nowadays they bear a name of Hamilton's diabolical points [START_REF] Berry | Chapter 2 Conical diffraction: Hamilton's diabolical point at the heart of crystal optics[END_REF]. The existence of the two different orientations of the eigenvalue cones in the Hamiltonian systems was established in [START_REF] Mackay | Stability of equilibria of Hamiltonian systems[END_REF]. This result is based on the works of [START_REF] Williamson | On the algebraic problem concerning the normal forms of linear dynamical systems[END_REF] and [START_REF] Krein | Topics in differential and integral equations and operator theory[END_REF], who introduced the signature of eigenvalues known as the symplectic signature in the Hamiltonian mechanics (MacKay & Sepulchre 1998) and as the Krein signature in a broader context of the theory of Krein spaces [START_REF] Kirillov | Determining role of Krein signature for three-dimensional Arnold tongues of oscillatory dynamos[END_REF].

To evaluate the symplectic signatures, we reduce equation (2.2) to ẏ = Ay, where

A = -ΩG I n -P -ΩG = J 2n A T J 2n , J 2n = 0 -I n I n 0 and y = x ẋ + ΩGx . (4.4)
The Hamiltonian symmetry of the matrix A implies its self-adjointness in a Krein space with the indefinite inner product

[a, b] = b T J 2n a, a, b ∈ C 2n . The matrix A
has the eigenvalues λ ± s given by the formulas (2.5) with the eigenvectors [START_REF] Mackay | Stability of equilibria of Hamiltonian systems[END_REF][START_REF] Mackay | Stability of discrete breathers[END_REF]. The signature of an eigenvalue in the Campbell diagram coincides with the sign of the doublet at Ω = 0, from which it is branched, and does not change with the variation of Ω. This implies αβ > 0 and near-vertically oriented cones of imaginary parts (4.1) at the crossings of eigenvalue branches with definite (positive) signature and αβ < 0 and near-horizontally oriented cones of imaginary parts (4.1) at the crossings with mixed signature (MacKay 1986).

a ++ s = u + s λ + s u + s + ΩGu + s and a +- s = u - s λ - s u - s + ΩGu - s , ( 4
[a -+ s , a -+ s ] = i[a -- s , a -- s ] = -4ω s < 0, have the opposite, negative symplectic (Krein) signature
The symplectic signature coincides with the sign of the second derivative of energy, which is a non-degenerate definite quadratic form on the real invariant space associated to a complex conjugate pair of simple pure imaginary nonzero eigenvalues (MacKay 1986). Interaction of waves with positive and negative energy is a well-known mechanism of instability of the moving fluids and plasmas [START_REF] Mackay | Stability of equilibria of Hamiltonian systems[END_REF][START_REF] Stepanyants | Propagation of waves in hydrodynamic shear flows[END_REF][START_REF] Hirota | Energy of hydrodynamic and magnetohydrodynamic waves with point and continuous spectra[END_REF]; in rotor dynamics, this yields flutter in the supercritical speed range, which is known as the mass and stiffness instabilities [START_REF] Mottershead | Flutter instability of circular discs with frictional follower loads[END_REF][START_REF] Genta | Dynamics of rotating systems[END_REF].

Therefore, in the case when anisotropy of the stator is caused by the stiffness modification only, the unfolding of the Campbell diagram is completely described by one-parameter slices of the two-parameter MacKay's eigenvalue cones. As there are only two possible spatial orientations of the cones corresponding to either definite or mixed symplectic signatures, all one has to do to predict the unfolding of the Campbell diagram into avoided crossings or into bubbles of instability is to calculate the signatures of the appropriate eigenvalues of the isotropic rotor. In the following, we develop MacKay's theory further and show that even in the presence of non-Hamiltonian perturbations, all the observed peculiarities of the Campbell diagrams and decay rate plots are one-parameter slices of the eigenvalue surfaces near a limited number of other singularities whose type is dictated by the definiteness of the symplectic signature of the double eigenvalues at the crossings.

Double coffee filter singularity near the crossings with definite symplectic (Krein) signature

Understanding general rules of unfolding the Campbell diagrams of weakly anisotropic rotor systems in the presence of dissipative and non-conservative perturbations is important for linear stability analysis and for interpretation of numerical data in both low-and high-speed applications [START_REF] Genta | Dynamics of rotating systems[END_REF]). In the latter, supercritical flutter and divergence instabilities are easily excited near the crossings with the mixed symplectic signature just by the Hamiltonian perturbations like stiffness modification. In low-speed applications, unfolding of the Campbell diagram is directly related to the onset of friction-induced oscillations in brakes, clutches, paper calendars and even in musical instruments like the glass harmonica [START_REF] Spurr | The ringing of wine glasses[END_REF][START_REF] Ouyang | Unstable travelling waves in the friction-induced vibration of discs[END_REF][START_REF] Chevillot | The destabilization paradox applied to friction-induced vibrations in an aircraft braking system[END_REF][START_REF] Hervé | Analysis of squeal noise and mode coupling instabilities including damping and gyroscopic effects[END_REF][START_REF] Kang | Dynamic instability of a thin circular plate with friction interface and its application to disc brake squeal[END_REF][START_REF] Kirillov | Subcritical flutter in the acoustics of friction[END_REF][START_REF] Ouyang | Prediction and assignment of latent roots of damped asymmetric systems by structural modifications[END_REF][START_REF] Spelsberg-Korspeter | In-and out-of-plane vibrations of a rotating plate with frictional contact: investigations on squeal phenomena[END_REF]. In contrast to the supercritical instabilities, excitation of the subcritical flutter near the crossings, with the definite symplectic signature by the Hamiltonian perturbations only is impossible. In this case, the non-Hamiltonian dissipative and circulatory forces are required for destabilization.

In general, dissipative, δD, and non-conservative positional, νN, perturbations unfold MacKay's eigenvalue cones (4.1) and (Re λ) 2 = -Re c into the surfaces Im λ(Ω, κ) and Re λ(Ω, κ), described by formulas (3.4). The new eigenvalue surfaces have singularities at the exceptional points [START_REF] Berry | The optical singularities of birefringent dichroic chiral crystals[END_REF][START_REF] Keck | Unfolding a diabolic point: a generalized crossing scenario[END_REF]. The latter correspond to the double eigenvalues with the Jordan chain that is born from the parent semisimple doublet iω 0 at Ω = Ω 0 . In some works, numerical methods were developed to find the coordinates of these singularities [START_REF] Jones | Multiple eigenvalues and mode classification in plane Poiseuille flow[END_REF][START_REF] Skorokhodov | Branch points of eigenvalues of the Orr-Sommerfeld operator[END_REF]. Perturbation of Hamilton's diabolical points is another efficient way to locate exceptional points [START_REF] Kirillov | Unfolding of eigenvalue surfaces near a diabolic point due to a complex perturbation[END_REF]. Indeed, condition c = 0 yields their approximate loci in the (Ω, κ)-plane

Ω ± EP = Ω 0 ± 4ω s ω t U -βω s tr K tt + αω t tr K ss 4ω s ω t (tσ -sε) N D and κ ± EP = ± N D , (5.1)
where

U = Re A 2 tr K st J εσ -Re B 2 tr K st I εσ αω s Im B 1 -βω t Im A 1 , D = U 2 + αβ tr K st J εσ 2 √ ω s ω t 2 + tr K st I εσ 2 √ ω s ω t 2 , N = αω s Im B 1 -βω t Im A 1 4ω s ω t 2 + αβ Re A 2 2 √ ω s ω t 2 + Re B 2 2 √ ω s ω t 2 . ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ (5.2)
The crossings with the definite symplectic signature (αβ > 0) always produce a pair of exceptional points. For example, for pure non-conservative (δ = 0) and pure dissipative (ν = 0) perturbation of the doublets at Ω 0 = 0, formulas (5.1) read

Ω ± EP,n = 0, κ ± EP,n = ± 2νn 2s-1,2s ρ 1 (K ss ) -ρ 2 (K ss ) , Ω ± EP,d = ±δ μ 1 (D ss ) -μ 2 (D ss ) 4s , κ ± EP,d = 0, ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ (5.3)
where ρ 1,2 (K ss ) are the eigenvalues of the block K ss of the matrix K and μ 1,2 (D ss ) are those of the block D ss of D. In the case of the mixed symplectic signature (αβ < 0), the two exceptional points exist when N /D > 0 and do not exist otherwise.

The strong influence of the exceptional points on the stability and their relation to Ziegler's destabilization paradox owing to small damping is well recognized [START_REF] Bottema | The Routh-Hurwitz condition for the biquadratic equation[END_REF][START_REF] Kirillov | A theory of the destabilization paradox in non-conservative systems[END_REF][START_REF] Kirillov | Gyroscopic stabilization of non-conservative systems[END_REF]Kirillov , 2007a,b;,b;[START_REF] Krechetnikov | Dissipation-induced instabilities in finite dimensions[END_REF][START_REF] Chevillot | The destabilization paradox applied to friction-induced vibrations in an aircraft braking system[END_REF]. In numerous applications in rotor dynamics [START_REF] Ono | Stability analysis for the head-disk interface in a flexible disk drive[END_REF][START_REF] Chen | Mathematical structure of modal interactions in a spinning diskstationary load system[END_REF][START_REF] Mottershead | Flutter instability of circular discs with frictional follower loads[END_REF][START_REF] Yang | Interactions between an idealized rotating string and stationary constraints[END_REF][START_REF] Tian | Self-excited vibration in flexible rotating discs subjected to various transverse interactive forces: a general approach[END_REF][START_REF] Xiong | Instability mechanism of a rotating disc subjected to various transverse interactive forces[END_REF][START_REF] Genta | Dynamics of rotating systems[END_REF][START_REF] Lesaffre | Contact analysis of a flexible bladed-rotor[END_REF]) as well as in hydrodynamics [START_REF] Or | On the behaviour of a pair of complex eigenmodes near a crossing[END_REF], crystal optics [START_REF] Berry | The optical singularities of birefringent dichroic chiral crystals[END_REF], acoustics [START_REF] Shuvalov | On singular features of acoustic wave propagation in weakly anisotropic thermoviscoelasticity[END_REF] and microwave billiards [START_REF] Keck | Unfolding a diabolic point: a generalized crossing scenario[END_REF], the generalized crossing scenario in the vicinity of the exceptional points has been observed (visible also in figure 1e,f ) when at the same values of the parameters the imaginary parts of the eigenvalues cross, whereas the real parts do not and vice versa. In our setting, the conditions for coincidence of imaginary parts of the eigenvalues (3.4) are Im c = 0 and Re c ≤ 0 and that for coincidence of the real parts are Im c = 0 and Re c ≥ 0. Both real and imaginary parts of the eigenvalues coincide only at the two exceptional points (Ω + EP , κ + EP ) and (Ω - EP , κ - EP ). The segment of the line Im c = 0 connecting the exceptional points is the projection of the branch cut of a singular eigenvalue surface Im λ(Ω, κ). The adjacent parts of the line correspond to the branch cuts of the singular eigenvalue surface Re λ(Ω, κ). As simultaneous intersection of the different segments of the line Im c = 0 in the (Ω, κ)-plane is not possible, one observes the generalized crossing scenario [START_REF] Keck | Unfolding a diabolic point: a generalized crossing scenario[END_REF][START_REF] Kirillov | Unfolding of eigenvalue surfaces near a diabolic point due to a complex perturbation[END_REF] in the planes (Ω, Im λ) and (Ω, Re λ) or (κ, Im λ) and (κ, Re λ).

For example, in the case of pure non-conservative positional perturbation, the real parts of the eigenvalues developing near the doublets at Ω 0 = 0 cross each other in the (Ω, Re λ)-plane at the points of the branch cuts

κ 2 > (κ ± EP,n ) 2 Re λ = ± 2νsn 2s-1,2s (ρ 1 (K ss ) -ρ 2 (K ss )) κ 2 -(κ ± EP,n ) 2 Ω + O(Ω 3 ), (5.4) whereas for κ 2 < (κ ± EP,n ) 2 they avoid crossing Re λ = ± ρ 1 (K ss ) -ρ 2 (K ss ) 4ω s (κ ± EP,n ) 2 -κ 2 + O(Ω 2 ).
(5.5)

At the exceptional points κ = κ ± EP,n , the eigenvalue branches touch each other

Re λ = ± 1 2 2νsn 2s-1,2s ω s Ω + O(Ω 3/2 ).
(5.6) The degenerate crossing (5.6) of the real parts has been observed in the model of a rotating circular string [START_REF] Yang | Interactions between an idealized rotating string and stationary constraints[END_REF][START_REF] Kirillov | Subcritical flutter in the acoustics of friction[END_REF]. Pure dissipative perturbation of the doublets at Ω 0 = 0 yields crossings of the real parts at the branch cuts Ω 2 > (Ω ± EP,d ) 2 in the (Re λ, κ)-plane and veering of the imaginary parts

Im λ = ω s ± s Ω 2 -(Ω ± EP,d ) 2 + O(κ), Re λ = - δ tr D ss 4 ± γ 16sω s Ω 2 -(Ω ± EP,d ) 2 δκ + O(κ 3 ), ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ (5.7)
where γ = 2 tr K ss D sstr K ss tr D ss . At the branch cut Ω 2 < (Ω ± EP,d ) 2 , the imaginary parts cross and the real parts avoid crossing

Im λ = ω s + tr K ss 4ω s κ ± γ 16sω s (Ω ± EP,d ) 2 -Ω 2 δκ + O(κ 2 ), Re λ = - δ tr D ss 4 ± s (Ω ± EP,d ) 2 -Ω 2 + O(κ 2 ). ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ (5.8)
At Ω = Ω ± EP,d , the crossings of both real and imaginary parts are degenerate

Re λ = - δ tr D ss 4 ± 1 4 -δκ γ ω s + O(κ 3/2 ), Im λ = ω s ± 1 4 -δκ γ ω s + tr K ss 4ω s κ + O(κ 3/2 ). ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ (5.9)
The evolving eigenvalue branches reconstruct the eigenvalue surfaces shown in figure 3. In the one-parameter slices of the surfaces, the transformation of the eigenvalue branches from the crossing to the avoided crossing owing to the variation of the parameters Ω and κ occurs after the passage through the exceptional points, where the branches touch each other and the eigenvalue surfaces have Whitney's umbrella singularities. The surface of the imaginary parts shown in figure 3a is formed by the two Whitney's umbrellas with the handles (branch cuts) glued when they are oriented towards each other. This singular surface is known in the physical literature on wave propagation in anisotropic media as the double coffee filter [START_REF] Berry | The optical singularities of birefringent dichroic chiral crystals[END_REF][START_REF] Keck | Unfolding a diabolic point: a generalized crossing scenario[END_REF]. The viaduct singular surface of the real parts results from the gluing of the roofs of two Whitney's umbrellas when their handles are oriented outwards (figure 3b). The double coffee filter singularity is a result of the deformation of MacKay's eigenvalue cone (shown by the dashed lines in figure 3a) by dissipative and non-conservative positional perturbations. The perturbations foliate the plane Re λ = 0 into the viaduct, which has self-intersections along two branch cuts and an ellipse-shaped arch between the two exceptional points, figure 3b. Both types of singular surfaces appear when non-Hermitian perturbation of Hermitian matrices is considered [START_REF] Kahan | Spectra of nearly Hermitian matrices[END_REF][START_REF] Kirillov | Unfolding of eigenvalue surfaces near a diabolic point due to a complex perturbation[END_REF].

Im λ Im λ Im λ Re λ Re λ Re λ (a) ( b) ( c) κ ω 0 ω 0 ω 0 κ κ κ κ κ Ω Ω Ω Ω Ω Ω
Therefore, in a weakly non-Hamiltonian system (2.1), the fundamental qualitative effect of the splitting of the doublets with the definite symplectic (Krein) signature is the origination of the double coffee filter of the imaginary parts and the viaduct of the real parts. Structural modification of the matrices of dissipative and non-conservative positional forces generically does not change the type of the surfaces, preserving the exceptional points and the branch cuts.

Unfolding MacKay's cones with mixed signature

The definite symplectic signature (αβ > 0) implies D > 0 and N > 0 and thus uniquely determines the type of the singular surface for the real and imaginary parts of the perturbed eigenvalues. The case of the mixed symplectic signature (αβ < 0) possesses several scenarios for the unfolding of MacKay's cones by the non-Hamiltonian perturbation because D and N can have different signs.

When D > 0 and N > 0, the imaginary parts of the eigenvalues form the double coffee filter singular surface whereas the real parts originate the viaduct (figure 3). For negative D and negative N , the type of the surfaces is interchanged: the imaginary parts form the viaduct and the real parts originate the double coffee filter (figure 4a).

Exceptional points are not created for negative values of N /D. In this case, the eigenvalue surfaces either intersect each other along the branch cut, which projects into the line Im c = 0 in the (Ω, κ)-plane, or do not cross at all. When N > 0, the surfaces of the imaginary parts Im λ(Ω, κ) cross and the surfaces Re λ(Ω, κ) avoid crossing (figure 4b). For N < 0, the surfaces of the imaginary parts are separated and those of the real parts cross (figure 4c).

Example 1: a rotating shaft

The simplest mechanical systems described by equations (2.1) and (2.2) are some models of rotating shafts [START_REF] Kimball | Internal friction theory of shaft whirling[END_REF][START_REF] Shieh | Some general principles of dynamic instability of solid bodies[END_REF][START_REF] Nagata | Bifurcations in gyroscopic systems with an application to rotating shafts[END_REF][START_REF] Genta | Dynamics of rotating systems[END_REF]. In [START_REF] Shieh | Some general principles of dynamic instability of solid bodies[END_REF], the shaft is modelled as the mass m, which is attached by two springs with the stiffness coefficients k 1 and k 2 = k 1 + κ and two dampers with the coefficients μ 1 and μ 2 to a coordinate system rotating at a constant angular velocity Ω (figure 5a). A non-conservative positional force βr acts on the mass. With u and v representing the displacements in the direction of the two rotating coordinate axes, respectively, the system is governed by the equations [START_REF] Shieh | Some general principles of dynamic instability of solid bodies[END_REF])

m ü + μ 1 u -2mΩ v + (k 1 -mΩ 2 )u + βv = 0, m v + μ 2 v + 2mΩ u + (k 2 -mΩ 2 )v -βu = 0. (7.1)
In figure 5b, we show a numerically found surface of frequencies for the shaft with m = 1 and k 1 = 4 in the absence of damping and non-conservative forces. The surface has four conical singularities corresponding to the doublets ±2i at Ω = 0 and to the double-zero eigenvalues at the critical speeds Ω = ±2. The cones in the subcritical speed range are near-vertically oriented while those at the critical speeds are near horizontal. Consequently, for small stiffness detuning κ, the system is stable in the subcritical speed range and unstable by divergence in the vicinity of the critical speeds, where the bubbles of instability in the decay rate plots originate.

Addition of the non-conservative forces with β = 0.2 and damping with μ 1 = 0.1 and μ 2 = 0.2 yields deformation of the conical surfaces with apexes at Ω = 0 into the double coffee filters. The real parts form the viaduct singular surfaces shown in figure 5c,d. In the absence of damping (μ 1,2 = 0), the gyroscopic system with the potential and non-conservative positional forces cannot be asymptotically stable in accordance with the theorem of [START_REF] Lakhadanov | On stabilization of potential systems[END_REF]. It is unstable almost everywhere in the space of parameters and can be only marginally stable on the set of measure zero in it. This is seen in figure 5c, which shows that the shaft is marginally stable at the points of the branch cuts, that form the set of measure zero and unstable at all other points of the parameter plane.

k 2 k 1 μ 2 μ 1 Re λ Re λ κ κ κ Ω Ω Ω (a) (b) (c) ( d )

Example 2: a rotating circular string

Consider a circular string of displacement W (ϕ, τ ), radius r and mass per unit length ρ that rotates with the speed γ and passes at ϕ = 0 through a massless eyelet, generating a constant frictional follower force F on the string [START_REF] Yang | Interactions between an idealized rotating string and stationary constraints[END_REF]. The circumferential tension P in the string is constant; the stiffness of the spring supporting the eyelet is K and the damping coefficient of the viscous damper is D; the velocity of the string in the ϕ direction has a constant value γ r.

With the non-dimensional variables and parameters

t = τ r P ρ , w = W r , Ω = γ r ρ P , k = Kr P , μ = F P and d = D √ ρP , (8.1)
the substitution of w(ϕ, t) = u(ϕ) exp(λt) into the governing equation and boundary conditions yields the boundary eigenvalue problem [START_REF] Yang | Interactions between an idealized rotating string and stationary constraints[END_REF]) (8.3) where = ∂ ϕ . The boundary eigenvalue problem (8.2) and ( 8.3) depends on the speed of rotation (Ω), and damping (d), stiffness (k) and friction (μ) coefficients. We note that the artificialness of the term, corresponding to the nonconservative positional forces, in the second of the boundary conditions (8.3), was discussed in the literature [START_REF] Yang | Interactions between an idealized rotating string and stationary constraints[END_REF][START_REF] Kirillov | Subcritical flutter in the acoustics of friction[END_REF]). We keep it, however, to show how the degeneracy of this operator is seen in the eigenvalue surfaces.

Lu = λ 2 u + 2Ωλu -(1 -Ω 2 )u = 0, (8.2) u(0) -u(2π) = 0 and u (0) -u (2π) = λd + k 1 -Ω 2 u(0) + μ 1 -Ω 2 u (0),
For d = 0, k = 0 and μ = 0, the eigenvalue problem (8.2) and ( 8.3) has the eigenvalues λ ε n = in(1 + εΩ), λ δ m = im(1 + δΩ), where ε, δ = ± 1 and n, m ∈ Z -{0}. In the (Ω, Im λ)-plane, the branches intersect each other at the node (Ω 0 , ω 0 ) with

Ω 0 = n -m mδ -nε and ω 0 = nm(δ -ε) mδ -nε , (8.4)
where the double eigenvalue λ 0 = iω 0 has two linearly independent eigenfunctions u ε n = cos(nϕ)εi sin(nϕ) and u δ m = cos(mϕ)δi sin(mϕ). (8.5)

Intersections of the branch with n = 1 and ε = 1 and the branches with m > 0 and δ < 0 in the subcritical range (|Ω| < 1) are marked in figure 6a by red dots.

Taking into account that δ =ε at all the crossings, excluding (Ω 0 = ± 1, ω 0 = 0) where δ = ε, we find approximations to the real and imaginary parts of the perturbed non-zero double eigenvalues [START_REF] Kirillov | Subcritical flutter in the acoustics of friction[END_REF])

Re λ = -d n + m 8π nm ω 0 ± |c| -Re c 2 , Im λ = ω 0 + ε n -m 2 Ω + n + m 8π nm k ± |c| + Re c 2 , ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ (8.6)
where Ω = Ω -Ω 0 , and for the complex coefficient c, we have

Im c = k 2dω 0 -εμ(n -m) 16π 2 nm -2 ε n + m 2 Ω + m -n 8π nm k ε 4π μ -d m -n 8π nm ω 0 , Re c = εn -δm 2 Ω + m -n 8π nm k 2 + k 2 16π 2 nm - [d(m + n)ω 0 ] 2 64π 2 n 2 m 2 . ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ (8.7)
Setting Re c = 0 and Im c = 0, we find the coordinates of the projections of the exceptional points of the surfaces Re λ(Ω, k) and Im λ(Ω, k) onto the (Ω, k)-plane

Ω EP = Ω 0 ± ε 8π nm (m + n)d 2 ω 2 0 nm(μ 2 nm + d 2 ω 2 0 )
, As in formulas (5.1), the existence of the exceptional points (8.8) depends on the symplectic (Krein) signature of the intersecting branches, i.e. on the sign of nm, where n, m ∈ Z -{0}. In the case of the rotating string, all the crossings in the subcritical speed range (|Ω| < 1) have definite Krein signature (nm > 0). For those in the supercritical speed range (|Ω| > 1), it is mixed with nm < 0. In the (Ω, κ)-plane, the exceptional points are situated on the line Im c = 0

κ EP = ± dω 0 (2εμnm -d(m -n)ω 0 ) 2 nm(μ 2 nm + d 2 ω 2 0 ) . ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ (8.8)
k = 2πε(n + m) 2εnmμ -dω 0 (m -n) dω 0 (m 2 + n 2 ) Ω.
(8.9)

In figure 6b, we show the exceptional points (8.8) of the string passing through the eyelet with the damping coefficient d = 0.3. The red open circles correspond to the exceptional points born after the splitting of the diabolical crossings with n = 1 and ε = 1, which are shown in figure 6a by red dots. The exceptional points in the (Ω, κ)-plane are distributed over a butterfly-shaped area, which preserves its form independently of the number of points involved. In comparison with the numerical methods of [START_REF] Jones | Multiple eigenvalues and mode classification in plane Poiseuille flow[END_REF] and [START_REF] Skorokhodov | Branch points of eigenvalues of the Orr-Sommerfeld operator[END_REF], our perturbation approach gives efficient explicit and interpretable expressions for the distribution of the exceptional points, for the branch cuts and for the very eigenvalue surfaces.

In figure 6c, we plot the exceptional points originated after the splitting of the diabolical points with n = 1 and ε = 1 together with the projections of the branch cuts (8.9) of the double coffee filters Im λ(Ω, k), which are shown by bold lines. The corresponding projections of the branch cuts (8.9) of the viaducts Re λ(Ω, k) are presented in figure 6d. Only exceptional points originated after the perturbation of the doublets with Ω 0 = 0 are situated on the Ω-axis. This explains why damping creates a perfect bubble of instability for the doublets with m = n and imperfect ones for the diabolical points with m = n [START_REF] Yang | Interactions between an idealized rotating string and stationary constraints[END_REF][START_REF] Kirillov | Subcritical flutter in the acoustics of friction[END_REF]. Approximations (8.6) to the eigenvalue surfaces of a string with μ = 0 and d = 0.3 are presented in figure 7 for different values of n, m, ε and δ. The smaller inclusions in figure 7 show the cross sections of the surfaces by the plane k = 0 for the convenience of comparing with the numerical data of [START_REF] Yang | Interactions between an idealized rotating string and stationary constraints[END_REF]. The results shown in figure 7 are in qualitative agreement with the developed theory for equations (2.1) and (2.2) and perfectly agree with the numerical modelling. In the engineering literature, it was observed that the exceptional points (strong modal resonances [START_REF] Dobson | Is strong modal resonance a precursor to power system oscillations?[END_REF]) are precursors to flutter instability because of their strong influence on the movement of eigenvalues in the complex plane. Figure 8 demonstrates the approximation of the 'dynamics' of eigenvalues in the vicinity of the exceptional points, calculated by formulas (8.6), which is in a good qualitative agreement with the known numerical results [START_REF] Sinou | Stability analysis and non-linear behaviour of structural systems using the complex non-linear modal analysis[END_REF]. Finally, we notice that pure non-conservative positional perturbation (d = 0) causes degeneration of the eigenvalue surfaces. Indeed, the line (8.9) reduces to Ω = Ω 0 and the two exceptional points merge into one at Ω EP = Ω 0 and κ EP = 0. As a consequence, the central arch of the viaduct and the branch cut of the double coffee filter shrink to a single point. At this exceptional point, the angle of crossing of the surfaces is zero in agreement with [START_REF] Yang | Interactions between an idealized rotating string and stationary constraints[END_REF] and [START_REF] Kirillov | Subcritical flutter in the acoustics of friction[END_REF]. This degeneration visualizes the artificialness of the term related to the friction force in equation (8.7) that was already pointed out in the literature by physical arguments.

Conclusion

We found that in a weakly anisotropic rotor system (2.1), the branches of the Campbell diagram and the decay rate plots in the subcritical speed range are the cross sections of the two companion singular eigenvalue surfaces. The double coffee filter and the viaduct are the imaginary and the real part of the unfolding of any double pure imaginary semisimple eigenvalue at the crossing of the Campbell diagram with the definite symplectic (Krein) signature. Generically, the structure of the perturbing matrices determines only the details of the geometry of the surfaces, such as the coordinates of the exceptional points and the spatial orientation of the branch cuts. It does not yield qualitative changes irrespective of whether dissipative and circulatory perturbations are applied separately or in a mixture. The two eigenvalue surfaces found unite seemingly different problems on friction-induced instabilities in rotating elastic continua because their existence does not depend on the specific model of the rotor-stator interaction and is dictated by the symplectic signature of the eigenvalues of the isotropic rotor and by the non-conservative nature of the forces originated at the frictional contact. The double coffee filter singularity and its viaduct companion are true symbols of instabilities causing the wine glass to sing and the brake to squeal that connect these phenomena of the wave propagation in rotating continua with the physics of non-Hermitian singularities associated with the wave propagation in stationary anisotropic chiral media [START_REF] Berry | Physics of non-Hermitian degeneracies[END_REF]).

The study has been supported by the research grant DFG HA 1060/43-1.

Figure 1 .

 1 Figure 1. (a) The Campbell diagram of the unperturbed system (2.2) with six degrees of freedom (d.f.) in case of ω 1 = 1, ω 2 = 3 and ω 3 = 6; (b) the Campbell diagram; (c) decay rate plots for the stiffness modification κK 1 with κ = 0.2; (d) unfolding the Campbell diagram owing to perturbation with the matrices K = K 1 , D = D 1 , N = N 1 and κ = 0.2, δ = 0.1 and ν = 0.2; (e) the corresponding Campbell diagram; and (f ) decay rate plots.

  5, k 55 = 4 and k 66 = 2, and for the matrices D = D 1 and N = N 1 , where

Figure 2 .

 2 Figure 2. Eigenvalue surfaces (MacKay 1986) and (bold lines) their cross sections in the plane κ = const. (grey): (a) a near-vertically oriented cone Im λ(Ω, κ) in the subcritical range (Re λ = 0); (b) imaginary parts forming a near-horizontally oriented cone (4.1) with the attached membrane (4.2); and (c) the real parts forming a near-horizontally oriented cone (Re λ) 2 = -Re c with the attached membrane Re λ = 0 in the supercritical range.

Figure 3 .

 3 Figure 3. (a) The 'double coffee filter' singular surface Im λ(Ω, κ) with the exceptional points (open circles) and branch cut (bold lines) originated from MacKay's cone (dashed lines) owing to mixed dissipative and circulatory perturbation at any crossing with the definite symplectic signature; (b) the corresponding 'viaduct' Re λ(Ω, κ).

Figure 4 .

 4 Figure 4. Mixed symplectic signature (αβ < 0): (a) The viaduct Im λ(Ω, κ) and the double coffee filter Re λ(Ω, κ) for D < 0 and N < 0; (b) the surfaces Im λ(Ω, κ) crossed along the branch cut (bold line) and the separated surfaces Re λ(Ω, κ) for D < 0, N > 0; (c) separated surfaces of imaginary parts and crossed surfaces of real parts for D > 0, N < 0.

Figure 5 .

 5 Figure 5. (a) A model of the rotating shaft; (b) four MacKay's cones owing to stiffness modification (μ 1 = 0, μ 2 = 0, β = 0); (c) the viaduct singular surface created by the circulatory force only (β = 0.2); and (d) by the damping only (μ 1 = 0.1, μ 2 = 0.2).

Figure 6 .

 6 Figure 6. (a) The Campbell diagram of the unperturbed rotating string with red dots marking the nodes with n = 1; (b) a butterfly distribution (8.8) of the exceptional points (open circles) in the subcritical speed range in the (Ω, k)-plane when μ = 0 and d = 0.3 (red open circles correspond to n = 1); (c) projections of the branch cuts (8.9) of the coffee filters Im λ(Ω, k) and the exceptional points for n = 1; (d) projections of the branch cuts (8.9) of the viaducts Re λ(Ω, k) and the exceptional points for n = 1.

Figure 7 .

 7 Figure 7. For d = 0.3 and μ = 0: (a) the double coffee filter singular surface Im λ(Ω, k) in the vicinity of the crossing (n = 1, m = 2); (b) the viaduct surface Re λ(Ω, k) corresponding to the crossing (n = 1, m = 2); (c) intersecting surfaces Im λ(Ω, k) in the vicinity of the crossing (n = 1, m = -2); and (d) the corresponding non-intersecting surfaces Re λ(Ω, k).

Figure 8 .

 8 Figure 8. Exceptional points (open circles) with the parent diabolical points with n = 1, ε = 1 in the complex plane when d = 0.3 and the trajectories λ(Ω) for k = 0.05.


Timothy Whitton

Ken

Boris

The familiar faces of political renewal at City Hall

Keywords: Greater London Authority, Livingstone, Johnson

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

On 3 May 2012 the fourth Greater London Authority (GLA) elections will be held in London to choose the 25 members of the capital city's assembly and above all, its executive mayor. In 2000, Ken Livingstone became London's first directly elected mayor after defying his own party, New Labour, which had rigged the selection process to prevent him from standing and stealing the limelight from their own candidate. Londoners preferred Livingstone's independent, maverick style and above all his desire to put the interests of London first rather than those of a national political party.

His first mandate saw him introduce the congestion charge which guaranteed the success he would need to be re-elected in 2004, all the more so as no other political party could find a suitable candidate to oppose him, especially on his own turf where personality politics had become more important than manifestos. Meanwhile, New Labour pulled every string in order to reintegrate him into the party so that he would defend their colours in this second election. The possibility of London being awarded the Olympic Games was looming large and both local and central government knew full well that they had to cooperate more openly in order to reassure the International Olympic Committee.

Despite his second electoral success, Livingstone's connivance with New Labour had somewhat reduced his grass-root London support and accusations of cronyism adroitly exploited by the press gradually eroded his popularity. The Conservatives had also come up with their own candidate capable of challenging the incumbent mayor in the field of personality politics. Boris Johnson did exactly that, won the 2008 elections and gave David Cameron the political stronghold he needed in London to bolster his national appeal.

In 2000, Livingstone had played the card of political independence and in 2004, that of burying the hatchet so that the GLA and central government should see the Olympic Games through together. In 2008, his opponent managed to underline the fact that it was perhaps Time for a Change and once elected, pursued many of the policies that Livingstone had initiated 1 . For the 2012 election Livingstone has seemingly reverted to a 2000 style campaign by clearly stating that one of the London mayor's main tasks will be to oppose central government's policies and above all the spending cuts being inflicted on public services. This openly belligerent attitude possibly stems from the fact that the 2012 election will not be won or lost according to any major GLA policy issue, all the more so since the Olympic Games seem to be going to plan. In light of this, the familiar faces of those who seek election in 2012 will have their work cut out to give their respective campaigns the twist enabling their particular brand name to stand out from the others. Failing this, and if there is no major Olympic hitch, a second Johnson victory could be a foregone conclusion.

2000: David against Goliath

Livingstone's victory in the first mayoral election can essentially be put down to his ability to portray himself as being the candidate for London and Londoners rather than a party apparatchik inclined to promote his party's politics rather than strategies in keeping with local policies 2 . This attitude was to cost him his membership of the Labour Party because he disagreed with the Public Private Partnership (PPP) they wanted to set up in order to renovate the London underground. Livingstone bitterly opposed this choice saying that it would bury relationships between London government and the private companies involved in reams of intricate contracts that armies of lawyers would have to unravel at the slightest snarl. Ideologically speaking, Livingstone believed that the London underground should be managed by Londoners themselves through their local government which tallied with his -as yet secret -plan to create a popular shareholding system to finance renovation work.

Yet the PPP had been written into the Labour Party's 1997 election manifesto which Livingstone was supposed to support given that he had fought and won his seat as a Labour candidate3 . It had also been included in the White Paper A Mayor and Assembly for London: the 2 The first lines of Livingstone's 2000 election manifesto state this very clearly: "...The direct election of a Mayor and Assembly for London will give Londoners back the right to govern their own affairs and decide upon their own priorities. I am standing as an independent candidate because I believe the job of the Mayor will be to stand up for London. If candidates and policies can be imposed centrally then devolution will mean nothing. (emphasis added).

Government's Proposals for Modernising the Governance of London4 . Despite the intense pressure brought upon him, Livingstone refused obstinately to support the PPP for the underground contending that the Party's General Election manifesto was one thing and the manifesto for London another. At this stage, it is worth wondering whether Livingstone was not being deliberately provocative in order to reap full benefit from being the (future) victim of the Party's political machinery5 .

As was to be expected, Livingstone fell foul of the Labour Party which used a particularly devious primary election system to make sure that their own candidate, Frank Dobson, would be elected6 . The press went to town7 over these rigged elections and a short time later, on March 6th, Livingstone announced his intention to run as an independent candidate: "I am standing as an independent solely in order to ensure that we have real devolution to London. I will not be setting up a new political party and I still hope one day to be able to return to the Labour Party". 8 Thus the "purple bus" campaign kicked off and Livingstone was duly excluded from the party for a period of five years9 . But Londoners appreciated his style, they were fond of "our Ken" as they would refer to him and were quite ready to punish New Labour politicians for excluding the local hero from being their official candidate, but also for their overall arrogant attitude towards government.

Try as they might, New Labour couldn't manage to sway opinion and on May 4 th 2000, Ken Livingstone became the first directly elected executive mayor of London, streaks ahead of the Conservative candidate, Steve Norris, Dobson coming in a sorry fourth behind the Liberal Democrats' candidate10 . Livingstone had won thanks to his maverick stance, his deliberately provocative campaign but above all by convincing Londoners that it was up to them to choose between the political juggernaut of central government and the defender of local policies. This first victory was going to give him the opportunity to show that his political acts were in keeping with his promises11 .

2004: the congestion charge victory

Officially, Livingstone was initially excluded from the Labour Party for a period of five years and on taking office, he immediately expressed his intention to challenge the decision to use a PPP in order to renovate the London underground. As if to reinforce this desire to impose what he considered to be the needs of London, he also declared his plan to introduce the congestion charge, a tax levied on vehicles entering the central part of London. This was enshrined in section 295 of the 1999 Greater London Authority Act which under the heading "Road User Charging" states that: a) Transport for London, b) any London borough council, or c) the Common Council12 , may establish or operate schemes for imposing charges in respect of the keeping or use of motor vehicles on roads in its area.

The charge was to become effective on February 17 th 2003 during the school holidays so as to reduce traffic during the first days of operation and while the private company, Capita, began setting up the complex system of cameras to be used in order to enforce it, Livingstone continued to fight the PPP plan tooth and nail. But he knew full well that the success of the congestion charge depended largely on an improved public transport system given that the basic aim was to coax commuters out of their cars and onto the buses, trains and of course the underground. In view of this, opposition to the PPP gradually dwindled even when the reintegration of Livingstone into the Labour Party was refused by the party's National Executive Council on July 24 th , 2002 13 . On January 26 th , 2003 an underground accident at Chancery Lane sparked off a bout of fierce opposition to Livingstone's congestion charge and he was asked to postpone it so that car traffic could for a while ease the extra burden on public transport. The mayor's answer was a categorical "no" given the financial penalties that the GLA would suffer from such a delay. Alone in the face of adversity, Livingstone weathered the storm and as the official opening of the charge grew nearer, braced himself to deal with the national and international media knowing full well that his political future was inextricably linked to the success of this major project. As it happened, very few of the problems that his opponents had brandished actually materialised and in spite of a few colourful anti congestion charge demonstrations and some minor technical hitches, the charge was a relative success 14 . At the end of April during his monthly press conference, the Prime Minister warmly congratulated the mayor of London for such an innovative policy, quite in keeping with New Labour's modern vision of politics, and prepared public opinion for the reintegration of Livingstone 15 . This was to take some months given that the party already had an official candidate, Nicky Gavron, but it was common knowledge that she had little chance either against Livingstone, if he chose to run as an independent again, or the other party candidates. On January 7 th , 2004 the ban on Livingstone was officially lifted and as provocative as ever, he declared that both Blair and Thatcher, two of the most dominant figures in recent political history had tried to crush him and both had failed.

Livingstone's victory in the June 10 th , 2004 election was less resounding than in 2000. This can surely be put down to his recent collaboration with New Labour who suffered during the local elections held on the same day. Yet both parties knew that they had more to gain 13 Ironically, the decision was taken on the very same day as City Hall, which houses the GLA, was being inaugurated by the Queen. 14 Perhaps the most famous anti congestion charge demonstration was organised by Terry Burrows, window cleaner by trade -and Guinness Book World Record holderwho to protest about the substantial increase in his overheads that the congestion charge would imply on a daily basis, drove into central London on a horse drawn carriage. These vehicles are exempt from the charge to allow the Queen and the Lord Mayor to circulate freely. 15 "'Livingstone right on car fees', says Blair", The Daily Telegraph, 28/04/2003. together than by constantly being at loggerheads especially with the Olympic agenda looming so large.

The second mandate: Livingstone losing touch

There is no denying that along with the success of the congestion charge, the Olympic Games provided a serious backdrop for the 2004 mayoral elections. Yet when on July 5 th , 2005 Jacques Rogge announced that they had been awarded to London, the decision came perhaps too early for the mayor to reap electoral success from a project that was bound to substantially modify the capital. To this end, Livingstone had openly admitted on various occasions that his interest in sport was limited but that the Games would enable the GLA to renovate and improve huge swathes of east London.

When a few hours following the announcement London was hit by several bomb attacks, Livingstone managed to reach a peak of popularity by stating in statesmanlike terms that the terrorists had failed in their attempt to weaken Londoners faith in their city 16 . But despite carrying the banner of "London United", his second mandate was to lack a major project that would continue to promote his particular brand name. Instead, Livingstone became bogged down in a series of statements about the cost of the Games that affected his already tainted credibility concerning the financial governance of the capital. At the same time, scandals and sleaze were gradually whittling away at his popular support and reinforcing the belief that he was turning London into what the popular press coined a "kenocracy" 17 . Public opinion had already been shocked when he had welcomed the Muslim preacher, Sheikh Yusuf al-Qaradawi to City Hall a year earlier and the Finegold affair had weakened his support within the Jewish community 18 . The mayor's grand projects including a new toll bridge across the Thames, 16 http://www.youtube.com/watch?v=6BSIBPsbL9c 17 This was described by The Evening Standard as "the fanatically loyal group of advisers who worked for Mr Livingstone long before he became Mayor and who imposed his will on City Hall", 04/08/2008 in " 'Ken's cronies', £1.6m PAYOFF; (1) Former Mayor's eight aides benefit from his rule change (2) Boris settles rather than waste taxes on long court battle (2) EXCLUSIVE (4) Merry-go-round of excess". 18 On February 8th, 2005, Livingstone had publicly compared Olivier Finegold, a journalist with Jewish origins, to a concentration camp guard. Livingstone had subsequently refused to apologise for his declarations and become embroiled in a complex court case to determine whether he was still fit for public office.

CrossRail 19 , hosting the first stage of the Tour de France or a system of bicycle rental like the Parisian "vélib" model were proving to be insufficient to maintain his popularity and prevent him from becoming "Blair's mayor". In the same vein, Londoners were beginning to tire somewhat from his outbursts on issues that had very little to do with his job of mayor 20 . In May, 2006 when the Venezuelan president Hugo Chavez paid a well publicised visit to his socialist friend Livingstone in what became known as the "brooms for oil" deal 21 , the mayor was criticised for lording it over London especially when the western extension of the congestion charge was introduced despite popular disagreement clearly shown in the surveys 22 .

The reason behind Livingstone's provocative behaviour even towards his traditionally faithful electorate can perhaps be explained by the lack of a credible challenger. This can be compounded by the fact that in the previous elections, the Liberal Democrat candidates, respectively Susan Kramer in 2000 and Simon Hughes in 2004, had failed to make any serious inroads into the main candidates' electorate despite hopes that they might just make it to the top two and then benefit from a considerable second preference vote count 23 . In this way, Brian Paddick, their candidate for the 2008 election, tried desperately to portray himself as the "third" man, an alternative not only to the two main candidates but also to the two main parties. 19 CrossRail is a colossal project to create a fast train line across London linking Maidenhead in the west to Shenfield in the east. 20 Livingstone constantly criticised George Bush, Ariel Sharon and certain Saudi Arabian leaders. He had also insulted the American ambassador in London for failing to pay the congestion charge as well as the Reuben brothers, two Jewish businessmen involved in the construction of the Olympic village. 21 London was to receive cheap oil from Venezuela in exchange for its expertise on urban government provided by the GLA. 22 "Where the streets are paved with gold. On Monday the London congestion-charge zone will double in size, while on Tuesday an anti-road-pricing petition of more than 1.5 million signatures will close on the Number 10 website. Dr Leon Mannings considers the connection", The Daily Telegraph (London issue), 17/02/2007. 23 The supplementary vote system is used to choose the mayor of London. Voters express a first choice and if they choose, a second choice. After the first round, if no candidate has received 50% of first choice votes, a second round is organised for the top two candidates. Voters whose first choice has been eliminated but whose second choice is one of the top two candidates have their second preference vote added to the first-round totals for the leading candidates.

Meanwhile, the Conservatives had looked somewhat frantically for a suitable candidate 24 but it was only in July 2007, having postponed their deadline for candidates to declare, that they turned to Boris Johnson. At this stage they felt that the least their Have I Got News For You 25 candidate could do would be to compete with Livingstone in the popularity ratings. London had certainly changed since Livingstone had become mayor in 2000, but the 2008 elections were definitely going to be a battle of personalities rather than programmes. In view of this, the relentless anti-Livingstone campaign organised by the Evening Standard cannot be overlooked especially when the incumbent and his campaign team became embroiled in sleaze and accusations of corruption in the last six months before the election 26 . While Livingstone had to face allegations of corruption, the Conservatives handed their candidate's campaign over to Lynton Crosby, the Australian political strategist. His scheme to wrest City Hall from Livingstone was based on what he called the "doughnut" strategy whereby the incumbent mayor was portrayed as being an essentially "zone-one" candidate: canvassing for the conservative candidate had therefore to be concentrated in the outer more Conservative-voting boroughs 27 . This time round, the mayoral election was to be fought along more traditional party lines linked to personality politics: Tory canvassers found 'very, very wide' recognition of their candidate. You only had to say on the doorstep, 'I'm campaigning for Boris', and people would know what you were talking about. Boris got people talking about politics, a subject which self-respecting Londoners often prefer to ignore. He and Ken were demonstrating that to get people to come out and vote, you needed interesting candidates on the ballot paper, not tedious party clones. 28 Johnson's most convincing argument was surely Time for a Change which had quickly become his official slogan. After eight years of Livingstone's reign over London the electorate were prepared to give 24 See Whitton T., "Over to you Mr Johnson: the defeat of Ken Livingstone in 2008", in Whitton T., directeur de publication, Londres: capitale internationale, multiculturelle et olympique, revue n°11, Observatoire de la Civilisation Britannique, décembre 2011, p. 25 A popular chat show where contestants were required to improvise in order to comment on current affairs. Both Livingstone and Johnson were regularly invited to attend and sometimes to chair contests. 26 Ibidem. 27 Ibid.

the helm of the city to another personality capable of reinforcing the reputation of London as an international capital. This they did in May 2008 not so much as to reject Livingstone as to usher in some political renewal at City Hall. As was to be expected, Brian Paddick came in third after the first count with an even lower score than his predecessors but this can be put down to 2008 having been polarised around the two main characters whose highly mediatised duel was undeniably behind the 10% increase in voter participation. This said, four years later confrontation will more than likely concern the same people whose task it will be to convince the electorate that despite their familiar faces, they do represent political renewal.

From red Ken to Boris blue

Johnson's administration got off to a difficult start with the resignation of a senior advisor, James McGrath, after remarks he was said to have made publicly about Caribbean immigrants. This was followed at the beginning of July by the departure of deputy mayor Ray Lewis for allegations of sleaze. Lewis' resignation came as a serious blow to Johnson's momentum given his plans to fight youth crimeespecially what had been labelled "knife" crime -which was to be his deputy's special remit. On August 19 th it was the turn of Tim Parker, Chief Executive (First Deputy Mayor) of the GLA to tender his resignation because of strained relations with other staff members. Yet, despite these setbacks which smacked of Livingstone's last six months in power, it was on August 24 th when Johnson was given the Olympic flag to wave six times in order to officially acknowledge the handover ceremony in Beijing, that the success of his first mandate was inextricably linked to the successful organisation of the 2012 Games: Boris Johnson had shed the mantle of celebrity to don that of statesman 29 . To this end, when Brian Coleman, one of the senior Conservative members of the GLA, accused British athletes present in Beijing of being "tainted" with blood over China's human rights records, his remarks brought not only the assembly but also his party into disrepute. Whichever "big idea" mayor Johnson might come up with after 2008, there was no avoiding the fact that for the next four years, London was going to be first and foremost an international Olympic city. In this respect, the electorate would be particularly sensitive to issues concerning the Olympic legacy despite the fact that "London won the Games in a time of economic plenty but [was] building them in a global recession" 30 . Nevertheless Londoners had not elected a merely Olympic mayor and it quickly became apparent that many of the bedrock policies 31 promoted during the 2008 campaign would be rolled out against an economic landscape that few had anticipated. Price rises in London transport, announced in September and implemented in January 2009 put an end to Johnson's political honeymoon at the GLA and turned his response to the overall downturn in the British economy into an issue itself. The economic crisis was indeed forcing Johnson to frame many of his policy choices as correctives rather than innovative choices for the capital, including counter-cyclical investment to boost overall recovery. But at the same time, the economy was the area where he would use his powerbase to fire the occasional broadside against Cameron and the Conservative Party who in the run up to the General Elections were being particularly careful about not doing too many favours for the rich. Concerning the 50p tax rate that Alistair Darling was to impose on people earning more than £150000 a year and which neither David Cameron nor the Shadow Chancellor George Osborne had said they would scrap once in power, Johnson declared: I just feel that as Mayor of London, I have to point out that London is full of fantastically creative businesses that need to attract the best talent from around the world. If you tell them you are going to take considerably more of their money away than they could expect in the competitor capitals, that's a poke in the eye for London. 32 His performance on October 5 th at the Conservative conference was equally provocative as he stole the show from the party's top politicians by explaining that what he could do in London, could be done 30 "Is the economic downturn going to blight the 2012 London Olympics? The big question", The Independent, 16/01/2009. 31 Housing, crime, youth and transport for example. 32 "All eyes on Boris as he calculates his future; Boris Johnson has made enough waves as the Mayor of London to be considered as an heir to David Cameron -but will he make the jump? Interview by Andy McSmith", The Independent, 27/04/2009. Johnson renewed his criticism on Labour's tax strategy in January 2010 when bankers' bonuses were targeted. See "Financial: tax attack, Johnson under fire", The Guardian, 16/01/2010. nationwide by the next -Conservative -government 33 . Johnson has always been suspected of nurturing the hope of leading his party and occupying Downing Street 34 but the coalition that emerged from the 2010 General Elections was not in keeping with his vision of power, especially in its shared form.

Ken versus the coalition: political renewal at the GLA?

The famous "doughnut" strategy organised by Lynton Crosby in 2008 had repeated itself in the London constituencies in 2010 and shown that Labour would need a candidate capable of reaching across these boundaries if it wanted to conquer mayoral power once again. Oona King was convinced that she could accomplish just this and soon after the General Elections started her bid to become Labour's official candidate. Her gender, ethnicity and youth certainly enabled her to stand out from Livingstone, the maverick warhorse of the London left whose critics claimed he had become too tired and obdurate to organise a serious rematch against Johnson. His supporters, on the contrary, were wary of Labour choosing an inexperienced candidate such as King for London while the party itself was in the difficult throes of selecting its own leader. The capital's mayor holds the biggest mandate of any politician in Britain and at this stage in the reorganisation of the party, putting the right people in the right places was vital. While King felt she could muster a higher proportion of the swing and suburb voters than Livingstone, those who had rallied behind Johnson in 2008, Livingstone was certainly the candidate who was quite openly basing his campaign on an anti-Tory and anti-cuts platform. His first words uttered in Croydon -one of the outer boroughs incidentally -on June 1 st to launch his bid clearly stated his campaign pitch: "I want to be Mayor for one overriding reason: if I am elected my focus will be to do everything I can to protect Londoners from the recession and the effects of the Government's policies" 35 . On September 11 th , 2010 Johnson officially announced he would be seeking a second term as mayor of London fully aware that his main rival would persuade Londoners to make him carry the can for central government spending cuts: "I welcome Boris Johnson's confirmation that he will run again because he now has a record that means he can be held to account. [...] The Government's cuts are his cuts" 36 . To make his case, time and again Livingstone would return to the hefty hikes in the cost of public transport that had been implemented since Johnson had come to power, while constantly reminding the electorate of the mayor's political affiliation [START_REF]See for example "Who's best for London?[END_REF] . This aggressive stance was perhaps the main reason for his being elected -once again -to the party's NEC in September and it certainly proved that one of Livingstone's main skills was to garner the support of those who had initially given up on him, people who believed that in high profile elections such as the mayoral election in 2008, the loser loses all. In this particular contest, what tipped the balance to give him a distinct advantage over King was the need for Labour to have a candidate capable not only of being mayor, but first and foremost of beating Johnson and then standing up to Cameron and Osborne. In short, despite King's other advantages, Livingstone came across as having an acute understanding of the levers of power that would have to be pushed and pulled in order to find a suitable alternative to the coalition's agenda.

Indeed, Livingstone's first words on learning that he had won the competition to become the Labour Party's candidate [START_REF]6% of the vote to King's 31[END_REF] were once again to underline his commitment to oppose central government's cuts and help Labour defeat the coalition:

The London election in 2012 is their [Londoners'] chance to send a message to Cameron and Osborne that we don't want these devastating cuts to our public services, fewer jobs and declining living standards. We will stand up for London and to do that we need to change the mayor. If you want to get them out, we start by getting out Boris Johnson. 39 Ever since, Livingstone's political message has remained fairly faithful to this line of action, for he realises that to win back the mayoralty in 2012, he will have to take the message into Johnson's own camp beyond the boundaries of the inner London boroughs where his traditional supporters are concentrated.

Conclusion

The loss of City Hall in May 2012 would seriously undermine Cameron's attempts to woo Londoners with his particular brand of modern, metropolitan conservatism. He might enjoy the sight of his arch rival being humiliated but his joy will be short for at this time, a period of midterm "blues" will have set in for the coalition government especially if the spending cuts have begun to bite. Cameron will therefore be looking more for a Boris "bounce" than a failure. Johnson needs to take this into account and perform the acrobatic trick of refraining from overstating his disagreements with central government merely to emphasise his interest in London rather than national politics 40 . Livingstone was in much the same situation in the run-up to the 2004 election when New Labour was low in the opinion polls, but contrary to Johnson, he had a history of contradicting his own party when his convictions dictated he had to 41 .

In May 2012, just like Livingstone in 2008, Johnson will have a record to defend. He won't be able to treat the mayoral election as some big joke that will enable him to guffaw his way to a second victory and be present on the evening of 27 th July 27 th to open the Olympic Games. Johnson's record is honourable but far less revolutionary than his predecessor's and indeed many of his projects were initiated by Livingstone [START_REF]Barclay's Cycle Hire is a perfect example. The bicycles are commonly referred to as "Boris Bikes" even though Livingstone drew up the initial plans to introduce a bicycle sharing system based on the Parisian "Vélib" model[END_REF] . Even so, the GLA precept has been frozen, the Olympic budget brought under control, likewise the western extension of the congestion charge. His gaffes have been few and far between but his declaration that the £250,000 a year he receives from the Daily Telegraph for a weekly column is "chickenfeed" has provided ammunition for his opponents to chip away at his credibility as a "Tory toff" out of tune with Londoners. This leaves essentially the Olympic Games which generally speaking are going to plan to the extent that on an inspection visit to mark the milestone of "one year to go", Jacques Rogge, president of the International Olympic Committee declared that "Preparations are excellent. They are on time, on budget. Quality-wise we have not the slightest concern. There is no doubt about that. London is very well organised, the team is very strong. We are very optimistic" 43 . This must have been music to the ears of Johnson who knows full well that the final competition will be between himself and Livingstone, with one other candidate perhaps topping 10% of the first count 44 .

Johnson can rely on Olympic success to boost his candidacy but be must beware of coming across as his party's stooge. Cameron and Osborne can expect a certain degree of solidarity, especially when the going gets tough: in May 2008 Labour were behind in the national polls and Johnson reaped substantial advantage from this. In 2012, the situation could be the reverse with Johnson suffering from his political allegiance with an unpopular coalition and Livingstone gleaning the extra votes that had failed to give him a third term in the last election. There will be political renewal in May, either for Johnson, who could reinforce his powerbase and popularity in order for Downing Street to be a distinct possibility, or for Livingstone, who will demonstrate his limpet like ability to win elections, often in the familiar face of considerable adversity.

43 "Olympic Games: IOC leader lauds London as 2012 preparations enter final straight: Countdown to London 2012: Rogge says progress is the best he's seen but fears influence of illegal betting", The Guardian, 26/07/2012. 44 Brian Paddick for the Liberal Democrats, will most certainly be the third man once again.

  33 "Conference Sketch; Boris devises his own plot, with a climax at No 10. After a cameo on EastEnders, the Mayor of London steals the show at the conference", The Daily Telegraph, 06/10/2012. 34 "Weekend: Next stop, Number 10?: Boris Johnson talks adultery, ambitions and why he's a 'monstrous zeppelin of self-confidence' with Simon Hattenstone", The Guardian, 16/04/2011.

	35	This	also	figures	clearly	on	his	"KenforLondon"	website:
	http://www.kenlivingstone.com/about

This was Johnson's

campaign slogan invented by the Australian political strategist, Lynton Crosby, drafted in by the Conservative Party in order to organise their candidate's campaign.

Labour plans a new public/private partnership to improve the Underground, safeguard its commitment to the public interest and guarantee value for money to taxpayers and passengers. Labour Party manifesto 1997, New Labour because Britain deserves better.

Cm. 3897, the Stationery Office, London, 1998.

See for example "London Mayor : citizen Ken sends Millbank's plans down the tube ; Labour's shortlist scheme in turmoil after former GLC leader refuses to toe the party line over privatisation of underground", The Independent, 17/11/1999.

The electoral college established to select the official candidate was divided into three parts and heavily weighted in favour of those most likely to support the New Labour project to the detriment of the rank and file. See Whitton, T., Ken "le Rouge" et la Mairie deLondres, du Greater London Council à la Greater London Authority, l'Harmattan, 2010, p. 61-68.

See for example, "Labours mayoral candidate: the vanquished Livingstone listens for the voices of London to tell him to turn again and stand as mayor", "Labour faces biggest split since 1980s after 'tainted' Dobson win", The Independent, 21/02/2000 & "Defeated Livingstone cries foul", The Times, 21/02/2012.

"I will run for mayor; exclusive. 'Red' Ken announces he will defy Blair and stand as an independent", The Evening Standard, 06/03/2000.

For their deliberately provocative and outlandish campaign, Livingstone's team purchased a Routemaster bus, painted it purple and roamed the streets of London. Livingstone spoke to Londoners via a megaphone and one of the messages on the bus asked Londoners to "Hoot4Ken".

Full details of the results can be found inWhitton T., op. cit., p. 202.

Livingstone's first words on being elected were "As I was saying before I was so rudely interrupted fourteen years ago...".

I.e. the GLA.

Gimson, A., Boris. The Rise of Boris Johnson, Simon & Schuster, 2008, p. 284.

For all this Johnson was remembered in the press for his flag waving as a "character created in a unique collaboration between JK Rowling and the Beano". See "BBC coverage needed Boris bombing into the Olympic pool", The Independent, 01/08/2011.