An application of the inch algebraization to the stability of non-linear normal vibration modes

A normal vibration mode stability in conservative non-linear systems is investigated. The algebraization by Ince (transition from linear equations with periodic coefficients to equations with singular points) is used. The normal mode stability in homogeneous systems, whose potential is an even homogeneous function of the variables and systems close to the homogeneous one, is investigated. Eigenvalues and eigenfunctions are obtained. Conditions when a number of instability zones in a non-linear system parameters space are finite (finite zoning or finite-gap conditions) are also obtained.


INTRODUCTION
Normal (principal) vibration modes can be used to construct a general solution in the linear theory. In non-linear systems the "normal vibration modes" (or "normal vibrations") are defined as such free motions where all coordinates of the system vary equiperiodically, reaching the extreme equipotential surface simultaneously [1][2][3][4]. All position coordinates can be well defined from any one of them by X;=p;(xd (i=2,3, ... ) p;(x) being analytical functions. Rosenberg [1,2] gets credit for being the first to introduce broad classes of essentially non-linear (not quasilinear) conservative systems allowing normal vibrations with rectilinear trajectories in a configurational space (similar normal modes). In his and other works, normal mode stability was computed by linearized, Mathieu-type analysis [1, 2, 5-7], but it cannot guarantee good precision.
The stability of the normal vibration modes is studied by deriving an approximation for the Poincare map by Month and Rand (via Birkhoff-Gustavson canonical transformation) [8] and Pak [9]. This method is presented as an alternative to the usuallinearized stability on Floquet-Liapunov theory. Some papers [9,10] are based on Synge's stability in the kinematico-statical sense. Some examples were considered. In Pecelli and Thomas's works [11,12] the Poincare map, Kolmogorov-Arnold-Moser theorems and Lame equation theory were used. The similar normal modes stability was investigated while the linearizing variational equations represent the Lame equation; criterions of a stability in the non-linear sense were obtained. The effect of a normal mode bifurcation is investigated in various papers, for example, [7,10]. This effect and resonance neighborhood are investigated by numerical and analytical techniques in [13].
In this work, the algebraization by Ince [14] and transition from linear equations with periodic coefficients to equations with singular points (Section 2) will be used. The normal vibration modes stability of conservative homogeneous systems whose potential is an even homogeneous function of the variables and systems close to the homogeneous one will be investigated (Section 3). Eigenvalues and eigenfunctions will be obtained. Conditions when a number of instability zones in a system parameters space are finite (finite zoning or finite-gap conditions) will be also obtained (Section 4). Finally, in Section 5, the results relating to non-linear stability analysis [11] will be reduced.

TRANSITION FROM LINEAR EQUATIONS WITH PERIODIC COEFFICIENTS TO EQUATIONS WITH SINGULAR POINTS
Let us now describe what is called the algebraization by Ince, that is the transfer from equations with periodic coefficients to equations with singular points, following [14], together with the relevant change in the stability problem. There are many methods of algebraization but only one of them is shown in this item. In the analytical treatment of normal vibrations, the algebraization is performed by choosing a new independent variable associated with the undisturbed motion. The variable t may be replaced by x, a variable that defines motion along a rectilinear trajectory, or by the velocity x, or by the kinetic energy for this solution etc.
We consider now the Hill equation y + (a 0 + 2a 1 cos2t + a 2 cos4t + ... )y = 0 Let us introduce a new independent variable z = cost. We obtain This equation has regular singularities at z = ± 1 (the indices for singular points are 0 and 1/2), and an irregular singular point at infinity.
Two fundamental solutions may be separated for the vicinity of the singular point z = 1 [14]: where f 1 and f 2 are analytical functions that converge within a circle 11 -z I : : : : : ; 2.
Within a common region of the two circles of convergence IX, p, y, b are constant.
The solution is even at et. = 1 and odd at et. = -1; it is regular in any finite part of the plane. Since z = cost, at rJ. = 1 the solution may be presented as an even power expansion in cosines that are multiples of t, and et. = -1, in the form of an odd power expansion in cosines that are multiples of t.
The solution is the product of y' 1 -z 2 by a regular function that changes its sign upon each revolution about the singular point z = 1 or z = -1. The regular function is even if 6 = 1 and odd if b = -1. Substituting z = cos t, one obtains an even power expansion in sines that are multiples of t(b = 1), and an odd power expansion in sines that are multiples of t(<> = -1).
Thus, for the Hill equation in algebraic form, "boundary" solutions of the Hill equation are obtained that separate the region in which all solutions are bounded from the one in which all solutions are unbounded.
Let us consider two examples which are special cases of the Hill equation.
Substituting z = (sin t ) 2 , one obtains an algebraic form of It has two regular singular points: z = 0 and z = 1 whose indices are 0 and 1/2, and an essential singular point z = oc. In view of the above, the solutions separating the stability region from the instability region should be sought in the following form: where each series converges in a domain containing both regular singular points. Using analytical or numerical methods, one can eventually obtain the Ince-Strett diagram. Here, a transfer to algebraic forms may be effected by a number of methods [17]. For example, substituting z = [sn(t, k)] 2 , one obtains the following equation with three finite regular singular points: Algebraic forms of the Lame equation are used in a number of problems considered in this article. (1) whose potential energy ll(x 1 , x 2 . . . . . x") is analytical.
Let us rotate the coordinate axes so that the axis x 1 is aligned with the rectilinear trajectory, with the remaining axes orthogonal to the trajectory. In the new coordinates, the trajectory is defined as xj = 0 (j = 2, 3, ... ,n), x 1 = x(t).
It was shown in [6] that upon such rotation of the axes in a system with two degrees of freedom the variational equations always "split". Denoting the variation along the rectilinear trajectory by u, and the variation in the orthogonal direction by v, we obtain: Let us turn to a system with n degrees of freedom. Introducing a vector of variations U = (u 1 , u 2 , ... , u.f: we obtain a set of variational equations in the matrix form D + (UV)Vll = 0, or D + AU = 0. (2) Let us expand the matrix A is a series according to x: where A; are symmetric matrices with constant coefficients. For a homogeneous system whose potential is a homogeneous function of variables to the power r + 1, equations (2) is "split" by the use of a non-degenerate transformation with constant coefficients. Indeed, the variational equations then have the form D + A,U = 0, and the symmetric matrix A, can be diagonalized usmg a non-degenerate orthogonal transformation [ 18]. Let us separate out another case: The two symmetric matrices A. and Am are always diagonalizable through one nondegenerate transformation [ 18], provided one of them is positively defined; moreover, if the matrices are commuting, the transformation may be made orthogonal. In the literature [19] one can find proof of the theorem on the diagonalization in the general system (2) under the conditions that all matrices A; commute between them and also have non-multiple proper values. Let the set of variational equations decompose into n independent equations. One of these will govern the variations along the rectilinear trajectory, and the remaining equations will refer to the orthogonal directions. It is with the variations in the orthogonal directions that the orbital stability of normal vibrations is associated.
Let us introduce a new independent variable x. Every variational equation then will have the following form where the primes refer to the derivation with respect to x: Here R is an analytical function that has no real roots whose moduli are smaller than a or b.
As shown in the previous section, the problem of defining T-and 2T-periodic solutions of the variational equations (1: is the period of the coefficients in the equations) that separate the regions of stability and instability is reduced to solving Sturm-Liouville problems for functions that are regular at x = a and x = b or have a singularity of the form ~ or ~at these points.
If the potential energy IT is an even analytical function, the coefficients of the variational equations have a period which is twice as short as that of the normal vibrations. The kinetic energy is proportional to h -IT and is an even function of x too. However, S in (3) is proportional to the kinetic energy, therefore here b =a. Without loss of generality, one thus can select a scale such that S = (1x 2 )R. The problem of defining the boundaries of 4 the stability and instability regions by the first approximation is now decomposed into four Sturm~Liouville problems for even and odd functions that are regular at x = ± 1 or have a singularity of the form J1x 2 at these points. Upon the diagonalization of the matrix A each of the equations can be written in the form

Homogeneous systems
Let us replace t by a new independent variable x. We obtain, instead of (5), or t:".
Let us introduce another variable z = Q 2 x'+ 1 /(r + 1). We then have Relationship (6) is a hypergeometric equation with regular singular points z = 0 (the indices equal 0 and 1/(r + 1)) and z = 1 (the indices equal 0 and 1/2). The problem of stability and branching, as suggested above, is now much simpler. It essentially reduces to finding the values of the parameter i. such that, on having traversed a closed contour containing singular points 0 and 1, a solution is multiplied by + 1 or by -1 (Section 2). Such solutions are named degenerate solutions of the hypergeometric equation and are given in [20]. Their form is where q.(z) is a polynome, Jll can take values of 0 and I /(r + I), J1 2 can take values of 0 and 1/2. Let us further denote the period of solution x(t) as T. By analogy with the Mathieu equation, let us denote the proper functions that are Gegenbauer polynomes as C 4 i(z) (the even T /2-periocic solution), C 41 + 1 (z) (the odd T -periodic solution), S 4 J+ 2 (z) (the even T -periodic solution), and S 4 J+ 3 (z) (the odd T /2-periodic solution). In the linear case, the trajectories of these solutions are the well-known Lissajou figures. We write the eigenvalues of these problems in the same order as above [21]: Note that so simple a form for the spectrum of a stability problem is not very often obtainable for the non-linear case.

Homogeneous chain system
For the sake of definiteness, let us consider a homogeneous chain system with two degrees of freedom: (7) An equation to determine the vibration forms is An examination of this relationship shows [3,21] that for positive values of the constraint parameter f1 there always exists one in phase vibration mode and one, two or three anti phase modes whose number depends on f1 and the homogeneity index r. For negative values of the constraint parameter f1 at any value of r there exists one anti phase and one or three inphase vibration modes.
For the system in question the eigenvalue ;, of the hypergeometric equation (6) . r(1 + xk)(ak'-xk) The plots of Yf = 4n-1 arctg({l· 2'/(r -1)) and ~ = arctg k shown in Fig. 1 allow one to find the relation between the number of vibration modes and the parameters of the chain system. Curve (a) corresponds to the case of r = 3, a = 1; between the points whose numbers are from 4j + 1 to 4j + 2 and from 4j + 3 and 4j + 4 (j = 0, 1, 2, ... }. In the asymmetrical case(). i= 1} some points of branching turn into limit points. Figure 2 shows plots of f1 versus arctg k for other values of the system parameters. Curve (e) corresponds to the limiting case r = 0, x = 1, a = 1; (f) r = 1/3, x = 1, a = 1;   Fig. 2. The dependence between the number of vibration modes and the parameters of the system (7) correspond to r < I and various values of other system parameters.
In order to find those periodic solution branching points that do not affect the stability of normal vibrations, let us write a general solution of the variational equation (6): (in the vicinity of the singular point z = 1), C 1 ,C 2 ,D 1 ,D 2 being arbitrary constants, u 1 (z), u 2 (1-z), u 5 (z), u 6 (1-z) known hypergeometric functions [20]. Let us denote the matrix that relates C, C 2 to D 1 , D 2 as N: One can formulate denumerably many boundary problems

Systems approaching homogeneity
The results can be used for systems close to homogeneous ones. The periodic solutions separating the regions of stability and instability should be sought for the variational equations by perturbation theory. In the zeroth approximation these solutions are described by the eigenfunctions of a hypergeometric equation that correspond to the generating homogeneous system. From the conditions of the solutions periodicity, the stability shift points are determined in the further approximations.
In this section, however, only an example is given. Let us consider a symmetrical chain system whose potential contains x 1 and x 1 to the second and fourth power: m.x1 + a 11 x 1 + a 13 xi + a 2 dx 1 -x 2 ) + a 23 (x 1 -x 2 ) 3 = 0, (9) The equation that governs the anti-phase vibration mode x 2 =x 1 and the variational equation that permits a judgement about the orbital stability of this vibration form are as follows: x + x(y + 2px 2 or, with the primes denote the derivation with respect to u, (10) This is the Lame equation. Let us use the perturbation theory in order to find the first two points of stability shift (corresponding to the branching of normal vibrations and periodic solutions with the period of the generating solution but displaced in phase by half-period).
Let the system be quasilinear. We assume that p and f3 are small. Let us introduce a formal small parameter c; and multiply the non-linear terms by c;: In the first stability shift point, normal vibrations branch: v 0 = kx (here x = cos wt) and y 0 = a-0 . Therefore, in the zeroth approximation the linear system is uncoupled, that is a 21 = 0. We may readily check that it is at this point that the periodic solutions with the period of the generating solution which are displaced in phase by a half-period branch too: ). Consequently, in the zeroth approximation the equality l'o = a-0 holds for the second stability shift point as well.
Let us write the first approximation with respect to e: After simple calculations one finds that For the second stability shift point, the periodicity condition is f[v0(ylx + 2plx 3 ) -vo(al + P1x 2 )]xdt = 0 Here v 0 = Cx = C sin wt, x =cos wt. After simple rearrangements one obtains 1 -4 Let us now consider a quasihomogeneous system. Assume that y and a are small. We introduce a formal small parameter c before each linear term: We use the following expansions: v=v 0 +cv 1 + ···, y=cy 1 + ···, a=ea 1 + In the zeroth approximation with respect to c The branching of normal vibrations (the first stability shift point) v 0 = Cx provided that Po = 2p 0 . For the second point, that is the branching point of solutions v 0 = Cx the conditions will be Po = 6po.
In the first approximation with respect to c we write From the periodicity conditions for the first and second points of stability shift we obtain, respectively: f [x(2p1x 3 + Y1X)-x(/1 1 x 2 + at)x] dt = 0.

Finite number of orbital stability zones
Let us consider a situation when the number of orbital stability zones of normal vibration modes is finite (it is about a first approximation of stability and instability). Then all other instability zones are contracted to lines ("collapse"). This situation may be realized under certain conditions imposed upon the systems parameters. If these conditions are not satisfied, the number of instability zones will obviously be infinite (e.g. as on the well known Ince-Strutt diagram for the Mathieu equation). However, the importance of finite zoning conditions (or finite-gap conditions) depends on the fact that all the instability zones, except a finite number of them, are "narrow" if the values of the parameters are "close" to values which guarantee a realization of the finite zoning conditions when these zones are contracted to lines. It is interesting that sometimes the first parametric resonance zone may be "narrow" in this sense though it is usually considered as wide (e.g. for the Mathieu equation). Moreover, the finite zoning conditions are of interest for the problem of soli tons in non-linear wave systems.
So, let us consider a conservative system having two degrees of freedom, which may be described by the following equations where n is the analytically positively determined potential energy.

(11)
One may assume the system to permit rectilinear normal vibration modes x 2 = kx1. Let us separate one of the modes and reduce it to x 2 = 0 by rotation of the coordinate axes.
where g;(x d are analytical functions. The condition of existence of the solution x 2 = 0 is obviously satisfied. In the normal vibration mode the system behaves like a system with one degree of freedom .x + Dx,(x, 0) = 0 (x = xd (13) with the energy integral x l + D(x, 0) =h. (14) The orbital stability of normal mode x 2 = 0 is related to y variations orthogonal to the trajectory. The variational equation is: The function nx,x,(x(t), 0) is polynomial with respect to X and periodical with respect tot. The finite zoning criterion for equations as (15) was given in [22,23]   Comparing (15) and (16) one may conclude that u-e =nx,x,(x, 0). Taking into consideration the initial expression ( 12) for potential energy D(x 1 , x 2 ), we have e = 2e 0 .

Finite zoning conditions
Let us consider an algebraic form of the equation (17) (algebraization in the Ince sense [14]) to obtain finite zoning conditions as some conditions imposed on factors of potential energy ll(x 1 , x 2 ) expansion. Namely, let us introduce a new independent variable x instead of t. Taking the following equalities into account [the last three equalities were obtained from (13) and (14)]. One may have an equation with regular singularities instead of equation (17) with periodic coefficients: where a prime symbol denotes differentiation with respect to x.
Then then-zoning condition means the solution of equation (18) to be a polynomial in e, the coefficients being functions of x: k=O (19) Substituting (19) into (18) and grouping the terms with equal powers of e, one may have a problem in the eigenvalues a;, e;.

Characteristic classes of analytical potential
This problem is absolutely boundless in general, so let us describe results for three characteristic classes of analytical potentials. Moreover, we should confine ourselves to the values n = 0, 1, 2.  (12) being zero. Note, that equation (11) in this case has linear and cubic terms with respect to x 1 ,x 2 .
Grouping the terms with identical powers of e, then grouping the terms with identical powers of x, we may obtain the finite zoning condition: (20) As shown in this section, the equation in variations is reduced to the Lame equation in this case, and the finite zoning conditions are known [11,17]. Hence, let us discuss other potentials for which finite zoning conditions were determined for the first time in [24] (12) being zero. Equation ( 11) in this case contains the terms of the first, third and fifth powers of x 1 , x 2 • Grouping the terms with identical powers of e, then grouping the terms with identical powers of x, we may obtain the finite zoning conditions: e 4 = 4(n + 1)na 6 ; e 2 = 2(n + 1)na 4 ; 4a 2 a 4 a 6 -a~ + 8ha~ = 0 (21) 4.3.3. a 2 -# 0, a 3 -# 0, a 4 -# 0, e 0 -# 0, e 1 -# 0, e 2 -# 0, the rest of the factors in (12) being zero. Equation (11) in this case contains the terms of the first, second and third powers of xl> x 2 • Grouping the terms with identical powers of e, then grouping the terms with identical powers of x, we may obtain the finite zoning conditions: The finite zoning conditions (20)  One may repeat all the calculations for n > 2 selecting the terms with senior powers of x to realize that the first two equations in (21) and (22) are valid again.

Eigenfunctions and eigenvalues
Now let us discuss the calculation of eigenfunctions and eigenvalues for a finite zoning case. At the same time conditions (20)-(22) will be obtained in another way for any n.
Let us transform the variational equation (15) to an algebraic form introducing a new independent variable x instead oft. Then equation (15) will be 2y"(hn(x, 0)) + y'(-Ox(x, 0)) + yllx,x,(x, 0) = 0. This is an equation of Fuchs class for the polynomial potentials given above, whose exponents of the finite singularities equal 0 or 1/2 (the singularities here are zeros of the expression h -O(x, 0) and x = oc ). Hence, one may search all the eigenfunctions as (the number of the eigenfunctions are 2n + 1 for the case of n-zoning): where p(x) and R(x) are analytical functions and x = ~ is a singular point of the equation.
Indeed, that way one may find all the eigenfunctions and eigenvalues for the potentials I, II, III when n = 0, 1, 2.
Let us consider another approach. First, let us write down the variational equations in the algebraic form for the three types of potentials: The following intervals of the A. axis correspond to instability zones: ( Hence the finite zoning conditions (21) for any positive integer n may be obtained. The eigenfunctions and eigenvalues are transformed accordingly.
It is remarkable that the first zone of parametric resonance is being contracted to a line in the case 11 when the finite zoning conditions are valid. Indeed, the eigenfunctions of equation (24) corresponding to the boundaries of the finite zone of instability, e.g. for n = 1 (b 2 < 0), are (Fig. 5) For a periodic mode: -2a:::; x:::; 2a. But the functions y 1 (x) and y 2 (x) turn to zero within the interval of [ -2a, 2a], i.e. their period with respect tot is T /2 (T being the period of x(t), x being changed within these limit (on a half of the period). Functions corresponding to the first zone of parametric resonance should have the period ofT).
If conditions (20)- (22) are not strictly met, the number of instability zones will be infinite. However all of them are "narrow" in the sense mentioned except the selected n zones, since the zones are contracted to lines when the finite zoning conditions are valid. This is also true for the first zone of parametric resonance which is usually assumed to be "wide".
Note once again that we will obtain the finite zoning conditions (20)- (22) for an arbitrary positive integer n transforming the Lame equation into variational equations as mentioned above (parameters being changed continuously).

Pendulum systems
Let us consider another class of non-linear systems-pendulum systems with the potential Linearization around the normal mode of vibration x 2 = 0, x 1 = x(t) (the equation of motion x + a sin x = 0) gives the variational equation y + 2(b + d cos x) y = 0. In this case a solution x(t) may be either periodic or aperiodic (oscillations and rotation of the pendulum). Introducing a new independent variable z = :X 2 /2, one may transform the variational equation to: being the Lame equation [17]. Here the finite zoning conditions are: 2d -= n(n + 1) (n = 0, 1, 2, ... ) a (n being a number of finite zones of instability within the space of system parameters).

General procedure of transformation
Let us describe the general procedure of transformation from the finite zoning Lame equation (26) to the variational equation (15) in an algebraic form. One should multiply (26) by an arbitrary function q(z) and perform the transformation z = f(x). Comparing the equation obtained with equation (15), one should have the following conditions: These equations may be used to solve the following problem. Let us assume the transformation z = f(x). Then one should select the value of potential energy TI(x 1 , x 2 ) to satisfy the equalities. The conditions imposed upon the coefficients of potential energy are the finite zoning conditions for a positive integer n.
Note that stability of some modes of oscillations has been investigated in [12] for a particle on a plane. In many cases the stability problem may be reduced to analysis of the finite zoning Lame equation.
The results presented in this paragraph are not obvious. Suffice it to mention the possibility of the .first zone of parametric resonance being contracted to a line.

NON-LINEAR STABILITY OF NORMAL VIBRATIONS
A problem of non-linear stability of normal vibrations in conservative systems may be solved only after the corresponding linear problem has been solved. The non-linear analysis is applicable to a system class, such that the variational equations are reduced to hypergeometric equations or Lame equations, etc. The problem of non-linear stability of normal vibrations was also discussed by Pecelli and Thomas [11,12] (using an approach first published by Arnol'd and Mozer) for a system with two degrees of freedom and a Hamiltonian H = 1(Yi + y~) + TI(xb Xz), x 1 , x 2 being positional variables, y 1 , y 2 the corresponding impulses. Assume that the system allows the normal form of vibrations x 2 = y 2 = 0.
What we have in view is the isoenergetic stability of the vibration form. Introducing, following [11], the variables R, r, x, y by a substitution Y1 = J2R sin r, x 1 = J2R cosr, x 2 = x, y 2 = y, one obtains R(x, y) from the equation H =h. (29) If T is a new independent variable, one obtains [25] dx eR dy eR dT oy' dT i,j=O i,j= 0 may be constructed. On substituting these series into (30·), one is able to find the coefficients aij(T), bii(T). In this case, a 10 , a 01 , b 10 , h 01 are solutions of the linear variational equation wherein Z= 11:11· The series (31) define the solution P(T) of (30). What is known as Poincare transformation is P(2n). Thus comparison is made between the values of x(O), y(O) and x(2n), y(2n). Stability is ensured when the linearized Poincare transformation P 1 (2n) defined by the coefficients a 10 , a 01 , h 10 , b 01 has eigenvalues /.T with a modulus equal to unity and when the condition If we now consider normal vibrations that are stable in the first approximation (which implies that £\ 2 < 4, for in this case the eigenvalues of the linearized transformation have moduli equal to unity) and assume A -:p 0, -1 and y =1= 0, it follows from the Arnold-Mozer-Ruisman theorem [26,27] that the normal form of vibrations possesses orbital stability in a 2n-dimensional phase space. It is evident that in the system parameter space, the stability regions A = 0, A = -1 and"/ = 0 have a smaller dimension than the instability region A 2 ;?: 4.
Once again it is highlighted that it is only the knowledge of the solution of the linearized variational equation that is necessary to calculate the values of A and y that define the instability regions obtained when taking into account non-linear terms.
This section deals with isoenergetic stability (for fixed values of the energy h) of normal vibrations; however, it was shown [26] that complete stability takes place when R = <l>(x, y) is continuously dependent on the energy h.
More detailed analysis to a specific Hamiltonian of the type is given in [11]. The problem of non-linear stability of normal vibrations was also discussed by Month and Rand [8].