On the stability of certain systems under resonance conditions

Abstract

WE SHALL be concerned with the stability of the stationary point of the system of differential equations dx dt = A(x) ; (I) here, x = ( x 1, …, x n ), A( x) = ( A 1( x), …, A n ( x)). The classical approach to this type of problem is to linearize the differential equations in the neighbourhood of the required stationary point x 0 = ( x 0 1, …, x 0 n ), A( x 0 = 0. The stability problem for the linearized equation dx i dt = A ijx j , A ij = ∂A i ∂x j ¦ x = x 0 (II) amounts to the purely algebraic problem of finding the eigenvalues of the matrix A ij . If the real parts of the eigenvalues of A ij are negative, system (II) is asymptotically stable. If one of the eigenvalues has a positive real part, the system (II) is unstable. Lyapunov's basis principle is that the asymptotic stability of the linearized system implies the asymptotic stability of the initial system. If the linear system is unstable, the initial system is also unstable. In these circumstances the stability is determined entirely within the framework of the linear theory. In many important cases, however, the linearized system is partially or completely neutral. This means that, either some eigenvalues of the system matrix are pure imaginary pairs, while the rest have negative real parts, or else all the eigenvalues are pure imaginary. In such cases the linear approximation tells us nothing about the stability of the initial system close to the equilibrium position, its behaviour being determined by higher order terms. A considerable amount of work has been devoted to the stability of systems which are neutral in the linear approximation. In [1], A.M. Lyapunov analysed the case when the matrix has a pair of pure imaginary eigenvalues and n eigenvalues with negative real parts. The cases of two pairs of pure imaginary eigenvalues, and one pair of imaginary and one non-zero eigenvalue, were considered in [2, 3], In [4], the case of 2 n pure imaginary eigenvalues and several with negative real parts was treated. Systems which are neutral in the linear approximation (e.g. Hamiltonian systems) are of special interest here. In complex systems, in addition to vanishing of the real parts of the eigenvalues, integer-valued relationships can occur between the eigenvalues λ j = μ j + iω j , i.e. Σ j=1 3 pjρj=0, where the p j are integers not all zero simultaneously. In particular, a relationship (II) for a system with pure imaginary eigenvalues denotes the existence of internal resonance in the system. Such systems are of great interest in mechanics, though they represent an exception from the formal view-point (since conditions in the form of equations are excepted). In the above-mentioned works, it was usually assumed that there are no resonance relations, apart from identities implied by the fact that the initial system is real (in the matrix of the linear approximation, a complex conjugate \\ ̄ gl corresponds to each complex eigenvalue λ). It is a well-known fact that, if resonance occurs, there can be an essential change in the behaviour of the system. As a result of resonance, the system can become unstable to the second order. In the present paper we try to see what role one internal second-order resonance has in regard to the stability of the equilibrium of a system which is neutral in the linear approximation. Our problem amounts to examining the simplest possibilities when there are precisely three pairs of pure imaginary roots (i.e. ± iω 1, 2, 3) giving a second-order internal resonance (i.e. ω 1 − ω 2 − ω 3 = 0). The existence of second-order internal resonance leads to an essentially new phenomenon, namely the possible instability of the system up to the second order. We find below the necessary and sufficient condition for retaining neutrality up to the second order, and the necessary and sufficient condition for monotonic stability to the third order. The minimal real system in which such resonance is possible is the sixth order system dν α dt = V β α ν β+V βγ α ν βν γ+V βγδ αν βν γν δ + O(¦ν¦ 4) ( α = 1, …, 6). (1) To investigate the resonant case, the sixth order system can be reduced to a fourth order system by a number of transformations. (We first use a linear transformation with complex coefficients to reduce the linear part of the system to the diagonal form. We then use a non-linear change of variables to separate the resonant terms. Finally, we make use of the structure of the resonant terms to lower the order by means of a special change of variables.) We show that resonance can lead to second order instability, and find the necessary and sufficient condition for retaining neutrality, namely: the necessary and sufficient condition for system (1) to be second order neutral is that the determinants D 1, D 2, D 3 are of the same sign (the elements of the determinants are evaluated here in terms of the coefficients of the quadratic terms of the initial system, see (3.1) and (2.4)). The sufficiency of the condition is proved by constructing a positive first integral, and the necessity by direct construction of an unstable solution, i.e. we show that the existence of a positive first integral is necessary as well as sufficient for second-order neutrality of system (1). Clearly, if the system is neutral in the linear approximation, and unstable to the second order, the stability of the system as a whole will have been determined; if it is still neutral to the second order, the stability will be determined by the terms of next higher order. A complete determination of the stability of system (1) to the third order has proved far from easy, and the question remains open at the present time. The present paper finds the necessary and sufficient conditions for third-order monotonic stability of system (1); these represent sufficient conditions for asymptotic stability to the third order. The concept of monotonic stability depends on the norm with respect to which the stability is established. In our case, the natural norm is the sum of the squares of the variables r 1( t), r 2( t), r 3( t) (these variables are expressible in terms of the initial variables). Monotonic stability means that, whatever the initial data, i.e. whatever the values of the variables, the derivative of r 2( t) ( Σ i=1 3 r i2(t)) must be negative by virtue of the equations of motion. Thus, to find if the system is monotonically stable, we have to find the sign of the derivative of r 2( t). We introduce spherical coordinates r, α, β, and write down the equation for the derivative dr 2 dt = −ε 2r 4(α, β) . To find if the system is monotonically stable, we also have to study the conditions on the coefficient of the function R( α, β) under which R( α, β) is strictly positive in the region D = 0 ⩽ α ⩽ π 2 , 0 ⩽ β ⩽ π 2 . Investigation of the positive minimum of R( α, β) on the boundary of D leads to some simple necessary conditions a < 0, b > 0, x ⩾ − 1, y ⩾ − 1, z ⩾ −1. Here, a, b, x, y, z are coefficients of the function R( α, β) and are evaluated by means of the coefficients of the initial system. The most (technically) laborious case occurs when the minimum lies inside D. We then have to find the signs of certain functions F 1( a, b, x, y, z), F 2( a, b, x, y, z), F 3( a, b, x, y, z) in five-dimensional space and the sign of F( x, y, z) in the three-dimensional space of the coefficients. We find that, if F 1, F 2, F 3, F are of the same sign, then R( α, β) has a positive minimum inside D, i.e. it is positive inside D. If two of F 1, F 2, F 3 are of opposite sign, then, whatever the sign of F, R( α, β) has no minimum inside D. In this case the sign of R( α, β) inside D is determined by its sign on the boundary of D. We prove the necessary and sufficient condition for monotonie stability of system (1), namely: the necessary and sufficient condition for system (1) to be monotonically stable to the third order is that one of the following be true: (1) The functions F 1, F 2, F 3, F are of the same sign for a, b, x, y, z ϵ K = a > 0, b > 0, x ⩾ − 1, y ⩾ − 1, z ⩾ − 1; (2) Two of the F 1, F 2, F 3 have opposite signs for a, b, x, y, z ϵ K. This necessary and sufficient condition for third-order monotonic stability is also a sufficient condition for asymptotic stability to the third order of system (1).