On approximate methods of analysing certain singularly-perturbed systems (Q1100614)

Suppose the perturbed motion of a system is described by a differential equation of the form \(dz/dt=Z(t,\mu,z,x),\quad dx_ 1/dt=P_ 1(\mu)x+X_ 1(t,\mu,z,x),\quad \mu \quad 2dx_ 2/dt=P_ 2(\mu)x+X_ 2(t,\mu,z,x),\quad dx_ 3/dt=\quad P_ 3(\mu)x+X_ 3(t,\mu,z,x).\) Here z is an m-dimensional vector which corresponds to the Lyapunov-critical variables, \(x_ i(i=1,2,3)\) are \(n_ i\)-dimension vectors corresponding to the non-critical variables, \(\mu\) is a small parameter, \(P_ i(\mu)\) are matrices of corresponding dimensions whose elements are continuous functions of \(\mu\), \(P_ i(\mu)=(P_{i1}(\mu)\), \(P_{i2}(\mu)\), \(P_{i3}(\mu))\) where \(P_{ij}(\mu)\) are \(n_ i\times n_ j\) submatrices, Z and \(X_ i\) are certain functions which are holomorphic with respect to the aggregate of variables z and x. Let Z, \(X_ i(i=1,2,3)\) vanish for \(x=0\) and let \(P_{21}(\mu)=\mu P'_{21}(\mu)\), \(P_{22}(\mu)=\mu P'_{22}(\mu)\). The characteristic equation for this system has m zero roots. Putting \(\mu =0\) one obtains from the above a degenerate system. Now the author studies the conditions under which the solution of a stability problem reduces to the same problem for a degenerate system. As an application in practice gyroscopic stabilizing systems with elastic elements of high stiffness are discussed. The conditions under which the solution of the problem of the stability of steady motion follows from the solution of this problem for an ideal system (with absolutely rigid elements) are obtained.