Asymptotic properties of formal fundamental matrices of systems of second-order linear differential equations that contain a parameter
Consider the second order linear system \(\epsilon^{p/q}\ddot x+A(t,\epsilon)x=0\) where \(\epsilon>0\) is a small parameter, p, q are arbitrary natural numbers and A(t,\(\epsilon)\) is a real \(n\times n\) matrix, represented by a convergent series \(A(t,\epsilon)=\sum^{\infty}_{0}\epsilon^ rA_ r(t).\) A formal fundamental matrix of the system is a matrix X(t,\(\epsilon)\) which verifies identically the differential equation in the sense of formal matrix series equality. It is shown that if the matrices \(A_ r(t)\) are \(C^{\infty}\) on [0,L] and the eigenvalues of \(A_ 0(t)\) are of constant multiplicity for 0\(\leq t\leq L\), their elementary divisors being also of constant multiplicity for 0\(\leq t\leq L\) then there exists a formal fundamental matrix \(X(t,\epsilon)=U(t,\epsilon)H(t,\epsilon)\) where \(0<\epsilon<\epsilon_ 0\), 0\(\leq t\leq L\) and U(t,\(\epsilon)\), H(t,\(\epsilon)\) can be computed as formal series with respect to the fractionary powers of \(\epsilon\).
second-order linear differential equations
scientific article; zbMATH DE number 3842311