Necessary and sufficient conditions for exponential dichotomy of an invariant torus matrix equation
by
V. A. Chiracalov
Kiev University

In this report we consider matrix linear equation on a torus

\fracdjdt=a(j),        \fracdXdt=A(j)X-XB(j)

(\theequation)

where a^T(j)=(a₁(j), a₂(j), ...a_m(j)), A=A_n1×n1, B=B_n2×n2, X=X_n1×n2 is matrix, j^T=(j₁, j₂, ...j_m), j_i in [0, 2\pi), i=[`1, m], the right hand side of which is defined and continuous in j. We shall make assumption that the torus X(j)=X_n1×n2(j)=0 , j in T_m is an exponentially dichotomous invariant torus of the system (1), i.e. the space of matrix M_n1×n2 can be decomposed into a direct sum of the spaces M_n1×n2^r , M_n1×n2^n-r, (n=n₁n₂). The conditions, under which the trivial torus X=0, j in T_m of system (1) is exponentially dichotomous, have found out. It is convenient to use the machinery of sign-changing Lyapunov functions that are quadratic form of the matrix variable X. We formulate this in the form of the following statement.

Theorem. Let a(j) in C_Lip(T_m) and A(j) in C_Lip(T_m), B(j) in C_Lip(T_m). The trivial torus of system of equations (1) is exponentially dichotomous if and only if there exists a non-singular quadratic form V(j, X), V in C¹(T_m) for which the form dV(j_t, X_t)/dt|_t=0 is negative for all j in T_m or if the Green`s function for sydtem (1) exists and is unique.

References.

[1] Samoilenko A.M. Elements of the mathematical theory of multy-frequency oscillation. -Dordrecht: Kluver Acad. Publ. -1991. -313 p.

Date received: March 16, 2000