(a) for every e>0 there exists 8>0 such that dist(a(t), 1o())<e for all t20 and all solutions c=.(t)of(9. 11)such that dist(r(0), To()<8, where dist(I, ro()=min i-To(t) (b)there exists e>0 such that dist(r(t), o()-0 as t-o for every solution of (9. 11) such that dist(=(0), o()<8 Note that a non-constant periodic solution o= o(t) of time-invariant ODE equations is never asymptotically stable, because, as d-0, the initial conditions for the solution 8(t)=co(t +8) approach the initial conditions for to(), but the difference s(t)-to(t does not converge to0 as t- o0 unless as To. Therefore, the notion of a stable limit cycle is a relaxed version of asymptotic stability of a solution Theorem 9.3 Let a: R"+R be a continuous(i)-periodic function. Let co: RHR be a non-constant(T, i)-periodic solution of(9.11). Assume that there exists e>0 such that a is continuously differentiable on the set X={∈R:|z-x0(t)< e for some t∈R Let A: 0,H R,n be defined by(9.9), (910).Then (a) if all eigenvalues of A(r) except one have absolute value less than 1, io( is a stable (b) if one eigenvalue of A(r has absolute value greater than 1, ro( is not a stable limit cycle5 (a) for every δ > 0 there exists � > 0 such that dist(x(t), x0(·)) < δ for all t ∀ 0 and all solutions x = x(t) of (9.11) such that dist(x(0), x0(·)) < �, where dist(¯x, x0(·)) = min |x¯ − x0(t)|; t�R (b) there exists δ > 0 such that dist(x(t), x0(·)) � 0 as t � → for every solution of (9.11) such that dist(x(0), x0(·)) < �. Note that a non-constant periodic solution x0 = x0(t) of time-invariant ODE equations is never asymptotically stable, because, as � � 0, the initial conditions for the solution x�(t) = x0(t + �) approach the initial conditions for x0(·), but the difference x�(t) − x0(t) does not converge to 0 as t � → unless x� ≥ x0. Therefore, the notion of a stable limit cycle is a relaxed version of asymptotic stability of a solution. Theorem 9.3 Let a : Rn ∞� Rn be a continuous (ˆx)-periodic function. Let x0 : R ∞� Rn be a non-constant (π, xˆ)-periodic solution of (9.11). Assume that there exists δ > 0 such that a is continuously differentiable on the set X = {x¯ ¯ ≤ Rn : |x − x0(t)| < δ for some t ≤ R. Let � : [0, π ] ∞� Rn,n be defined by (9.9),(9.10). Then (a) if all eigenvalues of �(π ) except one have absolute value less than 1, x0(·) is a stable limit cycle; (b) if one eigenvalue of �(π ) has absolute value greater than 1, x0(·) is not a stable limit cycle