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Extra resources for Differential Equations: Stability, Oscillations, Time Lags
T: : E. This means that the matrix C(t;to) is nonsingular, and its inverse is q t o ; t). It is useful to consider in some problems the adjoint system of system (3). Namely, we shall call the adjoint system of system ( 3 ) the system 3 = --yA(t), dt where y is a row vector. 1. STABILITY THEORY 42 Proposition. If x is a solution of the given system and y a solution of the adjoint system, then the product y x is constant. Proof. We have d dt dY dt --yx = -x dx +y = --yA(t) x +yA(t)x = 0 ; dt hence y x is constant.
STABILITY T H E O R Y 16 Remarks. and we have 1. If V is differentiable, the function V * ( t ) is differentiable since x is the solution of the system of differential equations. T h e function is called the total derivative of the function Vwith respect to the system. Condition (c) of the theorem is evidently satisfied if V is differentiable and W ( t , x) 0. 2. T h e stability of the trivial solution obviously implies that the solutions with initial conditions in a neighborhood of x = 0 are defined for every t 3 t o , since the fact that I x ( t ; t o , x,)j < c assures the possibility of extension, the solution remaining in the domain in which the conditions of the existence theorem are fulfilled.
For to > 0 arbitrary, we have kw to < (k 1)w; hence 0 to kw < w and 1 x(t; to , xo)l = I x(t - kw, to - kw, xo)l < ~ ( 6if) I x, I < 6; hence I x ( t ; t o ,xo)l < E if I xo I < a(€). < + < < < + Proposition 2. 2’ is periodic of period w . + + + + Proof. >I = V(t, 4A particular case of periodic systems is constituted by systems in which f does not depend on t. It follows that for such systems the stability is always uniform and the function V may be chosen independent of t. + Definition. A solution x ( t ) of system (1) is said to be unstable if it is not stable.
Differential Equations: Stability, Oscillations, Time Lags by Halanay