Autonomous System (mathematics) - Properties

Properties

Let be a unique solution of the initial value problem for an autonomous system

.

Then solves

.

Indeed, denoting we have and, thus

\frac{d}{dt}x_2(t)=\frac{d}{dt}x_1(t-t_0)=\frac{d}{ds}x_1(s)= f(x_1(s))=f(x_2(t)).

For the initial condition, the verification is trivial,

.

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