a)$, we can splice $u$ and $v$ and obtain a (possibly new) solution $w = u \underset{t_1}{\bowtie} v$ starting at $a$: w(t) = u v(t) =

u(t, a), & 0tt\_1,\
v(t - t\_1, u(t\_1, a)), & tt\_1.

If solutions are unique (for every $a\in X$ there is a unique solution $u(t, a)$), then $v(t, u(t_1, a)) = u(t + t_1, a)$. In general, in the case of uniqueness, the solutions of enjoy the semigroup property, i.e., for every $a\in X$, $u(0, a) = a$ and \[sg\] u(t\_2, u(t\_1, a)) = u(t\_1 + t\_2, a),t\_1, t\_2 0. This allows us to define the semigroup $U(t): X\to X$ by the formula $U(t)(a) = u(t, a)$.

For non-autonomous differential equations the situation is similar and different. Consider an archetypical non-autonomous equation \[n-aut\] = g(t, u). Now, in addition to the initial position/state $a\in X$, it is important to specify the initial moment of time, $t_0$. The solution(s) will depend on $a$ and $t_0$; we write $u(t; t_0, a)$ to denote a solution of for $t\ge t_0$ that equals $a$ when $t = t_0$. If we follow the solution $u(t; t_0, a)$ until the moment $t = t_1$ and then follow a solution $v(t; 