The local flow of an autonomous first-order system

Let $n\ge 1$, let $U\subseteq {\mathbb{R}}^n$ be open, and let $F:U\to {\mathbb{R}}^n$ be continuous and locally Lipschitz in the state variable in the sense of Local Lipschitz continuity in the state variable, locally uniform in time and parameters. Read the autonomous system

$x^{\mathrm{\prime }}(t)=F(x(t))$

as the first-order system of First-order systems, initial value problems, and solutions on intervals with domain $D=\mathbb{R}\times U\subseteq \mathbb{R}\times {\mathbb{R}}^n$ and right-hand side $(t,x)\mapsto F(x)$. Its initial value problems satisfy the Picard-Lindelöf hypotheses, so by Every Picard–Lindelöf initial value problem has one maximal solution on an open interval every $a\in U$ has a unique maximal solution.

For $a\in U$ denote by

$x(\cdot ;a):I_a⟶U$

the unique maximal solution with $x(0;a)=a$, where $I_a\subseteq \mathbb{R}$ is its maximal interval of existence (Extensions, maximal solutions, maximal intervals, and global solutions); $I_a$ is an open interval containing $0$, and every solution of $x^{\mathrm{\prime }}=F(x)$ through $(0,a)$ is a restriction of $x(\cdot ;a)$.

The flow domain of $F$ is the set

$\mathrm{\Omega }:=\{(t,a)\in \mathbb{R}\times U:t\in I_a\},$

and the local flow (also phase flow) of $F$ is the map

$\mathrm{\Phi }:\mathrm{\Omega }⟶U,\mathrm{\Phi }(t,a):=x(t;a)\mathrm{.}$

For $t\in \mathbb{R}$ the time-$t$ map is the partial map

${\mathrm{\Phi }}_t:{\mathrm{\Omega }}_t⟶U,{\mathrm{\Omega }}_t:=\{a\in U:(t,a)\in \mathrm{\Omega }\},{\mathrm{\Phi }}_t(a):=\mathrm{\Phi }(t,a)\mathrm{.}$

Thus ${\mathrm{\Phi }}_0(a)=a$ whenever it is defined, and $(t,a)\in \mathrm{\Omega }$ exactly when the maximal solution of $x^{\mathrm{\prime }}=F(x)$ through $a$ is defined at time $t$.

• The word local refers to the domain: $\mathrm{\Omega }$ need not equal $\mathbb{R}\times U$. Whether $\mathrm{\Omega }=\mathbb{R}\times U$ (completeness of the field) is a separate property of $F$ and is not asserted by this definition. Every statement about $\mathrm{\Phi }$ has to carry its domain condition $(t,a)\in \mathrm{\Omega }$ explicitly.

• The local flow is the solution family of an autonomous field. It is a different object from the stroboscopic map of a time-periodic system (The stroboscopic (time-$P$) map of a $P$-periodic first-order system), although for an autonomous field the stroboscopic map of period $P>0$ is the restriction of ${\mathrm{\Phi }}_P$ to the set of initial states whose maximal solutions exist for a full period.

• The identities ${\mathrm{\Phi }}_0(a)=a$ and ${\mathrm{\Phi }}_{t+s}(a)={\mathrm{\Phi }}_t({\mathrm{\Phi }}_s(a))$ are not part of the definition; they are proved from maximality and uniqueness in The local flow is a partial one-parameter group: identity, composition and inverse-time laws. The definition records only the assignment of solutions, so that no uniqueness statement is smuggled into a definition.