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The local flow is a partial one-parameter group: identity, composition and inverse-time laws

Statement

Let $n \geq 1$ , let $U \subseteq R^{n}$ be open, let $F : U \to R^{n}$ be continuous and locally Lipschitz in the state variable (Local Lipschitz continuity in the state variable, locally uniform in time and parameters), and let $Φ : Ω \to U$ be the local flow of $x^{'} = F (x)$ as in The local flow of an autonomous first-order system, with maximal solutions $x (\cdot; a) : I_{a} \to U$ and flow domain $Ω = {(t, a) : t \in I_{a}}$ . Then:

$(0, a) \in Ω$ and $Φ (0, a) = a$ for every $a \in U$ ;
if $(s, a) \in Ω$ and $(t, Φ (s, a)) \in Ω$ , then $(s + t, a) \in Ω$ and $Φ (s + t, a) = Φ (t, Φ (s, a))$ ;
if $(t, a) \in Ω$ , then $(- t, Φ (t, a)) \in Ω$ and $Φ (- t, Φ (t, a)) = a$ .

Facts & Assumptions

Given: The open set $U$ , the field $F$ , and the local flow $Φ$ built from the unique maximal solutions $x (\cdot; a)$ through $(0, a)$ .

[F1]

For every $a \in U$ , $x (\cdot; a) : I_{a} \to U$ is a solution of $x^{'} = F (x)$ with $x (0; a) = a$ on the open interval $I_{a}$ containing $0$ , it is maximal, and every solution of that initial value problem is a restriction of it; moreover this maximal solution is unique (The local flow of an autonomous first-order system, Extensions, maximal solutions, maximal intervals, and global solutions, Every Picard–Lindelöf initial value problem has one maximal solution on an open interval, First-order systems, initial value problems, and solutions on intervals).

[F2]

If $y : J \to U$ is a solution of $x^{'} = F (x)$ and $s \in J$ , then the translate $y_{s} (σ) : = y (σ + s)$ , defined for $σ \in J - s : = {σ : σ + s \in J}$ , is again a solution of $x^{'} = F (x)$ on the open interval $J - s$ , with $y_{s} (0) = y (s)$ : this is the chain rule applied to the translation $σ \mapsto σ + s$ , which has derivative $1$ (First-order systems, initial value problems, and solutions on intervals).

Proof

technique · direct; use uniqueness and maximality of solutions to glue translated solutions

1.1givenF1

Clause 1 is definitional: $0 \in I_{a}$ because $I_{a}$ is the maximal interval of the solution through $(0, a)$ , and $Φ (0, a) = x (0; a) = a$ .

1.2givenF2

Fix $(s, a) \in Ω$ and put $b : = Φ (s, a) = x (s; a)$ . Define $J : = I_{a} - s$ and $y : J \to U$ by $y (σ) : = x (σ + s; a)$ . By [F2], $y$ solves $x^{'} = F (x)$ on the open interval $J$ , and $y (0) = x (s; a) = b$ .

2.1givenstep 1.2F1

The curve $x (\cdot; b)$ is the unique maximal solution through $(0, b)$ and $y$ is a solution through $(0, b)$ ; by [F1] $y$ is the restriction of $x (\cdot; b)$ to $J$ , so $J \subseteq I_{b}$ and $y = x (\cdot; b) ∣_{J}$ . If $J$ were a proper subinterval of $I_{b}$ , the curve $z (r) : = x (r - s; b)$ on $I_{b} + s$ would, by [F2], be a solution extending $x (\cdot; a)$ from $I_{a} = J + s$ to a strictly larger interval, contradicting its maximality in [F1]. Thus $I_{b} = J$ . Now let $t \in I_{b}$ ; then $t \in J$ , that is $s + t \in I_{a}$ , and $Φ (s + t, a) = x (s + t; a) = y (t) = x (t; b) = Φ (t, Φ (s, a))$ . This is clause 2.

3.1givenstep 2.1F1F2∎

For clause 3, fix $(t, a) \in Ω$ and put $b : = Φ (t, a) = x (t; a)$ . Define $K : = I_{a} - t$ and $w : K \to U$ by $w (σ) : = x (σ + t; a)$ . By [F2] (with $s = t$ ) $w$ solves $x^{'} = F (x)$ on the open interval $K$ , and $w (0) = x (t; a) = b$ . By [F1] $w$ is the restriction of the unique maximal solution $x (\cdot; b)$ to $K$ ; hence $- t \in K \subseteq I_{b}$ and $Φ (- t, b) = x (- t; b) = w (- t) = x (0; a) = a$ .

Remarks

Clause 2 is the cocycle or semigroup identity for the partial flow; it is proved, not assumed. Its hypothesis $(t, Φ (s, a)) \in Ω$ cannot be dropped: for a field whose solutions blow up in finite time, the solution through $a$ may be defined at time $s$ but not for a further time $t$ from $Φ (s, a)$ .
Clause 3 says that $Φ_{t}$ is injective on its domain with inverse $Φ_{- t}$ restricted to $Φ_{t} (Ω_{t})$ ; this is a consequence of uniqueness and maximality, and it is used in Finite-time tangent growth rates and Lyapunov exponents of a C¹ flow.

The local flow is a partial one-parameter group: identity, composition and inverse-time laws

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources