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Benji
Benji
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Fix $f: \mathbb{R}\times \mathbb{R}^n \mapsto \mathbb{R}^n$ and $v_0:\mathbb{R}^n \mapsto \mathbb{R}^n$. Let $X_t$ be the solution to the second-order ODE

$$\frac{d^2}{dt^2}X_t = f(t,X_t), \quad X_0 = id, \frac{d}{dt}X_t|_{t = 0}=v_0.$$

If $f$ is bounded and globally Lipschitz and $v_0$ is bounded and continuous, then $X_t$ is a homeomorphism for each $t$. Under what conditions on $f$ and $v_0$ is $X_t$ a diffeomorphism?

Fix $f: \mathbb{R}\times \mathbb{R}^n \mapsto \mathbb{R}^n$ and $v_0:\mathbb{R}^n \mapsto \mathbb{R}^n$. Let $X_t$ be the solution to the second-order ODE

$$\frac{d^2}{dt^2}X_t = f(t,X_t), \quad X_0 = id, \frac{d}{dt}X_t|_{t = 0}=v_0.$$

If $f$ is bounded and globally Lipschitz and $v_0$ is bounded and continuous, then $X_t$ is a homeomorphism for each $t$. Under what conditions on $f$ and $v_0$ is $X_t$ a diffeomorphism?

Fix $f: \mathbb{R}\times \mathbb{R}^n \mapsto \mathbb{R}^n$ and $v_0:\mathbb{R}^n \mapsto \mathbb{R}^n$. Let $X_t$ be the solution to the second-order ODE

$$\frac{d^2}{dt^2}X_t = f(t,X_t), \quad X_0 = id, \frac{d}{dt}X_t|_{t = 0}=v_0.$$

Under what conditions on $f$ and $v_0$ is $X_t$ a diffeomorphism?

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Benji
Benji
  • 317
  • 1
  • 3
  • 13

Diffeomorphisms as solutions to second-order ODEs

Fix $f: \mathbb{R}\times \mathbb{R}^n \mapsto \mathbb{R}^n$ and $v_0:\mathbb{R}^n \mapsto \mathbb{R}^n$. Let $X_t$ be the solution to the second-order ODE

$$\frac{d^2}{dt^2}X_t = f(t,X_t), \quad X_0 = id, \frac{d}{dt}X_t|_{t = 0}=v_0.$$

If $f$ is bounded and globally Lipschitz and $v_0$ is bounded and continuous, then $X_t$ is a homeomorphism for each $t$. Under what conditions on $f$ and $v_0$ is $X_t$ a diffeomorphism?