1. ## Autonomous system

Consider the autonomous system

$\displaystyle \frac{dx}{dt}=F(x, y),\ \frac{dy}{dt}=G(x, y)$

in a region where the functions $\displaystyle F$ and $\displaystyle G$ are continuously differentiable.

Suppose that $\displaystyle (x(t), y(t))$ is a solution of the system and that $\displaystyle \gamma\neq 0$. Define $\displaystyle \phi(t)=x(t+\gamma)$ and $\displaystyle \psi(t)=y(t+\gamma)$. Then show that $\displaystyle (\phi(t), \psi(t))$ is also a solution of the system.

My attempt:

$\displaystyle (x(t), y(t))$ is a solution of the autonomous system.

So, $\displaystyle x'(t)=F(x(t), y(t))$ and $\displaystyle y'(t)=G(x(t), y(t))$.

$\displaystyle \frac{d\phi(t)}{dt}=x'(t+\gamma)$

$\displaystyle \frac{d\psi(t)}{dt}=y'(t+\gamma)$

How do I proceed?

2. ## Re: Autonomous system

Originally Posted by alexmahone
How do I proceed?
$\displaystyle \frac{d\phi(t)}{dt}=x'(t+\gamma)=F(x(t+\gamma),y(t +\gamma))=F(\phi(t),\psi(t))$

$\displaystyle \frac{d\psi(t)}{dt}=y'(t+\gamma)=G(x(t+\gamma),y(t +\gamma))=G(\phi(t),\psi(t))$

3. ## Re: Autonomous system

Originally Posted by FernandoRevilla
$\displaystyle \frac{d\phi(t)}{dt}=x'(t+\gamma)=F(x(t+\gamma),y(t +\gamma))=F(\phi(t),\psi(t))$

$\displaystyle \frac{d\psi(t)}{dt}=y'(t+\gamma)=G(x(t+\gamma),y(t +\gamma))=G(\phi(t),\psi(t))$
I'm not sure that I completely understand how $\displaystyle x'(t+\gamma)=F(x(t+\gamma),y(t+\gamma))$ and $\displaystyle y'(t+\gamma)=G(x(t+\gamma),y(t+\gamma))$.

I know $\displaystyle x'(t)=F(x(t),y(t))$ and $\displaystyle y'(t)=G(x(t),y(t))$, but how do we know that substituting $\displaystyle t+\gamma$ for $\displaystyle t$ will lead to a valid equation?

4. ## Re: Autonomous system

For example, suppose $\displaystyle x'(t)=F(x(t),y(t))$ and $\displaystyle y'(t)=G(x(t),y(t))$ for all $\displaystyle t\in\mathbb{R}$ . What is $\displaystyle x'(u)$ and $\displaystyle y'(u)$ for all $\displaystyle u\in\mathbb{R}$ ?

5. ## Re: Autonomous system

Originally Posted by FernandoRevilla
For example, suppose $\displaystyle x'(t)=F(x(t),y(t))$ and $\displaystyle y'(t)=G(x(t),y(t))$ for all $\displaystyle t\in\mathbb{R}$ . What is $\displaystyle x'(u)$ and $\displaystyle y'(u)$ for all $\displaystyle u\in\mathbb{R}$ ?
$\displaystyle x'(u)=F(x(u),y(u))$ and $\displaystyle y'(u)=G(x(u),y(u))$.

6. ## Re: Autonomous system

Originally Posted by alexmahone
$\displaystyle x'(u)=F(x(u),y(u))$ and $\displaystyle y'(u)=G(x(u),y(u))$.
Right, now use the substitution $\displaystyle u=t+\gamma$ .

7. ## Re: Autonomous system

Originally Posted by FernandoRevilla
Right, now use the substitution $\displaystyle u=t+\gamma$ .
Hmm...makes sense. Thanks!