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Thread: Autonomous system

  1. #1
    MHF Contributor alexmahone's Avatar
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    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?
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Autonomous system

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

    $\displaystyle \frac{d\psi(t)}{dt}=y'(t+\gamma)=G(x(t+\gamma),y(t +\gamma))=G(\phi(t),\psi(t))$
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  3. #3
    MHF Contributor alexmahone's Avatar
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    Re: Autonomous system

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

    $\displaystyle \frac{d\psi(t)}{dt}=y'(t+\gamma)=G(x(t+\gamma),y(t +\gamma))=G(\phi(t),\psi(t))$
    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?
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    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 $\displaystyle u\in\mathbb{R}$ ?
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  5. #5
    MHF Contributor alexmahone's Avatar
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    Re: Autonomous system

    Quote Originally Posted by FernandoRevilla View Post
    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 $\displaystyle y'(u)=G(x(u),y(u))$.
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  6. #6
    MHF Contributor FernandoRevilla's Avatar
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    Re: Autonomous system

    Quote Originally Posted by alexmahone View Post
    $\displaystyle x'(u)=F(x(u),y(u))$ and $\displaystyle y'(u)=G(x(u),y(u))$.
    Right, now use the substitution $\displaystyle u=t+\gamma$ .
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  7. #7
    MHF Contributor alexmahone's Avatar
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    Re: Autonomous system

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