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Two functions $x(t)$ and $y(t)$ are coupled via: $$\dot x(t) = a y(t)-b,y(t+x(t))=x(t)$$

where $a<0$, $b\neq 0$ is some constant.

I am mostly confused with the second equation. What is the mathematical term for this kind of delay differential equation? What kind of initial condition do I need to specify?

It is obvious that, if $x(0)=y(0)=b/a$, then the system will stay stable for all $t>0$.

Through doing some simulation also back of envelope thinking, it seems that when $x(0)=y(0)=0$, then the system may never converge to an equilibrium/stable solution. However, I have no idea what type of proof technique is required to show that.

Thanks!

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  • $\begingroup$ So you have functions $x(t)$, $y(t)$, $X(t)$ and $Y(t)$. I presume $x=X$ and $y=Y$? I changed that. But tou also have "$a \lt 0,b$ is a constant". Does this mean "$a\lt 0$ and $a\lt b$", or "$a\lt 0$ is a constant, and $b$ is a constant"? I can't tell, please edit to make it unambiguous. $\endgroup$
    – David Roberts ♦
    Jan 15, 2019 at 7:25
  • $\begingroup$ @DavidRoberts, Thank you! I just did. I hope this is more clear now. $\endgroup$
    – JYL
    Jan 15, 2019 at 18:28
  • $\begingroup$ Now you have $x(0)=x(0)=b/a$... $\endgroup$
    – David Roberts ♦
    Jan 15, 2019 at 21:11
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    $\begingroup$ Fixed. Sorry about that! $\endgroup$
    – JYL
    Jan 16, 2019 at 19:16
  • $\begingroup$ @DavidRoberts, any advice on how I can go about proving the non-convergence? Thanks! $\endgroup$
    – JYL
    Jan 22, 2019 at 22:19

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