I have an ordinary differential equation in the form of dp/dt=p-f(p). Since p and f(p) are bounded, any equilibrium of this ODE should satisfy p*=f(p*). I have another ODE in the form of dw/dt=g(w). I know this ODE diverges and because w is bounded between zero and one, at the equilibrium w* is zero or one. Now I consider a system that has p and w and both of them are changing which gives me two ODEs, dp/dt=p-f(p,w) and dw/dt=g(w,p). I know for any fixed w, we have  p*=f(p*,w) at the equilibrium and for any fixed p, w* equals zero or one. Does anyone know if I can claim there is any equlibrium for general case and at the equilibrium p*=f(p*,w*) and w* equals zero or one?

1 hour ago, Faeze said:

I have an ordinary differential equation in the form of dp/dt=p-f(p). Since p and f(p) are bounded, any equilibrium of this ODE should satisfy p*=f(p*). I have another ODE in the form of dw/dt=g(w). I know this ODE diverges and because w is bounded between zero and one, at the equilibrium w* is zero or one. Now I consider a system that has p and w and both of them are changing which gives me two ODEs, dp/dt=p-f(p,w) and dw/dt=g(w,p). I know for any fixed w, we have  p*=f(p*,w) at the equilibrium and for any fixed p, w* equals zero or one. Does anyone know if I can claim there is any equlibrium for general case and at the equilibrium p*=f(p*,w*) and w* equals zero or one?

 

So would this be a pair of autonomous equations with

[math]\varphi \left( p \right) = p - f\left( p \right)[/math]

 

17 minutes ago, studiot said:

 

So would this be a pair of autonomous equations with

φ(p)=p−f(p)

 

Not sure. I am not familiar with this area that much. If we can write them as you said, what would we get?

Well you haven't siad what f(p) is and I don't know why you have separated p from f(p), since phi (p) is still a function of p alone and makes the ODE autonomous.

Here is a simple explanation of what is a single autonomous ODE

https://sites.math.washington.edu/~aloveles/Math307Spring2016/m307Review2-5.pdf

Here is some stuff about solving pairs of them

https://math.stackexchange.com/questions/1405497/solving-a-pair-of-odes?rq=1

 

35 minutes ago, studiot said:

 

So would this be a pair of autonomous equations with

φ(p)=p−f(p)

 

Now I think yes they are autonomous equations

16 minutes ago, studiot said:

Well you haven't siad what f(p) is and I don't know why you have separated p from f(p), since phi (p) is still a function of p alone and makes the ODE autonomous.

Here is a simple explanation of what is a single autonomous ODE

https://sites.math.washington.edu/~aloveles/Math307Spring2016/m307Review2-5.pdf

Here is some stuff about solving pairs of them

https://math.stackexchange.com/questions/1405497/solving-a-pair-of-odes?rq=1

 

Thank you. The problem is f(p) is not an explicit function of p. And I do not want to solve this ODE.  My question is if I can say something about the equilibriun of a system with these two ODEs when I have information that I mentioned?