Qualitative implications of parameter shift in non-autonomous ODE

[tjc]

Hi everyone,

I've got a one-dimensional non-autonomous ODE of the following form:
dy / dx = f(x,y;w)
x_{0} = g(w)
y_{0} = h(x_{0};w)
--- i.e., w is a parameter that influences both the derivative dy/dx along with both coordinates in the initial condition (x_{0},y_{0}). I basically want to learn if an increase in w will raise or lower y at some given x.

Can anyone recommend a good reading? I studied qualitative ODEs a while ago, but it was mainly about the stability and long-run behaviour of solutions and always in the context of autonomous systems.

Thanks very much!


T

 

[chiro]

Hi everyone,

I've got a one-dimensional non-autonomous ODE of the following form:
dy / dx = f(x,y;w)
x_{0} = g(w)
y_{0} = h(x_{0};w)
--- i.e., w is a parameter that influences both the derivative dy/dx along with both coordinates in the initial condition (x_{0},y_{0}). I basically want to learn if an increase in w will raise or lower y at some given x.

Can anyone recommend a good reading? I studied qualitative ODEs a while ago, but it was mainly about the stability and long-run behaviour of solutions and always in the context of autonomous systems.

Thanks very much!


T

Hey tjc and welcome to the forums.

Are you talking about a general non-autonomous DE? If you are, I think the question is too broad to be answered with any real clarity.

If I were to give a few ideas about what I would look for, it would be with regard to some kind of long term convergence properties of the actual DE itself rather than short-term convergence properties. Also you can look at things like drift and treat it like you were considering a distribution of the possible outputs with a mean and a variance.

If the DE has the habit of converging to a small subset of values long term (like say a Bessel function), then what you can do is find out whether initial conditions change the value of long-term convergence. For some DE's if this attribute exists, then it might or it may not. If it has this behaviour and the long term behaviour doesn't change then you're done. If it has this behaviour but it changes, then you will need further analyses.

If you have some kind of non-zero drift then you will need to consider what the initial conditions do to the drift. If there is no drift and the new conditions create drift then that will complicate things. If the equation is unchanged with respect to drift with new initial conditions, then this needs to be taken into account as well.

In terms of figuring out explicitly what the change will be at a specific point x, this is equivalent to solving the DE analytically or evaluating it numerically with given accuracy. If your statement is correct then you might as well just do this for the specific DE. If you want something a little less specific in terms of general function behaviours then hopefully the above can give you some ideas to start you off.