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Hi all,

If I have an autonomous system:

dx/dt=f(x)

The k-dimensional state vectorxlives in a k-dimensional phase space. Does each point in the k-dimensional phase space have a UNIQUE time associated with it?

I don't think it does, and this is why. Lets say we choose some pointx0, we can iterate the solution forward using a runge-kutta routine. (I'm going to drop the bold vector notation.) That is we find x1 = O(x0, j), where O() is an operator that evolves the solution (our runge-kutta), and j is the step. In our case we use j=1, so we iterate one time step.

Now, if we let the initial condition x0 exist at time t=0, then when we iterate forward via O() we get x1 which exists at time t=0+1*step. Now, we could just as easily have chosen x0 to exist at time t=100. So it seems to me, that "we" pick a reference time, and tracing out the trajectory returns phase space points from our initial (chosen) reference time!

So is it just that all points in phase space are unique, like we don't have intersecting trajectories? BUT, we could have a point x0 at time t=0, a point x1 at time t=0, x2 at time t=0, etc... where we picked the t=0 point.

Could someone explain this to me? I'm trying to read and understand, but I think I'm confusing myself. Thank you!

P.S. Sorry for the sloppy notation. I hope it can be understood.

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# Phase Space - does each point have a unique time associated with it?

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