Can 3D Phase Space Surfaces Arise from Differential Equations?

[marellasunny]

Is it possible for a surface to be obtained as the solution for a system of differential equations in a 3d phase space?
Almost all 3d phase plots I have observed are curves,example the Lorenz system.

Is there a general formula that says that a system of n differential equations would produce a n-dimensional curve in a n-dimensional phase space?

Problem at hand:
I have been provided the individual 2d solution plots for a 3 variable,2 parameter system.Now,with these 2d solution plots,I want to make the 3d phase plot.In the process,I also want to create a system of 3 differential equations.Any suggestions?

 

[pasmith]

Is it possible for a surface to be obtained as the solution for a system of differential equations in a 3d phase space?

The solution of a 3D (ordinary) differential equation is a function from the reals to $\mathbb{R}^3$, in other words a curve (or a fixed point). If there is a conserved quantity, then the solution curve must lie on a surface on which that quantity is constant, but it's still a curve.

It may be possible for the solution curve to be dense on some surface.

 

[marellasunny]

My problem is a math modelling problem.It is a system of 3 variables and I have been provided 2 solution curve graphs for 1 variable with another.(X vs Y, Y vs Z). I am to now arrive at a 3 eq autonomous system differential equation w.r.t time i.e dX/dt,dY/dt,dZ/dt.
To top all of this complexity,one of my solution graphs is a set of implicit functions(inverse parabola).
So,I would like finally want to arrive at a 3system autonomous d.e and plot the solution curve and hope that it is dense enough to form a surface(another constraint).
I am struggling to go from the solution curves to the autonomous case.Suggestions/helpful keywords in context would help greatly.