Final answer:
To find an invariant manifold for the system of differential equations X′=X and Y′=Y, consider the solutions X(t) = X_0e^t and Y(t) = Y_0e^t. The manifold M = y = cx is shown to be invariant as any point on it remains on it after evolving over time according to the equations of motion.
Step-by-step explanation:
Finding an Invariant Manifold
To find an invariant manifold for the set of differential equations given by X′=X and Y′=Y, we first consider particular solutions to these equations. The general form of the solutions to these equations can be found to be:
- X(t) = X_0e^t
- Y(t) = Y_0e^t
where X_0 and Y_0 are constants determined by the initial conditions. A manifold that is invariant under the flow of these equations would be one where if a point lies on the manifold at some initial time, it will always lie on the manifold as it evolves according to the flow.
To prove that a manifold is invariant, we need to show that any point on the manifold, after evolving for a time t according to the equations of motion, still satisfies the equation defining the manifold. Let us define a manifold:
M = (x,y)
As a trial solution, let's take the case where f(x) = cx for some constant c. This gives us the manifold M as:
M = y = cx
If (x(t), y(t)) is a point on the manifold, then we must have y(t) = cx(t). Differentiating both sides with respect to t, we get:
Y′(t) = cX′(t)
Since X′(t)=X and Y′(t)=Y, our previous relation becomes Y = cX, which is the same as our manifold equation. Therefore, any point that starts on the manifold will evolve in time to remain on the manifold, proving that it's invariant.