When does a Lipschitz condition fail for a DE?

[marellasunny]

could you give an example where the Lipschitz condition fails,like when there is a periodic forcing function?
I'm thinking the Lipschitz condition would fail for a non-autonomous differential system because period-2 orbits exist for 2D non-autonomous continuous dynamical systems,which means the uniqueness condition that the Lipschitz criterion so vehemently describes in DE does not hold for non-auto systems.

How does one explain this mathematically? Intuitively, I could say uniqueness of solution curve exists because of phase change due to the periodic forcing function. But,which theorem states this mathematically?

for example,take the non-autonomous differential system:
$$\frac{\mathrm{dx} }{\mathrm{d} t}=x^3 + aSin(\omega t)$$
$$|f(t,u)-f(t,v))|\leq L|u-v|$$
$$u^3 - aSin(\omega t)-v^3-aSin(\omega t)$$
$$|u^2+uv+v^2||u-v|\Rightarrow |u^2+uv+v^2|\leq L$$

 

[HallsofIvy]

Are you clear on what the "Lipschitz" condition is? It is a condition on functions, not differential equations: A function has the "Lipschitz" property, or "is Lipschitz", on interval [a, b] if and only if there exist a number, c, such that |f(x)- f(y)|\le c|x- y| for all x and y in the interval.

The "Fundamental Existence and Uniqueness Theorem" for differential equations says that the if f(x,y) is continuous and is Lipschitz in y (x constant) in some neighborhood of (x_0, y_0), then equation dy/dx= f(x, y), has a unique solution satisfying y(x_0)= y_0,

I don't know where you got the idea that "non-autonomous" equations do not have unique solutions. That is certanly NOT true. Such equations have unique solutions in the same conditions as autonomous equations.

It is true that to apply the "existence and uniqueness" theorem to systems like dx/dt= f(x,y,t), dy/dt= g(x,y,t), you have to look at the equation
\frac{dy}{dx}= \frac{g(x,y,t)}{f(x,y,t)}
so the theorem will not apply where f(x, y, t)= 0.

 

[marellasunny]

Are you clear on what the "Lipschitz" condition is? It is a condition on functions, not differential equations: A function has the "Lipschitz" property, or "is Lipschitz", on interval [a, b] if and only if there exist a number, c, such that |f(x)- f(y)|\le c|x- y| for all x and y in the interval.

The "Fundamental Existence and Uniqueness Theorem" for differential equations says that the if f(x,y) is continuous and is Lipschitz in y (x constant) in some neighborhood of (x_0, y_0), then equation dy/dx= f(x, y), has a unique solution satisfying y(x_0)= y_0,

I don't know where you got the idea that "non-autonomous" equations do not have unique solutions. That is certanly NOT true. Such equations have unique solutions in the same conditions as autonomous equations.

It is true that to apply the "existence and uniqueness" theorem to systems like dx/dt= f(x,y,t), dy/dt= g(x,y,t), you have to look at the equation
\frac{dy}{dx}= \frac{g(x,y,t)}{f(x,y,t)}
so the theorem will not apply where f(x, y, t)= 0.

But HallsofIvy, in the diagram I have attached,I show a period-2 orbit in a 2D continuous phase space. Notice that the orbit loops onto itself. This kind-of orbit cannot exist in autonomous systems because there will be no uniqueness as time progresses. So,my question is whether such a orbit exists in non-autonomous systems? I can intuitively guess that they do exist in N.A systems because there will be a phase change in the solution and so one can say that the solution curve is unique when it loops onto itself. How can I explain this uniqueness in N.A systems using Lipschitz condition? I wrote an example of Lipschitz uniqueness for non-autonomous systems and the coefficient is bounded.Does this suggest something?