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marellasunny
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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:
$$\frac{\mathrm{dx} }{\mathrm{d} t}=x^3 + aSin(\omega t)$$
$$|f(t,u)-f(t,v))|\leq L|u-v|$$
$$u^3 - aSin(\omega t)-v^3-aSin(\omega t)$$
$$|u^2+uv+v^2||u-v|\Rightarrow |u^2+uv+v^2|\leq L$$
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:
$$\frac{\mathrm{dx} }{\mathrm{d} t}=x^3 + aSin(\omega t)$$
$$|f(t,u)-f(t,v))|\leq L|u-v|$$
$$u^3 - aSin(\omega t)-v^3-aSin(\omega t)$$
$$|u^2+uv+v^2||u-v|\Rightarrow |u^2+uv+v^2|\leq L$$
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