# Two ODE proofs

1. Apr 22, 2004

### kitsch_22

Hello and thanks in advance for anyone who can help at all. I have two problems that have stumped me.. I'm in an advanced ODE class. Here they are:

1) Consider the first order ODE f_a(x) where a is a parameter; let f_a(x0) = 0
for some solution x0 and also let f'_a(x0) != 0. Prove that the ODE
f_a+e(x) has an equlibrium point x0(e) where e -> x0(e) is a smooth function satisfying x0(0) = x0 for e sufficiently small.

2) Consider the system X' = F(X) where X is in R_n. Suppose F has an equilbrium point at X0. Show that there exists a change of coordinates that moves X0 to the origin and converts the system to X' = AX + G(X) where A is an nxn matrix which is the canonical form of DF_X0 and where G(X) satifies

lim (|G(X)| / |X|) = 0.
|X|->0

I am so lost on these...can anyone help pleeeeeeeeeease?

Michelle

2. Aug 25, 2009

### Reb

Ok let's see...

For 1), you are given the ODE with the parameter shifted by a small $$\epsilon$$, and you are required to show that this new ODE with $$\epsilon$$ will have an equilibrium, which is "close" to the original one. Since $$f_a$$ has nonzero gradient, continuity implies that $$f'_{a+\epsilon}$$ will also be nonzero for small $$\epsilon$$. Invoking the existence theorem, there is a smooth equilibrium that depends on $$\epsilon$$. Call this $$x_0(\epsilon,\cdot)$$. By the dependence on parameters theorem, $$x_0(\epsilon,\cdot)\rightarrow x_0(\cdot)$$ as $$\epsilon\rightarrow0$$.

For 2), note that $$DF_{x_0}$$ being nonzero, implies that $$F$$ is a local diffeomorphism in a neighbouhood of $$x_0$$. This grants us the validity of a local change of variables to $$y=F(x)$$. Under $$F$$, the equilibrium is mapped to the origin. For the last part, note that $$F^{-1}$$ will have a similar Taylor expansion as $$F$$, and that a Taylor expansion for $$F$$ gives $$F(x)=F(x_0)+DF_{x_0}(x)+G(x)=0+A\cdot x+G(x)$$, where $$G$$ will contain higher order terms than $$|x|$$, and so $$G(x)=0(|x|)$$.