I have the second order autonomous system
$$x_1 '= ax_1 - bx_1x_2 \\ x_2 '= -cx_2 + dx_1x_2$$
How can I apply the Lipschitz condition to this second order system?
Can anyone explain step by step how to apply the condition in this case?
Regards
Keiver
Hints: You can write the right hand side of the system as:
$$f(x,y)= (ax-bxy,-cy+dxy)^T$$
Think about what the Lipschitz condition means for a function $f:\mathbb{R}^2 \rightarrow \mathbb{R}^2$. Remember that in the one variable case, it suffices to check that $\bigg{|}\cfrac{\partial f}{\partial x}\bigg{|}\leq L_0$ on some compact and convex set $D_0$, i.e. the derivative of $f$ with respect to $x$ is bounded on some subset of the real line. This follows from the mean value theorem (MVT), as you can check.
Now for two variables, what is the derivative of $f$ with respect to the vector $(x,y)^T$? This is the Jacobian matrix.
To show that $f$ is Lipschitz, it is sufficient to find a uniform bound given your favorite norm of the Jacobian matrix. Any two norms on a finite dimensional vector space are equivalent, so you're free to choose any norm.
Computint the Jacobian we get:
$$Df(x)=\begin{pmatrix}a-by&-bx \\ dy &-c+dx \end{pmatrix} $$
Now use this to find a uniform bound for $Df$. Another hint is to recall the definition of the operator norm of $A:E \rightarrow F$:
$$||A|| = \sup_{||x||_{E}=1}{||A(x)||_{F}}$$
Here, $E$ and $F$ are the the "source" and "target" vector spaces of the operator $A$, which in this case are both $\mathbb{R}^2$.
Pick any matrix (as an operator on $\mathbb{R}^2$) norm you want to find a suitable upper bound, e.g. the $\ell_1$ matrix norm. That is, $$||A||_{\ell_1}= \max_{1 \leq j \leq n} \sum_{i=1}^{m}|a_{ij}|$$
This is the maximum value of the sum of absolute values of the columns of $A$. Analogously, you can define the $\ell_{\infty}$ matrix norm as the maximum value of the sum of absolute values of the rows of $A$.
I hope this helps in at least getting started. You should be able to take it from here.
Note: The function may not be globally Lipschitz, but only locally Lipschitz. In other words, it is Lipschitz only on some open and connected set $D \subset \mathbb{R}^2$.