We have the following differential equation $F(t,c)$, where as $c=(x,y)$
\begin{cases} \dot{x}(t) = |x(t)| - y(t) \\ \dot{y}(t) = 3 - 2x(t) - y(t)^2 \end{cases}
How do I determine if $F(t,c)$ is Lipschitz continuous regarding $c$ within some closed interval $[-\delta,\delta]$ with $\delta>0$.
In fact, suppose $x_1,x_2,y_1,y_2\in(-\delta,\delta)$. Then, for $c_1=(x_1,y_1),c_2=(x_2,y_2)$, one has $$\begin{eqnarray} |F(t,c_1)-F(t,c_2)|&=&|(|x_1|-y_1,3-2x_1-y_1^2)-(|x_2|-y_2,3-2x_2-y_2^2)|\\ &=&|(|x_1|-|x_2|-(y_1-y_2),-2(x_1-x_2)-(y_1^2-y_2^2)|\\ &=&||x_1|-|x_2|-(y_1-y_2)|+|-2(x_1-x_2)-(y_1^2-y_2^2)| \\ &\le&||x_1|-|x_2||+|y_1-y_2|+2|x_1-x_2|+|y_1+y_2||y_1-y_2| \\ &\le&|x_1-x_2|+|y_1-y_2|+2|x_1-x_2|+2\delta|y_1-y_2| \\ &=&3|x_1-x_2|+(2\delta+1)|y_1-y_2|\\ &\le&(4+2\delta)|x_1-x_2|+|y_1-y_2|\\ &=&(4+2\delta)|c_1-c_2| \end{eqnarray}$$ and hece $F$ is Lipschitz.