If $X$ is a linear focus in $\mathbb{R^2}$, then exists $\delta>0$ that if $Y$ is $C^1$ in $\mathbb{R^2}$ with $\sup_{x\in \mathbb{R^2}}\|DY(x)\|\leq \delta$, then $X+Y$ has no periodic orbits.
I am trying to apply Bendixon's Critery. So, if $\lambda=a+ib$ is an eigenvalue and X is a focus, then $a\neq 0$ and $div X = \lambda + \overline{\lambda}=2a$. So, if $\delta=2a/3$ and $Y$ is a field that $\sup_{x\in \mathbb{R^2}}\|DY(x)\|\leq \frac{2a}{3}$, what can i say about the divergence of $Y=(Y_1,Y_2)$? I need to concluded that $div X+Y \neq 0$ for all $x\in \mathbb{R^2}$.
edit: Actually, i just need that the trDY can not be $-2a$.