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It is not hard to concoct such an example in sufficiently high degree. For an example of degree $5$, take $$ \begin{align} x' &= x\,(1-x^2-y^2)(x^2+y^2-3) - y\,(2-x^2-y^2)\\ y' &= y\,(1-x^2-y^2)(x^2+y^2-3) + x\,(2-x^2-y^2). \end{align} $$ The circles $x^2+y^2=1$ and $x^2+y^2=3$ are integral curves of this vector field with opposite orientations. The only zero of the vector field is at the origin $(x,y) = (0,0)$. In the annulus $1 < x^2+y^2 < 3$, the flow always increases $x^2+y^2$ and it moves in the counterclockwise direction when $1 < x^2+y^2 < 2$ but in the clockwise direction when $2 < x^2+y^2 < 3$.

I don't know whether there is a cubic system that has such a property, but it wouldn't surprise me.

It is not hard to concoct such an example in sufficiently high degree. For an example of degree $5$, take $$ \begin{align} x' &= x\,(1-x^2-y^2)(x^2+y^2-3) - y\,(2-x^2-y^2)\\ y' &= y\,(1-x^2-y^2)(x^2+y^2-3) + x\,(2-x^2-y^2). \end{align} $$ The circles $x^2+y^2=1$ and $x^2+y^2=3$ are integral curves of this vector field with opposite orientations. The only zero of the vector field is at the origin $(x,y) = (0,0)$. In the annulus $1 < x^2+y^2 < 3$, the flow always increases $x^2+y^2$ and it moves in the counterclockwise direction when $1 < x^2+y^2 < 2$ but in the clockwise direction when $2 < x^2+y^2 < 3$.

I don't know whether there is a system of degree 3 or 4 that has such a property, but it wouldn't surprise me.

It is not hard to concoct such an example in sufficiently high degree. For an example of degree $5$, take $$ \begin{align} x' &= x\,(1-x^2-y^2)(x^2+y^2-3) - y\,(2-x^2-y^2)\\ y' &= y\,(1-x^2-y^2)(x^2+y^2-3) + x\,(2-x^2-y^2). \end{align} $$ The circles $x^2+y^2=1$ and $x^2+y^2=3$ are integral curves of this vector field with opposite orientations. The only zero of the vector field is at the origin $(x,y) = (0,0)$. In the annulus $1 < x^2+y^2 < 3$, the flow always increases $x^2+y^2$ and it moves in the counterclockwise direction when $1 < x^2+y^2 < 2$ but in the clockwise direction when $2 < x^2+y^2 < 3$.

I don't know whether there is a cubic or quartic system that has such a property, but it wouldn't surprise me.

It is not hard to concoct such an example in sufficiently high degree. For an example of degree $5$, take $$ \begin{align} x' &= x\,(1-x^2-y^2)(x^2+y^2-3) - y\,(2-x^2-y^2)\\ y' &= y\,(1-x^2-y^2)(x^2+y^2-3) + x\,(2-x^2-y^2). \end{align} $$ The circles $x^2+y^2=1$ and $x^2+y^2=3$ are integral curves of this vector field with opposite orientations. The only zero of the vector field is at the origin $(x,y) = (0,0)$. In the annulus $1 < x^2+y^2 < 3$, the flow always increases $x^2+y^2$ and it moves in the counterclockwise direction when $1 < x^2+y^2 < 2$ but in the clockwise direction when $2 < x^2+y^2 < 3$.

I don't know whether there is a cubic system that has such a property, but it wouldn't surprise me.

It is not hard to concoct such an example. For an example of degree $5$, take $$ \begin{align} x' &= x\,(1-x^2-y^2)(x^2+y^2-3) - y\,(2-x^2-y^2)\\ y' &= y\,(1-x^2-y^2)(x^2+y^2-3) + x\,(2-x^2-y^2). \end{align} $$ The circles $x^2+y^2=1$ and $x^2+y^2=3$ are integral curves of this vector field with opposite orientations. The only zero of the vector field is at the origin $(x,y) = (0,0)$. In the annulus $1 < x^2+y^2 < 3$, the flow always increases $x^2+y^2$ and it moves in the counterclockwise direction when $1 < x^2+y^2 < 2$ but in the clockwise direction when $2 < x^2+y^2 < 3$.

I don't know whether there is a cubic or quartic system that has such a property, but it wouldn't surprise me.

It is not hard to concoct such an example in sufficiently high degree. For an example of degree $5$, take $$ \begin{align} x' &= x\,(1-x^2-y^2)(x^2+y^2-3) - y\,(2-x^2-y^2)\\ y' &= y\,(1-x^2-y^2)(x^2+y^2-3) + x\,(2-x^2-y^2). \end{align} $$ The circles $x^2+y^2=1$ and $x^2+y^2=3$ are integral curves of this vector field with opposite orientations. The only zero of the vector field is at the origin $(x,y) = (0,0)$. In the annulus $1 < x^2+y^2 < 3$, the flow always increases $x^2+y^2$ and it moves in the counterclockwise direction when $1 < x^2+y^2 < 2$ but in the clockwise direction when $2 < x^2+y^2 < 3$.

I don't know whether there is a cubic or quartic system that has such a property, but it wouldn't surprise me.