Hi I would like to know the best way in which to find the direction of the trajectories in a phase portrait.
Here is the system I am working with:
$$\dot{x_1}=x_2-x_2^3~;~~~~~\dot{x_2}=-x_1-3x_2^2+x_1^2x_2+x_2$$
In this system there is, for example, a stable focus at the point (2,1) and I would like to know if this spirals in a clockwise or anti-clockwise direction.
I have been told of two ways to do this: Find when $\dot{x_1}>0,\dot{x_2}>0, \dot{x_1}<0 \text{ and }\dot{x_2}<0 $. or to just sub in points and see in which direction the vector points at this point. For example at the point (a,b) I would look at the vector $(\dot{x_1}(a,b),\dot{x_2}(a,b))$
I don't understand how you the first method can help get the directions of the arrows.
In the second method you are only getting the gradient at the point, but could the direction be going either way? For example at the point $(3,2)$ the gradient vector is $(-6,5)$ but how do I know that it is not heading from top to bottom or bottom to top? is it always the direction of the vector?
Cheers.