Sketching a Phase Portrait for a System with Fixed Points: Tips and Techniques

[MathematicalPhysics]

Hello, I'm considering the system:

dx/dt=y-x

dy/dt=x-y

I've found the fixed points and shown that this is a first integral of the system:

f(x,y)=x+y

How can I (using this information only) sketch a phase portrait? I know the graph of fixed points looks like the x=y graph, but I am not sure how this relates to a phase portrait.

Any help? Thanks.

 

[arildno]

From my slight acquaintance with phase portraits( among lots of other stuff..), I'd say the
equations f(x,y)=x+y=const. for different constants defines your phase portrait.
In particular, it would be important to graph whether solutions converge towards some fixed point at x=y (i.e stable fixed points), or if the fixed points are unstable/saddle-like.

 

[M98Ranger]

To sketch a phase portrait for this system, we can use the fixed points and the first integral to guide our sketch. First, let's recall that fixed points are points where the system remains constant, meaning that the derivatives of both x and y are equal to 0. In this case, the fixed points are where x=y. So, we can start by drawing a line on our phase plane where x=y, which will intersect the x and y axes at the origin (0,0). This represents the fixed point of our system.

Next, we can use the first integral f(x,y)=x+y to determine the behavior of the system around the fixed point. Since the first integral is constant along the trajectories of the system, we can draw level curves of f(x,y) on our phase plane. These level curves will be parallel to the line x=y and will intersect it at different points. This will give us an idea of how the system behaves around the fixed point.

For example, if we choose a level curve with a positive value for f(x,y), it will intersect the line x=y in the first quadrant of the phase plane. This means that the system will move away from the fixed point in a positive direction, either towards the top right or bottom left quadrant. On the other hand, if we choose a level curve with a negative value for f(x,y), it will intersect the line x=y in the third quadrant, indicating that the system will move away from the fixed point in a negative direction, towards the bottom right or top left quadrant.

We can continue to draw multiple level curves to get a better understanding of the behavior of the system around the fixed point. The closer the level curves are to the fixed point, the slower the system is moving away from it. Similarly, the farther away the level curves are, the faster the system is moving away from the fixed point.

Overall, the phase portrait will consist of a series of level curves intersecting the line x=y at different points, giving us a visual representation of the behavior of the system around the fixed point. By using this technique, we can also determine the stability of the fixed point. If the level curves are converging towards the fixed point, it is a stable fixed point. If they are diverging, it is an unstable fixed point.

In summary, to sketch a phase portrait for a system with fixed points, we can use the fixed points and the first integral to draw level curves