Question

For each of the following systems, find the fixed points, classify them, sketch the neighboring trajectories, and try to fill in the rest of the phase portrait.

Answer 1

In system (a), the fixed points are (2,2), (-2,-2), and (0,0). In system (b), the fixed points can be found and classified using the Jacobian matrix. The sketch of the neighboring trajectories and phase portraits can be made based on the stability of the fixed points.

Step-by-step explanation:

System (a):

Given the system ˙x=x−y and ˙y=x^2−4.

To find the fixed points, we set ˙x = 0 and ˙y = 0.

For ˙x = x−y = 0, we get x = y.

For ˙y = x^2−4 = 0, we get x = ±2.

The fixed points are (2,2), (-2,-2), and (0,0).

To classify the fixed points, we compute the Jacobian matrix J:

J = [∂˙x/∂x, ∂˙x/∂y; ∂˙y/∂x, ∂˙y/∂y]

Using J, we compute the eigenvalues of the matrix at each fixed point.

The sketch of the neighboring trajectories and the phase portrait can be made based on the eigenvalues and the stability of each fixed point.

System (b):

Similarly, for the system ˙x = 1+y−e^−x and ˙y = x^3−y, we find the fixed points by setting ˙x = 0 and ˙y = 0.

Then we classify the fixed points using the Jacobian matrix J and find the eigenvalues.

Lastly, we can sketch the neighboring trajectories and complete the phase portrait based on the stability of each fixed point.

Complete Question

For each of the following systems, find the fixed points, classify them, sketch the neighboring trajectories, and try to fill in the rest of the phase portrait.

(a)˙x=x−y, ˙y=x2 −4

(b)˙x= 1 +y−e−x, ˙y=x3 −y