P$1.(10$ points$)$

In the last a few lectures, we talked about the oscillatory networks. Often, when dealing with complex systems, the emergence of interesting behavior like hysteresis or oscillations is dependent on the parameters. Usually, we are interested in the changing point of system behavior, such as bifurcation points and Hopf points.

Task

For the following system, try to use Matlab to numerically solve it with different values of $k.$ Then plot the bifurcation diagram $($for example, k vs X$).[$Hint: I talked about how to use for loop to get a coarse grain bifurcation diagram$].$

$x^{\text{'}}=kx+5y-x(x^2+y^2)$

$y^{\text{'}}=5x+ky-y(x^2+y^2)$

Deliverables

Your matlab code and figures, along with some text describing your observations.