a. Give an example of a one-dimensional system with a transcritical bifurcation. Write down the equation for the system explicitly. Plot the phase portrait before, at, and after the bifurcation point. Plot the bifurcation diagram.
b. What is the definition of a limit cycle in the context of dynamical systems?
c. Give an example of a two-dimensional system with a saddle-node of limit cycles bifurcation. Write down the equations for the system explicitly. What are the amplitude and the period of the limit cycle at the bifurcation point? Plot the phase portrait before, at, and after the bifurcation point. Plot the bifurcation diagram.
d. Explain the concept of a trapping region and formally state the Poincare Bendixson theorem.