Introduction to the Modeling and Analysis of Complex Systems

3.3. WHAT CAN WE LEARN? 33Another thing you can learn from phase space visualizations is the stability of thesystem’s states. If you see that trajectories are converging to a certain point or area inthe phase space, that means the system’s state is stable in that area. But if you seetrajectories are diverging from a certain point or area, that means the system’s state isunstable in that area. Knowing system stability is often extremely important to understand,design, and/or control systems in real-world applications. The following chapters will puta particular emphasis on this stability issue.Exercise 3.4 Where are the attractor(s) in the phase space of the bouncing ballexample created in Exercise 3.3? Assume that every time the ball bounces it losesa bit of its kinetic energy.Exercise 3.5of attraction.For each attractor obtained in Exercise 3.4 above, identify its basinExercise 3.6 For each of the phase spaces shown below, identify the following:• attractor(s)• basin of attraction for each attractor• stability of the system’s state at several locations in the phase spaceExercise 3.7 Consider a market where two equally good products, A and B, arecompeting with each other for market share. There is a customer review website