The solution you found in the indicated exercise depends on the parameters k and b. Describe the qualitative behavior of each solution in the following situations in terms of the motion of the "mass-spring with handle" model in this section (assume ):
(a) k large, b and small but positive.
(b) b large, k and small but positive.
If a nonlinear system depends on a parameter, then the equilibrium points can change as the parameter varies. In other words, as the parameter changes, a bifurcation can occur. Consider the one-parameter system family of systems
where a is the parameter.
(a) Show that the system has no equilibrium points if a < 0.
(b) Show that the system has two equilibrium points if a > 0.
(c) Show that the system has exactly one equilibrium point if a = 0.
(d) Find the linearization of the equilibrium point for a = 0 and compute the eigenvalues of this linear system.
Remark: The system changes from having no equilibrium points to having two equilibrium points as the parameter a is increased through a = 0. We say that the system has a bifurcation at a = 0, and that a = 0 is a bifurcation value of the parameter.
Continuing the study of the nonlinear system given in the indicated exercise,
(a) use the direction field to sketch the phase portrait for the system if a = -1,
(b) use the direction field and the linearization at the equilibrium point to sketch the phase portrait for a = 0, and
(c) use the direction field and the linearization at the equilibrium points to sketch the phase portrait for a = 1.
Remark: The transition from a system with no equilibrium points to a system with one saddle and one sink via a system with one equilibrium point with zero as an eigenvalue is typical of bifurcations that create equilibria.
In the following exercise, each system depends on the parameter a. In the given exercise,
(a) find all equilibrium points,
(b) determine all values of a at which a bifurcation occurs, and
(c) in a short paragraph complete with pictures, describe the phase portrait at, before, and after each bifurcation value.
In the following predator-prey population models, represents the prey, and represents the predators. (i)
(a) In which system does the prey reproduce more quickly when there are no predators (when ) and equal numbers of prey? (b) In which system are the predators more successful at catching prey? In other words, if the number of predators and prey are equal for the two systems, in which system do the predators have a greater effect on the rate of change of the prey? (c) Which system requires more prey for the predators to achieve a given growth rate (assuming identical numbers of predators in both cases)?
Consider a unit mass sliding on a frictionless table attached to a spring, with spring constant k = 16. Suppose the mass is lightly tapped by a hammer every T seconds.
Suppose that the first tap occurs at time t = 0 and before that time the mass is at rest. Describe what happens to the motion of the mass for the following choices of the tapping period T:
(a) T = 1
(b) T = 3/2
(c) T = 2
(d) T = 5/2
(e) T = 3