Question

(a) Consider the one-parameter family of nonlinear ordinary differential equations

dt/dx ==(x+1)(x−a−1)(x+a−1), where a is a real parameter. i. Find all equilibrium points. ii. Sketch the bifurcation diagram, and indicate the behaviour of nonequilibrium solutions using appropriate arrows. iii. Find all bifurcation points and classify them.

Answer 1

To find the equilibrium points, set the derivative of the function equal to zero and solve for x. Sketch the bifurcation diagram by plotting the equilibrium points on the x-axis and indicating the behavior of nonequilibrium solutions. Bifurcation points occur when the number of equilibrium points changes.

Step-by-step explanation:

The equilibrium points can be found by setting the derivative of the function dt/dx = (x+1)(x-a-1)(x+a-1) equal to zero and solving for x. This will give us the critical points where the derivative is undefined or where the derivative changes sign.

To sketch the bifurcation diagram, we need to plot the equilibrium points on the x-axis and indicate the behavior of the nonequilibrium solutions using appropriate arrows. We can label the equilibrium points as stable or unstable based on the behavior of the solution around them.

The bifurcation points occur when the number of equilibrium points changes. To find these points, we can look at the values of the parameter a where the number of equilibrium points changes, and classify them based on the behavior of the equilibrium points.