Consider the following Duffing oscillator:

m (d²x/dt^˜2) - k₀x + µx³ = 0                             (1)

where m > 0, k0>0, µ>0

1. Find the fixed points.

2. If the units of x are meters and of t˜ are seconds find the units of m, k₀, and µ.

3. Transform the equation into a dimensionless form and normalize it.

4. Convert the second order ordinary differential equation (1) into a system of two first order ordinary differential equations.

5. Find the equation for the solution curves on the phase diagram.

6. Plot these curves on the phase diagram y = x^. vs. x or their normalized equivalent symbols (include the curve that passes via the origin, i.e. the homoclinic orbit).

7. Solve the normalized system numerically for different initial conditions, and plot the solution as a function of time as well as on the phase diagram. Compare the numerical results with the analytical ones.