Convert the second order ordinary differential equation 1

Question

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.

Answer 1

For the problem based on Duffing oscillator in the assignment above, answer the following question on the basis of the information provided. 1) Determine the fixed points. 2) If the units of x are meters and of t˜ are seconds determine the units of m, k0, and µ. 3) Convert the equation to a dimensionless form and normalize it. 4) Transform the second order ordinary differential equation to a system of two first order ordinary differential equations. 5) Determine the equation for solution curves on the phase diagram. 6) Plot such curves on the phase diagram y = x. vs. x or their normalized equivalent symbols. 7) Resolve the normalized system numerically for dissimilar initial conditions and plot the solution as a function of time and also on the phase diagram. Compare the numerical outcomes having the analytical ones.