Analysis

Dealing with the autonomous case, gamma = 0

-Steady states

 There are three steady states solutions to this model.

    1. x = 0, y = 0, corresponding to the steel beam at rest.
    2. x = 1, y = 0, which corresponds to the beam settling near one of the magnets
    3. x = -1, y = 0, which corresponds to the beam settling near the other magnet

-Steady State Stability

The Jacobian matrix for this system is:

0

1

1-3x2

-d

 

When the three steady states are entered into the Jacobian, you get the following:

  1. At x = 0, y = 0
  2. 0

    1

    1

    -d

    The eigenvalues of the matrix are r1 = (-d + sqrt(d2 + 4))/2 and r2= (-d - sqrt(d2 + 4))/2. Since delta is always positive, r1 will always be positive and r2 will always be negative so the steady states is always a saddle.

  3. At x = 1, y = 0
  4. 0

    1

    -2

    -d

    The eigenvalues of the matrix are r1 = (-d + sqrt(d2 - 8))/2 and r2= (-d - sqrt(d2 - 8))/2. Since delta is always positive, the real parts of r1 and r2 will always be negative so the steady state will always be a sink.

  5. At x = -1, y = 0
  6. 0

    1

    -2

    -d

    The eigenvalues of this matrix are the same as the ones for solution 2 so this steady state is also a sink.

This Direction field shows the steady states graphically.

The steady states (1,0) and (-1,0) in this case, represent the magnets on either side of the beam. The initial velocity starts the beam in motion, which will be attracted by one of the magnets. (Figure 2 in maple worksheet) The beam will oscillate around one of the magnets until it comes to rest. As the velocity increases past 1 (Figure 3 in maple worksheet), it will overpower the magnetic pull of the first magnet and become attracted by the other magnet. In general, the greater the velocity, the beam will make more oscillations between the magnets until it finally comes to rest at one of the steady states (magnets). (Figure 4 in maple worksheet)

 

Non-autonomous case:

By fixing omega = 1, delta = .2 and varying gamma, the initial conditions approach different steady states. When gamma = .1, smaller initial velocities attract to one of the magnets and settles into a fixed oscillation. Larger velocities overcome the magnetic attraction and oscillate between magnets until the speed of the beam decreases enough for the magnetic attraction to take over. The beam approaches a steady oscillation around one of the magnets. (Figure 8 in maple worksheet) At gamma = .2, the same outcome occurs; however, a greater time period is needed. (Figure 9 in maple worksheet)

When the amplitude of the forcing term is .3, the behavior of the model becomes chaotic. The beam oscillates through both magnets and never approaches a steady oscillation.(See Xrk output) When gamma = .5, regardless of the velocity, the beam finds a steady position about both magnets(Figure 10 in maple worksheet).

With parameters, gamma = .1, delta = .25 and omega = 1, the solutions around the stable steady states makes a closed path at a period of 6.267. This value is consistent with the period of the forcing term, which is 2pi. (Figures 5 and 6 in maple worksheet) Changing gamma to.3 and delta to .22, led to a periodic solution with period of 6pi. (figure 7 in maple worksheet) This period is 3 times the period of the forcing term. These periodic trajectories are symmetric about the x1 = x2 line.

 


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