ESAM 438

Problem Set 6

For Discussion Section November 8

1. An Interesting Dynamical System¹

   Consider the dynamical system
   

   d x dt = y, (1) d y dt = ax + by + a x3 - x2 y. (2)

   In this problem you will study this system analytically and numerically. For the numerical simulations you can use pplane5² or write your own code.

   1. Determine analytically the stability of the fixed point (x,y) = (0,0) as a function of a and b. Based on this information, what type of behaviors may be expected as a and/or b is varied?

   2. Consider the case a = -1.
      1. Solve (1,2) numerically. What kind of nonlinear behavior do you find if you vary a across the instability identified above for b = -1?
         For a = 0 and b = -1 identify the center manifold of the attractor numerically by starting from a number of different initial conditions. Increase now a to 0.05, 0.1, and 1. What happens to the center manifold?

      2. For small a perform analytically a center-manifold reduction of (1,2), i.e. determine a center manifold that describes the dynamics of this system for small amplitudes x, y, and for small values of the bifurcation parameter a. Go to cubic order. Obtain the evolution equation for the dynamics of the system on that manifold. Compare the analytically obtained center manifold and the fixed points on it quantitatively with your numerical results.

   3. Consider now the case a = +1.
      1. Choose a = -1 and increase b across the instability identified above. What happens in the numerical simulation?
         Measure the amplitude A and the frequency w of the oscillations as a function of b, increasing b up to b = 0.2. Is there a range in b over which the dependence of A and w on b agrees with your expectation for such a bifurcation?
         What happens for b > 0.2?

   4. Go back to the case a = -1.
      1. Study numericaly how the phase portrait changes for a = 0.1 when b is varied from b = -1 to b = 0.2. What happens to the center manifold?

Footnotes:

¹This is one of the interesting situations that can arise at a codimension-2 bifurcation, i.e. when two modes go unstable at the same parameter value (cf. Guckenheimer & Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Ch. 7.3)

²You may want to set the solver to ODE23 and set the calculation window to 10 times the display window.