Methods of Nonlinear Analysis 412-1


Winter 2002

Hermann Riecke


Homework Assignment 1

due Thursday, January 31, 2002


  1. Tricritical Point

    Consider the dynamical system
    tu = f(u,l)
    		(1)
    with f(-u,l) = -f(u,l). For u = 0 and l = 0 assume that u f = 0 and 3uuuf is small. For small l, identify the leading-order terms in u and l and obtain an equation for the fixed points of this system in the vicinity of u = 0. Sketch the various bifurcation diagrams that arise. For 3uuuf = 0 the system is said to be at a tricritical point. Why is this point of particular interest if one wants to discuss hysteretic (jump) transitions?

  2. Trajectories in a Simple Dynamical System

    Consider the dynamical system
    tx
    =
    -ax - x2,
    		(2)
    ty
    =
    -by
    		(3)
    with b positive.

    1. Determine the solution (x(t),y(t)) of (2,3) and the trajectory y = y(x) for a > 0.
    2. Determine the solution (x(t),y(t)) of (2,3) and the trajectory y = y(x) for a = 0.
    3. Compare the trajectories in the two cases. In particular, comment on the effect of the initial conditions on the trajectories as they approach the fixed point (0,0).
    4. The case a = 0 demonstrates the non-uniqueness of the center manifold. Comment on the distance between the different center manifolds and whether they can be distinguished in a power series expansion around the fixed point (0,0).

  3. Center Manifold Reduction for a Steady Bifurcation

    Consider the dynamical system
    tx
    =
    y - x3,
    		(4)
    ty
    =
    ax -x3-b  y
    1+x2
    		 .
    		(5)

    1. Perform a linear stability analysis of the fixed point (x,y) = (0,0). Depending on a and b what kind of instabilities can you identify? Determine the stable, unstable, and center linear eigenspaces in each of the cases.
    2. Identify the symmetries of the system (4,5). Based on this information together with the linear stability analysis, what types of bifurcations do you expect in this system?
    3. For b = 1 and a small, perform a center manifold reduction to obtain an equation for a center manifold and an equation for the slow evolution on that manifold. Keep only the leading-order terms in this expansion.
    4. For b = 1 and a small, obtain the equivalent result by performing a multiple-scale calculation. Identify clearly the right and left eigenvectors of the linear operator (matrix) obtained in the linear stability analysis in (a).
    5. Solve (4,5) numerically (using Matlab/pplane5 or dstool; see web site) in the regime for which your center manifold reduction is supposed to be valid. How well does the analytical approach do near the bifurcation point? (For a quantitative comparison with the center manifold you would have to write your own code (using ode45, for instance, in matlab) to be able to print the trajectories.)

  4. Center Manifold Equations for a Hopf Bifurcation

    Consider the dynamical system
    tx
    =
    -x-y+z2,
    		(6)
    ty
    =
    2x+(1+e) y-z2,
    		(7)
    tz
    =
    x + 2y -z.
    		(8)
    It exhibits a Hopf bifurcation at e = 0 leading to a limit cycle (periodic orbit).

    1. Determine a center manifold that describes the dynamics of this system for small amplitudes x, y, and z and for small values of the bifurcation parameter e (which is not necessarily 0).
    2. Determine the evolution equations describing the dynamics on the center manifold. Express them also in terms of the complex amplitude A = u+iv and compare the resulting equation with the normal form for a Hopf bifurcation
      tA = aA + g|A|2 A.
      		(9)
      Is the bifurcation forward or backward (i.e. super- or subcritical)?

    3. Solve the equations numerically (e.g. using dstool or write your own matlab etc. code). Is the result qualitatively consistent with your analytic result? Explore what happens to the limit cycle (periodic orbit) when e is increased.

File translated from TEX by TTH, version 2.60.
On 20 Jan 2002, 18:42.