Math 4325: Homework Notes

Assignment 4: Bifurcations in Scaler Systems

• For problems 3.1.3 and 3.4.4, "vector fields" is the same as the phase line. Also compute the Normal Form equations apppropriate for the particular bifurcation.
• For 3.3.2:
  a) Determine P and D in terms of E by solving the approximate steady-state equations (Set dP/dt =0 and dD/dt=0). Substitute the results into the equation for E to get a single first order ODE with just E (and lambda). After you have a single equation for E, rescaled E with the following change of variables: E = F/|lambda|^(1/2). In other words, make the substitution for E and simplify by canceling lambda wherever possible. The resulting equation for the evolution of F is easier to analyze as a function of lambda.
  (b) Solve for the equilibrium for F. Should be F=0 and F^2 = lambda. Sketching the two looks with respect to lambda suggest a pitchfork bifurcation.
  (c) Computer the linear stability for F=0. Taking the derivative requires the product rule but once F=0 is substituted in it simplifies. This is enough to infer a supercritical pitchfork.
• For: 3.7.4 (no part (e)):
  (b) First let tau = r t so that dN/dt = dN/dtau (dtau/dt) = r dN/dtau. Then let X = N/K. Rearrange terms and set h = H / (kr) and a = H/k.
  (c) The condition is found by deciding when two of the roots are real vs. complex. If you did the normal form analysis (you don't have to), it would determine a saddle-node bifurcation, i.e., 2 solutions or 0 solutions
  (d) Computer linear stability of x = 0. Neutral stability is at h = a. Now derive the Normal Form equations. There are two different results for a = 1 and a not equal to 1. (e) You don't have to do part (e).

Assignment 6: Linear 2D systems

• This is not a short assignment. Give yourself plenty of time!
• Feel free to use pplane to double check your work and make accurate sketches. However, you should make your sketches by hand using the information that you compute. pplane is just a double check.
• 5.2.2 a not b: Find the E-value and E-vector and write the solution in complex form (like we did in class). You do not have to do part (b). If you want a real solution you simply take the complex form and expand all the real and imaginary parts.
• 5.2.4: You can calculate the E-vector if you like but it is not necessary. The result is an unstable focus going clockwise. For a more accurate plot note that if x2=0, then dx1/dt = 5x1 and dx2/dt = -x1. Thus, if x1 > 0 the trajectory must be going to the right and down. Thus, instead of a circular spiral it is elliptical. Alternatively, or in addition to, you could plot the nullclines where the flow must be horz or vert.
• 5.2.5: There is only one eigenvector and is an improper node like Fig. 5.2.6. Use it and the nullclines to create a sketch.
• 5.2.7: Same advice as 5.2.4 to make the centers more accurate... if you like... not really necessary since we have the pplane plot.

Assignment 7: Phase Plane

Problem (1)
Using MATLAB and the routine pplane analyze the equation below for the case when C = 1 (these equations are preprogrammed in the "Gallery" menu).

x' = ( 1 - x - y) x
y' = ( A - B x - C y) y

• Analyse the phase plane when (i) A < 1, B < A, (ii) A < 1, B > A, (iii) A > 1, B < A, (iv) A > 1, B > A.
• Plot the nullclines, the vector field, and locate equilibrium points. Examine the linear behavior of each fixpoint (this can be done numerically using pplane) and label it's type (either on the plot or separately).
• For each case what is predicted as for the long term population of species x and y. Why?
• For each case/plot, make sure you understand why the trajectories and solutions are behaving as they do. You should understand how the nullclines and the types of fix points determine the flow that you observe. In other words, don't just make plots to turn them in. You must understand why the plots look the way they do.

• Plot trajectories for initial conditions close to the origin and away from the origin. Then plot the solutions as a function of time corresponding to those trajectories.
• For each case, make sure you understand how the trajectory in the phase plane relates to the solution as a function of time. When and why is the solution harmonic? When and why is the solution pulsating?

• Phase Plane
• Harmonic "Both"
• Harmonic "Composite"
• Pulsating "Both"
• Pulsating "Composite"

• 6.5.19 (b): Let T = a t so that d/dt = (d/dT)(dT/dt) = a d/dT. Then let x = (d/c) R and y = (b/a) F.
  6.5.19 (d): Don't worry about part (d)

Assignment 9: Bifurcations in Planer Systems

• 8.1.6
  b) One way to find the bifurcation is of course to use linear stability of the EP. Alternatively, we know that the nullclines have to cross to have EP. The point at which they just become tangent is when the bifurcation must occur. Thus, you can find the slope of one nullcline, the slope of the second nullcline, and compute when they will be the same, ie, the nullclines will have the same slope.
  c) You can use PPLANE to do this computationally.
• 8.1.10
  a) Each equation is a generalization of the logistic equation with a net growth rate and competition terms. Apply your understanding of the logistic equation to interpret the terms in this model.
  b) Let x = S/Ks, y = E/Ke and T = rs t such that d/dt = rs d/dT.
  The result should be
  x' = x (1 - x/y), y' = gamma y (1- y) - beta/x.
  Find gamma and beta.
  c) Plotting the nullclines for beta large and small is enough to determine the type of bifurcation.
  d) Use PPLANE. No need for heavy analysis.
• Numerically simulate (pplane) the following problem:
  x'= y-x-x^2,
  y'=rx-y-2y^2.
  Examine the nullclines and equilibrium points for r<1, r=1 and r>1. What type of bifurcation occurs? For the case r>1, add some trajectories, print and turn in a plot.

Assignment 10: Hopf Bifurcation

• (a) Show that there are three E.P. at (0,0), (1,0) and (a, a(1-a))
• (b) Analyze the linear stability of the three E.P. as a function of a. In other words, indicate how stability changes as a changes.
  • (b.1) It turns out that (0,0) is singular in that one eigenvalue is always 0. Numerical analysis (PPlane) shows that it has stable and unstable directions but doesn't behave like a linear saddle. It's "odd", i.e., singular.
  • (b.2) Analyzing the stability of (a,a(1-a)) verify that there are bifurcations at a = 0, 1/2 and 1.
• (c) Sketch the bifurcation diagram x vs a of the three steady states. This alone is enough to see that at a = 0 and a = 1 there are transcritical bifurcations.
• (d) It turns out that a ~ 0.33 is also an important value. Use PPlane to analyze the system at a = -0.5, 0.2, 0.45, 0.7 and 1.2. Print the PPlane in each case.
  • Comparing a = 0.5 to 0.2 and then 0.7 to 1.2, confirms that there is an exchange of stability between two E.P. and, hence, a transcritical bifuration.
  • Verify that there is limit cycle for a = 0.45 that is gone when a = 0.7.
• 8.2.8 solutions
• Additional figures for 8.2.8
  Fig for a = 0.25
  Fig for a = 0.33
  Fig for a = 0.45
  Fig for a = 0.75

• 8.2.9