Mathematics 345Homework Assignment 5Due Thursday 15 March 2018

Bensdixson-Dulac criterion, Poincare-Bendixson Theorem, and 2D Parameter Space

1. Consider the system in the plane R2, given by

x = y, y = −x3 + x4 − δy,

where δ ≥ 0 is a constant.

(a) Find all fixed points.

(b) Linearize at each fixed point and classify each fixed point (as hyperbolic attractor,repeller, or saddle point; or non-hyperbolic fixed point with a simple or doublezero eigenvalue or with pure imaginary eigenvalues). Consider both i) δ = 0 andii) δ > 0.

(c) Sketch the nullclines and direction fields by hand, for both i) δ = 0 and ii) δ > 0.

(d) Use Bendixson’s criterion (identical to Dulac’s criterion with g(x, y) = 1) to provethat there can be no closed orbits in the phase plane when δ > 0.

(e) Find a continuously differentiable function V (x, y) such that: i) if δ = 0 then Vis nonincreasing along all trajectories; and ii) if δ > 0 then carefully show thisV decreases along all trajectories except at fixed points. (Hint: When δ = 0 thesystem is conservative.) Use the function V to: i) determine the stability of anynon-hyperbolic fixed points when δ = 0 and when δ > 0 ; and ii) give an alternateproof that there can be no closed orbits when δ > 0.

(f) Sketch the global phase portrait of the system by hand, for: i) δ = 0; andii) 0 < δ � 1, using the results of parts (a)–(e) as justification. Indicate therelationship between the level sets of V and the trajectories of the system. (Itmay be helpful to use XPP with the file hw5-1.ode.) For i) δ = 0, indicate theglobal stable manifold of any saddle point, and for ii) 0 < δ � 1, indicate theglobal stable manifold of any saddle point, and also the global basin of attractionof any attractor.

2. Consider a simplified model of a chemical reaction, in dimensionless form, given by

x = a− x+ x2y, y = b− x2y,

where x ≥ 0, y ≥ 0 are dimensionless concentrations of certain chemicals, and a > 0,b > 0 are parameters (concentrations of other chemicals that are assumed to remainconstant).

(a) Find the unique fixed point, and classify it using linearization (hyperbolic attrac-tor, repeller or saddle; non-hyperbolic with a simple or double zero eigenvalue orwith pure imaginary eigenvalues).

(b) Fix a > 0 and consider b > 0 as a control parameter. Show that, provided a is fixedat a small enough value (determine exactly how small), then Hopf bifurcationsoccur at two distinct parameter values b = bc1 and b = bc2, with bc1 < bc2. To dothis, plot the curve in the ab-plane that represents the parameter values wherethe linearization has pure imaginary eigenvalues. Show that this curve can beexpressed as

a =1

2x∗(

1− (x∗)2), b =

1

2x∗(

1 + (x∗)2),

where x∗ > 0 is the x-coordinate of the fixed point, then plot the curve usingthese parametric equations.

(c) Sketch the nullclines and direction field by hand. Then construct a trapping regionR that contains the fixed point, carefully verifying that R is trapping. (Hint: Forthe left-hand boundary of R, consider a vertical line segment in the first quadrant,between the nullclines.)

(d) Use the Poincare-Bendixson Theorem and explain whether a limit cycle exists for0 < b < bc1 or bc1 < b < bc2 or bc2 < b < ∞, when a is fixed at a small enoughvalue.

(e) Get the file hw5-2.ode from the course web page, and use XPP to plot phaseportraits for a = 0.1365 and b = 1.00000, 0.67916, 0.47916, 0.16350 and 0.14350.Hand in the plots, and discuss the correspondence between the plots and youranalysis above. Are there closed orbits in any of these cases? Explain.

(f) Use AUTO to plot a bifurcation diagram of y vs. b, with a fixed at 0.1365. To dothis, begin in the main XPPAUT window (see also Homework Assignment 2):

• Select File, Auto, then in the AUTO window select Axes, hI-lo, and lookin the AutoPlot window. Observe that the variable y will be represented bythe vertical axis and b is the main parameter, represented by the horizontalaxis, with 0 ≤ y ≤ 3 and 0 ≤ b ≤ 1. These values were set in the hw5-2.ode

file and could be changed, if necessary. You don’t need to change anything,though, so click Ok.

• From the main XPP window (turquoise buttons Param and ICs) set the pa-rameter values to b = 1, a = 0.1365, and set the initial conditions to theappropriate fixed point (you could compute the coordinates of the fixed pointby hand, or use Sing pts in the main XPP window, as described in Home-work Assignment 3, to compute the coordinates numerically). AUTO usesNewton’s method to find fixed points, so it is important to give it accurate val-ues for the fixed point at the starting parameter values. AUTO then changesthe parameter values and recomputes the fixed point. It detects bifurcationsby monitoring the eigenvalues of the linearization at the fixed point as theparameter value is changed.

• Returning to the AUTO window, select Numerics and look in the AutoNumwindow. Note especially that Ds has been set (via the hw5-2.ode file) to anegative value. This instructs AUTO to decrease the main parameter fromb = 1 to b = 0. You don’t need to change anything, click Ok.

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• In the AUTO window, select Run, Steady state to plot the branch offixed points (both the x- and y-coordinates are calculated, but only the y-coordinate is plotted).

• AUTO can be used to find the periodic orbits, starting from the Hopf bifur-cation point. Select Grab, then use the < Tab > key to move the cross-hairson the bifurcation diagram to the bifurcation point with largest b-value. Thisis bc2. Look at the information line (below the bifurcation diagram in theAUTO window) as you do this. You should see the label HB for a Hopf bi-furcation. When the correct point is in the information window, press the< Enter > key to accept the point. Then select Run, Periodic and youshould get a plot showing the minimum and maximum y-values of the peri-odic orbit for different values of the parameter b (if something goes wrong,select ABORT).

Save the bifurcation diagram, print it and hand it in. Write down (from thexterm window or the information line in the AUTO window) the values of theparameter b, and of the fixed point coordinates x∗ and y∗ computed by AUTO atthe Hopf bifurcation points. Discuss the correspondence between AUTO’s results(the diagram and specific numerical values) and your previous analytical and XPPresults.

3. Show that {x = −x+ 2y3 − 2y4,y = −x− y + xy.

has no closed orbits by constructing a Lyapunov function of the form V = xm + ayn

with suitable values of a,m, n. The existence of a Lyapunov function also impliesthat the single fixed point (0, 0) is asymptotically stable, verify this fact using linearstability analysis.

4. Show that the following version of a predator-pray model{x = rx (1− x)− xy

1+x,

y = xy1+x− y, (r > 0, x, y > 0)

has no closed orbits by using Dulac’s criterion with function g(x, y) = 1+xxyα−1 for a

suitable choice of α. What are the choices of α?

5. Show that the following system has a periodic solution using Poincare-Bendixson The-orem. {

x = x− y − x3,y = x+ y − y3.

(Hint: find a square area −b ≤ x ≤ b, −b ≤ y ≤ b (b > 0) in the phase space that isa trapping region). Determine the smallest possible number bm > 0 such that for allb ≥ bm this square region is trapping but not if b < bm.

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