Mathematics 345
Homework Assignment 5
Due Tuesday 15 March 2016
Bensdixson-Dulac criterion, Poincaré-Bendixson Theorem, and 2D Parameter Space
1. Consider the system in the plane R2 , given by
ẋ = y,
ẏ = −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 double
zero eigenvalue or with pure imaginary eigenvalues). Consider both i) δ = 0 and
ii) 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 prove
that 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 V
is nonincreasing along all trajectories; and ii) if δ > 0 then carefully show this
V decreases along all trajectories except at fixed points. (Hint: When δ = 0 the
system is conservative.) Use the function V to: i) determine the stability of any
non-hyperbolic fixed points when δ = 0 and when δ > 0 ; and ii) give an alternate
proof 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; and
ii) 0 < δ 1, using the results of parts (a)–(e) as justification. Indicate the
relationship between the level sets of V and the trajectories of the system. (It
may be helpful to use XPP with the file hw5-1.ode.) For i) δ = 0, indicate the
global stable manifold of any saddle point, and for ii) 0 < δ 1, indicate the
global stable manifold of any saddle point, and also the global basin of attraction
of any attractor.
2. Consider a simplified model of a chemical reaction, in dimensionless form, given by
ẋ = a − x + x2 y,
ẏ = b − x2 y,
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 remain
constant).
(a) Find the unique fixed point, and classify it using linearization (hyperbolic attractor, repeller or saddle; non-hyperbolic with a simple or double zero eigenvalue or
with pure imaginary eigenvalues).
(b) Fix a > 0 and consider b > 0 as a control parameter. Show that, provided a is fixed
at a small enough value (determine exactly how small), then Hopf bifurcations
occur at two distinct parameter values b = bc1 and b = bc2 , with bc1 < bc2 . To do
this, plot the curve in the ab-plane that represents the parameter values where
the linearization has pure imaginary eigenvalues. Show that this curve can be
expressed as
1 ∗
1 x 1 − (x∗ )2 ,
b = x∗ 1 + (x∗ )2 ,
2
2
∗
where x > 0 is the x-coordinate of the fixed point, then plot the curve using
these parametric equations.
a=
(c) Sketch the nullclines and direction field by hand. Then construct a trapping region
R that contains the fixed point, carefully verifying that R is trapping. (Hint: For
the left-hand boundary of R, consider a vertical line segment in the first quadrant,
between the nullclines.)
(d) Use the Poincaré-Bendixson Theorem and explain whether a limit cycle exists for
0 < b < bc1 or bc1 < b < bc2 or bc2 < b < ∞, when a is fixed at a small enough
value.
(e) Get the file hw5-2.ode from the course web page, and use XPP to plot phase
portraits 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 your
analysis 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 do
this, begin in the main XPPAUT window (see also Homework Assignment 2):
• Select File, Auto, then in the AUTO window select Axes, hI-lo, and look
in the AutoPlot window. Observe that the variable y will be represented by
the vertical axis and b is the main parameter, represented by the horizontal
axis, 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 parameter values to b = 1, a = 0.1365, and set the initial conditions to the
appropriate fixed point (you could compute the coordinates of the fixed point
by hand, or use Sing pts in the main XPP window, as described in Homework Assignment 3, to compute the coordinates numerically). AUTO uses
Newton’s method to find fixed points, so it is important to give it accurate values for the fixed point at the starting parameter values. AUTO then changes
the parameter values and recomputes the fixed point. It detects bifurcations
by monitoring the eigenvalues of the linearization at the fixed point as the
parameter value is changed.
• Returning to the AUTO window, select Numerics and look in the AutoNum
window. Note especially that Ds has been set (via the hw5-2.ode file) to a
negative value. This instructs AUTO to decrease the main parameter from
b = 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 of
fixed points (both the x- and y-coordinates are calculated, but only the ycoordinate is plotted).
• AUTO can be used to find the periodic orbits, starting from the Hopf bifurcation point. Select Grab, then use the < Tab > key to move the cross-hairs
on the bifurcation diagram to the bifurcation point with largest b-value. This
is bc2 . Look at the information line (below the bifurcation diagram in the
AUTO window) as you do this. You should see the label HB for a Hopf bifurcation. When the correct point is in the information window, press the
< Enter > key to accept the point. Then select Run, Periodic and you
should get a plot showing the minimum and maximum y-values of the periodic 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 the
xterm window or the information line in the AUTO window) the values of the
parameter b, and of the fixed point coordinates x∗ and y ∗ computed by AUTO at
the Hopf bifurcation points. Discuss the correspondence between AUTO’s results
(the diagram and specific numerical values) and your previous analytical and XPP
results.
3. Show that
(
ẋ = −x + 2y 3 − 2y 4 ,
ẏ = −x − y + xy.
has no closed orbits by constructing a Lyapunov function of the form V = xm + ay n
with suitable values of a, m, n. The existence of a Lyapunov function also implies
that the single fixed point (0, 0) is asymptotically stable, verify this fact using linear
stability analysis.
4. Show that the following version of a predator-pray model
(
ẋ = rx (1 − x) −
xy
ẏ = 1+x
− y,
xy
,
1+x
(r > 0, x, y > 0)
has no closed orbits by using Dulac’s criterion with function g(x, y) =
suitable choice of α. What are the choices of α?
1+x α−1
y
x
for a
5. Show that the following system has a periodic solution using Poincaré-Bendixson Theorem.
(
ẋ = x − y − x3 ,
ẏ = x + y − y 3 .
(Hint: find a square area −b ≤ x ≤ b, −b ≤ y ≤ b (b > 0) in the phase space that is
a trapping region). Determine the smallest possible number bm > 0 such that for all
b ≥ bm this square region is trapping but not if b < bm .
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