Mathematics 345
Homework Assignment 3
Due Thursday 4 February 2016
In this assignment, XPP commands for phase plane plots are introduced. ( XPP refers to the part of XPPAUT that does not use AUTO.)
From the course web page get the hw3-1.ode
file, for the two-dimensional system
φ
˙
= ν, ν ˙ = − δν + γ sin φ cos φ − sin φ, ( φ, ν ) ∈ S
1 × R , (1) that corresponds to the second-order differential equation (3) below. Note the default values of the parameters δ = 0 and γ = 0 specified in hw3-1.ode
. When γ = 0, the equations are the same as for the pendulum.
Open an xterm window, and start XPPAUT with hw3-1.ode
. The file hw3-1.ode
sets up the window to plot trajectories in phase space, ν vs.
φ . Select Dir.field/flow , (D)irect
Field , and keep the value for Grid at 10. This plots the vector field at points on a 10 × 10 grid. The length of each vector is proportional to the speed at that point.
Erase the picture and select Dir.field/flow , (S)caled Dir.Fld
, with the same grid. This time the vectors are normalized (scaled) to have all the same lengths. Select Dir.field/flow , (F)low , and change the value for Grid to 5. This computes solutions (from t = 0 to t = 20) for initial conditions on a 5 × 5 grid. The trajectories cannot be saved for plotting, but the picture gives a reasonable idea of the phase portrait.
To get a picture of the phase portrait that you can save as a Postscript file and eventually print, Erase the picture, and choose a single initial condition. A convenient way to do this is to select Initialconds , (M)ouse , then choose an initial condition with the mouse. To save the trajectory for a plot, select Graphic stuff , (F)reeze , (F)reeze as done in previous assignments. Repeat this until you get enough trajectories for a plot of the phase portrait.
(You could also select Graphic stuff , (F)reeze , (O)n freeze to automatically freeze a number of trajectories, but don’t forget to select Graphic stuff , (F)reeze , (O)ff freeze when you are finished plotting the picture.) Nullclines can be plotted by selecting Nullcline
New , and frozen, if desired, so they show up in the Postscript file.
Try locating the fixed point at the origin numerically. Specify an initial guess near the origin through the turquoise button ICs , then on the Main Menu select Sing pts , (G)o .
Select YES to print eigenvalues. The Equilibria window pops up (you may have to move other windows around) showing the number of positive real eigenvalues r +, the number of negative real eigenvalues r − , the number of complex eigenvalues with positive real parts c +, the number of complex eigenvalues with negative real parts c − , and the number of pure imaginary eigenvalues im . The coordinates of the fixed point (computed numerically) appear at the bottom. Values for the eigenvalues appear in the xterm window. You can also find fixed points by selecting Sing pts , (M)ouse and then using the mouse.
Try locating a saddle point numerically, first editing the Initial Data window to give a reasonable guesses for the saddle point coordinates (or use the mouse). Select YES to print eigenvalues, and YES to draw invariant sets (stable and unstable manifolds). Type ./ to end the invariant set computations. You may have to do this a few times before you can use the
Main Menu again. On the plot you get a picture of the stable and unstable manifolds of the saddle point. Notice on the diagram there are two representatives of the same saddle point in the phase cylinder S
1 × R .
XPP treats φ as belonging to R , and you have interpret the

results yourself for φ ∈ S
1
. Locate the other representative of the saddle point numerically, and plot its stable and unstable manifolds.
This is the assignment:
Consider the equation describing a bead on a rotating hoop, mr d 2 φ dt 2
+ b dφ dt
− mrω
2 sin φ cos φ + mg sin φ = 0 , φ ∈ S
1
, (2) that was mentioned in the lectures. The phase variable φ belongs to the circle S
1
, so values of φ are identified if they differ by an integer multiple of 2 π .
1. Show that (2) can be rewritten in dimensionless form as d 2 φ dτ 2
+ δ dφ dτ
− γ sin φ cos φ + sin φ = 0 , φ ∈ S
1
, (3) for suitably defined dimensionless quantities τ , δ and γ . Determine how the dimensionless quantities in (3) are related to the original quantities in (2). The two-dimensional system of first-order equations (1) is equivalent to (3).
(a) For all γ ≥ 0 find all fixed points of the system (1) in the phase cylinder ( θ, ν ) ∈
S 1 × R . There are a number of different cases i ), ii ), etc. as the number of fixed points may vary, depending on the value of γ .
(b) For each of the cases i ), ii ), etc. in part (a), for all δ ≥ 0 linearize at each fixed point and classify each the fixed point according to its topological type as: hyperbolic saddle point, hyperbolic attractor, hyperbolic repeller, non-hyperbolic fixed point with pure imaginary eigenvalues (linear centre), or non-hyperbolic fixed point with zero eigenvalue. Note that for each fixed γ there may be further different cases as the classification of fixed points may vary, depending on the value of δ .
(c) For δ = 0 .
3 and γ = 0, use XPP to plot the phase portrait. Indicate the global stable and unstable manifolds of the saddle point in the phase cylinder S 1 × R .
(d) Consider the undamped case δ = 0, for any γ . This system is conservative: find a conserved quantity. Sketch representative global phase portraits for all γ > 0
(there are several cases – this may be done by hand, or using XPP ). Identify all homoclinic orbits.
(e) For the damped case δ > 0, let V ( φ, ν ) be the quantity that was conserved for the undamped case in part (e). Show that ˙ ≤ 0, and explain why the value of V along any trajectory that is not a fixed point is decreasing (not just nonincreasing) as a function of τ . For γ = 2, plot by hand some representative level curves of V using dashed lines. On the same plot, draw using solid lines a phase portrait of
(1) for γ = 2 and small positive δ .
(f) For γ = 2 and: i ) δ = 0 .
5 ; ii ) δ = 1, use XPP to plot the nullclines and phase portraits of system (1), and indicate the global stable and unstable manifolds of any saddle point (do this for all representatives in the plot of the same saddle point in S 1 × R ). (Draw the stable and unstable manifolds on your two XPP plots by hand, in contrasting colours, remembering to identify φ -values as is appropriate for φ ∈ S 1 .)
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2. Recall from the lectures that (2) can be written in a different dimensionless form as d 2 φ dτ 2
+ dφ dτ
− γ sin φ cos φ + sin φ = 0 , φ ∈ S
1
, (4) for suitably defined dimensionless quantities τ , and γ (a different τ than for Question
1). Let dφ/dτ = ν , and give the two-dimensional system of first-order equations that is equivalent to (4). Label this system (5).
If we set = 0 in the second-order equation (4), we get the first-order equation (6)
(below) on the circle, discussed in class as the “overdamped bead on a rotating hoop”.
However, the limit → 0 is singular . This is especially obvious in the system (5). In this question you explore this singular limit.
(a) For all γ ≥ 0 and > 0 find and classify all the fixed points of the system (5), similarly to what was done for Question 1, parts (a) and (b). (In fact, you could just use the results of Question 1 parts (a) and (b), if you can determine the relationship between the parameters of the two different dimensionless forms (3) and (4).)
(b) Get the hw3-2.ode
file from the course web page and start XPPAUT with it.
Hand in accurate phase portraits for γ = 2 and the four cases i ) = 1; ii )
= 0 .
5; iii ) = 0 .
1; iv ) = 0 .
01.
For the last case the system is quite stiff (behaviour occurs on very different time scales). Do you believe the computed results? Try changing the step size for RK4 by selecting nUmerics , Dt .
Also try changing the numerical method used by selecting nUmerics , Method and (S)tiff for a method suited to stiff systems (for more information, see http://www.math.pitt.edu/ ∼ bard/xpp/help/xppnumerics.html#method ). Try both methods with various (min and max) step sizes (and tolerances) until you are reasonably sure you are computing solutions accurately. Explain why you believe the plot for case iv ) that you finally hand in is accurate!
(c) Explain in what sense solutions of the initial value problem for (5) with the initial conditions φ (0) = φ
0
, ν (0) = ν
0 for very small but positive are approximated by solutions of the initial value problem for the first-order equation for the overdamped (zero ) case
φ
˙
= γ sin φ cos φ − sin φ, φ ∈ S
1
, (6) with the initial condition φ (0) = φ
0
. What happens to ν ( τ ) in (5)?
3. Consider the following predator-prey model in which x represents the prey population and y denotes the population of the predator.
x
0 y
0
= x (1 − γx − 0 .
5 y )
= y ( − 0 .
25 + 0 .
5 x ) .
Let γ = 0 .
5 .
in parts (a), (b), (c) and (d).
(a) In the phase plane, sketch the nullclines and find all the critical points (steady states). Represent these points with a black dot.
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(b) For each critical point, discuss its local linear properties and classify its type.
Determine the trajectory flows near each one of these points (i.e. sketch a local phase portrait near each one).
(c) Draw a rough phase portrait for the system using the provided phase plane (please try it on a separate draft before putting down the final picture).
(d) Determine the long-term or limiting behaviour of x ( t ) and y ( t ) as t → ∞ for any initial condition x (0) , y (0) > 0. Interpret the results in terms of the populations of the two species.
(e) For γ = 0, determine how the number, location, and type of the critical points change. Without attempting a detailed analysis, what can you say about the change in the limiting behaviour of x ( t ) and y ( t ) for x (0) , y (0) > 0?
(f) Use the ode file provided, do the phase plane analysis using the XPP software.
(1) Generate the nullclines; (2) locate all the steady states; (3) determine the stability of each steady state by finding out the eigenvalues using and draw all the invariant sets when exist; (4) draw the direction filed (grad=10); (5) draw the flows in the phase plane (grad=10).
(g) Compare your hand-sketched phase portrait to the one you obtained by using
XPP.
Instructions for doing 3(f):
(1) To prepare, click ”nUmerics” (or hit the only capitalized letter key ”u” found in
”nUmerics”), then click on ”Ncline ctrl” (hit the letter key ”n”), on top you can see ”ncline mesh:40”, change the number 40 to 120, hit enter and then the ”esc” key. Now you are ready to go. Click on ”Nullcline” (or hit the key ”n”) and then click on ”(N)ew” (or hit the key ”n”) , you well see nullclines drawn in two different colors orange for the 1st eqn and green for the 2nd.
(2) Each crossing point between the nullclines of different colors is a steady state.
(3) Click ”Sing pts” (or hit the key ”s”), then click ”(M)ouse” (or hit the key ”m”).
You see the message ”Click on guess”, you move the curser as close as possible to one singular point (s.s.) that you want to study and click. You see the message ”Print eigenvalues?”, click ”yes”. Sometimes, you are asked ”Draw Invariant
Sets?”, you say ”yes” and click ”ok” twice. Then, hit the ”esc” key and click ”ok” twice again. You will first see yellow ”invariant manifolds” and then blue ones. At the end, you see a small window in which the location (x,y values), the stability property, and the eigenvalues distribution are all listed. On the X11 window, you see the corresponding eigenvalues. For a spiral, no invariant sets will be drawn.
Close the stability window after you finish.
(4) For the direction field, click ”Dir.field/flow” or hit the key ”d”, then click ”(D)irect
Field” or hit the key ”d”, you see on the top ”Grid:10”. You can change the number to another one if you want, then hit enter.
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(5) For the flow, click ”Dir.field/flow” or hit the key ”d”, then click ”(F)low” or hit the key ”f”, you see on the top ”Grid:10”. You can change the number to another one if you want, then hit enter.
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