Homework #3
- For each of the following vector fields, find and classify all the fixed points, and sketch the phase portrait on the circle.
- θ˙ = 1 + 2 cos θ.
- θ˙ = sin θ + cos θ.
- θ˙ = sin 4θ.
- (You can read more about the background of this problem in Strogatz, Section 4.5).
Fireflies are known to synchronize their flashing in certain settings. Before begin- ning this problem, I highly recommend watching a video of this incredible phe- nomena. The synchronization does not occur immediately or randomly. When the fireflies first group, they are out of sync. As the night progresses, they begin to sync up and flash in unison. The key to this behavior is that the fireflies influence one another. When a firefly synchronizes its flashing to an external stimulus (poten- tially another firefly), this is called entrainment. Hanson (1978) showed that fireflies would become entrained to external stimulus (a flashing light) if that stimulus was close enough to the firefly’s natural flashing period (about 0.9 seconds).
Suppose we have an external periodic stimulus whose phase θs satisfies
θ˙s = Ω,
for some constant Ω. The stimulus flashes when θs = 0. If the stimulus is flashing before the firefly, the firefly speeds up its flashing to match the stimulus. If the stimulus is flashing after the firefly, the firefly slows down to try to match the stimulus. In this problem we model the phase of the firefly’s flashing, θf , as
θ˙f = ω + Af (θs − θf ),
where A > 0, ω is the natural flashing frequency of the firefly, and f is a somewhat arbitrary function for us to define. In this problem we will take θs ∈ [−π, π) with
−π and π being identified to form a circle. This allows the flashing to occur in the
middle of our symmetric domain. For |ϕ| < π, define
f (ϕ) = 2ϕ, |ϕ| ≤ π/2
2sgn(ϕ)π − 2ϕ, |ϕ| > π/2,
(1)
with f (ϕ) defined periodically outside of the interval [−π, π).
- Graph f (ϕ) on the domain −π ≤ ϕ < π.
- Write the dynamical system for the phase difference ϕ = θs − θf in terms of the dimensionless time τ = At and the dimensionless parameter µ = (Ω − ω)/A.
- Find the values of µ for which the firefly will be phase-locked to the stimulus
i.e. any values for which the firefly approaches the same frequency as the stimulus. Note that the phase does not need to be the same in order to be phase-locked, only the frequency needs to be the same (e.g. the firefly does not need to flash at the same time as the stimulus, but it does need to flash with the same frequency).
- Using the definition of µ and your answer from part (c), find the range of frequencies of the stimulus Ω for which the firefly will be phase-locked to the stimulus. This is called the range of entrainment.
- What kind of bifurcation occurs at µ = ±π? Does this look like the usual form for this type of bifurcation? If not, why not?
- Assuming the firefly is phase-locked to the stimulus, find a formula for the phase difference ϕ⋆ (i.e. the stable fixed point).
- Plot the phase portrait and classify the fixed point of the following linear systems. Put the system in matrix form.
(a)
x˙ = y,
y˙ = −2x − 3y.
(b)
(c)
x˙ = 3x − 4y,
x¨ + 2x˙ − x = 0.
y˙ = x − y.
- Consider an LRC circuit, which contains a capacitor, resistor, and inductor. The voltage across a capacitor is V = Q/C where Q is the charge on the capacitor, and C > 0 is the capacitance. Similarly, the voltage across an inductor is given by V = LdI/dt, where L > 0 is the inductance. Finally, recall Ohm’s law, namely that, for current flowing through a resistor, V = IR, where R ≥ 0 is the resistance
of that resistor. Putting all of this together with Kirchoff’s loop rule (sum of voltage changes over any closed path is zero), we get to
Q/C + IR + LdI/dt = 0.
If we take a time-derivative of the above equation, and recall that dQ/dt = I
(current is flowing charge), we get to the equation of interest in this problem
LI¨ + I˙R + I/C = 0.
- Rewrite the second-order differential equation as a 2-D linear dynamical system
- Show that the origin is asymptotically stable if R > 0 and neutrally stable if
R = 0.
- Classify the fixed point at the origin depending on whether R2C − 4L is posi- tive, negative, or zero.
- (You can read more about the background of this problem in Strogatz, Section 5.3)
In this problem we are going to model the feelings of affection between R and J. Let
R(t) = R’s feelings for J at time t, J(t) = J’s feelings for R at time t.
When R or J is positive, the feeling is a positive feeling (endearment). When R or J is negative, the feeling is a negative feeling (animus). Consider the following dynamical system governing the relationship between R and J,
R˙ = aR + J
J˙ = −R − aJ,
where a > 0 is a constant.
- (Not graded) Try to explain how R and J respond to their own feelings and to each other’s feelings. What role does the parameter a play? This should be a text-response, not calculations.
- Do the following for a > 1 and a < 1.
- Determine whether the origin is asymptotically stable, neutrally stable, unstable and attracting, or unstable and not attracting. Also classify the origin as a saddle point, node, center, or spiral. If the origin is a saddle point, identify (either by highlighting or explaining with words/formulae) the stable and unstable manifolds.
- Verify what you found in part (i) with a plot of the phase portrait (using pplane). You may choose a particular value of a for plotting. Interpret the long-time behavior of the system in terms of R and J’s relationship.
- Summarize and interpret what you found in part (b). How does the size of a
play out in J and R’s relationship?
