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Math 8501 — Homework II

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Math 8501 — Homework II
Math 8501 — Homework II
due Wednesday, October 27
1. (V.I. Arnold, V.V. Beletskii) During a spacewalk, an astronaut in a circular orbit around the earth throws
the lens cap of his camera directly at the earth below him. Describe the subsequent motion of the lens cap.
Consider the Kepler problem in the plane, given by the second-order ODE q̈ = − |q|q 3 where q ∈ R2
is the position. Using polar coordinates (r, θ) in the plane one gets the differential equations
r̈ − rθ̇2 = −
1
r2
rθ̈ + 2ṙθ̇ = 0.
a. Show that there are circular solutions where r(t) = r0 is constant and θ(t) = c t. Find the relation
between r0 and c for such solutions. We will assume our astronaut is in the circular orbit with r0 = c = 1.
b. Write the ODE as a first-order system with variables (r, v, θ, w) = (r, ṙ, θ, θ̇). Find the variational ODE
of this first-order system along the circular orbit from part a. This will be a linear ODE for the approximate
behavior of the difference vector (δr, δv, δθ, δw) between the lens cap and the astronaut.
c. Use this to solve the variational equations with appropriate initial conditions to find the (rather surprising)
approximate motion of the lens cap. In particular, find δr(t) and δθ(t) and try to visualize the motion of
¨ + δv = 0.
the lens cap. Hint: Show that the variational equations lead to the simple scalar ODE δv
2.(Equilibria) Consider an autonomous ODE ẋ = f (x), where f : Rn → Rn is locally Lipschitz. Let φ(t, x)
denote the corresponding local flow and φt the time-t map.
a. A point x0 ∈ Rn is called an equilibrium or a restpoint if f (x0 ) = 0. Show that in this case φt (x0 ) = x0
for all t ∈ R (in other words x0 is a fixed point for all of the time-t maps).
b. Show that x0 is an equilibrium point if and only if there is a sequence of times tn 6= 0, tn → 0 such that
φtn (x0 ) = x0 .
c. Suppose that for some initial condition x0 , the corresponding solution converges to a limit as t → ∞, i.e.,
φ(t, x0 ) → L as t → ∞, for some L ∈ Rn . Show that L is an equilibrium point.
3.(Periodic Solutions) Same hypotheses and notation as problem 2.
a. Let T > 0. A point x0 ∈ Rn is called periodic of period T if φT (x0 ) = x0 , i.e., it is a fixed point of the
time-T map. Show that if x0 is such a periodic point, then the solution φ(t, x0 ) is defined for all t ∈ R and
satisfies φ(t + T, x0 ) = φ(t, x0 ) for all t ∈ R, i.e., the corresponding solution is a periodic function of time.
b. Show that if x0 is periodic but is not an equilibrium point then there is some minimal period T0 > 0 such
that φT0 (x0 ) = x0 but φT (x0 ) 6= x0 for 0 < T < T0 . Moreover, any period T for x0 is an integer multiple of
T0 .
c. Suppose x0 is periodic with minimal period T0 > 0. By definition, the orbit or phase curve of x0 is the
set {φt (x0 ) : t ∈ R} ⊂ Rn (call this a periodic orbit). Show that every point x1 on the orbit of x0 is periodic
with the same minimal period T0 .
d. Show that a periodic orbit is homeomorphic to the unit circle. Conversely, show that if some orbit is
homeomorphic to the circle, the corresponding solution is periodic.
4. Consider the ODE ẋ = 1 + x2 = f (x), x ∈ R1 with initial condition x(0) = 0.
a. Show by separation of variables, that the solution is x(t) = tan t. Next, substitute a formal power series
x(t) = c1 t + c2 t2 + . . . into the ODE to find a recursion formula giving ck+1 uniquely in terms of c1 , . . . , ck .
Find c1 , . . . , c7 .
1
is obviously a majorant for f (x) so the solution X(t)
b. The geometric series g(x) = 1 + x + x2 + . . . = 1−x
of the ODE Ẋ = g(X), X(0) = 0 will be a majorant for the series x(t). Find the solution X(t) and the
coefficients C1 , . . . , C7 of its Taylor series.
c. The radius of convergence of X(t) will be a lower bound for that of x(t). What are the exact radii of
convergence ?
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