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Math 645: Problem set #4
1. (a) Consider the linear inhomogenous equation x0=Ax +f(t) where f(t) is continuous
and periodic with period p. Show that the system has a unique solution xp(t) of
period pif Ahas no eigenvalue which is a multiple 2iπ/p.
Hnt: Use Duhamel’s formula to reduce the existence of a solution to the equation
x0=epAx0+Wfor the initial condition x0.
(b) Consider the second order equation x00 +bx +kx =g(t) with b≥0 and k≥0 and
g(t) periodic of period p. Determine for which values of band kthe equation has a
periodic solution of period p.
(c) Show that if all the eigenvalues of Ahave negative real part then every solution y(t) of
x0=Ax+f(t) converges to periodic solution found in A, i.e. limt→∞ y(t)−xp(t) = 0.
Hint: Use Duhamel’s formula.
2. Consider the scalar equation (i.e. n= 1) x0=f(t)xwhere f(t) is continuous and periodic
of period p.
(a) Determine P(t) and Rin the decomposition of the resolvent given by Floquet Theo-
rem.
(b) Give necessary and sufficient conditions in terms of f(t) for the solutions to be
bounded as t→ ±∞ or to be periodic.
3. Compute the resolvant R(t,0) (in real representation) for the ODE
x0= cos(t)x−sin(t)y ,
y0= sin(t)x+ cos(t)y . (1)
and determine P(t) and Rin Floquet Theorem Hint: Find an equation for z=x+iy.
4. Consider the equation x0=A(t)xwhere Ais periodic of period p. Show that a solution
x(t) is asymptotically stable if and only if the Floquet multipliers have absolute value less
than 1.
5. Consider the differential equation x00 +f(t)x= 0, where f(t) is periodic of period 2πand
f(t) = (1 if 0 ≤t<π
0 if π < t ≤2π.(2)
For both = 1/4 and = 4
(a) Consider the fundamental solution Φ(t) which satisfies Φ(0) = Iand compute the
corresponding transition matrix C=epR.
(b) Compute the Floquet multipliers (the eigenvalues of C).
(c) Describe the behavior of solution.
6. Let A(t) be periodic of period pand consider ODE x0=A(t)x.
(a) Show that the transition matrix Cdepends on the fundamental solution, but that
the eigenvalues of C=epR are independent of this choice.
