Complex Eigenvalues — Question 10

Question 10 – Solution

Strategy. A full state repeats only when every active rotating block repeats at the same time.

Step 1: Solve each oscillatory block. The eigenvalues are $\pm i$ and $\pm i\gamma$. For the stated initial data, $$\boxed{X(t)=(\cos t,\sin t,\cos\gamma t,\sin\gamma t)^T.}$$ The two projections are unit circles with respective least periods $2\pi$ and $2\pi/\gamma$. Their closure as plane curves alone does not settle repetition of the combined state.

Step 2: Exclude a common period for irrational frequency ratio. At $\gamma=\sqrt 2$, any period must satisfy $T=2\pi m$ and $\sqrt 2T=2\pi n$ for positive integers $m,n$. This would imply $\sqrt 2=n/m$, impossible. Hence the four-dimensional state is not periodic. Its norm nevertheless remains $\sqrt 2$, so the obstruction is incompatible timing, not growth or damping.

Step 3: Classify all initial states in the irrational case. Each initially zero two-component block remains zero. A nonzero first block repeats exactly at integer multiples of $2\pi$; a nonzero second block repeats exactly at integer multiples of $2\pi/\sqrt 2$. Thus precisely the states with at least one block zero give periodic solutions. If only the first block is nonzero, the least period is $2\pi$; if only the second is nonzero, it is $\sqrt 2\pi$. If both vanish, the solution is constant. If both are nonzero, the same irrationality argument rules out every positive period, regardless of their initial phases.

Step 4: Recover a common period for rational frequency ratio. With $\gamma=3/2$ and both blocks active, $T=2\pi m$ must also satisfy $3m/2\in\mathbb Z$. The least positive choice is $m=2$, giving $\boxed{T=4\pi}$. In these independent rotation systems all states are bounded, but common periodicity requires commensurate active frequencies. Two planar centers therefore need not combine into a periodic full state; purely imaginary eigenvalues alone do not provide the missing frequency ratio.