Continuous time

Hill’s equation

Discrete time

Hill’s equation

Is a Hamiltonian system?

yes

yes

Its Monodromy matrix is symplectic?

yes

yes

Numbers of Arnold tongues

generically $\infty$

$K+1$

Arnold Tongues begin at

$\alpha ={\left(\frac{k\text{π}}{T}\right)}^2$

$k=0,1,2,\cdots$

$\left|{\displaystyle \sum _{j=0}^{\frac{K-i}{2}}}{l}_j{\left(-\alpha \right)}^{K-2j}\right|=2$

$l_j$ and $i$ as in theorem 12

Periodic/anti-periodicsolution alternation in the boundaries of the Arnold Tongues?

yes

yes

0-th Arnold tongue has a nonoscillatory solution?

yes

yes