Reducible Linear Quasi-periodic Systems with Positive Lyapunov Exponent and Varying Rotation Number

✍ Scribed by H.W Broer; C Simó

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 111 KB
• Volume: 168
• Category: Article
• ISSN: 0022-0396
• DOI: 10.1006/jdeq.2000.3877

✦ Synopsis

dedicated to professor jack hale on the occasion of his 70th birthday A linear system in two dimensions is studied. The coefficients are 2?-periodic in three angles, % j , j=1, 2, 3, and these angles are linear with respect to time, with incommensurable frequencies. The system has positive Lyapunov coefficients and the rotation number changes in a continuous way when some parameter moves. A lift to T 3 _R 2 , however, is only of class L p , for any p<2.

2000 Academic Press

Consider a class of linear differential equations

where M # Sl(2, R), A(t) # sl(2, R) and * =dÂdt. Corresponding evolutions frequently occur as fundamental solutions of second order equations of Hill (or Schro dinger) type

x +q(t) x=0.

(2)

Generically also the converse is true: the equation ( 1) can be associated to a second order equation of the form (2). Compare [4] for a discrete analogue with 2 frequencies.