TY - JOUR

T1 - Rigorous numerics in floquet theory

T2 - Computing stable and unstable bundles of periodic orbits

AU - Castelli, Roberto

AU - Lessard, Jean Philippe

PY - 2013

Y1 - 2013

N2 - In this paper, a rigorous method to compute Floquet normal forms of fundamental matrix solutions of nonautonomous linear differential equations with periodic coefficients is introduced. The Floquet normal form of a fundamental matrix solution F(t) is a canonical decomposition of the form F(t) = Q(t)eRt, where Q(t) is a real periodic matrix and R is a constant matrix. To rigorously compute the Floquet normal form, the idea is to use the regularity of Q(t) and to simultaneously solve for R and Q(t) with the contraction mapping theorem in a Banach space of rapidly decaying coefficients. The explicit knowledge of R and Q can then be used to construct, in a rigorous computer-assisted way, stable and unstable bundles of periodic orbits of vector fields. The new proposed method does not require rigorous numerical integration of the ODE.

AB - In this paper, a rigorous method to compute Floquet normal forms of fundamental matrix solutions of nonautonomous linear differential equations with periodic coefficients is introduced. The Floquet normal form of a fundamental matrix solution F(t) is a canonical decomposition of the form F(t) = Q(t)eRt, where Q(t) is a real periodic matrix and R is a constant matrix. To rigorously compute the Floquet normal form, the idea is to use the regularity of Q(t) and to simultaneously solve for R and Q(t) with the contraction mapping theorem in a Banach space of rapidly decaying coefficients. The explicit knowledge of R and Q can then be used to construct, in a rigorous computer-assisted way, stable and unstable bundles of periodic orbits of vector fields. The new proposed method does not require rigorous numerical integration of the ODE.

KW - Contraction mapping theorem

KW - Floquet theory

KW - Fundamental matrix solutions

KW - Periodic orbits

KW - Rigorous numerics

KW - Tangent bundles

UR - http://www.scopus.com/inward/record.url?scp=84877597324&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84877597324&partnerID=8YFLogxK

U2 - 10.1137/120873960

DO - 10.1137/120873960

M3 - Article

AN - SCOPUS:84877597324

SN - 1536-0040

VL - 12

SP - 204

EP - 245

JO - SIAM Journal on Applied Dynamical Systems

JF - SIAM Journal on Applied Dynamical Systems

IS - 1

ER -