Strongly Exponentially Separated Linear Systems
- 84 Downloads
In the study of linear differential systems, an important concept is that of exponential separation. It is closely related to the concept of exponential dichotomy and has played a key role in the theory of Lyapunov exponents. Usually in the study of exponential separation, it is assumed that the coefficient matrix A(t) is bounded in norm. Our first aim here is to develop a theory of exponential separation which applies to unbounded systems. It turns that in order to have a reasonable theory it is necessary to add the assumption that the angle between the two separated subspaces is bounded below (note this follows automatically for bounded systems). Our second aim is to show that if a bounded linear Hamiltonian system is exponentially separated into two subspaces of the same dimension, then it must have an exponential dichotomy.
KeywordsExponential separation Exponential dichotomy Symplectic matrices Iwasawa decomposition
Mathematics Subject Classification34D05 34C25
- 4.Bylov, B.F., Vinograd, R.E., Grobman, D.M., Nemyckii, V.V.: The Theory of Lyapunov Exponents and its Application to Problems of Stability, Izdat. “Nauka”, Moscow (1966). (Russian) Google Scholar
- 5.Bylov, B.F., Izobov, N.: Necessary and sufficient stability conditions for the characteristic indices of a linear system. Differ. Uravn. 5, 1794–1803 (1969)Google Scholar
- 9.Hartman, P.: Ordinary Differential Equations, Classics in Applied Mathematics (Book 38), 2nd edn. SIAM, Philadelphia (2002)Google Scholar