Strongly Exponentially Separated Linear Systems
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 coe...
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| Published in | Journal of dynamics and differential equations Vol. 31; no. 2; pp. 573 - 600 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
New York
Springer US
01.06.2019
Springer Nature B.V |
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| Abstract | 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. |
|---|---|
| AbstractList | 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. 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. |
| Author | Battelli, Flaviano Palmer, Kenneth J. |
| Author_xml | – sequence: 1 givenname: Flaviano surname: Battelli fullname: Battelli, Flaviano organization: Department of Industrial Engineering and Mathematical Sciences, Marche Polytechnic University – sequence: 2 givenname: Kenneth J. surname: Palmer fullname: Palmer, Kenneth J. email: palmer@math.ntu.edu.tw organization: Department of Mathematics, National Taiwan University |
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| Keywords | 34C25 Exponential separation Exponential dichotomy Symplectic matrices Iwasawa decomposition 34D05 |
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| References | MaizelADOn stability of solutions of systems of differential equationsUral. Politehn. Inst. Tr.195451205075384 BattelliFPalmerKJCriteria for exponential dichotomy for triangular systemsJ. Math. Anal. Appl.2015428525543332700210.1016/j.jmaa.2015.03.0291318.34072 PerronODie Stabilitätsfrage bei DifferentialgleichungenMath. Z.193032703728154519410.1007/BF0119466256.1040.01 SackerRJSellGRA spectral theory for linear differential systemsJ. Differ. Equ.1978732035850118210.1016/0022-0396(78)90057-80372.34027 JuNWigginsSOn roughness of exponential dichotomyJ. Math. Anal. Appl.20012623949185721310.1006/jmaa.2001.74960990.34047 PalmerKJExponential dichotomy, exponential separation and spectral theory for linear systems of ordinary differential equationsJ. Differ. Equ.19824632434510.1016/0022-0396(82)90098-50466.34025 SackerRJSellGRExistence of dichotomies and invariant splittings for linear differential systems. IIJ. Differ. Equ.19762247849644062010.1016/0022-0396(76)90042-50339.58013 PalmerKJExponential dichotomy, integral separation and diagonalizability of linear systems of ordinary differential equationsJ. Differ. Equ.19824318420364706210.1016/0022-0396(82)90090-00443.34007 SackerRJSellGRExistence of dichotomies and invariant splittings for linear differential systems. IJ. Differ. Equ.19741542945834145810.1016/0022-0396(74)90067-90294.58008 BylovBFIzobovNNecessary and sufficient stability conditions for the characteristic indices of a linear systemDiffer. Uravn.1969517941803 HartmanPOrdinary Differential Equations, Classics in Applied Mathematics (Book 38)20022PhiladelphiaSIAM BenziMRazoukNOn the Iwasawa decomposition of a symplectic matrixAppl. Math. Lett.200720260265229255610.1016/j.aml.2006.04.0041118.15015 CoppelWADichotomies and Stability, Springer Lecture Notes1978BerlinSpringer10.1007/BFb0067780 MasseraJLSchäfferJJLinear Differential Equations and Function Spaces1966New YorkAcademic Press0243.34107 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) DaleckiiJLKreinMGStability of Solutions of Differential Equations in Banach Space, Translations of Mathematical Monographs (Book 43)2002ProvidenceAmerican Mathematical Society10.1090/mmono/043 AdrianovaLYIntroduction to Linear Systems of Differential Equations, Translations of Mathematical Monographs1995ProvidenceAMS10.1090/mmono/146 EliaCFabbriRRotation number and exponential dichotomy for linear Hamiltonian systems: from theoretical to numerical resultsJ. Dyn. Differ. Equ.20132595120302763510.1007/s10884-013-9290-91272.37036 PujalsERSambarinoMOn the dynamics of dominated splittingAnn. Math.2009169675740248061610.4007/annals.2009.169.6751178.37032 C Elia (9695_CR8) 2013; 25 JL Massera (9695_CR12) 1966 O Perron (9695_CR15) 1930; 32 AD Maizel (9695_CR11) 1954; 51 P Hartman (9695_CR9) 2002 RJ Sacker (9695_CR19) 1978; 7 M Benzi (9695_CR3) 2007; 20 F Battelli (9695_CR2) 2015; 428 KJ Palmer (9695_CR13) 1982; 43 N Ju (9695_CR10) 2001; 262 LY Adrianova (9695_CR1) 1995 9695_CR4 BF Bylov (9695_CR5) 1969; 5 WA Coppel (9695_CR6) 1978 KJ Palmer (9695_CR14) 1982; 46 JL Daleckii (9695_CR7) 2002 ER Pujals (9695_CR16) 2009; 169 RJ Sacker (9695_CR17) 1974; 15 RJ Sacker (9695_CR18) 1976; 22 |
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| SubjectTerms | Applications of Mathematics Hamiltonian functions Liapunov exponents Linear systems Mathematics Mathematics and Statistics Ordinary Differential Equations Partial Differential Equations Separation Subspaces |
| Title | Strongly Exponentially Separated Linear Systems |
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