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February  2016, 36(2): 763-784. doi: 10.3934/dcds.2016.36.763

## Hill-type formula and Krein-type trace formula for $S$-periodic solutions in ODEs

 1 Department of Mathematics, Shandong University, Jinan, Shandong 250100, China, China

Received  June 2014 Published  August 2015

The present paper is devoted to studying the Hill-type formula and Krein-type trace formula for ODE, which is a continuous work of our previous work for Hamiltonian systems [5]. Hill-type formula and Krein-type trace formula are given by Hill at 1877 and Krein in 1950's separately. Recently, we find that there is a closed relationship between them [5]. In this paper, we will obtain the Hill-type formula for the $S$-periodic orbits of the first order ODEs. Such a kind of orbits is considered naturally to study the symmetric periodic and quasi-periodic solutions. By some similar idea in [5], based on the Hill-type formula, we will build up the Krein-type trace formula for the first order ODEs, which can be seen as a non-self-adjoint version of the case of Hamiltonian system.
Citation: Xijun Hu, Penghui Wang. Hill-type formula and Krein-type trace formula for $S$-periodic solutions in ODEs. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 763-784. doi: 10.3934/dcds.2016.36.763
##### References:
 [1] S. V. Bolotin and D. V. Treschev, Hill's formula,, Russian Math. Surveys, 65 (2010), 191. doi: 10.1070/RM2010v065n02ABEH004671. Google Scholar [2] R. Denk, On Hilbert-Schmidt operators and determinants corresponding to periodic ODE systems,, Differential and integral operators (Regensburg, 102 (1998), 57. Google Scholar [3] G. W. Hill, On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and moon, Cambridge, Wilson, 1877;, reprinted with some additions at Acta Math., 8 (1886), 1. doi: 10.1007/BF02417081. Google Scholar [4] X. Hu and Y. Ou, An Estimation for the Hyperbolic Region of Elliptic Lagrangian Solutions in the Planar Three-body Problem,, Regular and Chaotic Dynamics, 18 (2013), 732. doi: 10.1134/S1560354713060129. Google Scholar [5] X. Hu, Y. Ou and P. Wang, Trace formula for linear Hamiltonian systems with its applications to elliptic Lagrangian solutions,, Arch. Ration. Mech. Anal., 216 (2015), 313. doi: 10.1007/s00205-014-0810-5. Google Scholar [6] X. Hu and P. Wang, Conditional Fredholm determinant for the S -periodic orbits in Hamiltonian systems,, J. Funct. Anal., 261 (2011), 3247. doi: 10.1016/j.jfa.2011.07.025. Google Scholar [7] M. G. Krein, On criteria of stable boundedness of solutions of periodic canonical systems,, PrikL Mat. Mekh., 19 (1955), 641. Google Scholar [8] M. G. Krein, The basic propositions of the theory of $\lambda$-zones of stability of a canonical system of linear differential equations with periodic coefficients, In Memoriam: A.A.Andronov,, Izdat.Akad.Nauk SSSR, (1955), 413. Google Scholar [9] Y. Long, Index Theory for Symplectic Paths with Applications,, Progress in Math., (2002). doi: 10.1007/978-3-0348-8175-3. Google Scholar [10] H. Poincaré, Sur les déterminants d'ordre infini,, Bull. Soc. math. France, 14 (1886), 77. Google Scholar [11] B. Simon, Trace Ideals and Their Applications, Second edition. Mathematical Surveys and Monographs, 120., American Mathematical Society, (2005). Google Scholar

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##### References:
 [1] S. V. Bolotin and D. V. Treschev, Hill's formula,, Russian Math. Surveys, 65 (2010), 191. doi: 10.1070/RM2010v065n02ABEH004671. Google Scholar [2] R. Denk, On Hilbert-Schmidt operators and determinants corresponding to periodic ODE systems,, Differential and integral operators (Regensburg, 102 (1998), 57. Google Scholar [3] G. W. Hill, On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and moon, Cambridge, Wilson, 1877;, reprinted with some additions at Acta Math., 8 (1886), 1. doi: 10.1007/BF02417081. Google Scholar [4] X. Hu and Y. Ou, An Estimation for the Hyperbolic Region of Elliptic Lagrangian Solutions in the Planar Three-body Problem,, Regular and Chaotic Dynamics, 18 (2013), 732. doi: 10.1134/S1560354713060129. Google Scholar [5] X. Hu, Y. Ou and P. Wang, Trace formula for linear Hamiltonian systems with its applications to elliptic Lagrangian solutions,, Arch. Ration. Mech. Anal., 216 (2015), 313. doi: 10.1007/s00205-014-0810-5. Google Scholar [6] X. Hu and P. Wang, Conditional Fredholm determinant for the S -periodic orbits in Hamiltonian systems,, J. Funct. Anal., 261 (2011), 3247. doi: 10.1016/j.jfa.2011.07.025. Google Scholar [7] M. G. Krein, On criteria of stable boundedness of solutions of periodic canonical systems,, PrikL Mat. Mekh., 19 (1955), 641. Google Scholar [8] M. G. Krein, The basic propositions of the theory of $\lambda$-zones of stability of a canonical system of linear differential equations with periodic coefficients, In Memoriam: A.A.Andronov,, Izdat.Akad.Nauk SSSR, (1955), 413. Google Scholar [9] Y. Long, Index Theory for Symplectic Paths with Applications,, Progress in Math., (2002). doi: 10.1007/978-3-0348-8175-3. Google Scholar [10] H. Poincaré, Sur les déterminants d'ordre infini,, Bull. Soc. math. France, 14 (1886), 77. Google Scholar [11] B. Simon, Trace Ideals and Their Applications, Second edition. Mathematical Surveys and Monographs, 120., American Mathematical Society, (2005). Google Scholar
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