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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, 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), 191-257. 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, 1995), Oper. Theory Adv. Appl., Birkhäuser, Basel, 102 (1998), 57-71.  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), 1-36. 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), 732-741. 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), 313-357. 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), 3247-3278. 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), 641-680.  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, Moscow, (1955), 413-498.  Google Scholar

[9]

Y. Long, Index Theory for Symplectic Paths with Applications, Progress in Math., Vol.207, Birkhäuser. Basel. 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), 77-90.  Google Scholar

[11]

B. Simon, Trace Ideals and Their Applications, Second edition. Mathematical Surveys and Monographs, 120. American Mathematical Society, Providence, RI, 2005.  Google Scholar

show all references

References:
[1]

S. V. Bolotin and D. V. Treschev, Hill's formula, Russian Math. Surveys, 65 (2010), 191-257. 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, 1995), Oper. Theory Adv. Appl., Birkhäuser, Basel, 102 (1998), 57-71.  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), 1-36. 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), 732-741. 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), 313-357. 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), 3247-3278. 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), 641-680.  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, Moscow, (1955), 413-498.  Google Scholar

[9]

Y. Long, Index Theory for Symplectic Paths with Applications, Progress in Math., Vol.207, Birkhäuser. Basel. 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), 77-90.  Google Scholar

[11]

B. Simon, Trace Ideals and Their Applications, Second edition. Mathematical Surveys and Monographs, 120. American Mathematical Society, Providence, RI, 2005.  Google Scholar

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