The iteration formula

of the Maslov-type index theory

with applications

to nonlinear Hamiltonian systems

Authors:
Di Dong and Yiming Long

Journal:
Trans. Amer. Math. Soc. **349** (1997), 2619-2661

MSC (1991):
Primary 58F05, 58E05, 34C25; Secondary 15A18, 15A21

DOI:
https://doi.org/10.1090/S0002-9947-97-01718-2

MathSciNet review:
1373632

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, the iteration formula of the Maslov-type index theory for linear Hamiltonian systems with continuous, periodic, and symmetric coefficients is established. This formula yields a new method to determine the minimality of the period for solutions of nonlinear autonomous Hamiltonian systems via their Maslov-type indices. Applications of this formula give new results on the existence of periodic solutions with prescribed minimal period for such systems, and unify known results under various convexity conditions.

**[AZ]**H. Amann and E. Zehnder,*Nontrivial solutions for a class of non-resonance problems and applications to nonlinear differential equations*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)**7**(1980), 539-603. MR**82b:47077****[AC]**A. Ambrosetti and V. Coti Zelati,*Solutions with minimal period for Hamiltonian systems in a potential well*, Ann. Inst. H. Poincaré, Anal. Non Linéaire**4**(1987), 275-296. MR**88i:58028****[AM]**A. Ambrosetti and G. Mancini,*Solutions of minimal period for a class of convex Hamiltonian systems*, Math. Ann.**255**(1981), 405-421. MR**82j:58043****[Bo]**R. Bott,*On the iteration of closed geodesics and the Sturm intersection theory*, Comm. Pure Appl. Math.**9**(1956), 171-206. MR**19:859f****[Ch1]**K. C. Chang,*Solutions of asymptotically linear operator equations via Morse theory*, Comm. Pure Appl. Math.**34**(1981), 693-712. MR**82m:58015****[Ch2]**-,*Infinite dimensional Morse theory and multiple solution problems*, Birkhäuser, Boston, 1993. MR**94e:58023****[CE1]**F. Clarke and I. Ekeland,*Hamiltonian trajectories having prescribed minimal period*, Comm. Pure Appl. Math.**33**(1980), 103-116. MR**81e:70017****[CE2]**-,*Nonlinear oscillations and boundary value problems for Hamiltonian systems*, Arch. Rational Mech. Anal.**78**(1982), 315-333. MR**83h:58038****[CZ]**C. Conley and E. Zehnder,*Morse-type index theory for flows and periodic solutions for Hamiltonian equations*, Comm. Pure Appl. Math.**37**(1984), 207-253. MR**86b:58021****[CD]**R. Cushman and J. J. Duistermaat,*The behavior of the index of a periodic linear Hamiltonian system under iteration*, Adv. Math.**23**(1977), 1-21. MR**55:11296****[De]**S. Deng,*Minimal periodic solutions for a class of Hamiltonian equations*, Acta Math. Sinica**27**(1984), 664-675. (Chinese) MR**87c:58038****[Ek1]**I. Ekeland,*Une théorie de Morse pour les systèmes hamiltoniens convexes*, Ann. Inst. H. Poincaré Anal. Non Linéaire**1**(1984), 19-78. MR**85f:58023****[Ek2]**-,*An index theory for periodic solutions of convex Hamiltonian systems*, Proc. Sympos. Pure Math.**45**(1986), 395-423. MR**87j:58023****[Ek3]**-,*Convexity methods in Hamiltonian mechanics*, Springer-Verlag, Berlin, 1990. MR**91f:58027****[EH]**I. Ekeland and H. Hofer,*Periodic solutions with prescribed period for convex autonomous Hamiltonian systems*, Invent. Math.**81**(1985), 155-188. MR**87b:58028****[GL]**I. M. Gel´fand and V. B. Lidskii,*On the structure of the regions of stability of linear canonical systems of differential equations with periodic coefficients*, Uspekhi Mat. Nauk**10**(1955), no. 1, 3-40; English transl., Amer. Math. Soc. Transl. (2)**8**(1958), 143-181. MR**17:482****[Gh]**N. Ghoussoub,*Location, multiplicity and Morse indices of min-max critical points*, J.Reine Angew Math.**417**(1991), 27-76. MR**92e:58040****[GM1]**M. Girardi and M. Matzeu,*Some results on solutions of minimal period to superquadratic Hamiltonian equations*, Nonlinear Anal.**7**(1983), 475-482. MR**84g:58092****[GM2]**-,*Solutions of minimal period for a class of nonconvex Hamiltonian systems and applications to the fixed energy problem*, Nonlinear Anal.**10**(1986), 371-382. MR**87j:58038****[GM3]**-,*Periodic solutions of convex autonomous Hamiltonian systems with a quadratic growth at the origin and superquadratic at infinity*, Ann. Mat. Pura Appl. (4)**147**(1987), 21-72. MR**89b:58074****[GM4]**-,*Dual Morse index estimates for periodic solutions of Hamiltonian systems in some nonconvex superquadratic case*, Nonlinear Anal.**17**(1991), 481-497. MR**92k:58224****[GM5]**-,*Essential critical points of linking type and solutions of minimal period to superquadratic Hamiltonian systems*, Nonlinear Anal.**19**(1992), 237-247. MR**93f:58030****[K1]**W. Klingenberg,*Lectures on closed geodesics*, Springer-Verlag, Berlin, 1978. MR**57:17563****[LS]**A. Lazer and S. Solimini,*Nontrivial solutions of operator equations and Morse indices of critical points of min-max type*, Nonlinear Anal.**12**(1988), 761-775. MR**89i:58018****[Lo1]**Y. Long,*Maslov-type index, degenerate critical points, and asymptotically linear Hamiltonian systems*, Sci. China Ser. A**33**(1990), 1409-1419. MR**92d:58171****[Lo2]**-,*The structure of the singular symplectic matrix set*, Sci. China Ser. A**34**(1991), 897-907. MR**93e:58065****[Lo3]**-,*Maslov-type index theory and asymptotically linear Hamiltonian systems*, Dynamical Systems and Related Topics (Nagoya, 1990, K. Shiraiwa, ed.), World Sci. Publ., Singapore, 1991, pp. 333-341. MR**93f:58202****[Lo4]**-,*Estimates on the minimal period for periodic solutions of autonomous superquadratic second order Hamiltonian systems*, In Nonlinear Analysis and Microlocal Analysis (K. C. Chang et al., eds.), World Sci. Publ., Singapore, 1992, pp. 168-175.**[Lo5]**-,*The minimal period problem of periodic solutions for autonomous superquadratic second order Hamiltonian systems*, J. Differential Equations**111**(1994), 147-174. MR**95k:58029****[Lo6]**-,*The minimal period problem of classical Hamiltonian systems with even potentials*, Ann. Inst. H. Poincaré Anal. Non Linéaire**10**(1993), 605-626. MR**94m:58190****[Lo7]**-,*The index theory of Hamiltonian systems with applications*, Science Press, Beijing, 1993. (Chinese)**[Lo8]**-,*Nonlinear oscillations for classical Hamiltonian systems with bi-even subquadratic potentials*, Nonlinear Anal.**24**(1995), 1665-1671. MR**96h:34079****[Lo9]**Y. Long,*A Maslov-type index theory for symplectic paths*, Nankai Inst. of Math., Nankai Univ. Preprint, 1977.**[LD]**Y. Long and D. Dong,*Normal forms of symplectic matrices*, Preprint, Nankai Inst. Math., Nankai Univ., 1995.**[LZ]**Y. Long and E. Zehnder,*Morse theory for forced oscillations of asymptotically linear Hamiltonian systems*, Stochastic Processes Physics and Geometry (S. Albeverio et al., eds.), World Sci. Publ., Singapore, 1990, pp. 528-563. MR**92j:58019****[Mo]**J. K. Moser,*New aspects in the theory of stability of Hamiltonian systems*, Comm. Pure Appl. Math.**11**(1958), 81-114. MR**20:3354****[Ra1]**P. Rabinowitz,*Periodic solutions of Hamiltonian systems*, Comm. Pure Appl. Math.**31**(1978), 157-184. MR**57:7674****[Ra2]**-,*Minimax methods in critical point theory with applications to differential equations*, CBMS Regional Conf. Ser. Math., no. 65, Amer. Math. Soc., Providence, RI, 1986. MR**87j:58024****[RS]**J. Robbin and D. Salamon,*The Maslov index for paths*, Topology**32**(1993), 827-844. MR**94i:58071****[So]**S. Solimini,*Morse index estimates in min-max theorems*, Manuscripta Math.**63**(1989), 421-453. MR**90f:58028****[YS]**V. A. Yakubovich and V. M. Starzhinskii,*Linear differential equations with periodic coefficients*, Wiley, New York, 1975. MR**51:994****[Zh]**S. Zhang,*Doctoral Thesis*, Nankai University, 1991.

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Additional Information

**Di Dong**

Affiliation:
Nankai Institute of Mathematics, Nankai University, Tianjin 300071, People’s Republic of China

Address at time of publication:
Department of Mathematics, State University of New York at Stony Brook, Stony Brook, New York 11794-3651

Email:
ddong@math.sunysb.edu

**Yiming Long**

Affiliation:
Nankai Institute of Mathematics, Nankai University, Tianjin 300071, People’s Republic of China

Email:
longym@sun.nankai.edu.cn

DOI:
https://doi.org/10.1090/S0002-9947-97-01718-2

Keywords:
Normal form of a symplectic matrix,
perturbation of eigenvalues,
symplectic path,
Maslov-type index,
iteration formula,
minimal period,
Hamiltonian systems

Received by editor(s):
May 3, 1994

Additional Notes:
The second author was partially supported by YTF of Edu. Comm., NNSF of China, and Qiu Shi Sci. and Tech. Foundations.

Article copyright:
© Copyright 1997
American Mathematical Society