American Institute of Mathematical Sciences

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May  2015, 35(5): 2227-2272. doi: 10.3934/dcds.2015.35.2227

Minimal period problems for brake orbits of nonlinear autonomous reversible semipositive Hamiltonian systems

Duanzhi Zhang 1,

1. 

School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071

Received  June 2012 Revised  January 2013 Published  December 2014

In this paper, for any positive integer $n$, we study the Maslov-type index theory of $i_{L_0}$, $i_{L_1}$ and $i_{\sqrt{-1}}^{L_0}$ with $L_0 = \{0\}\times \mathbf{R}^n\subset \mathbf{R}^{2n}$ and $L_1=\mathbf{R}^n\times \{0\} \subset \mathbf{R}^{2n}$. As applications we study the minimal period problems for brake orbits of nonlinear autonomous reversible Hamiltonian systems. For first order nonlinear autonomous reversible Hamiltonian systems in $\mathbf{R}^{2n}$, which are semipositive, and superquadratic at zero and infinity， we prove that for any $T>0$, the considered Hamiltonian systems possesses a nonconstant $T$ periodic brake orbit $X_T$ with minimal period no less than $\frac{T}{2n+2}$. Furthermore if $\int_0^T H''_{22}(x_T(t))dt$ is positive definite, then the minimal period of $x_T$ belongs to $\{T,\;\frac{T}{2}\}$. Moreover, if the Hamiltonian system is even, we prove that for any $T>0$, the considered even semipositive Hamiltonian systems possesses a nonconstant symmetric brake orbit with minimal period belonging to $\{T,\;\frac{T}{3}\}$.
Keywords: Symmetric, semipositive and reversible, Maslov-type index, minimal period, Hamiltonian systems., brake orbit.
Mathematics Subject Classification: 58E05, 70H05, 34C2.
Citation: Duanzhi Zhang. Minimal period problems for brake orbits of nonlinear autonomous reversible semipositive Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2227-2272. doi: 10.3934/dcds.2015.35.2227
References:
[1]

A. Ambrosetti, V. Benci and Y. Long, A note on the existence of multiple brake orbits,, Nonlinear Anal. T. M. A., 21 (1993), 643.  doi: 10.1016/0362-546X(93)90061-V.  Google Scholar

[2]

A. Ambrosetti and V. Coti Zelati, Solutions with minimal period for Hamiltonian systems in a potential well,, Ann. I. H. P. Anal. non linéaire, 4 (1987), 275.   Google Scholar

[3]

A. Ambrosetti and G. Mancini, Solutions of minimal period for a class of convex Hamiltonian systems,, Math. Ann., 255 (1981), 405.  doi: 10.1007/BF01450713.  Google Scholar

[4]

T. An and Y. Long, Index theories of second order Hamiltonian systems,, Nonlinear Anal., 34 (1998), 585.  doi: 10.1016/S0362-546X(97)00572-5.  Google Scholar

[5]

V. Benci, Closed geodesics for the Jacobi metric and periodic solutions of prescribed energy of natural Hamiltonian systems,, Ann. I. H. P. Analyse Nonl., 1 (1984), 401.   Google Scholar

[6]

V. Benci and F. Giannoni, A new proof of the existence of a brake orbit. In "Advanced Topics in the Theory of Dynamical Systems",, Notes Rep. Math. Sci. Eng., 6 (1989), 37.   Google Scholar

[7]

S. Bolotin, Libration motions of natural dynamical systems,, Vestnik Moskov Univ. Ser. I. Mat. Mekh., 6 (1978), 72.   Google Scholar

[8]

S. Bolotin and V. V. Kozlov, Librations with many degrees of freedom,, J. Appl. Math. Mech., 42 (1978), 245.   Google Scholar

[9]

B. Booss and K. Furutani, The Maslov-type index - a functional analytical definition and the spectral flow formula,, Tokyo J. Math., 21 (1998), 1.  doi: 10.3836/tjm/1270041982.  Google Scholar

[10]

B. Booss and C. Zhu, General spectral flow formula for fixed maximal domain,, Central Eur. J. Math., 3 (2005), 558.  doi: 10.2478/BF02475923.  Google Scholar

[11]

S. E. Cappell, R. Lee and E. Y. Miller, On the Maslov-type index,, Comm. Pure Appl. Math., 47 (1994), 121.  doi: 10.1002/cpa.3160470202.  Google Scholar

[12]

C. Conley and E. Zehnder, Maslov-type index theory for flows and periodix solutions for Hamiltonian equations,, Commu. Pure. Appl. Math., 37 (1984), 207.  doi: 10.1002/cpa.3160370204.  Google Scholar

[13]

D. Dong and Y. Long, The Iteration Theory of the Maslov-type Index Theory with Applications to Nonlinear Hamiltonian Systems,, Trans. Amer. Math. Soc., 349 (1997), 2619.  doi: 10.1090/S0002-9947-97-01718-2.  Google Scholar

[14]

J. J. Duistermaat, Fourier Integral Operators,, Birkhäuser, (1996).   Google Scholar

[15]

I. Ekeland, Convexity Methods in Hamiltonian Mechanics,, Spring-Verlag. Berlin, (1990).  doi: 10.1007/978-3-642-74331-3.  Google Scholar

[16]

I. Ekeland and E. Hofer, Periodic solutions with percribed period for convex autonomous Hamiltonian systems,, Invent. Math., 81 (1985), 155.  doi: 10.1007/BF01388776.  Google Scholar

[17]

G. Fei and Q. Qiu, Minimal period solutions of nonlinear Hamiltonian systems,, Nonlinear Anal., 27 (1996), 811.  doi: 10.1016/0362-546X(95)00077-9.  Google Scholar

[18]

G. Fei, S.-K. Kim and T. Wang, Minimal Period Estimates of Period Solutions for Superquadratic Hamiltonian Syetems,, J. Math. Anal. Appl., 238 (1999), 216.  doi: 10.1006/jmaa.1999.6527.  Google Scholar

[19]

G. Fei, S.-K. Kim and T. Wang, Solutions of minimal period for even classical Hamiltonian systems,, Nonlinear Anal., 43 (2001), 363.  doi: 10.1016/S0362-546X(99)00199-6.  Google Scholar

[20]

M. Girardi and M. Matzeu, Some results on solutions of minimal period to superquadratic Hamiltonian equations,, Nonlinear Anal., 7 (1983), 475.  doi: 10.1016/0362-546X(83)90039-1.  Google Scholar

[21]

M. M. Girardi and M. Matzeu, Solutions of minimal period for a class of nonconvex Hamiltonian systems and applications to the fixed energy problem,, Nonlinear Anal. TMA., 10 (1986), 371.  doi: 10.1016/0362-546X(86)90134-3.  Google Scholar

[22]

M. Girardi and M. Matzeu, Periodic solutions of convex Hamiltonian systems with a quadratic growth at the origin and superquadratic at infinity,, Ann. Math. Pura ed App., 147 (1987), 21.  doi: 10.1007/BF01762410.  Google Scholar

[23]

M. Girardi and M. Matzeu, Dual Morse index estimates for periodic solutions of Hamiltonian systems in some nonconvex superquadratic case,, Nonlinear Anal. TMA., 17 (1991), 481.  doi: 10.1016/0362-546X(91)90143-O.  Google Scholar

[24]

H. Gluck and W. Ziller, Existence of periodic solutions of conservtive systems,, Seminar on Minimal Submanifolds, (1983), 65.   Google Scholar

[25]

E. W. C. van Groesen, Analytical mini-max methods for Hamiltonian brake orbits of prescribed energy,, J. Math. Anal. Appl., 132 (1988), 1.  doi: 10.1016/0022-247X(88)90039-X.  Google Scholar

[26]

K. Hayashi, Periodic solution of classical Hamiltonian systems,, Tokyo J. Math., 6 (1983), 473.  doi: 10.3836/tjm/1270213886.  Google Scholar

[27]

C. Liu, A note on the monotonicity of Maslov-type index of Linear Hamiltonian systems with applications,, Proceedings of the royal Society of Edinburg, 135 (2005), 1263.  doi: 10.1017/S0308210500004364.  Google Scholar

[28]

C. Liu, Maslov-type index theory for symplectic paths with Lagrangian boundary conditions,, Adv. Nonlinear Stud., 7 (2007), 131.   Google Scholar

[29]

C. Liu, Asymptotically linear hamiltonian systems with largrangian boundary conditions,, Pacific J. Math., 232 (2007), 233.  doi: 10.2140/pjm.2007.232.233.  Google Scholar

[30]

C. Liu, Minimal period estimates for brake orbits of nonlinear symmetric Hamiltonian systems,, Discrete Contin. Dyn. Syst., 27 (2010), 337.  doi: 10.3934/dcds.2010.27.337.  Google Scholar

[31]

C. Liu and Y. Long, An optimal increasing estimate for iterated Maslov-type indices,, Chinese Sci. Bull., 42 (1997), 2275.   Google Scholar

[32]

C. Liu and Y. Long, Iteration inequalities of the Maslov-type index theory with applications,, J. Diff. Equa., 165 (2000), 355.  doi: 10.1006/jdeq.2000.3775.  Google Scholar

[33]

C. Liu and D. Zhang, An iteration theory of Maslov-type index for symplectic paths associated with a Lagranfian subspace and Multiplicity of brake orbits in bounded convex symmetric domains,, , ().   Google Scholar

[34]

Y. Long, Maslov-type index, degenerate critical points, and asymptotically linear Hamiltonian systems,, Science in China, 7 (1990), 673.   Google Scholar

[35]

Y. Long, The minimal period problem of classical Hamiltonian systems with even potentials,, Ann. I. H. P. Anal. non linéaire, 10 (1993), 605.   Google Scholar

[36]

Y. Long, The minimal period problem of period solutions for autonomous superquadratic second Hamiltonian systems,, J. Diff. Equa., 111 (1994), 147.  doi: 10.1006/jdeq.1994.1079.  Google Scholar

[37]

Y. Long, On the minimal period for periodic solution problem of nonlinear Hamiltonian systems,, Chinese Ann. of math., 18 (1997), 481.   Google Scholar

[38]

Y. Long, Bott formula of the Maslov-type index theory,, Pacific J. Math., 187 (1999), 113.  doi: 10.2140/pjm.1999.187.113.  Google Scholar

[39]

Y. long, Index Theory for Symplectic Paths with Applications,, Birkhäuser, (2002).  doi: 10.1007/978-3-0348-8175-3.  Google Scholar

[40]

Y. Long, D. Zhang and C. Zhu, Multiple brake orbits in bounded convex symmetric domains,, Advances in Math., 203 (2006), 568.  doi: 10.1016/j.aim.2005.05.005.  Google Scholar

[41]

P. H. Rabinowitz, Periodic solution of Hamiltonian systems,, Commu. Pure Appl. Math., 31 (1978), 157.  doi: 10.1002/cpa.3160310203.  Google Scholar

[42]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations,, CBMS Regional Conf. Ser. in Math., 45 (1986), 287.   Google Scholar

[43]

P. H. Rabinowitz, On the existence of periodic solutions for a class of symmetric Hamiltonian systems,, Nonlinear Anal. T. M. A., 11 (1987), 599.  doi: 10.1016/0362-546X(87)90075-7.  Google Scholar

[44]

J. Robbin and D. Salamon, The Maslov indices for paths,, Topology, 32 (1993), 827.  doi: 10.1016/0040-9383(93)90052-W.  Google Scholar

[45]

H. Seifert, Periodische Bewegungen mechanischer Systeme,, Math. Z., 51 (1948), 197.  doi: 10.1007/BF01291002.  Google Scholar

[46]

A. Szulkin, Cohomology and Morse theory for strongly indefinite functions,, Math. Z., 209 (1992), 375.  doi: 10.1007/BF02570842.  Google Scholar

[47]

Y. Xiao, Periodic Solutions with Prescribed Minimal Period for Second Order Hamiltonian Systems with Even Potentials,, Acta Math. Sinica, 26 (2010), 825.  doi: 10.1007/s10114-009-8305-2.  Google Scholar

[48]

D. Zhang, Symmetric period solutions with prescribed period for even autonomous semipositive hamiltonian systems,, Sci. China Math., 57 (2014), 81.  doi: 10.1007/s11425-013-4598-9.  Google Scholar

[49]

D. Zhang, Maslov-type index and brake orbits in nonlinear Hamiltonian systems,, Science in China, 50 (2007), 761.  doi: 10.1007/s11425-007-0034-3.  Google Scholar

[50]

C. Zhu and Y. Long, Maslov index theory for symplectic paths and spectral flow(I),, Chinese Ann. of Math., 20 (1999), 413.  doi: 10.1142/S0252959999000485.  Google Scholar

show all references

References:
[1]

A. Ambrosetti, V. Benci and Y. Long, A note on the existence of multiple brake orbits,, Nonlinear Anal. T. M. A., 21 (1993), 643.  doi: 10.1016/0362-546X(93)90061-V.  Google Scholar

[2]

A. Ambrosetti and V. Coti Zelati, Solutions with minimal period for Hamiltonian systems in a potential well,, Ann. I. H. P. Anal. non linéaire, 4 (1987), 275.   Google Scholar

[3]

A. Ambrosetti and G. Mancini, Solutions of minimal period for a class of convex Hamiltonian systems,, Math. Ann., 255 (1981), 405.  doi: 10.1007/BF01450713.  Google Scholar

[4]

T. An and Y. Long, Index theories of second order Hamiltonian systems,, Nonlinear Anal., 34 (1998), 585.  doi: 10.1016/S0362-546X(97)00572-5.  Google Scholar

[5]

V. Benci, Closed geodesics for the Jacobi metric and periodic solutions of prescribed energy of natural Hamiltonian systems,, Ann. I. H. P. Analyse Nonl., 1 (1984), 401.   Google Scholar

[6]

V. Benci and F. Giannoni, A new proof of the existence of a brake orbit. In "Advanced Topics in the Theory of Dynamical Systems",, Notes Rep. Math. Sci. Eng., 6 (1989), 37.   Google Scholar

[7]

S. Bolotin, Libration motions of natural dynamical systems,, Vestnik Moskov Univ. Ser. I. Mat. Mekh., 6 (1978), 72.   Google Scholar

[8]

S. Bolotin and V. V. Kozlov, Librations with many degrees of freedom,, J. Appl. Math. Mech., 42 (1978), 245.   Google Scholar

[9]

B. Booss and K. Furutani, The Maslov-type index - a functional analytical definition and the spectral flow formula,, Tokyo J. Math., 21 (1998), 1.  doi: 10.3836/tjm/1270041982.  Google Scholar

[10]

B. Booss and C. Zhu, General spectral flow formula for fixed maximal domain,, Central Eur. J. Math., 3 (2005), 558.  doi: 10.2478/BF02475923.  Google Scholar

[11]

S. E. Cappell, R. Lee and E. Y. Miller, On the Maslov-type index,, Comm. Pure Appl. Math., 47 (1994), 121.  doi: 10.1002/cpa.3160470202.  Google Scholar

[12]

C. Conley and E. Zehnder, Maslov-type index theory for flows and periodix solutions for Hamiltonian equations,, Commu. Pure. Appl. Math., 37 (1984), 207.  doi: 10.1002/cpa.3160370204.  Google Scholar

[13]

D. Dong and Y. Long, The Iteration Theory of the Maslov-type Index Theory with Applications to Nonlinear Hamiltonian Systems,, Trans. Amer. Math. Soc., 349 (1997), 2619.  doi: 10.1090/S0002-9947-97-01718-2.  Google Scholar

[14]

J. J. Duistermaat, Fourier Integral Operators,, Birkhäuser, (1996).   Google Scholar

[15]

I. Ekeland, Convexity Methods in Hamiltonian Mechanics,, Spring-Verlag. Berlin, (1990).  doi: 10.1007/978-3-642-74331-3.  Google Scholar

[16]

I. Ekeland and E. Hofer, Periodic solutions with percribed period for convex autonomous Hamiltonian systems,, Invent. Math., 81 (1985), 155.  doi: 10.1007/BF01388776.  Google Scholar

[17]

G. Fei and Q. Qiu, Minimal period solutions of nonlinear Hamiltonian systems,, Nonlinear Anal., 27 (1996), 811.  doi: 10.1016/0362-546X(95)00077-9.  Google Scholar

[18]

G. Fei, S.-K. Kim and T. Wang, Minimal Period Estimates of Period Solutions for Superquadratic Hamiltonian Syetems,, J. Math. Anal. Appl., 238 (1999), 216.  doi: 10.1006/jmaa.1999.6527.  Google Scholar

[19]

G. Fei, S.-K. Kim and T. Wang, Solutions of minimal period for even classical Hamiltonian systems,, Nonlinear Anal., 43 (2001), 363.  doi: 10.1016/S0362-546X(99)00199-6.  Google Scholar

[20]

M. Girardi and M. Matzeu, Some results on solutions of minimal period to superquadratic Hamiltonian equations,, Nonlinear Anal., 7 (1983), 475.  doi: 10.1016/0362-546X(83)90039-1.  Google Scholar

[21]

M. M. Girardi and M. Matzeu, Solutions of minimal period for a class of nonconvex Hamiltonian systems and applications to the fixed energy problem,, Nonlinear Anal. TMA., 10 (1986), 371.  doi: 10.1016/0362-546X(86)90134-3.  Google Scholar

[22]

M. Girardi and M. Matzeu, Periodic solutions of convex Hamiltonian systems with a quadratic growth at the origin and superquadratic at infinity,, Ann. Math. Pura ed App., 147 (1987), 21.  doi: 10.1007/BF01762410.  Google Scholar

[23]

M. Girardi and M. Matzeu, Dual Morse index estimates for periodic solutions of Hamiltonian systems in some nonconvex superquadratic case,, Nonlinear Anal. TMA., 17 (1991), 481.  doi: 10.1016/0362-546X(91)90143-O.  Google Scholar

[24]

H. Gluck and W. Ziller, Existence of periodic solutions of conservtive systems,, Seminar on Minimal Submanifolds, (1983), 65.   Google Scholar

[25]

E. W. C. van Groesen, Analytical mini-max methods for Hamiltonian brake orbits of prescribed energy,, J. Math. Anal. Appl., 132 (1988), 1.  doi: 10.1016/0022-247X(88)90039-X.  Google Scholar

[26]

K. Hayashi, Periodic solution of classical Hamiltonian systems,, Tokyo J. Math., 6 (1983), 473.  doi: 10.3836/tjm/1270213886.  Google Scholar

[27]

C. Liu, A note on the monotonicity of Maslov-type index of Linear Hamiltonian systems with applications,, Proceedings of the royal Society of Edinburg, 135 (2005), 1263.  doi: 10.1017/S0308210500004364.  Google Scholar

[28]

C. Liu, Maslov-type index theory for symplectic paths with Lagrangian boundary conditions,, Adv. Nonlinear Stud., 7 (2007), 131.   Google Scholar

[29]

C. Liu, Asymptotically linear hamiltonian systems with largrangian boundary conditions,, Pacific J. Math., 232 (2007), 233.  doi: 10.2140/pjm.2007.232.233.  Google Scholar

[30]

C. Liu, Minimal period estimates for brake orbits of nonlinear symmetric Hamiltonian systems,, Discrete Contin. Dyn. Syst., 27 (2010), 337.  doi: 10.3934/dcds.2010.27.337.  Google Scholar

[31]

C. Liu and Y. Long, An optimal increasing estimate for iterated Maslov-type indices,, Chinese Sci. Bull., 42 (1997), 2275.   Google Scholar

[32]

C. Liu and Y. Long, Iteration inequalities of the Maslov-type index theory with applications,, J. Diff. Equa., 165 (2000), 355.  doi: 10.1006/jdeq.2000.3775.  Google Scholar

[33]

C. Liu and D. Zhang, An iteration theory of Maslov-type index for symplectic paths associated with a Lagranfian subspace and Multiplicity of brake orbits in bounded convex symmetric domains,, , ().   Google Scholar

[34]

Y. Long, Maslov-type index, degenerate critical points, and asymptotically linear Hamiltonian systems,, Science in China, 7 (1990), 673.   Google Scholar

[35]

Y. Long, The minimal period problem of classical Hamiltonian systems with even potentials,, Ann. I. H. P. Anal. non linéaire, 10 (1993), 605.   Google Scholar

[36]

Y. Long, The minimal period problem of period solutions for autonomous superquadratic second Hamiltonian systems,, J. Diff. Equa., 111 (1994), 147.  doi: 10.1006/jdeq.1994.1079.  Google Scholar

[37]

Y. Long, On the minimal period for periodic solution problem of nonlinear Hamiltonian systems,, Chinese Ann. of math., 18 (1997), 481.   Google Scholar

[38]

Y. Long, Bott formula of the Maslov-type index theory,, Pacific J. Math., 187 (1999), 113.  doi: 10.2140/pjm.1999.187.113.  Google Scholar

[39]

Y. long, Index Theory for Symplectic Paths with Applications,, Birkhäuser, (2002).  doi: 10.1007/978-3-0348-8175-3.  Google Scholar

[40]

Y. Long, D. Zhang and C. Zhu, Multiple brake orbits in bounded convex symmetric domains,, Advances in Math., 203 (2006), 568.  doi: 10.1016/j.aim.2005.05.005.  Google Scholar

[41]

P. H. Rabinowitz, Periodic solution of Hamiltonian systems,, Commu. Pure Appl. Math., 31 (1978), 157.  doi: 10.1002/cpa.3160310203.  Google Scholar

[42]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations,, CBMS Regional Conf. Ser. in Math., 45 (1986), 287.   Google Scholar

[43]

P. H. Rabinowitz, On the existence of periodic solutions for a class of symmetric Hamiltonian systems,, Nonlinear Anal. T. M. A., 11 (1987), 599.  doi: 10.1016/0362-546X(87)90075-7.  Google Scholar

[44]

J. Robbin and D. Salamon, The Maslov indices for paths,, Topology, 32 (1993), 827.  doi: 10.1016/0040-9383(93)90052-W.  Google Scholar

[45]

H. Seifert, Periodische Bewegungen mechanischer Systeme,, Math. Z., 51 (1948), 197.  doi: 10.1007/BF01291002.  Google Scholar

[46]

A. Szulkin, Cohomology and Morse theory for strongly indefinite functions,, Math. Z., 209 (1992), 375.  doi: 10.1007/BF02570842.  Google Scholar

[47]

Y. Xiao, Periodic Solutions with Prescribed Minimal Period for Second Order Hamiltonian Systems with Even Potentials,, Acta Math. Sinica, 26 (2010), 825.  doi: 10.1007/s10114-009-8305-2.  Google Scholar

[48]

D. Zhang, Symmetric period solutions with prescribed period for even autonomous semipositive hamiltonian systems,, Sci. China Math., 57 (2014), 81.  doi: 10.1007/s11425-013-4598-9.  Google Scholar

[49]

D. Zhang, Maslov-type index and brake orbits in nonlinear Hamiltonian systems,, Science in China, 50 (2007), 761.  doi: 10.1007/s11425-007-0034-3.  Google Scholar

[50]

C. Zhu and Y. Long, Maslov index theory for symplectic paths and spectral flow(I),, Chinese Ann. of Math., 20 (1999), 413.  doi: 10.1142/S0252959999000485.  Google Scholar

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