# American Institute of Mathematical Sciences

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

 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 $\{T,\;\frac{T}{3}\}$.
Citation: Duanzhi Zhang. Minimal period problems for brake orbits of nonlinear autonomous reversible semipositive Hamiltonian systems. Discrete and Continuous Dynamical Systems, 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), 643-649. doi: 10.1016/0362-546X(93)90061-V. [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), 275-296. [3] A. Ambrosetti and G. Mancini, Solutions of minimal period for a class of convex Hamiltonian systems, Math. Ann., 255 (1981), 405-421. doi: 10.1007/BF01450713. [4] T. An and Y. Long, Index theories of second order Hamiltonian systems, Nonlinear Anal., 34 (1998), 585-592. doi: 10.1016/S0362-546X(97)00572-5. [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), 401-412. [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), 37-49. [7] S. Bolotin, Libration motions of natural dynamical systems, Vestnik Moskov Univ. Ser. I. Mat. Mekh., 6 (1978), 72-77 (in Russian). [8] S. Bolotin and V. V. Kozlov, Librations with many degrees of freedom, J. Appl. Math. Mech., 42 (1978), 245-250 (in Russian). [9] B. Booss and K. Furutani, The Maslov-type index - a functional analytical definition and the spectral flow formula, Tokyo J. Math., 21 (1998), 1-34. doi: 10.3836/tjm/1270041982. [10] B. Booss and C. Zhu, General spectral flow formula for fixed maximal domain, Central Eur. J. Math., 3 (2005), 558-577. doi: 10.2478/BF02475923. [11] S. E. Cappell, R. Lee and E. Y. Miller, On the Maslov-type index, Comm. Pure Appl. Math., 47 (1994), 121-186. doi: 10.1002/cpa.3160470202. [12] C. Conley and E. Zehnder, Maslov-type index theory for flows and periodix solutions for Hamiltonian equations, Commu. Pure. Appl. Math., 37 (1984), 207-253. doi: 10.1002/cpa.3160370204. [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), 2619-2661. doi: 10.1090/S0002-9947-97-01718-2. [14] J. J. Duistermaat, Fourier Integral Operators, Birkhäuser, Basel, 1996. [15] I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Spring-Verlag. Berlin, 1990. doi: 10.1007/978-3-642-74331-3. [16] I. Ekeland and E. Hofer, Periodic solutions with percribed period for convex autonomous Hamiltonian systems, Invent. Math., 81 (1985), 155-188. doi: 10.1007/BF01388776. [17] G. Fei and Q. Qiu, Minimal period solutions of nonlinear Hamiltonian systems, Nonlinear Anal., 27 (1996), 811-820. doi: 10.1016/0362-546X(95)00077-9. [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), 216-233. doi: 10.1006/jmaa.1999.6527. [19] G. Fei, S.-K. Kim and T. Wang, Solutions of minimal period for even classical Hamiltonian systems, Nonlinear Anal., 43 (2001), 363-375. doi: 10.1016/S0362-546X(99)00199-6. [20] M. Girardi and M. Matzeu, Some results on solutions of minimal period to superquadratic Hamiltonian equations, Nonlinear Anal., 7 (1983), 475-482. doi: 10.1016/0362-546X(83)90039-1. [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), 371-382. doi: 10.1016/0362-546X(86)90134-3. [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), 21-72. doi: 10.1007/BF01762410. [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), 481-497. doi: 10.1016/0362-546X(91)90143-O. [24] H. Gluck and W. Ziller, Existence of periodic solutions of conservtive systems, Seminar on Minimal Submanifolds, Princeton University Press, (1983), 65-98. [25] E. W. C. van Groesen, Analytical mini-max methods for Hamiltonian brake orbits of prescribed energy, J. Math. Anal. Appl., 132 (1988), 1-12. doi: 10.1016/0022-247X(88)90039-X. [26] K. Hayashi, Periodic solution of classical Hamiltonian systems, Tokyo J. Math., 6 (1983), 473-486. doi: 10.3836/tjm/1270213886. [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), 1263-1277. doi: 10.1017/S0308210500004364. [28] C. Liu, Maslov-type index theory for symplectic paths with Lagrangian boundary conditions, Adv. Nonlinear Stud., 7 (2007), 131-161. [29] C. Liu, Asymptotically linear hamiltonian systems with largrangian boundary conditions, Pacific J. Math., 232 (2007), 233-255. doi: 10.2140/pjm.2007.232.233. [30] C. Liu, Minimal period estimates for brake orbits of nonlinear symmetric Hamiltonian systems, Discrete Contin. Dyn. Syst., 27 (2010), 337-355. doi: 10.3934/dcds.2010.27.337. [31] C. Liu and Y. Long, An optimal increasing estimate for iterated Maslov-type indices, Chinese Sci. Bull., 42 (1997), 2275-2277. [32] C. Liu and Y. Long, Iteration inequalities of the Maslov-type index theory with applications, J. Diff. Equa., 165 (2000), 355-376. doi: 10.1006/jdeq.2000.3775. [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,, , (). [34] Y. Long, Maslov-type index, degenerate critical points, and asymptotically linear Hamiltonian systems, Science in China, Series A., 7 (1990), 673-682 (Chinese edition), 33 (1990), 1409-1419.(English edition) [35] Y. Long, The minimal period problem of classical Hamiltonian systems with even potentials, Ann. I. H. P. Anal. non linéaire, 10 (1993), 605-626. [36] Y. Long, The minimal period problem of period solutions for autonomous superquadratic second Hamiltonian systems, J. Diff. Equa., 111 (1994), 147-174. doi: 10.1006/jdeq.1994.1079. [37] Y. Long, On the minimal period for periodic solution problem of nonlinear Hamiltonian systems, Chinese Ann. of math., 18 (1997), 481-484. [38] Y. Long, Bott formula of the Maslov-type index theory, Pacific J. Math., 187 (1999), 113-149. doi: 10.2140/pjm.1999.187.113. [39] Y. long, Index Theory for Symplectic Paths with Applications, Birkhäuser, Basel, 2002. doi: 10.1007/978-3-0348-8175-3. [40] Y. Long, D. Zhang and C. Zhu, Multiple brake orbits in bounded convex symmetric domains, Advances in Math., 203 (2006), 568-635. doi: 10.1016/j.aim.2005.05.005. [41] P. H. Rabinowitz, Periodic solution of Hamiltonian systems, Commu. Pure Appl. Math., 31 (1978), 157-184. doi: 10.1002/cpa.3160310203. [42] P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conf. Ser. in Math., A.M.S., Providence, 45 (1986), 287-306. [43] P. H. Rabinowitz, On the existence of periodic solutions for a class of symmetric Hamiltonian systems, Nonlinear Anal. T. M. A., 11 (1987), 599-611. doi: 10.1016/0362-546X(87)90075-7. [44] J. Robbin and D. Salamon, The Maslov indices for paths, Topology, 32 (1993), 827-844. doi: 10.1016/0040-9383(93)90052-W. [45] H. Seifert, Periodische Bewegungen mechanischer Systeme, Math. Z., 51 (1948), 197-216. doi: 10.1007/BF01291002. [46] A. Szulkin, Cohomology and Morse theory for strongly indefinite functions, Math. Z., 209 (1992), 375-418. doi: 10.1007/BF02570842. [47] Y. Xiao, Periodic Solutions with Prescribed Minimal Period for Second Order Hamiltonian Systems with Even Potentials, Acta Math. Sinica, English Series, 26 (2010), 825-830. doi: 10.1007/s10114-009-8305-2. [48] D. Zhang, Symmetric period solutions with prescribed period for even autonomous semipositive hamiltonian systems, Sci. China Math., 57 (2014), 81-96. doi: 10.1007/s11425-013-4598-9. [49] D. Zhang, Maslov-type index and brake orbits in nonlinear Hamiltonian systems, Science in China, 50 (2007), 761-772. doi: 10.1007/s11425-007-0034-3. [50] C. Zhu and Y. Long, Maslov index theory for symplectic paths and spectral flow(I), Chinese Ann. of Math., 20 (1999), 413-424. doi: 10.1142/S0252959999000485.

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##### 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), 643-649. doi: 10.1016/0362-546X(93)90061-V. [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), 275-296. [3] A. Ambrosetti and G. Mancini, Solutions of minimal period for a class of convex Hamiltonian systems, Math. Ann., 255 (1981), 405-421. doi: 10.1007/BF01450713. [4] T. An and Y. Long, Index theories of second order Hamiltonian systems, Nonlinear Anal., 34 (1998), 585-592. doi: 10.1016/S0362-546X(97)00572-5. [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), 401-412. [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), 37-49. [7] S. Bolotin, Libration motions of natural dynamical systems, Vestnik Moskov Univ. Ser. I. Mat. Mekh., 6 (1978), 72-77 (in Russian). [8] S. Bolotin and V. V. Kozlov, Librations with many degrees of freedom, J. Appl. Math. Mech., 42 (1978), 245-250 (in Russian). [9] B. Booss and K. Furutani, The Maslov-type index - a functional analytical definition and the spectral flow formula, Tokyo J. Math., 21 (1998), 1-34. doi: 10.3836/tjm/1270041982. [10] B. Booss and C. Zhu, General spectral flow formula for fixed maximal domain, Central Eur. J. Math., 3 (2005), 558-577. doi: 10.2478/BF02475923. [11] S. E. Cappell, R. Lee and E. Y. Miller, On the Maslov-type index, Comm. Pure Appl. Math., 47 (1994), 121-186. doi: 10.1002/cpa.3160470202. [12] C. Conley and E. Zehnder, Maslov-type index theory for flows and periodix solutions for Hamiltonian equations, Commu. Pure. Appl. Math., 37 (1984), 207-253. doi: 10.1002/cpa.3160370204. [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), 2619-2661. doi: 10.1090/S0002-9947-97-01718-2. [14] J. J. Duistermaat, Fourier Integral Operators, Birkhäuser, Basel, 1996. [15] I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Spring-Verlag. Berlin, 1990. doi: 10.1007/978-3-642-74331-3. [16] I. Ekeland and E. Hofer, Periodic solutions with percribed period for convex autonomous Hamiltonian systems, Invent. Math., 81 (1985), 155-188. doi: 10.1007/BF01388776. [17] G. Fei and Q. Qiu, Minimal period solutions of nonlinear Hamiltonian systems, Nonlinear Anal., 27 (1996), 811-820. doi: 10.1016/0362-546X(95)00077-9. [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), 216-233. doi: 10.1006/jmaa.1999.6527. [19] G. Fei, S.-K. Kim and T. Wang, Solutions of minimal period for even classical Hamiltonian systems, Nonlinear Anal., 43 (2001), 363-375. doi: 10.1016/S0362-546X(99)00199-6. [20] M. Girardi and M. Matzeu, Some results on solutions of minimal period to superquadratic Hamiltonian equations, Nonlinear Anal., 7 (1983), 475-482. doi: 10.1016/0362-546X(83)90039-1. [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), 371-382. doi: 10.1016/0362-546X(86)90134-3. [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), 21-72. doi: 10.1007/BF01762410. [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), 481-497. doi: 10.1016/0362-546X(91)90143-O. [24] H. Gluck and W. Ziller, Existence of periodic solutions of conservtive systems, Seminar on Minimal Submanifolds, Princeton University Press, (1983), 65-98. [25] E. W. C. van Groesen, Analytical mini-max methods for Hamiltonian brake orbits of prescribed energy, J. Math. Anal. Appl., 132 (1988), 1-12. doi: 10.1016/0022-247X(88)90039-X. [26] K. Hayashi, Periodic solution of classical Hamiltonian systems, Tokyo J. Math., 6 (1983), 473-486. doi: 10.3836/tjm/1270213886. [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), 1263-1277. doi: 10.1017/S0308210500004364. [28] C. Liu, Maslov-type index theory for symplectic paths with Lagrangian boundary conditions, Adv. Nonlinear Stud., 7 (2007), 131-161. [29] C. Liu, Asymptotically linear hamiltonian systems with largrangian boundary conditions, Pacific J. Math., 232 (2007), 233-255. doi: 10.2140/pjm.2007.232.233. [30] C. Liu, Minimal period estimates for brake orbits of nonlinear symmetric Hamiltonian systems, Discrete Contin. Dyn. Syst., 27 (2010), 337-355. doi: 10.3934/dcds.2010.27.337. [31] C. Liu and Y. Long, An optimal increasing estimate for iterated Maslov-type indices, Chinese Sci. Bull., 42 (1997), 2275-2277. [32] C. Liu and Y. Long, Iteration inequalities of the Maslov-type index theory with applications, J. Diff. Equa., 165 (2000), 355-376. doi: 10.1006/jdeq.2000.3775. [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,, , (). [34] Y. Long, Maslov-type index, degenerate critical points, and asymptotically linear Hamiltonian systems, Science in China, Series A., 7 (1990), 673-682 (Chinese edition), 33 (1990), 1409-1419.(English edition) [35] Y. Long, The minimal period problem of classical Hamiltonian systems with even potentials, Ann. I. H. P. Anal. non linéaire, 10 (1993), 605-626. [36] Y. Long, The minimal period problem of period solutions for autonomous superquadratic second Hamiltonian systems, J. Diff. Equa., 111 (1994), 147-174. doi: 10.1006/jdeq.1994.1079. [37] Y. Long, On the minimal period for periodic solution problem of nonlinear Hamiltonian systems, Chinese Ann. of math., 18 (1997), 481-484. [38] Y. Long, Bott formula of the Maslov-type index theory, Pacific J. Math., 187 (1999), 113-149. doi: 10.2140/pjm.1999.187.113. [39] Y. long, Index Theory for Symplectic Paths with Applications, Birkhäuser, Basel, 2002. doi: 10.1007/978-3-0348-8175-3. [40] Y. Long, D. Zhang and C. Zhu, Multiple brake orbits in bounded convex symmetric domains, Advances in Math., 203 (2006), 568-635. doi: 10.1016/j.aim.2005.05.005. [41] P. H. Rabinowitz, Periodic solution of Hamiltonian systems, Commu. Pure Appl. Math., 31 (1978), 157-184. doi: 10.1002/cpa.3160310203. [42] P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conf. Ser. in Math., A.M.S., Providence, 45 (1986), 287-306. [43] P. H. Rabinowitz, On the existence of periodic solutions for a class of symmetric Hamiltonian systems, Nonlinear Anal. T. M. A., 11 (1987), 599-611. doi: 10.1016/0362-546X(87)90075-7. [44] J. Robbin and D. Salamon, The Maslov indices for paths, Topology, 32 (1993), 827-844. doi: 10.1016/0040-9383(93)90052-W. [45] H. Seifert, Periodische Bewegungen mechanischer Systeme, Math. Z., 51 (1948), 197-216. doi: 10.1007/BF01291002. [46] A. Szulkin, Cohomology and Morse theory for strongly indefinite functions, Math. Z., 209 (1992), 375-418. doi: 10.1007/BF02570842. [47] Y. Xiao, Periodic Solutions with Prescribed Minimal Period for Second Order Hamiltonian Systems with Even Potentials, Acta Math. Sinica, English Series, 26 (2010), 825-830. doi: 10.1007/s10114-009-8305-2. [48] D. Zhang, Symmetric period solutions with prescribed period for even autonomous semipositive hamiltonian systems, Sci. China Math., 57 (2014), 81-96. doi: 10.1007/s11425-013-4598-9. [49] D. Zhang, Maslov-type index and brake orbits in nonlinear Hamiltonian systems, Science in China, 50 (2007), 761-772. doi: 10.1007/s11425-007-0034-3. [50] C. Zhu and Y. Long, Maslov index theory for symplectic paths and spectral flow(I), Chinese Ann. of Math., 20 (1999), 413-424. doi: 10.1142/S0252959999000485.
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