August 2019, 39(8): 4895-4928. doi: 10.3934/dcds.2019200

Prescribed energy connecting orbits for gradient systems

1. 

Dipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica delle Marche, Via Brecce Bianche, I-60131 Ancona, Italy

2. 

Dipartimento di Ingegneria Civile, Edile e Architettura, Università Politecnica delle Marche, Via Brecce Bianche, I-60131 Ancona, Italy

3. 

CEREMADE (CNRS UMR n° 7534), Université Paris-Dauphine, PSL Research University, Place de Lattre de Tassigny, 75775 Paris Cedex 16, France

* Corresponding author: Andres Zuniga

Received  January 2019 Revised  February 2019 Published  May 2019

Fund Project: The third author is supported by a public grant overseen by the French National Research Agency (ANR) as part of the Investissement d'Avenir project, reference ANR-10-LABX-0098, LabEx SMP, and also supported by the project EFI ANR-17-CE40-0030 of the ANR

We are concerned with conservative systems $ \ddot q = \nabla V(q) $, $ q\in{\mathbb R}^{N} $ for a general class of potentials $ V\in C^1({\mathbb R}^N) $. Assuming that a given sublevel set $ \{V\leq c\} $ splits in the disjoint union of two closed subsets $ \mathcal{V}^{c}_{-} $ and $ \mathcal{V}^{c}_{+} $, for some $ c\in{\mathbb R} $, we establish the existence of bounded solutions $ q_{c} $ to the above system with energy equal to $ -c $ whose trajectories connect $ \mathcal{V}^{c}_{-} $ and $ \mathcal{V}^{c}_{+} $. The solutions are obtained through an energy constrained variational method, whenever mild coerciveness properties are present in the problem. The connecting orbits are classified into brake, heteroclinic or homoclinic type, depending on the behavior of $ \nabla V $ on $ \partial \mathcal{V}^{c}_{\pm} $. Next, we illustrate applications of the existence result to double-well potentials $ V $, and for potentials associated to systems of duffing type and of multiple-pendulum type. In each of the above cases we prove some convergence results of the family of solutions $ (q_{c}) $.

Citation: Francesca Alessio, Piero Montecchiari, Andres Zuniga. Prescribed energy connecting orbits for gradient systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4895-4928. doi: 10.3934/dcds.2019200
References:
[1]

F. Alessio, Stationary layered solutions for a system of Allen-Cahn type equations, Indiana Univ. Math. J., 62 (2013), 1535-1564. doi: 10.1512/iumj.2013.62.5108.

[2]

F. AlessioM. L. Bertotti and P. Montecchiari, Multibump solutions to possibly degenerate equilibria for almost periodic Lagrangian systems, Z. Angew. Math. Phys., 50 (1999), 860-891. doi: 10.1007/s000330050184.

[3]

F. Alessio and P. Montecchiari, Entire solutions in $\mathbb{R}^{2}$ for a class of Allen-Cahn equations, ESAIM Control Optim. Calc. Var., 11 (2005), 633-672. doi: 10.1051/cocv:2005023.

[4]

_____, Multiplicity of entire solutions for a class of almost periodic Allen-Cahn type equations, Adv. Nonlinear Stud., 5 (2005), 515-549. doi: 10.1515/ans-2005-0404.

[5]

_____, Brake orbits type solutions to some class of semilinear elliptic equations, Calc. Var. Partial Differential Equations, 30 (2007), 51-83. doi: 10.1007/s00526-006-0078-1.

[6]

_____, An energy constrained method for the existence of layered type solutions of NLS equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 725-749. doi: 10.1016/j.anihpc.2013.07.003.

[7]

_____, Brake orbit solutions for semilinear elliptic systems with asymmetric double well potential, J. Fixed Point Theory Appl., 19 (2017), 691-717. doi: 10.1007/s11784-016-0370-4.

[8]

N. D. Alikakos and G. Fusco, On the connection problem for potentials with several global minima, Indiana Univ. Math. J., 57 (2008), 1871-1906. doi: 10.1512/iumj.2008.57.3181.

[9]

A. AmbrosettiV. Benci and Y. Long, A note on the existence of multiple brake orbits, Nonlinear Anal., 21 (1993), 643-649. doi: 10.1016/0362-546X(93)90061-V.

[10]

A. Ambrosetti and M.L. Bertotti, Homoclinics for second order conservative systems, in Partial Differential Equations and Related Subjects (Trento, 1990), Pitman Res. Notes in Math. Ser., 269 (1992), 21–37.

[11]

P. Antonopoulos and P. Smyrnelis, On minimizers of the Hamiltonian system $u'' = \nabla W(u)$ and on the existence of heteroclinic, homoclinic and periodic orbits, Indiana Univ. Math. J., 65 (2016), 1503-1524. doi: 10.1512/iumj.2016.65.5879.

[12]

V. Benci, Closed geodesics for the Jacobi metric and periodic solutions of prescribed energy of natural Hamiltonian systems, Ann. Inst. H. Poincaré Anal. Non Linéare, 1 (1984), 401-412. doi: 10.1016/S0294-1449(16)30420-6.

[13]

V. Benci and F. Giannoni, A new proof of the existence of a brake orbit, in Advanced Topics in the Theory of Dynamical Systems (Trento 1987), Notes Rep. Math. Sci. Eng., 6, Academic Press, (1989), 37–49.

[14]

M. L. Bertotti and P. Montecchiari, Connecting orbits for some classes of almost periodic Lagrangian systems, J. Differential Equations, 145 (1998), 453-468. doi: 10.1006/jdeq.1998.3415.

[15]

S. Bolotin and V. V. Kozlov, Librations with many degrees of freedom (Russian), Prikl. Mat. Mekh., 42 (1978), 245-250.

[16]

J. ByeonP. Montecchiari and P. H. Rabinowitz, A double well potential system, Anal. PDE, 9 (2016), 1737-1772. doi: 10.2140/apde.2016.9.1737.

[17]

V. Coti Zelati and E. Serra, Multiple brake orbits for some classes of singular Hamiltonian systems, Nonlinear Anal., 20 (1993), 1001-1012. doi: 10.1016/0362-546X(93)90090-F.

[18]

G. FuscoG. F. Gronchi and M. Novaga, On the existence of connecting orbits for critical values of the energy, J. Differential Equations, 263 (2017), 8848-8872. doi: 10.1016/j.jde.2017.08.067.

[19]

______, On the existence of heteroclinic connections, São Paulo J. Math. Sci., 12 (2018), 68-81. doi: 10.1007/s40863-017-0080-x.

[20]

R. GiambòF. Giannoni and P. Piccione, Orthogonal geodesic chords, brake orbits and homoclinic orbits in Riemannian manifolds, Adv. Differential Equations, 10 (2005), 931-960.

[21]

______, Multiple brake orbits and homoclinics in Riemannian manifolds, Arch. Ration. Mech. Anal., 200 (2011), 691-724. doi: 10.1007/s00205-010-0371-1.

[22]

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.

[23]

N. Katzourakis, On the loss of compactness in the heteroclinic connection problem, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 595-608. doi: 10.1017/S0308210515000700.

[24]

C. Li, The study of minimal period estimates for brake orbits of autonomous subquadratic Hamiltonian systems, Acta Math. Sin. (Engl. Ser.), 31 (2015), 1645-1658. doi: 10.1007/s10114-015-4421-3.

[25]

Y. LongD. Zhang and C. Zhu, Multiple brake orbits in bounded convex symmetric domains, Adv. Math., 203 (2006), 568-635. doi: 10.1016/j.aim.2005.05.005.

[26]

A. Monteil and H. Santambrogio, Metric methods for heteroclinic connections, Math. Methods Appl. Sci., 41 (2018), 1019-1024. doi: 10.1002/mma.4072.

[27]

P. H. Rabinowitz and K. Tanaka, Some results on connecting orbits for a class of Hamiltonian systems, Math. Z., 206 (1991), 473-479. doi: 10.1007/BF02571356.

[28]

P. H. Rabinowitz, Homoclinic and heteroclinic orbits for a class of Hamiltonian systems, Calc. Var. Partial Differential Equations, 1 (1993), 1-36. doi: 10.1007/BF02163262.

[29]

______, On a theorem of Strobel, Calc. Var. Partial Differential Equations, 12 (2001), 399-415. doi: 10.1007/PL00009919.

[30]

H. Seifert, Periodische bewegungen mechanischer systeme, (German) [Periodic movements of mechanical systems], Math. Z., 51 (1948), 197-216. doi: 10.1007/BF01291002.

[31]

P. Sternberg and A. Zuniga, On the heteroclinic connection problem for multi-well potentials with several global minima, J. Differential Equations, 261 (2016), 3987-4007. doi: 10.1016/j.jde.2016.06.010.

[32]

A. Weinstein, Periodic orbits for convex Hamiltonian systems, Ann. of Math.(2), 108 (1978), 507-518. doi: 10.2307/1971185.

[33]

A. Zuniga, Geometric Problems in the Calculus of Variations, Ph.D thesis, Indiana University in Bloomington, 2018.

show all references

References:
[1]

F. Alessio, Stationary layered solutions for a system of Allen-Cahn type equations, Indiana Univ. Math. J., 62 (2013), 1535-1564. doi: 10.1512/iumj.2013.62.5108.

[2]

F. AlessioM. L. Bertotti and P. Montecchiari, Multibump solutions to possibly degenerate equilibria for almost periodic Lagrangian systems, Z. Angew. Math. Phys., 50 (1999), 860-891. doi: 10.1007/s000330050184.

[3]

F. Alessio and P. Montecchiari, Entire solutions in $\mathbb{R}^{2}$ for a class of Allen-Cahn equations, ESAIM Control Optim. Calc. Var., 11 (2005), 633-672. doi: 10.1051/cocv:2005023.

[4]

_____, Multiplicity of entire solutions for a class of almost periodic Allen-Cahn type equations, Adv. Nonlinear Stud., 5 (2005), 515-549. doi: 10.1515/ans-2005-0404.

[5]

_____, Brake orbits type solutions to some class of semilinear elliptic equations, Calc. Var. Partial Differential Equations, 30 (2007), 51-83. doi: 10.1007/s00526-006-0078-1.

[6]

_____, An energy constrained method for the existence of layered type solutions of NLS equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 725-749. doi: 10.1016/j.anihpc.2013.07.003.

[7]

_____, Brake orbit solutions for semilinear elliptic systems with asymmetric double well potential, J. Fixed Point Theory Appl., 19 (2017), 691-717. doi: 10.1007/s11784-016-0370-4.

[8]

N. D. Alikakos and G. Fusco, On the connection problem for potentials with several global minima, Indiana Univ. Math. J., 57 (2008), 1871-1906. doi: 10.1512/iumj.2008.57.3181.

[9]

A. AmbrosettiV. Benci and Y. Long, A note on the existence of multiple brake orbits, Nonlinear Anal., 21 (1993), 643-649. doi: 10.1016/0362-546X(93)90061-V.

[10]

A. Ambrosetti and M.L. Bertotti, Homoclinics for second order conservative systems, in Partial Differential Equations and Related Subjects (Trento, 1990), Pitman Res. Notes in Math. Ser., 269 (1992), 21–37.

[11]

P. Antonopoulos and P. Smyrnelis, On minimizers of the Hamiltonian system $u'' = \nabla W(u)$ and on the existence of heteroclinic, homoclinic and periodic orbits, Indiana Univ. Math. J., 65 (2016), 1503-1524. doi: 10.1512/iumj.2016.65.5879.

[12]

V. Benci, Closed geodesics for the Jacobi metric and periodic solutions of prescribed energy of natural Hamiltonian systems, Ann. Inst. H. Poincaré Anal. Non Linéare, 1 (1984), 401-412. doi: 10.1016/S0294-1449(16)30420-6.

[13]

V. Benci and F. Giannoni, A new proof of the existence of a brake orbit, in Advanced Topics in the Theory of Dynamical Systems (Trento 1987), Notes Rep. Math. Sci. Eng., 6, Academic Press, (1989), 37–49.

[14]

M. L. Bertotti and P. Montecchiari, Connecting orbits for some classes of almost periodic Lagrangian systems, J. Differential Equations, 145 (1998), 453-468. doi: 10.1006/jdeq.1998.3415.

[15]

S. Bolotin and V. V. Kozlov, Librations with many degrees of freedom (Russian), Prikl. Mat. Mekh., 42 (1978), 245-250.

[16]

J. ByeonP. Montecchiari and P. H. Rabinowitz, A double well potential system, Anal. PDE, 9 (2016), 1737-1772. doi: 10.2140/apde.2016.9.1737.

[17]

V. Coti Zelati and E. Serra, Multiple brake orbits for some classes of singular Hamiltonian systems, Nonlinear Anal., 20 (1993), 1001-1012. doi: 10.1016/0362-546X(93)90090-F.

[18]

G. FuscoG. F. Gronchi and M. Novaga, On the existence of connecting orbits for critical values of the energy, J. Differential Equations, 263 (2017), 8848-8872. doi: 10.1016/j.jde.2017.08.067.

[19]

______, On the existence of heteroclinic connections, São Paulo J. Math. Sci., 12 (2018), 68-81. doi: 10.1007/s40863-017-0080-x.

[20]

R. GiambòF. Giannoni and P. Piccione, Orthogonal geodesic chords, brake orbits and homoclinic orbits in Riemannian manifolds, Adv. Differential Equations, 10 (2005), 931-960.

[21]

______, Multiple brake orbits and homoclinics in Riemannian manifolds, Arch. Ration. Mech. Anal., 200 (2011), 691-724. doi: 10.1007/s00205-010-0371-1.

[22]

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.

[23]

N. Katzourakis, On the loss of compactness in the heteroclinic connection problem, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 595-608. doi: 10.1017/S0308210515000700.

[24]

C. Li, The study of minimal period estimates for brake orbits of autonomous subquadratic Hamiltonian systems, Acta Math. Sin. (Engl. Ser.), 31 (2015), 1645-1658. doi: 10.1007/s10114-015-4421-3.

[25]

Y. LongD. Zhang and C. Zhu, Multiple brake orbits in bounded convex symmetric domains, Adv. Math., 203 (2006), 568-635. doi: 10.1016/j.aim.2005.05.005.

[26]

A. Monteil and H. Santambrogio, Metric methods for heteroclinic connections, Math. Methods Appl. Sci., 41 (2018), 1019-1024. doi: 10.1002/mma.4072.

[27]

P. H. Rabinowitz and K. Tanaka, Some results on connecting orbits for a class of Hamiltonian systems, Math. Z., 206 (1991), 473-479. doi: 10.1007/BF02571356.

[28]

P. H. Rabinowitz, Homoclinic and heteroclinic orbits for a class of Hamiltonian systems, Calc. Var. Partial Differential Equations, 1 (1993), 1-36. doi: 10.1007/BF02163262.

[29]

______, On a theorem of Strobel, Calc. Var. Partial Differential Equations, 12 (2001), 399-415. doi: 10.1007/PL00009919.

[30]

H. Seifert, Periodische bewegungen mechanischer systeme, (German) [Periodic movements of mechanical systems], Math. Z., 51 (1948), 197-216. doi: 10.1007/BF01291002.

[31]

P. Sternberg and A. Zuniga, On the heteroclinic connection problem for multi-well potentials with several global minima, J. Differential Equations, 261 (2016), 3987-4007. doi: 10.1016/j.jde.2016.06.010.

[32]

A. Weinstein, Periodic orbits for convex Hamiltonian systems, Ann. of Math.(2), 108 (1978), 507-518. doi: 10.2307/1971185.

[33]

A. Zuniga, Geometric Problems in the Calculus of Variations, Ph.D thesis, Indiana University in Bloomington, 2018.

Figure 1.  Possible configurations in Duffing like systems
Figure 2.  Possible configurations in pendulum like systems
[1]

Fei Liu, Jaume Llibre, Xiang Zhang. Heteroclinic orbits for a class of Hamiltonian systems on Riemannian manifolds. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1097-1111. doi: 10.3934/dcds.2011.29.1097

[2]

E. Canalias, Josep J. Masdemont. Homoclinic and heteroclinic transfer trajectories between planar Lyapunov orbits in the sun-earth and earth-moon systems. Discrete & Continuous Dynamical Systems - A, 2006, 14 (2) : 261-279. doi: 10.3934/dcds.2006.14.261

[3]

Daniel Wilczak. Abundance of heteroclinic and homoclinic orbits for the hyperchaotic Rössler system. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 1039-1055. doi: 10.3934/dcdsb.2009.11.1039

[4]

B. Buffoni, F. Giannoni. Brake periodic orbits of prescribed Hamiltonian for indefinite Lagrangian systems. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 217-222. doi: 10.3934/dcds.1995.1.217

[5]

Chungen Liu. Minimal period estimates for brake orbits of nonlinear symmetric Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 337-355. doi: 10.3934/dcds.2010.27.337

[6]

Ying Lv, Yan-Fang Xue, Chun-Lei Tang. Homoclinic orbits for a class of asymptotically quadratic Hamiltonian systems. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2855-2878. doi: 10.3934/cpaa.2019128

[7]

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

[8]

Addolorata Salvatore. Multiple homoclinic orbits for a class of second order perturbed Hamiltonian systems. Conference Publications, 2003, 2003 (Special) : 778-787. doi: 10.3934/proc.2003.2003.778

[9]

Flaviano Battelli. Saddle-node bifurcation of homoclinic orbits in singular systems. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 203-218. doi: 10.3934/dcds.2001.7.203

[10]

Qinqin Zhang. Homoclinic orbits for discrete Hamiltonian systems with indefinite linear part. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1929-1940. doi: 10.3934/cpaa.2015.14.1929

[11]

Chen-Chang Peng, Kuan-Ju Chen. Existence of transversal homoclinic orbits in higher dimensional discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1181-1197. doi: 10.3934/dcdsb.2010.14.1181

[12]

Juntao Sun, Jifeng Chu, Zhaosheng Feng. Homoclinic orbits for first order periodic Hamiltonian systems with spectrum point zero. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3807-3824. doi: 10.3934/dcds.2013.33.3807

[13]

Boris Buffoni, Laurent Landry. Multiplicity of homoclinic orbits in quasi-linear autonomous Lagrangian systems. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 75-116. doi: 10.3934/dcds.2010.27.75

[14]

Qinqin Zhang. Homoclinic orbits for discrete Hamiltonian systems with local super-quadratic conditions. Communications on Pure & Applied Analysis, 2019, 18 (1) : 425-434. doi: 10.3934/cpaa.2019021

[15]

Jun Wang, Junxiang Xu, Fubao Zhang. Homoclinic orbits for a class of Hamiltonian systems with superquadratic or asymptotically quadratic potentials. Communications on Pure & Applied Analysis, 2011, 10 (1) : 269-286. doi: 10.3934/cpaa.2011.10.269

[16]

Tiantian Wu, Xiao-Song Yang. A new class of 3-dimensional piecewise affine systems with homoclinic orbits. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5119-5129. doi: 10.3934/dcds.2016022

[17]

Cyril Joel Batkam. Homoclinic orbits of first-order superquadratic Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3353-3369. doi: 10.3934/dcds.2014.34.3353

[18]

Flaviano Battelli, Ken Palmer. Transversal periodic-to-periodic homoclinic orbits in singularly perturbed systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 367-387. doi: 10.3934/dcdsb.2010.14.367

[19]

Héctor E. Lomelí. Heteroclinic orbits and rotation sets for twist maps. Discrete & Continuous Dynamical Systems - A, 2006, 14 (2) : 343-354. doi: 10.3934/dcds.2006.14.343

[20]

Christopher K. R. T. Jones, Siu-Kei Tin. Generalized exchange lemmas and orbits heteroclinic to invariant manifolds. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 967-1023. doi: 10.3934/dcdss.2009.2.967

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (17)
  • HTML views (35)
  • Cited by (0)

[Back to Top]