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. Google Scholar

[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. Google Scholar

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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. Google Scholar

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_____, 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. Google Scholar

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_____, 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[15]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

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

[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. Google Scholar

[21]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

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

[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. Google Scholar

[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. Google Scholar

[29]

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

[30]

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

[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. Google Scholar

[32]

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

[33]

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

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[15]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

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

[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. Google Scholar

[21]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

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

[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. Google Scholar

[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. Google Scholar

[29]

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

[30]

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

[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. Google Scholar

[32]

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

[33]

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

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