June  2008, 21(2): 551-569. doi: 10.3934/dcds.2008.21.551

Generic properties of Lagrangians on surfaces: The Kupka-Smale theorem

1. 

Departamento de Matemática, Universidade Federal do Rio Grande do Sul, Av. Bento Goncalves, 9500, 91509-900, Porto Alegre, RS, Brazil

Received  June 2007 Revised  January 2008 Published  March 2008

We consider generic properties of Lagrangians. Our main result is a Kupka-Smale Theorem for the Lagrangian setting. We show that for convex and superlinear Lagrangians defined on a compact surface and $k\in \mathbb{R}$, then generically, in Mañé's sense, the energy level $k$ is regular and all periodic orbits in this level are nondegenerate at all orders (the linearized Poincaré map, restricted to this energy level, does not have roots of unity as eigenvalues). Moreover, all heteroclinic intersections in this level are transversal. The results that we present are true in dimension $n \geq 2$, with the exception of Theorem 4.5, which we are only able to prove in dimension 2.
Citation: Elismar R. Oliveira. Generic properties of Lagrangians on surfaces: The Kupka-Smale theorem. Discrete and Continuous Dynamical Systems, 2008, 21 (2) : 551-569. doi: 10.3934/dcds.2008.21.551
[1]

Armengol Gasull, Víctor Mañosa. Periodic orbits of discrete and continuous dynamical systems via Poincaré-Miranda theorem. Discrete and Continuous Dynamical Systems - B, 2020, 25 (2) : 651-670. doi: 10.3934/dcdsb.2019259

[2]

Viktor L. Ginzburg, Başak Z. Gürel. On the generic existence of periodic orbits in Hamiltonian dynamics. Journal of Modern Dynamics, 2009, 3 (4) : 595-610. doi: 10.3934/jmd.2009.3.595

[3]

Mário Jorge Dias Carneiro, Alexandre Rocha. A generic property of exact magnetic Lagrangians. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4183-4194. doi: 10.3934/dcds.2012.32.4183

[4]

Weigu Li, Kening Lu. Takens theorem for random dynamical systems. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 3191-3207. doi: 10.3934/dcdsb.2016093

[5]

Roberta Fabbri, Carmen Núñez, Ana M. Sanz. A perturbation theorem for linear Hamiltonian systems with bounded orbits. Discrete and Continuous Dynamical Systems, 2005, 13 (3) : 623-635. doi: 10.3934/dcds.2005.13.623

[6]

Alain Jacquemard, Weber Flávio Pereira. On periodic orbits of polynomial relay systems. Discrete and Continuous Dynamical Systems, 2007, 17 (2) : 331-347. doi: 10.3934/dcds.2007.17.331

[7]

Daniel Franco, J. R. L. Webb. Collisionless orbits of singular and nonsingular dynamical systems. Discrete and Continuous Dynamical Systems, 2006, 15 (3) : 747-757. doi: 10.3934/dcds.2006.15.747

[8]

Chao Ma, Baowei Wang, Jun Wu. Diophantine approximation of the orbits in topological dynamical systems. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2455-2471. doi: 10.3934/dcds.2019104

[9]

Doan Thai Son. On analyticity for Lyapunov exponents of generic bounded linear random dynamical systems. Discrete and Continuous Dynamical Systems - B, 2017, 22 (8) : 3113-3126. doi: 10.3934/dcdsb.2017166

[10]

Mahdi Khajeh Salehani. Identification of generic stable dynamical systems taking a nonlinear differential approach. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4541-4555. doi: 10.3934/dcdsb.2018175

[11]

Daniel Lear, David N. Reynolds, Roman Shvydkoy. Grassmannian reduction of cucker-smale systems and dynamical opinion games. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5765-5787. doi: 10.3934/dcds.2021095

[12]

Alessio Figalli, Vito Mandorino. Fine properties of minimizers of mechanical Lagrangians with Sobolev potentials. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1325-1346. doi: 10.3934/dcds.2011.31.1325

[13]

Vadim Kaloshin, Maria Saprykina. Generic 3-dimensional volume-preserving diffeomorphisms with superexponential growth of number of periodic orbits. Discrete and Continuous Dynamical Systems, 2006, 15 (2) : 611-640. doi: 10.3934/dcds.2006.15.611

[14]

Weigu Li, Kening Lu. A Siegel theorem for dynamical systems under random perturbations. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 635-642. doi: 10.3934/dcdsb.2008.9.635

[15]

Răzvan M. Tudoran. On the control of stability of periodic orbits of completely integrable systems. Journal of Geometric Mechanics, 2015, 7 (1) : 109-124. doi: 10.3934/jgm.2015.7.109

[16]

Francesco Fassò, Simone Passarella, Marta Zoppello. Control of locomotion systems and dynamics in relative periodic orbits. Journal of Geometric Mechanics, 2020, 12 (3) : 395-420. doi: 10.3934/jgm.2020022

[17]

Mariko Arisawa, Hitoshi Ishii. Some properties of ergodic attractors for controlled dynamical systems. Discrete and Continuous Dynamical Systems, 1998, 4 (1) : 43-54. doi: 10.3934/dcds.1998.4.43

[18]

Nitha Niralda P C, Sunil Mathew. On properties of similarity boundary of attractors in product dynamical systems. Discrete and Continuous Dynamical Systems - S, 2022, 15 (2) : 265-281. doi: 10.3934/dcdss.2021004

[19]

P.E. Kloeden, Desheng Li, Chengkui Zhong. Uniform attractors of periodic and asymptotically periodic dynamical systems. Discrete and Continuous Dynamical Systems, 2005, 12 (2) : 213-232. doi: 10.3934/dcds.2005.12.213

[20]

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

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (48)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]