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January  2013, 33(1): 345-357. doi: 10.3934/dcds.2013.33.345

Instability of periodic minimals

Antonio J. Ureña 1,

1. 

Departamento de Matematica Aplicada, Facultad de Ciencias, Universidad de Granada, E-18071, Granada, Spain

Received  September 2011 Revised  December 2011 Published  September 2012

We consider second-order Euler-Lagrange systems which are periodic in time. Their periodic solutions may be characterized as the stationary points of an associated action functional, and we study the dynamical implications of minimizing the action. Examples are well-known of stable periodic minimizers, but instability always holds for periodic solutions which are minimal in the sense of Aubry-Mather.
Keywords: Periodic minimizers, quasi-asymptotic solutions, minimals, instability..
Mathematics Subject Classification: 34D20, 37J45, 37J5.
Citation: Antonio J. Ureña. Instability of periodic minimals. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 345-357. doi: 10.3934/dcds.2013.33.345
References:
[1]

S. V. Bolotin and V. V. Kozlov, Asymptotic solutions of the equations of dynamics, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 102 (1980), 84-89.  Google Scholar

[2]

C. Carathéodory, "Calculus of Variations and Partial Differential Equations ofthe First Order," Chelsea, New York, 1989. Google Scholar

[3]

E. N. Dancer and R. Ortega, The index of Lyapunov stable fixed points in two dimensions, J. Dynam. Differential Equations, 6 (1994), 631-637. doi: 10.1007/BF02218851.  Google Scholar

[4]

P. Hagedorn, Die Umkehrung der Stabilitätssätze von Lagrange-Dirichlet und Routh, Arch. Rational Mech. Anal., 42 (1971), 281-316. doi: 10.1007/BF00282334.  Google Scholar

[5]

P. Hagedorn and J. Mawhin, A simple variational approach to a converse of the Lagrange-Dirichlet theorem, Arch. Rational Mech. Anal., 120 (1992), 327-335. doi: 10.1007/BF00380318.  Google Scholar

[6]

J. N. Mather, Variational construction of orbits of twist diffeomorphisms, J. Amer. Math. Soc., 4 (1991), 207-263. doi: 10.1090/S0894-0347-1991-1080112-5.  Google Scholar

[7]

J. Moser, "Selected Chapters in the Calculus of Variations," Lecture notes by Oliver Knill. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2003.  Google Scholar

[8]

R. Ortega, The number of stable periodic solutions of time-dependentHamiltonian systems with one degree of freedom, Ergodic TheoryDynam. Systems, 18 (1998), 1007-1018. doi: 10.1017/S0143385798108362.  Google Scholar

[9]

R. Ortega, Instability of periodic solutions obtained by minimization, 'The first 60 years of Nonlinear Analysis of Jean Mawhin', World Scientific, (2004), 189-197. .  Google Scholar

[10]

P. H. Rabinowitz, Heteroclinics for a reversible Hamiltonian system, Ergodic Theory Dynam. Systems, 14 (1994), 817-829. doi: 10.1017/S0143385700008178.  Google Scholar

[11]

P. H. Rabinowitz, A note on a class of reversibleHamiltonian systems, Adv. Nonlinear Stud., 9 (2009), 815-823.  Google Scholar

[12]

A. J. Ureña, Isolated periodic minima are unstable, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 877-889.  Google Scholar

[13]

A. J. Ureña, All periodic minimizers are unstable, Arch. Math., 91 (2008), 63-75. doi: 10.1007/s00013-008-2693-x.  Google Scholar

show all references

References:
[1]

S. V. Bolotin and V. V. Kozlov, Asymptotic solutions of the equations of dynamics, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 102 (1980), 84-89.  Google Scholar

[2]

C. Carathéodory, "Calculus of Variations and Partial Differential Equations ofthe First Order," Chelsea, New York, 1989. Google Scholar

[3]

E. N. Dancer and R. Ortega, The index of Lyapunov stable fixed points in two dimensions, J. Dynam. Differential Equations, 6 (1994), 631-637. doi: 10.1007/BF02218851.  Google Scholar

[4]

P. Hagedorn, Die Umkehrung der Stabilitätssätze von Lagrange-Dirichlet und Routh, Arch. Rational Mech. Anal., 42 (1971), 281-316. doi: 10.1007/BF00282334.  Google Scholar

[5]

P. Hagedorn and J. Mawhin, A simple variational approach to a converse of the Lagrange-Dirichlet theorem, Arch. Rational Mech. Anal., 120 (1992), 327-335. doi: 10.1007/BF00380318.  Google Scholar

[6]

J. N. Mather, Variational construction of orbits of twist diffeomorphisms, J. Amer. Math. Soc., 4 (1991), 207-263. doi: 10.1090/S0894-0347-1991-1080112-5.  Google Scholar

[7]

J. Moser, "Selected Chapters in the Calculus of Variations," Lecture notes by Oliver Knill. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2003.  Google Scholar

[8]

R. Ortega, The number of stable periodic solutions of time-dependentHamiltonian systems with one degree of freedom, Ergodic TheoryDynam. Systems, 18 (1998), 1007-1018. doi: 10.1017/S0143385798108362.  Google Scholar

[9]

R. Ortega, Instability of periodic solutions obtained by minimization, 'The first 60 years of Nonlinear Analysis of Jean Mawhin', World Scientific, (2004), 189-197. .  Google Scholar

[10]

P. H. Rabinowitz, Heteroclinics for a reversible Hamiltonian system, Ergodic Theory Dynam. Systems, 14 (1994), 817-829. doi: 10.1017/S0143385700008178.  Google Scholar

[11]

P. H. Rabinowitz, A note on a class of reversibleHamiltonian systems, Adv. Nonlinear Stud., 9 (2009), 815-823.  Google Scholar

[12]

A. J. Ureña, Isolated periodic minima are unstable, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 877-889.  Google Scholar

[13]

A. J. Ureña, All periodic minimizers are unstable, Arch. Math., 91 (2008), 63-75. doi: 10.1007/s00013-008-2693-x.  Google Scholar

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