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December  2012, 32(12): 4429-4443. doi: 10.3934/dcds.2012.32.4429

Variational destruction of invariant circles

Lin Wang 1,

1. 

Department of Mathematics, Nanjing University, No. 22 Hankou Road, Nanjing, 210093, China

Received  June 2011 Revised  November 2011 Published  August 2012

We construct a sequence of generating functions $(h_n)_{n\in\mathbb{N}}$, arbitrarily close to an integrable system in the $C^r$ topology with $r<4$ for $n$ large enough. With the variational method, we prove that for a given rotation number $\omega$ and $n$ large enough, the exact monotone area-preserving twist maps generated by $(h_n)_{n\in\mathbb{N}}$ admit no invariant circles with rotation number $\omega$.
Keywords: variational method., rotation number, Invariant circle, Peierls barrier, generating function.
Mathematics Subject Classification: Primary: 58F27, 58F05; Secondary: 58F3.
Citation: Lin Wang. Variational destruction of invariant circles. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4429-4443. doi: 10.3934/dcds.2012.32.4429
References:
[1]

V. Bangert, Mather sets for twist maps and geodesics ontori, Dynamics Reported, 1 (1988), 1-56.

[2]

G. Forni, Analytic destruction of invariant circles, Ergod. Th. & Dynam. Sys., 14 (1994), 267-298.

[3]

M. R. Herman, Sur la conjugation différentiable des difféomorphismes du cercle à des rotations, Publ. Math. IHES, 49 (1979), 5-233. doi: 10.1007/BF02684798.

[4]

M. R. Herman, Sur les courbes invariantes par les difféomorphismes de l'anneau, Astérisque, 103-104 (1983), 1-221.

[5]

M. R. Herman, Sur les courbes invariantes par les difféomorphismes de l'anneau, Astérisque, 144 (1986), 1-248.

[6]

J. N. Mather, Existence of quasi periodic orbits for twist homeomorphisms of the annulus, Topology, 21 (1982), 457-467. doi: 10.1016/0040-9383(82)90023-4.

[7]

J. N. Mather, A criterion for the non-existence of invariant circle, Publ. Math. IHES, 63 (1986), 153-204. doi: 10.1007/BF02831625.

[8]

J. N. Mather, Modulus of continuity for Peierls's barrier, Periodic Solutions of Hamiltonian Systems and Related Topics, NATO ASI Series C, 209, Reidel, Dordrecht, (1987), 177-202.

[9]

J. N. Mather, Destruction of invariant circles, Ergod. Th. & Dynam. Sys., 8 (1988), 199-214.

[10]

D. Salamon, The Kolmogorov-Arnold-Moser theorem, Math. Phys. Eletron. J., 10 (2004), Paper 3, 37 pp. (electronic).

show all references

References:
[1]

V. Bangert, Mather sets for twist maps and geodesics ontori, Dynamics Reported, 1 (1988), 1-56.

[2]

G. Forni, Analytic destruction of invariant circles, Ergod. Th. & Dynam. Sys., 14 (1994), 267-298.

[3]

M. R. Herman, Sur la conjugation différentiable des difféomorphismes du cercle à des rotations, Publ. Math. IHES, 49 (1979), 5-233. doi: 10.1007/BF02684798.

[4]

M. R. Herman, Sur les courbes invariantes par les difféomorphismes de l'anneau, Astérisque, 103-104 (1983), 1-221.

[5]

M. R. Herman, Sur les courbes invariantes par les difféomorphismes de l'anneau, Astérisque, 144 (1986), 1-248.

[6]

J. N. Mather, Existence of quasi periodic orbits for twist homeomorphisms of the annulus, Topology, 21 (1982), 457-467. doi: 10.1016/0040-9383(82)90023-4.

[7]

J. N. Mather, A criterion for the non-existence of invariant circle, Publ. Math. IHES, 63 (1986), 153-204. doi: 10.1007/BF02831625.

[8]

J. N. Mather, Modulus of continuity for Peierls's barrier, Periodic Solutions of Hamiltonian Systems and Related Topics, NATO ASI Series C, 209, Reidel, Dordrecht, (1987), 177-202.

[9]

J. N. Mather, Destruction of invariant circles, Ergod. Th. & Dynam. Sys., 8 (1988), 199-214.

[10]

D. Salamon, The Kolmogorov-Arnold-Moser theorem, Math. Phys. Eletron. J., 10 (2004), Paper 3, 37 pp. (electronic).

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