An Extension of Greene's Criterion for Conformally Symplectic Systems and a Partial Justification

Greene's criterion for twist mappings asserts the existence of smooth invariant circles with preassigned rotation number if and only if the periodic trajectories with frequency approaching that of the quasi-periodic orbit are at the border of linear stability. We formulate an extension of this criterion for conformally symplectic systems in any dimension and prove one direction of the implication, namely, that if there is a smooth invariant attractor, we can predict the eigenvalues of the periodic orbits whose frequencies approximate that of the torus for values of the parameters close to that of the attractor. The proof of this result is very different from the proof in the area-preserving case, since in the conformally symplectic case the existence of periodic orbits requires adjusting parameters. Also, as shown in [R. Calleja, A. Celletti, and R. de la Llave, J. Dynam. Differential Equations, 55 (2013), pp. 821--841], in the conformally symplectic case there are no Birkhoff invariants giving obstructio...

[1]  John M. Greene,et al.  A method for determining a stochastic transition , 1979, Hamiltonian Dynamical Systems.

[2]  A. Celletti,et al.  Periodic and quasi--periodic orbits of the dissipative standard map , 2011 .

[3]  R. MacKay.  Greene's residue criterion , 1992 .

[4]  A. Banyaga.  Some properties of locally conformal symplectic structures , 2002 .

[5]  Moser stability for locally conformally symplectic structures , 2008, 0811.0621.

[6]  J. Gillis,et al.  Asymptotic Methods in the Theory of Non‐Linear Oscillations , 1963 .

[7]  Sylvain Crovisier.  Ensembles de torsion nulle des applications déviant la verticale , 2003 .

[8]  J. Moser.  A rapidly convergent iteration method and non-linear partial differential equations - I , 1966 .

[9]  R. Llave,et al.  Cohomology equations near hyperbolic points and geometric versions of sternberg linearization theorem , 1996 .

[10]  R. de la Llave,et al.  Time-dependent scattering theory for ODEs and applications to reaction dynamics , 2011 .

[11]  George R. Sell,et al.  Existence of dichotomies and invariant splittings for linear differential systems, II☆ , 1976 .

[12]  Shlomo Sternberg,et al.  Convexity properties of the moment mapping. II , 1982 .

[13]  G. R. Hall.  A topological version of a theorem of Mather on twist maps , 1984, Ergodic Theory and Dynamical Systems.

[14]  L. Arbogast.  Du calcul des dérivations , 1800 .

[15]  V. Arnold,et al.  Mathematical aspects of classical and celestial mechanics , 1997 .

[16]  Harold Levine,et al.  Singularities of differentiable mappings , 1971 .

[17]  Adam M. Fox,et al.  Greene’s residue criterion for the breakup of invariant tori of volume-preserving maps , 2012, 1205.6143.

[18]  R. Llave,et al.  Fast numerical computation of quasi-periodic equilibrium states in 1D statistical mechanics, including twist maps , 2009 .

[19]  W. D. Melo,et al.  ONE-DIMENSIONAL DYNAMICS , 2013 .

[20]  Michael Atiyah,et al.  Convexity and Commuting Hamiltonians , 1982 .

[21]  Rafael de la Llave,et al.  Numerical calculation of domains of analyticity for perturbation theories in the presence of small divisors , 1992 .

[22]  Neil Fenichel.  Persistence and Smoothness of Invariant Manifolds for Flows , 1971 .

[23]  R. McLachlan,et al.  Conformal Hamiltonian systems , 2001 .

[24]  Rafael de la Llave,et al.  A numerically accessible criterion for the breakdown of quasi-periodic solutions and its rigorous justification , 2010 .

[25]  M. Casdagli.  Periodic orbits for dissipative twist maps , 1987, Ergodic Theory and Dynamical Systems.

[26]  R. de la Llave,et al.  The obstruction criterion for non-existence of invariant circles and renormalization , 2006 .

[27]  Stathis Tompaidis,et al.  Numerical Study of Invariant Sets of a Quasiperiodic Perturbation of a Symplectic Map , 1996, Exp. Math..

[28]  Jordi-Lluís Figueras,et al.  Collision of invariant bundles of quasi-periodic attractors in the dissipative standard map. , 2012, Chaos.

[29]  J. Moser.  On the volume elements on a manifold , 1965 .

[30]  R. Llave,et al.  Local Behavior Near Quasi-Periodic Solutions of Conformally Symplectic Systems , 2013 .

[31]  H. Rüssmann.  On a new proof of Moser's twist mapping theorem , 1976 .

[32]  Ralph Abraham,et al.  Foundations Of Mechanics , 2019 .

[33]  R. Llave,et al.  A rigorous partial justification of Greene's criterion , 1992 .

[34]  R. Llave,et al.  Singularity Theory for Non-Twist Kam Tori , 2014 .

[35]  Stathis Tompaidis,et al.  Approximation of Invariant Surfaces by Periodic Orbits in High-Dimensional Maps: Some Rigorous Results , 1996, Exp. Math..

[36]  R. Llave,et al.  Canonical perturbation theory of Anosov systems and regularity results for the Livsic cohomology equation , 1986 .

[37]  J. Meiss.  The destruction of tori in volume-preserving maps , 2011, 1103.0050.

[38]  I. Vaisman.  LOCALLY CONFORMAL SYMPLECTIC MANIFOLDS , 1985 .

[39]  ESITheErwinSchrodingerInternational InstituteforMathematicalPhysics Boltzmanngasse9 A-1090Wien,Austria OntheGroupofDieomorphismsPreserving aLocallyConformalSymplecticStructure , 2022 .

[40]  Carlangelo Liverani,et al.  Conformally Symplectic Dynamics and Symmetry of the Lyapunov Spectrum , 1998 .

[41]  Morriss,et al.  Proof of Lyapunov exponent pairing for systems at constant kinetic energy. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[42]  W. A. Coppel.  Dichotomies in Stability Theory , 1978 .

[43]  T. M. Seara,et al.  GEOMETRIC PROPERTIES OF THE SCATTERING MAP OF A NORMALLY HYPERBOLIC INVARIANT MANIFOLD , 2008 .

[44]  A. Celletti,et al.  On the break-down threshold of invariant tori in four dimensional maps , 2004 .

[45]  G. Floquet,et al.  Sur les équations différentielles linéaires à coefficients périodiques , 1883 .

[46]  A. Celletti,et al.  Quasi-Periodic Attractors in Celestial Mechanics , 2009 .

[47]  N N Nekhoroshev,et al.  AN EXPONENTIAL ESTIMATE OF THE TIME OF STABILITY OF NEARLY-INTEGRABLE HAMILTONIAN SYSTEMS , 1977 .

[48]  Dressler,et al.  Symmetry property of the Lyapunov spectra of a class of dissipative dynamical systems with viscous damping. , 1988, Physical review. A, General physics.

[49]  Alan Weinstein,et al.  Lectures on Symplectic Manifolds , 1977 .

[50]  A. Celletti,et al.  Breakdown of invariant attractors for the dissipative standard map. , 2010, Chaos.

[51]  Yakov Pesin,et al.  Lectures on Partial Hyperbolicity and Stable Ergodicity , 2004 .

[52]  Helmut Rüssmann,et al.  On optimal estimates for the solutions of linear partial differential equations of first order with constant coefficients on the torus , 1975 .

[53]  T. M. Seara,et al.  A Geometric Approach to the Existence of Orbits with Unbounded Energy in Generic Periodic Perturbations by a Potential of Generic Geodesic Flows of ?2} , 2000 .

[54]  R. Llave,et al.  A KAM theory for conformally symplectic systems: Efficient algorithms and their validation , 2013 .