Is the Hamiltonian geometrical criterion for chaos always reliable

Abstract It is found that the application of a newly developed geometrical criterion, in which negative eigenvalues of the associated matrix determined by the dynamical curvature of a conformal metric for a Hamiltonian system are used to identify the onset of local instability or chaos, is somewhat problematic in some circumstances. In fact, this criterion is neither necessary nor sufficient for the prediction of instability of orbits on a same energy hypersurface because it is not in good agreement with information on unstable or chaotic behavior given by the maximal Lyapunov exponent in general.

[1]  Geodesic Deviation Equation Approach to Chaos , 1999, chao-dyn/9901007.

[2]  K. Maeda,et al.  Chaos in static axisymmetric spacetimes: I. Vacuum case , 1996 .

[3]  Clementi,et al.  Riemannian theory of Hamiltonian chaos and Lyapunov exponents. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[4]  M. Pettini,et al.  On the Riemannian description of chaotic instability in Hamiltonian dynamics. , 1995, Chaos.

[5]  Curvature and chaos in general relativity , 1996, gr-qc/9609003.

[6]  Marcelo Schiffer,et al.  Geometry of Hamiltonian chaos. , 2007, Physical review letters.

[7]  G. Savvidy.  The Yang-Mills classical mechanics as a Kolmogorov K-system , 1983 .

[8]  Xin Wu,et al.  Revisit on ``Ruling out chaos in compact binary systems'' , 2007, 1004.5057.

[9]  Xin Wu,et al.  Lyapunov indices with two nearby trajectories in a curved spacetime , 2006, 1006.5251.

[10]  Xin Wu,et al.  Resurvey of order and chaos in spinning compact binaries , 2008, 1004.5317.

[11]  Fernando Roig,et al.  A COMPARISON BETWEEN METHODS TO COMPUTE LYAPUNOV EXPONENTS , 2001 .

[12]  V. Arnold.  Mathematical Methods of Classical Mechanics , 1974 .

[13]  Xin Wu,et al.  Computation of Lyapunov exponents in general relativity , 2003 .

[14]  Russberg,et al.  Existence of stable orbits in the x2y2 potential. , 1990, Physical review letters.

[15]  Alberto Saa.  On the viability of local criteria for chaos , 2004 .

[16]  Chaos in M(atrix) theory , 1997, hep-th/9710032.

[17]  L. Horwitz,et al.  Detecting order and chaos in three-dimensional Hamiltonian systems by geometrical methods. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  Marco Pettini,et al.  Geometric approach to Hamiltonian dynamics and statistical mechanics , 2000 .

[19]  Pollner,et al.  New method for computing topological pressure. , 1996, Physical review letters.

[20]  Pettini,et al.  Geometric description of chaos in two-degrees-of-freedom Hamiltonian systems. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.