Why Instantaneous Values of the “Covariant” Lyapunov Exponents Depend upon the Chosen State-Space Scale
Ruby Valley Research Institute
Highway Contract 60, Box 601
Ruby Valley, Nevada 89833
E-mail: hooverwilliam@yahoo.com
Received:
Received: 20 November 2013; accepted: 9 December 2013; published online: 31 December 2013
DOI: 10.12921/cmst.2014.20.01.5-8
Abstract:
We explore a simple example of a chaotic thermostated harmonic-oscillator system which exhibits qualitatively different local Lyapunov exponents for simple scale-model constant-volume transformations of its coordinate q and momentum p: { q, p } → { (Q/s), (sP ) }. The time-dependent thermostat variable ζ(t) is unchanged by such scaling. The original (qpζ) motion and the scale-model (QP ζ) version of the motion are physically identical. But both the local Gram-Schmidt Lyapunov exponents and the related local “covariant” exponents change with the change of scale. Thus this model furnishes a clearcut chaotic time-reversible example showing how and why both the local Lyapunov exponents and covariant exponents vary with the scale factor s.
Key words:
References:
[1] H. Bosetti, H.A. Posch, Ch. Dellago and Wm.G. Hoover,
Time-Reversal Symmetry and Covariant Lyapunov Vectors
for Simple Particle Models in and out of Thermal Equilib-
rium, arXiv:1004.4473, Version 1 (2010); Physical Review E
82, 046218 (2010).
[2] H.A. Posch, Symmetry Properties of Orthogonal and Covari-
ant Lyapunov Vectors and Their Exponents, arXiv:1107.4032
(2012); Journal of Physics A: Mathematical and Theoretical
46, 254006 (2013).
[3] H-L. Yang and G. Radons, Comparison between Covariant
and Orthogonal Lyapunov Vectors, Physical Review E 82,
046204 (2010).
[4] F. Ginelli, P. Poggi, A. Turchi, H. Chaté, R. Livi, and A.
Politi, Characterizing Dynamics with Covariant Lyapunov
Vectors Physical Review Letters 99, 130601 (2007).
[5] M. Romero-Bastida, D. Pazó, J.M. López, and M.A. Ro-
dríguez, Structure of Characteristic Lyapunov Vectors in
Anharmonic Hamiltonian Lattices, Physical Review E 82,
036205 (2010).
[6] C.L. Wolfe and R.M. Samelson. An Efficient Method for
Recovering Lyapunov Vectors from Singular Vectors, Tellus
59A, 355-366 (2007).
[7] W.G. Hoover and H.A. Posch, Direct Measurement of Lya-
punov Exponents, Physics Letters A 113, 82-84 (1985).
[8] Wm.G. Hoover and C.G. Hoover, Local Gram-Schmidt and
Covariant Lyapunov Vectors and Exponents for Three Har-
monic Oscillator Problems, Communications in Nonlinear
Science and Numerical Simulation 17, 1043-1054 (2012).
[9] Wm.G. Hoover, C.G. Hoover, and H.A. Posch, Lyapunov In-
stability of Pendulums, Chains, and Strings, Physical Review
A 41, 2999-3004 (1990).
[10] H.A. Posch and Wm.G. Hoover, Time-Reversible Dissipative
Attractors in Three and Four Phase-Space Dimensions, Phys-
ical Review E 55, 6803-6810 (1997).
[11] Wm.G. Hoover, Canonical Dynamics: Equilibrium Phase-
Space Distributions, Physical Review A 31, 1695-97 (1985).
[12] H.A. Posch, Wm.G. Hoover, and F.J. Vesely, Canonical Dy-
namics of the Nosé Oscillator: Stability, Order, and Chaos,
Physical Review A 33, 4253-4265 (1986).
[13] Wm.G. Hoover, Remark on ‘Some Simple Chaotic Flows’,
Physical Review E 51, 759-760 (1995).
[14] J.C. Sprott, Some Simple Chaotic Flows, Physical Review E
50, R647-R650 (1994).
[15] Wm.G. Hoover and C.G. Hoover, Time-Symmetry Breaking
in Hamiltonian Mechanics, arXiv 1302.2533 (2013); Com-
putational Methods in Science and Technology 19, 77-87
(2013).
We explore a simple example of a chaotic thermostated harmonic-oscillator system which exhibits qualitatively different local Lyapunov exponents for simple scale-model constant-volume transformations of its coordinate q and momentum p: { q, p } → { (Q/s), (sP ) }. The time-dependent thermostat variable ζ(t) is unchanged by such scaling. The original (qpζ) motion and the scale-model (QP ζ) version of the motion are physically identical. But both the local Gram-Schmidt Lyapunov exponents and the related local “covariant” exponents change with the change of scale. Thus this model furnishes a clearcut chaotic time-reversible example showing how and why both the local Lyapunov exponents and covariant exponents vary with the scale factor s.
Key words:
References:
[1] H. Bosetti, H.A. Posch, Ch. Dellago and Wm.G. Hoover,
Time-Reversal Symmetry and Covariant Lyapunov Vectors
for Simple Particle Models in and out of Thermal Equilib-
rium, arXiv:1004.4473, Version 1 (2010); Physical Review E
82, 046218 (2010).
[2] H.A. Posch, Symmetry Properties of Orthogonal and Covari-
ant Lyapunov Vectors and Their Exponents, arXiv:1107.4032
(2012); Journal of Physics A: Mathematical and Theoretical
46, 254006 (2013).
[3] H-L. Yang and G. Radons, Comparison between Covariant
and Orthogonal Lyapunov Vectors, Physical Review E 82,
046204 (2010).
[4] F. Ginelli, P. Poggi, A. Turchi, H. Chaté, R. Livi, and A.
Politi, Characterizing Dynamics with Covariant Lyapunov
Vectors Physical Review Letters 99, 130601 (2007).
[5] M. Romero-Bastida, D. Pazó, J.M. López, and M.A. Ro-
dríguez, Structure of Characteristic Lyapunov Vectors in
Anharmonic Hamiltonian Lattices, Physical Review E 82,
036205 (2010).
[6] C.L. Wolfe and R.M. Samelson. An Efficient Method for
Recovering Lyapunov Vectors from Singular Vectors, Tellus
59A, 355-366 (2007).
[7] W.G. Hoover and H.A. Posch, Direct Measurement of Lya-
punov Exponents, Physics Letters A 113, 82-84 (1985).
[8] Wm.G. Hoover and C.G. Hoover, Local Gram-Schmidt and
Covariant Lyapunov Vectors and Exponents for Three Har-
monic Oscillator Problems, Communications in Nonlinear
Science and Numerical Simulation 17, 1043-1054 (2012).
[9] Wm.G. Hoover, C.G. Hoover, and H.A. Posch, Lyapunov In-
stability of Pendulums, Chains, and Strings, Physical Review
A 41, 2999-3004 (1990).
[10] H.A. Posch and Wm.G. Hoover, Time-Reversible Dissipative
Attractors in Three and Four Phase-Space Dimensions, Phys-
ical Review E 55, 6803-6810 (1997).
[11] Wm.G. Hoover, Canonical Dynamics: Equilibrium Phase-
Space Distributions, Physical Review A 31, 1695-97 (1985).
[12] H.A. Posch, Wm.G. Hoover, and F.J. Vesely, Canonical Dy-
namics of the Nosé Oscillator: Stability, Order, and Chaos,
Physical Review A 33, 4253-4265 (1986).
[13] Wm.G. Hoover, Remark on ‘Some Simple Chaotic Flows’,
Physical Review E 51, 759-760 (1995).
[14] J.C. Sprott, Some Simple Chaotic Flows, Physical Review E
50, R647-R650 (1994).
[15] Wm.G. Hoover and C.G. Hoover, Time-Symmetry Breaking
in Hamiltonian Mechanics, arXiv 1302.2533 (2013); Com-
putational Methods in Science and Technology 19, 77-87
(2013).
