Instantaneous Pairing of Lyapunov Exponents in Chaotic Hamiltonian Dynamics and the 2017 Ian Snook Prizes
Ruby Valley Research Institute
Highway Contract 60, Box 601, Ruby Valley, Nevada 89833, USA
E-mail: hooverwilliam@yahoo.com
Received:
Received: 11 February 2017; accepted: 04 March 2017; published online: 24 March 2017
DOI: 10.12921/cmst.2017.0000011
Abstract:
The time-averaged Lyapunov exponents, {λi}, support a mechanistic description of the chaos generated in and by nonlinear dynamical systems. The exponents are ordered from largest to smallest with the largest one describing the exponential growth rate of the (small) distance between two neighboring phase-space trajectories. Two exponents, λ1 + λ2, describe the rate for areas defined by three nearby trajectories. λ1 + λ2 + λ3 is the rate for volumes defined by four nearby trajectories, and so on. Lyapunov exponents for Hamiltonian systems are symmetric. The time-reversibility of the motion equations links the growth and decay rates together in pairs. This pairing provides a more detailed explanation than Liouville’s for the conservation of phase volume in Hamiltonian mechanics. Although correct for long-time averages, the dependence of trajectories on their past is responsible for the observed lack of detailed pairing for the instantaneous “local” exponents, {λi(t)}. The 2017 Ian Snook Prizes will be awarded to the author(s) of an accessible and pedagogical discussion of local Lyapunov instability in small systems. We desire that this discussion build on the two nonlinear models described here, a double pendulum with Hooke’s-Law links and a periodic chain of Hooke’s-Law particles tethered to their lattice sites. The latter system is the φ4 model popularized by Aoki and Kusnezov. A four-particle version is small enough for comprehensive numerical work and large enough to illustrate ideas of general validity.
Key words:
References:
[1] Wm.G. Hoover, C. G. Hoover, H. A. Posch, Lyapunov Instability of Pendula, Chains and Strings, Physical Review A 41, 2999–3004 (1990).
[2] H.A. Posch, Symmetry Properties of Orthogonal and Covariant Lyapunov Vectors and Their Exponents, Journal of Physics A 46, 254006 (2013).
[3] I. Shimada, T. Nagashima, A Numerical Approach to Ergodic Problems of Dissipative Dynamical Systems, Progress of The- oretical Physics 61, 1605–1616 (1979).
[4] G. Benettin, L. Galgani, A. Giorgilli, J.-M. Strelcyn, Lyapunov Characteristic Exponents for Smooth Dynamical Systems and for Hamiltonian Systems; a Method for Computing All of Them, Parts I and II: Theory and Numerical Application, Mec- canica 15, 9–20 and 21–30 (1980).
[5] J.-P. Eckmann, D. Ruelle, Ergodic Theory of Chaos and Strange Attactors, Reviews of Modern Physics 57, 617–656 (1985).
[6] B.A. Bailey, Local Lyapunov Exponents; Predictability Depends on Where You Are, in Nonlinear Dynamics and Economics, W. A. Barnett, A. P. Kirman, M. Salmon, editors (Cambridge University Press, 1996) pages 345–359.
[7] Hong-Liu Yang, G. Radons, Comparison of Covariant and Orthogonal Lyapunov Vectors, Physical Review E 82, 046204 (2010) = arχiv:1008.1941.
[8] H.A. Posch, R. Hirschl, Simulation of Billiards and of Hard Body Fluids in Hard Ball Systems and the Lorentz Gas, En- cyclopedia of the Mathematical Sciences 101, edited by D. Szász (Springer Verlag, Berlin, 2000), pages 269–310.
[9] K. Aoki, D. Kusnezov, Nonequilibrium Statistical Mechanics of Classical Lattice φ4 Field Theory, Annals of Physics 295, 50–80 (2002).
[10] K. Aoki, Stable and Unstable Periodic Orbits in the One- Dimensional Lattice φ4 Theory, Physical Review E 94, 042209 (2016).
[11] Wm.G. Hoover and K. Aoki, Order and Chaos in the One- Dimensional φ4 Model: N-Dependence and the Second Law of Thermodynamics, Communications in Nonlinear Science and Numerical Simulation 49, 192–201 (2017), arχiv 1605.07721.
[12] Wm.G. Hoover, J.C. Sprott, C.G. Hoover, Adaptive Runge-Kutta Integration for Stiff Systems: Comparing Nosé and Nosé- Hoover Dynamics for the Harmonic Oscillator, American Journal of Physics 84, 786–794 (2016).
[13] Wm. G. Hoover and C. G. Hoover, Time-Symmetry Break- ing in Hamiltonian Mechanics, Computational Methods in Science and Technology 19, 77–87 (2013) = arχiv 1302.2533.
[14] Wm.G. Hoover, C.G. Hoover, What is Liquid? Lyapunov Instability Reveals Symmetry-Breaking Irreversibilities Hid- den Within Hamilton’s Many-Body Equations of Motion, Con- densed Matter Physics 18, 1–13 (2015) = arχiv 1405.2485.
[15] C. P. Dettmann and G. P Morriss, Proof of Lyapunov Expo- nent Pairing for Systems at Constant Kinetic Energy, Physical Review E 53, R5545-R5548 (1996).
The time-averaged Lyapunov exponents, {λi}, support a mechanistic description of the chaos generated in and by nonlinear dynamical systems. The exponents are ordered from largest to smallest with the largest one describing the exponential growth rate of the (small) distance between two neighboring phase-space trajectories. Two exponents, λ1 + λ2, describe the rate for areas defined by three nearby trajectories. λ1 + λ2 + λ3 is the rate for volumes defined by four nearby trajectories, and so on. Lyapunov exponents for Hamiltonian systems are symmetric. The time-reversibility of the motion equations links the growth and decay rates together in pairs. This pairing provides a more detailed explanation than Liouville’s for the conservation of phase volume in Hamiltonian mechanics. Although correct for long-time averages, the dependence of trajectories on their past is responsible for the observed lack of detailed pairing for the instantaneous “local” exponents, {λi(t)}. The 2017 Ian Snook Prizes will be awarded to the author(s) of an accessible and pedagogical discussion of local Lyapunov instability in small systems. We desire that this discussion build on the two nonlinear models described here, a double pendulum with Hooke’s-Law links and a periodic chain of Hooke’s-Law particles tethered to their lattice sites. The latter system is the φ4 model popularized by Aoki and Kusnezov. A four-particle version is small enough for comprehensive numerical work and large enough to illustrate ideas of general validity.
Key words:
References:
[1] Wm.G. Hoover, C. G. Hoover, H. A. Posch, Lyapunov Instability of Pendula, Chains and Strings, Physical Review A 41, 2999–3004 (1990).
[2] H.A. Posch, Symmetry Properties of Orthogonal and Covariant Lyapunov Vectors and Their Exponents, Journal of Physics A 46, 254006 (2013).
[3] I. Shimada, T. Nagashima, A Numerical Approach to Ergodic Problems of Dissipative Dynamical Systems, Progress of The- oretical Physics 61, 1605–1616 (1979).
[4] G. Benettin, L. Galgani, A. Giorgilli, J.-M. Strelcyn, Lyapunov Characteristic Exponents for Smooth Dynamical Systems and for Hamiltonian Systems; a Method for Computing All of Them, Parts I and II: Theory and Numerical Application, Mec- canica 15, 9–20 and 21–30 (1980).
[5] J.-P. Eckmann, D. Ruelle, Ergodic Theory of Chaos and Strange Attactors, Reviews of Modern Physics 57, 617–656 (1985).
[6] B.A. Bailey, Local Lyapunov Exponents; Predictability Depends on Where You Are, in Nonlinear Dynamics and Economics, W. A. Barnett, A. P. Kirman, M. Salmon, editors (Cambridge University Press, 1996) pages 345–359.
[7] Hong-Liu Yang, G. Radons, Comparison of Covariant and Orthogonal Lyapunov Vectors, Physical Review E 82, 046204 (2010) = arχiv:1008.1941.
[8] H.A. Posch, R. Hirschl, Simulation of Billiards and of Hard Body Fluids in Hard Ball Systems and the Lorentz Gas, En- cyclopedia of the Mathematical Sciences 101, edited by D. Szász (Springer Verlag, Berlin, 2000), pages 269–310.
[9] K. Aoki, D. Kusnezov, Nonequilibrium Statistical Mechanics of Classical Lattice φ4 Field Theory, Annals of Physics 295, 50–80 (2002).
[10] K. Aoki, Stable and Unstable Periodic Orbits in the One- Dimensional Lattice φ4 Theory, Physical Review E 94, 042209 (2016).
[11] Wm.G. Hoover and K. Aoki, Order and Chaos in the One- Dimensional φ4 Model: N-Dependence and the Second Law of Thermodynamics, Communications in Nonlinear Science and Numerical Simulation 49, 192–201 (2017), arχiv 1605.07721.
[12] Wm.G. Hoover, J.C. Sprott, C.G. Hoover, Adaptive Runge-Kutta Integration for Stiff Systems: Comparing Nosé and Nosé- Hoover Dynamics for the Harmonic Oscillator, American Journal of Physics 84, 786–794 (2016).
[13] Wm. G. Hoover and C. G. Hoover, Time-Symmetry Break- ing in Hamiltonian Mechanics, Computational Methods in Science and Technology 19, 77–87 (2013) = arχiv 1302.2533.
[14] Wm.G. Hoover, C.G. Hoover, What is Liquid? Lyapunov Instability Reveals Symmetry-Breaking Irreversibilities Hid- den Within Hamilton’s Many-Body Equations of Motion, Con- densed Matter Physics 18, 1–13 (2015) = arχiv 1405.2485.
[15] C. P. Dettmann and G. P Morriss, Proof of Lyapunov Expo- nent Pairing for Systems at Constant Kinetic Energy, Physical Review E 53, R5545-R5548 (1996).
