Time-Reversible Ergodic Maps and the 2015 Ian Snook Prizes
Ruby Valley Research Institute
Highway Contract 60, Box 601
Ruby Valley, Nevada 89833
E-mail: hooverwilliam@yahoo.com
Received:
Received: 03 July 2015; accepted: 06 July 2015; published online: 24 August 2015
DOI: 10.12921/cmst.2015.21.03.003
Abstract:
The time reversibility characteristic of Hamiltonian mechanics has long been extended to nonHamiltonian
dynamical systems modeling nonequilibrium steady states with feedback-based thermostats and ergostats. Typical solutions
are multifractal attractor-repellor phase-space pairs with reversed momenta and unchanged coordinates, (q, p) ←→ (q, −p).
Weak control of the temperature, ∝ p2 and its fluctuation, resulting in ergodicity, has recently been achieved in a three-
dimensional time-reversible model of a heat-conducting harmonic oscillator. Two-dimensional cross sections of such
nonequilibrium flows can be generated with time-reversible dissipative maps yielding æsthetically interesting attractor-
repellor pairs. We challenge the reader to find and explore such time-reversible dissipative maps. This challenge is the 2015
Snook-Prize Problem.
Key words:
algorithms, chaos, ergodicity, maps, mapsergodicity, time-reversible flows
References:
[1] W. G. Hoover and C. G. Hoover, Simulation and Control of Chaotic Nonequilibrium Systems (World Scientific Publishers, Singapore, 2015).
[2] H. A. Posch and Wm. G. Hoover, Time-Reversible Dissipative Attractors in Three and Four Phase-Space Dimensions, Physical Review E 55, 6803-6810 (1997).
[3] W. G. Hoover, J. C. Sprott, and C. G. Hoover, Nonequilibrium Molecular Dynamics and Dynamical Systems Theory for Small Systems with Time-Reversible Motion Equations, Molecular Simulation (in press, 2015).
[4] W. G. Hoover, O. Kum, and H. A. Posch, Time-Reversible Dissipative Ergodic Maps, Physical Review E 53, 2123-2129 (1996).
[5] J. Kumicák, Irreversibility in a Simple Reversible Model, Physical Review E 71, 016115 (2005), arχiv nlin/0510016.
[6] L. Ermann and D. L. Shepelyansky, Arnold Cat Map, Ulam Method, and Time Reversal, arχiv1107.0437.
[7] W. G. Hoover and C. G. Hoover, Comparison of Very Smooth Cell-Model Trajectories Using Five Symplectic and Two Runge-Kutta Integrators, Computational Methods in Science and Technology 21 (to appear, 2015), arχiv 1504.00620.
[8] D. Faranda, Analysis of Roundoff Errors with Reversibility Test as a Dynamical Indicator, arχiv 1205.3060.
The time reversibility characteristic of Hamiltonian mechanics has long been extended to nonHamiltonian
dynamical systems modeling nonequilibrium steady states with feedback-based thermostats and ergostats. Typical solutions
are multifractal attractor-repellor phase-space pairs with reversed momenta and unchanged coordinates, (q, p) ←→ (q, −p).
Weak control of the temperature, ∝ p2 and its fluctuation, resulting in ergodicity, has recently been achieved in a three-
dimensional time-reversible model of a heat-conducting harmonic oscillator. Two-dimensional cross sections of such
nonequilibrium flows can be generated with time-reversible dissipative maps yielding æsthetically interesting attractor-
repellor pairs. We challenge the reader to find and explore such time-reversible dissipative maps. This challenge is the 2015
Snook-Prize Problem.
Key words:
algorithms, chaos, ergodicity, maps, mapsergodicity, time-reversible flows
References:
[1] W. G. Hoover and C. G. Hoover, Simulation and Control of Chaotic Nonequilibrium Systems (World Scientific Publishers, Singapore, 2015).
[2] H. A. Posch and Wm. G. Hoover, Time-Reversible Dissipative Attractors in Three and Four Phase-Space Dimensions, Physical Review E 55, 6803-6810 (1997).
[3] W. G. Hoover, J. C. Sprott, and C. G. Hoover, Nonequilibrium Molecular Dynamics and Dynamical Systems Theory for Small Systems with Time-Reversible Motion Equations, Molecular Simulation (in press, 2015).
[4] W. G. Hoover, O. Kum, and H. A. Posch, Time-Reversible Dissipative Ergodic Maps, Physical Review E 53, 2123-2129 (1996).
[5] J. Kumicák, Irreversibility in a Simple Reversible Model, Physical Review E 71, 016115 (2005), arχiv nlin/0510016.
[6] L. Ermann and D. L. Shepelyansky, Arnold Cat Map, Ulam Method, and Time Reversal, arχiv1107.0437.
[7] W. G. Hoover and C. G. Hoover, Comparison of Very Smooth Cell-Model Trajectories Using Five Symplectic and Two Runge-Kutta Integrators, Computational Methods in Science and Technology 21 (to appear, 2015), arχiv 1504.00620.
[8] D. Faranda, Analysis of Roundoff Errors with Reversibility Test as a Dynamical Indicator, arχiv 1205.3060.
