Abstract
We generalize Nambu dynamics to describe self-excited, active systems that evolve on attractors and account for the up-take of energy and activation processes, on the one hand, and damping and inhibitory processes, on the other. An application to rod wielding for haptic length perception and a model for self-propagating systems on two-spheres are discussed.
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References
V.I. Arnold, Mathematical methods of classical mechanics (Springer, Berlin, 1978)
A. Isihara, Statistical physics (Academic Press, New York, 1971)
Y. Nambu, Phys. Rev. D 7, 2405 (1973)
L. Takhtajan, Commun. Math. Phys. 160, 295 (1994)
Y. Nutku, J. Phys. A 36, 7559 (2003)
A. Tegmen, Czech. J. Phys. 54, 749 (2004)
S.A. Pandit, A.D. Gangal, J. Phys. A 31, 2899 (1998)
T.L. Curtright, C. Zachos, New J. Phys. 4, 831 (2002)
T.L. Curtright, C. Zachos, Phys. Rev. D 68, 085001 (2002)
C. Zachos, Phys. Lett. B 570, 82 (2003)
D. Baleanu, Proc. Inst. Math. NAS of Ukraine 50, 611 (2004)
N.G. Pletnev, Siberian Electronic Mathematical Reports 6, 272 (2009)
M. Hirayama, Phys. Rev. D 16, 530 (1977)
C. Gonera, Y. Nuktu, Phys. Lett. A 285, 301 (2001)
A. Tegmen, A. Vercin, Int. J. Mod. Physics A 19, 393 (2004)
O. Babelon, D. Bernard, M. Talon, Introduction to classical integrable systems (Cambridge University Press, Cambridge, 2003)
U. Erdmann, W. Ebeling, L. Schimansky-Geier, F. Schweitzer, Eur. Phys. J. B 15, 105 (2000)
L. Schimansky-Geier, W. Ebeling, U. Erdmann, Acta Phys. Pol. B 36, 1757 (2005)
U. Erdmann, W. Ebeling, Int. J. Bif. Chaos 15, 3623 (2005)
B. Lindner, E.M. Nicola, Eur. Phys. J. ST 157, 43 (2008)
J.W. Rayleigh, Theory of sound (Dover, New York, 1945), first edition published 1894
F. Verhulst, Nonlinear differential equations and dynamical systems, 2nd edn. (Springer, Berlin, 1996)
B.A. Kay, J.A.S. Kelso, E.L. Saltzman, G. Schöner, J. Exp. Psychol. Hum. Percept. Perform. 13, 178 (1987)
B.A. Kay, E.L. Saltzman, J.A.S. Kelso, J. Exp. Psychol. Hum. Percept. Perform. 17, 183 (1987)
P.J. Beek, R.C. Schmidt, A.W. Morris, M. Sim, M.T. Turvey, Biol. Cybern. 73, 499 (1995)
P.J. Beek, W.E.I. Rikkert, P.C.W. van Wieringen, J. Exp. Psychol. Hum. Percept. Perform. 22, 1077 (1996)
P. Silva, M. Moreno, S. Fonseca, M.T. Turvey, Neurosci. Lett. 429, 64 (2009)
F. Schweitzer, W. Ebeling, B. Tilch, Phys. Rev. Lett. 80, 5044 (1998)
F. Schweitzer, Brownian agents and active particles (Springer, Berlin, 2003)
U. Erdmann, W. Ebeling, Fluc. Noise Lett. 2, L145 (2003)
F. Schweitzer, W. Ebeling, B. Tilch, Phys. Rev. E 64, 021110 (2001)
U. Erdmann, W. Ebeling, A. Mikhailov, Phys. Rev. E 71, 051904 (2005)
W. Ebeling, L. Schimansky-Geier, Eur. Phys. J. ST 157, 17 (2008)
F. Schweitzer, L. Schimansky-Geier, Physica A 206, 359 (1994)
L. Schimansky-Geier, M. Meith, H. Rose, H. Malchow, Phys. Lett. A 207, 140 (1995)
A. Czirok, A.L. Barabasi, T. Vicsek, Phys. Rev. Lett. 82, 209 (1999)
A. Czirok, T. Vicsek, Physica A 281, 17 (2000)
H. Haken, Z. Physik 263, 267 (1973)
R. Graham, in Quantum statistics in optics and solid-state physics. Springer tracts in modern physics, Vol. 66, edited by G. Höhler (Springer, Berlin, 1973), pp. 1–97
M. Hongler, D.M. Ryter, Z. Physik B 31, 333 (1978)
R. Feistel, W. Ebeling, Evolution of complex system: self-organization, entropy and development (VEB Verlag, Berlin, 1989)
W. Ebeling, I.M. Sokolov, Statistical thermodynamics and stochastic theory of nonequilibrium systems (World Scientific, Singapore, 2004)
W. Ebeling, Condens. Matt. Physics 3, 285 (2000)
V.A. Makarov, W. Ebeling, M.G. Velarde, Int. J. Bif. Chaos 10, 1075 (2000)
H.U. Bödeker, T.D. Frank, R. Friedrich, H.G. Purwins, in Anomalous fluctuation phenomena in complex systems: plasmas, fluids and financial markets, edited by C. Riccardi, H.E. Roman (Research Signpost, Kerala, 2008), pp. 145–184
T.D. Frank, D. Dotov, M.T. Turvey, A canonical-dissipative approach to control and coordination in the complex system called the agent-task-environment, submitted
T.D. Frank, On a moment-based data analysis method for canonical-dissipative oscillatory systems, Fluc. Noise Lett., in press
T.D. Frank, A. Daffertshofer, Physica A 292, 392 (2001)
T.D. Frank, Phys. Lett. A 319, 173 (2003)
T.D. Frank, Phys. Lett. A 337, 224 (2005)
T.D. Frank, Nonlinear Fokker-Planck equations: Fundamentals and applications (Springer, Berlin, 2005)
J. LaSalle, S. Lefschetz, Stability by Lyapunov's direct method (Academic Press, New York, 1961)
N. Goldenfeld, Sensation and perception (Wadsworth, Pacific Grove, CA, 2002)
H.Y. Solomon, M.T. Turvey, J. Exp. Psychol. Hum. Percept. Perform. 14, 404 (1988)
H.Y. Solomon, M.T. Turvey, G. Burton, J. Exp. Psychol. Hum. Percept. Perform. 15, 58 (1989)
C.C. Pagano, J.M. Kinsella-Shaw, P.E. Cassidy, M.T. Turvey, J. Exp. Psychol. Hum. Percept. Perform. 20, 276 (1994)
C.C. Pagano, K.C. Donahue, Percept. Psychophys. 61, 1336 (1999)
P. Fitzpatrick, C. Carello, M.T. Turvey, Neuroscience 60, 551 (1994)
M.T. Turvey, G. Burton, E.L. Amazeen, M. Butwill, C. Carello, J. Exp. Psychol. Hum. Percept. Perform. 24, 35 (1998)
C. Carello, P. Fitzpatrick, I. Flascher, M.T. Turvey, Percept. Psychophys. 60, 89 (1998)
C. Neumann, Crelle Journal 56, 46 (1859)
A. Ramsey, Dynamics, Part II (Cambridge University Press, London, 1937)
A.R. Plastino, A. Plastino, Physica A 232, 458 (1996)
G.B. Roston, A.R. Plastino, M. Casas, A. Plastino, L.R. da Silva, Eur. Phys. J. B 48, 87 (2005)
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Frank, T. Active systems with Nambu dynamics: with applications to rod wielding for haptic length perception and self-propagating systems on two-spheres. Eur. Phys. J. B 74, 195–203 (2010). https://doi.org/10.1140/epjb/e2010-00083-8
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DOI: https://doi.org/10.1140/epjb/e2010-00083-8