Reversing k-symmetries in dynamical systems

We generalize the concept of (reversing) symmetries of a dynamical system, i.e. we study dynamical systems that possess symmetry properties only if considered on a proper time scale. In particular (considering dynamical systems with discrete time), the kth iterate of a map may possess more (reversing) symmetries than the map itself. In this way the concepts of (reversing) symmetries and (reversing) symmetry groups are generalized to (reversing) k-symmetries and (reversing) k-symmetry groups. Furthermore, a method is studied for finding orbits that are (k-) symmetric with respect to reversing (k-)symmetries. Firstly an existing method for finding orbits that are symmetric with respect to one reversing symmetry is extended to the case of more than one reversing symmetry and secondly a generalization of this method to the case of reversing k-symmetries is introduced. Some physically relevant examples of dynamical systems possessing reversing k-symmetries are discussed briefly.

[1]  D'Ariano,et al.  Classical and quantum structures in the kicked-top model. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[2]  P. Olver.  Applications of Lie Groups to Differential Equations , 1986 .

[3]  André Vanderbauwhede,et al.  Local bifurcation and symmetry , 1982 .

[4]  H. C. Corben,et al.  Classical Mechanics (2nd ed.) , 1961 .

[5]  J. Lamb,et al.  A natural class of generalized Fibonacci chains , 1994 .

[6]  M. Baake,et al.  Trace maps as 3D reversible dynamical systems with an invariant , 1994 .

[7]  G. Quispel,et al.  Cyclic reversing k-symmetry groups , 1995 .

[8]  R. MacKay.  Equivariant universality classes , 1984 .

[9]  J. Lamb.  Stochastic webs with fourfold rotation symmetry , 1994 .

[10]  Michael Baake,et al.  TRACE MAPS, INVARIANTS, AND SOME OF THEIR APPLICATIONS , 1993 .

[11]  J. Lamb.  Crystallographic symmetries of stochastic webs , 1993 .

[12]  E. Piña,et al.  On the symmetry lines of the standard mapping , 1987 .

[13]  A. L. Loeb,et al.  Color and symmetry , 1972 .

[14]  G. R. W. Quispel,et al.  Chaos and time-reversal symmetry. Order and chaos in reversible dynamical systems , 1992 .

[15]  David H. Sattinger,et al.  Group theoretic methods in bifurcation theory , 1979 .

[16]  Symmetry in regular motion; Relation to chaos , 1990 .

[17]  Symplectic reversible maps, tiles and chaos , 1992 .

[18]  J. Lamb,et al.  Symmetries and reversing symmetries in kicked systems , 1994 .

[19]  C. Eugene Wayne,et al.  Weak Chaos and Quasi-Regular Patterns. By G. M. ZASLAVSKY, R. Z. SAGDEEV, D. A. USIKOV and A. A CHERNIKOV. Cambridge University Press, 1991. 253 pp. £40 or $75 , 1992, Journal of Fluid Mechanics.

[20]  Vladimir Igorevich Arnolʹd,et al.  Problèmes ergodiques de la mécanique classique , 1967 .

[21]  J. W. Humberston.  Classical mechanics , 1980, Nature.

[22]  E. Piña,et al.  Symmetries of the quasi-crystal mapping , 1989 .

[23]  J. Craggs.  Applied Mathematical Sciences , 1973 .

[24]  F. Haake,et al.  Classical and quantum chaos for a kicked top , 1987 .

[25]  M. Golubitsky,et al.  Singularities and groups in bifurcation theory , 1985 .

[26]  Martin Golubitsky,et al.  Symmetry-increasing bifurcation of chaotic attractors , 1988 .

[27]  B. M. Fulk.  MATH , 1992 .

[28]  P. Richter,et al.  A breathing chaos , 1990 .

[29]  Jeroen S. W. Lamb,et al.  Reversing symmetries in dynamical systems , 1992 .

[30]  B. Chirikov.  A universal instability of many-dimensional oscillator systems , 1979 .

[31]  A critical Ising model on the Labyrinth , 1994, solv-int/9902009.

[32]  Chao Tang,et al.  Localization Problem in One Dimension: Mapping and Escape , 1983 .

[33]  Tadatsugu Hatori,et al.  Stochasticity and Symmetry of the Standard Map , 1989 .

[34]  G. Quispel,et al.  Continuous symmetries of differential-difference equations: the Kac-van Moerbeke equation and Painlevé reduction , 1992 .

[35]  J. Lamb,et al.  Local bifurcations on the plane with reversing point group symmetry , 1995 .

[36]  G. Quispel,et al.  Dynamics and K-Symmetries , 1994 .