Visual explorations of dynamics: The standard map

The Macintosh application StdMap allows easy exploration of many of the phenomena of area-preserving mappings. This tutorial explains some of these phenomena and presents a number of simple experiments centered on the use of this program.

[1]  Michael Frazier,et al.  Studies in Advanced Mathematics , 2004 .

[2]  H. Poincaré,et al.  Les méthodes nouvelles de la mécanique céleste , 1899 .

[3]  S. Smale,et al.  Finding a horseshoe on the beaches of Rio , 1998 .

[4]  John M. Greene,et al.  A method for determining a stochastic transition , 1979, Hamiltonian Dynamical Systems.

[5]  Rafael de la Llave,et al.  A Tutorial on Kam Theory , 2003 .

[6]  Charles F. F. Karney Long-time correlations in the stochastic regime , 1983, nlin/0501023.

[7]  R. MacKay,et al.  Linear Stability of Periodic Orbits in Lagrangian Systems , 1983, Hamiltonian Dynamical Systems.

[8]  J. Bialek,et al.  Fractal Diagrams for Hamiltonian Stochasticity , 1982, Hamiltonian Dynamical Systems.

[9]  J. Pöschel,et al.  A lecture on the classical KAM theorem , 2009, 0908.2234.

[10]  A. Lichtenberg,et al.  Regular and Chaotic Dynamics , 1992 .

[11]  R. Llave,et al.  Smooth Ergodic Theory and Its Applications , 2001 .

[12]  R. MacKay Greene's residue criterion , 1992 .

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

[14]  J. M. Greene THE CALCULATION OF KAM SURFACES * , 1980 .

[15]  James D. Meiss,et al.  Differential dynamical systems , 2007, Mathematical modeling and computation.

[16]  V. Gelfreich,et al.  Splitting of separatrices: perturbation theory and exponential smallness , 2001 .

[17]  Rafael de la Llave,et al.  KAM Theory and a Partial Justification of Greene's Criterion for Nontwist Maps , 2000, SIAM J. Math. Anal..

[18]  Henri Poincaré,et al.  méthodes nouvelles de la mécanique céleste , 1892 .

[19]  I. Hargittai,et al.  The mathematical intelligencer , 2008 .

[20]  Robert S. MacKay,et al.  Renormalisation in Area-Preserving Maps , 1993 .

[21]  Seng Hu A Variational Principle Associated to Positive Tilt Maps , 1998 .

[22]  I. Percival,et al.  Critical function and modular smoothing , 1990 .

[23]  James D. Meiss,et al.  Transport in Hamiltonian systems , 1984 .

[24]  Characterization of fat fractals in nonlinear dynamical systems. , 1986, Physical review letters.

[25]  J. Meiss Symplectic maps, variational principles, and transport , 1992 .

[26]  D. Chillingworth DYNAMICAL SYSTEMS: STABILITY, SYMBOLIC DYNAMICS AND CHAOS , 1998 .

[27]  J. Stark,et al.  Locally most robust circles and boundary circles for area-preserving maps , 1992 .

[28]  Robert W. Easton,et al.  Geometric methods for discrete dynamical systems , 1998 .

[29]  Dana D. Hobson,et al.  An efficient method for computing invariant manifolds of planar maps , 1993 .

[30]  James D. Meiss,et al.  Algebraic decay in self-similar Markov chains , 1985 .

[31]  P. Boyland Rotation sets and Morse decompositions in twist maps , 1988, Ergodic Theory and Dynamical Systems.

[32]  J. Meiss Transient measures in the standard map , 1994 .

[33]  J. Mather Variational construction of orbits of twist diffeomorphisms , 1991 .

[34]  S. Aubry The twist map, the extended Frenkel-Kontorova model and the devil's staircase , 1983 .

[35]  V. F. Lazutkin,et al.  Splitting of separatrices for standard and semistandard mappings , 1989 .

[36]  M. Feigenbaum,et al.  Universal Behaviour in Families of Area-Preserving Maps , 1981, Hamiltonian Dynamical Systems.

[37]  A. Veselov,et al.  What Is an Integrable Mapping , 1991 .

[38]  A. Apte,et al.  Meanders and reconnection-collision sequences in the standard nontwist map. , 2004, Chaos.

[39]  J. Lamb,et al.  Time-reversal symmetry in dynamical systems: a survey , 1998 .

[40]  Y. Suris,et al.  Integrable mappings of the standard type , 1989 .

[41]  A. Veselov,et al.  New integrable deformations of the Calogero-Moser quantum problem , 1996 .

[42]  Christophe Golé Symplectic Twist Maps: Global Variational Techniques , 2001 .

[43]  A. Andrianov Composition of solutions of quadratic Diophantine equations , 1991 .

[44]  S. Croucher,et al.  Surveys , 1965, Understanding Communication Research Methods.

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

[46]  James D. Meiss,et al.  Heteroclinic orbits and Flux in a perturbed integrable Suris map , 1996, math/9604231.

[47]  J. M. Greene Two‐Dimensional Measure‐Preserving Mappings , 1968 .

[48]  J. Mather,et al.  Existence of quasi-periodic orbits for twist homeomorphisms of the annulus , 1982 .