-
7
-
-
84926801345
-
-
The argument is reproduced in E. A. Jackson, Perspectives of Nonlinear Dynamics (Cambridge University, Cambridge, England, 1989), Vol. 2, p. 41ff.
-
-
-
-
14
-
-
84926866766
-
-
P. Collet and J.-P. Eckmann, Iterated Maps on the Interval as Dynamical Systems, Progress in Physics Vol. 1 (Birkhauser-Verlag, Basel, 1980).
-
-
-
-
15
-
-
84926866764
-
-
J. Gleick, Chaos: Making a New Science (Viking, New York, 1987); E. A. Jackson, Perspectives of Nonlinear Dynamics (Cambridge University Press, Cambridge, England, 1989); A. J. Lichtenberg and M. A. Lieberman, Regular and Stochastic Motion (Springer-Verlag, New York, 1983); R.H.G. Helleman, in Fundamental Problems in Statistical Mechanics, edited by E. G. D. Cohen (North-Holland, Amsterdam, 1980), Vol. 5, p. 165; in Long Time Prediction in Dynamics, edited by W. Horton, L. Reichl, and V. Szebehely (Wiley, New York, 1982).
-
-
-
-
17
-
-
84926866765
-
-
see also A. M. Ozorio de Almeida and M. Alfredo, Hamiltonian Systems: Chaos and Quantization (Cambridge University, Cambridge, England, 1988);
-
-
-
-
19
-
-
84926866763
-
-
The symmetric case they studied is different from ours, so the special bifurcations they found are different from those described here.
-
-
-
-
23
-
-
84926801344
-
-
The scaling transformation is written in Eq. (2.4) of the paper by J. B. Delos, S. K. Knudson, and D. W. Noid. The transformation used here differs from that one by a factor of 2 in the definition of r prime. This makes the Hamiltonian identical to that used by Edmonds [15].
-
-
-
-
27
-
-
84926866762
-
-
R. Abraham and J. E. Marsden, Foundations of Mechanics (Benjamin-Cummings, Reading, MA, 1978); D. Ruelle, Elements of Differentiable Dynamics and Bifurcation Theory (Academic, San Diego, 1989); J. Guckenheimar and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, New York, 1983).
-
-
-
-
28
-
-
84926844882
-
-
A trivial example is the two-dimensional anisotropic harmonic oscillator; it has two real periodic orbits at positive energy, and no real orbits at all at negative energy; the trace need not equal 2 at the boundary, curlep =0.
-
-
-
-
29
-
-
84926866761
-
-
T. Poston and I. Stewart, Catastrophe Theory and its Applications (Pitman, London, 1978), especially p. 107ff.
-
-
-
-
30
-
-
84926866760
-
-
It follows that, for example, all typical one-dimensional oscillators are harmonic for small displacements; quartic oscillators exist, but they are rare—one would expect to find them only in systems with special symmetries.
-
-
-
-
33
-
-
84926844881
-
-
Our labels are quasidescriptive: B means balloon, S means snake, and Pk means Pacman with k lips. Combinations like B1S1 can be understood with some imagination. A more definitive labeling scheme was created by Eckhardt and Wintgen [18].
-
-
-
-
35
-
-
84926844880
-
-
The new periodic orbits come ``out of nowhere'' because our dynamics is restricted to real values of coordinates and momenta. If the dynamics is extended to complex coordinates and momenta and the map (2.9) is analytic, then the new periodic orbits come down onto real coordinates from complex phase space.
-
-
-
|
