Abstract
In this chapter we study a larger class of dynamical systems that include but go beyond Hamiltonian systems. We are interested, on the one hand, in dissipative systems , i.e. systems that lose energy through frictional forces or into which energy is fed from exterior sources, and, on the other hand, in discrete, or discretized, systems such as those generated by studying flows by means of the Poincaré mapping.
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Notes
- 1.
In Sect. 5.5.4 symplectic transformations are defined without reference to coordinates, see definition SYT. If these are chosen to be infinitesimal, \(F=\mathrm{i}d+\varepsilon {\scriptstyle {\mathbf {\mathsf{{A}}}}}\), then to first order in \(\varepsilon \) relation (6.21) is obtained in the coordinate-free form \(\omega (Ae,e^\prime )+\omega (e, Ae^\prime )=0\).
- 2.
In systems with \(f=1\), i.e. with a two-dimensional phase space, and keeping clear from saddle point equilibria, the deformation can be no more than linear in time, cf. Exercise 6.3.
- 3.
H, \(I_1\), and \(I_2\) are in involution, for definitions cf. Sect. 2.37.2.
- 4.
The codimension of a bifurcation is defined to be the smallest dimension of a parameter space \(\{\mu _1,\ldots ,\mu _k\}\) for which this bifurcation does occur.
- 5.
In early Greek cosmology chaos meant “the primeval emptiness of the universe” or, alternatively, “the darkness of the underworld”. The modern meaning is derived from Ovid , who defined chaos as “the original disordered and formless mass from which the maker of the Cosmos produced the ordered universe” (The New Encyclopedia Britannica). Note that the loan-word gas is derived from the word chaos. It was introduced by J.B. von Helmont , a 17th-century chemist in Brussels.
- 6.
In proving this formula one makes use of the “orthogonality relation”
$$ {1 \over n} \sum ^n_{\sigma = 1} \mathrm{e}^{\mathrm{i}2\pi m \sigma /n} = \delta _{m0}\; , $$$$ m = 0, 1, \ldots , n - 1\; . $$(see also Exercise 6.15).
- 7.
Earlier it was held that chaotic motion would occur only in systems with very many degrees of freedom, such as gases in macroscopic vessels.
- 8.
See R.S. Shaw : “Strange attractors, chaotic behavior and information flow”, Z. Naturforschung A36, (1981) 80.
- 9.
This equation takes its name from its use in modeling the evolution of, e.g., animal population over time, as a function of fecundity and of the physical limitations of the surroundings. The former would lead to an exponential growth of the population, the latter limits the growth, the more strongly the bigger the population. If \(A_n\) is the population in the year n, the model calculates the population the following year by an equation of the form \(A_{n+1}=r A_n (1-A_n)\) where r is the growth rate, and \((1-A_n)\) takes account of the limitations imposed by the environmental conditions. See e.g. hypertextbook.com/chaos/42.shtml.
- 10.
I thank Peter Beckmann for providing these impressive figures and for his advise regarding the presentation of this system.
- 11.
The Golden Mean is a well-known concept in the fine arts, in the theory of proportions. For example, a column of height H is divided into two segments of heights \(h_1\) and \(h_2\), with \(H = h_1 + h_2\) such that the proportion of the shorter segment to the longer is the same as that of the longer to the column as a whole, i.e. \(h_1/h_2 = h_2/H = h_2/(h_1 + h_2)\). The ratio \(h_1/h_2 = \bar{r} = (\sqrt{5} - 1)/2\) is the Golden Mean. This very irrational number has a remarkable continued fraction representation: \(r = 1/(1 + 1/(1 + \ldots \).
- 12.
The observations of Voyager 2 are consistent with this prediction, since it found Hyperion in a position clearly out of the vertical. More recently, Hyperion’s tumbling was positively observed from the earth (J. Klavetter et al., Science 246 (1989) 998, Astron. J. 98 (1989) 1855).
- 13.
The reader will find hints to the original literature describing these methods in Wisdom’s review (1987).
- 14.
Analogous investigations of the 2:1 and 3:2 resonances indicate that there is chaotic behavior at the former while there is none at the latter. This is in agreement with the observation that there is a gap at 2:1 but not at 3:2.
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Scheck, F. (2018). Stability and Chaos. In: Mechanics. Graduate Texts in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55490-6_6
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