Abstract
Having discussed the phenomena of chaos and the routes leading to it in ‘simple’ one-dimensional settings, we continue with the exposition of chaos in dynamical systems of two or more dimensions. This is the relevant case for models in the natural sciences since very rarely can processes be described by only one single state variable. One of the main players in this context is the notion of strange attractors.
Never in the annals of science and engineering has there been a phenomenon so ubiquitous, a paradigm so universal, or a discipline so multidisciplinary as that of chaos. Yet chaos represents only the tip of an awesome iceberg, for beneath it lies a much finer structure of immense complexity, a geometric labyrinth of endless convolutions, and a surreal landscape of enchanting beauty. The bedrock which anchors these local and global bifurcation terrains is the omnipresent nonlinearity that was once wantonly linearized by the engineers and applied scientists of yore, thereby forfeiting their only chance to grapple with reality.
Leon O. Chua1
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Reference
From D. Ruelle, Strange Attractors, Math. Intelligencer 2 (1980) 126–137.
Li, T. Y. and Yorke, J. A., Period 3 Implies Chaos, American Mathematical Monthly 82 (1975) 985–992.
D. Ruelle and F. Takens, On the nature of turbulence, Comm. Math. Phys. 20 (1971) 167–192 and 23 (1971) 343–344.
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields Springer-Verlag, New York, 1983. See section 5.7 therein.
Hao, B. L.,Chaos II World Scientific, Singapore, 1990. For an even larger bibliography on chaos containing over 7000 references see Zhang Shu-yu Bibliography on Chaos — Directions in Chaos Vol. 5World Scientific, Singapore, 1991.
See M. Hénon, A two-dimensional mapping with a strange attractor, Comm. Math. Phys. 50 (1976) 69–77.
Pictures of this sort were first published in S. D. Feit, Characteristic exponents and strange attractors, Comm. Math. Phys. 61 (1978) 249–260.
Except for part 3, which we added here, this definition has been given in D. Ruelle, Strange Attractors, Math. Intelligencer 2 (1980) 126–137.
See the discussion on pages 255–259 in J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983.
M. Misiurewicz, Strange Attractors for the Lozi Mappings,in Nonlinear Dynamics,R. H. G. Helleman (ed.), Annals of the New York Academy of Sciences 357 (1980) 348–358.
R. Lozi, Un attracteur étrange (?) du type attracteur de Hénon, J. Phys. (Paris) 39 (Coll. C5 ) (1978) 9–10.
The table is adapted from B. Derrida, A. Gervois, Y. Pomeau, Universal metric properties of bifurcations of endomorphisms, J. Phys. A: Math. Gen. 12, 3 (1979) 269–296.
A very extensive analysis of the scenario of appearing and disappearing strange attractors and periodic orbits has been carried out in C. Simd, On the Hénon-Pomeau attractor, Journal of Statistical Physics 21,4 (1979) 465–494. See also F. R. Marotto, Chaotic behavior in the Hénon mapping, Comm. Math. Phys. 68 (1979) 187–194.
M. Benedicks and L. Carleson, The dynamics of the Hénon map, Annals of Mathematics 133,1 (1991) 73–169.
For most figures and computations presented in this book, we have used the code in Numerical Recipes in C by W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Cambridge University Press, 1988.
The basic tool for understanding planar dynamical systems is the Poincaré-Bendixson theory (see chapter 11 in M. W. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press, New York, 1974 ).
Rössler, O. E., An equation for continuous chaos, Phys. Lett. 57A (1976) 397–398.
Lord Rayleigh, On convective currents in a horizontal layer of fluid when the higher temperature is on the under side Phil. Mag. 32 (1916) 529–546.
B. Saltzman, Finite amplitude free convection as an initial value problem — I, J. atmos. Sci. 19 (1962) 329–341.
For more details and references see the original paper of E. N. Lorenz, Deterministic nonperiodic flow,J. Atmos. Sci. 20 (1963) 130–141. The book by J. M. T. Thompson and H. B. Stewart, Nonlinear Dynamics and Chaos,Wiley, Chichester, 1986, contains a broad introduction and much more material about the geometry underlying the attractor and its route to chaos.
The dimension has been estimated as 2.073. See E. N. Lorenz, The local structure of a chaotic attractor in four dimensions, Physica 13D (1984) 90–104.
For example, see the book by C. Sparrow, The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, Springer-Verlag, New York, 1982.
V. Franceschini, A Feigenbaum sequence of bifurcations in the Lorenz model, Jour. Stat. Phys. 22 (1980) 397–406.
This procedure was suggested by David Ruelle, see N. H. Packard, J. R. Crutchfield, J. D. Farmer, R. S. Shaw, Geometry from a time series, Phys. Rev. Lett. 45 (1980) 712–716.
Mané, R., On the dimension of the compact invariant set of certain nonlinear maps,in: Dynamical Systems and Turbulence, Warwick 1980,Lecture Notes in Mathematics 898, Springer-Verlag (1981) 230–242. Takens, F., Detecting strange attractors in turbulence,in: Dynamical Systems and Turbulence, Warwick 1980,Lecture Notes in Mathematics 898, Springer-Verlag (1981) 366–381.
Doyne Farmer presents an in-depth study of reconstructions of attractors in an infinite dimensional space with calculations of dimensions and Ljapunov exponents. See D. Farmer, Chaotic attractors of an infinite-dimensional system, P.ysica 4D (1982) 366–393. This study is continued in P. Grassberger and I. Procaccia, Measuring the strangeness of strange attractors, Physica 9D (1983) 189–208.
M. Faraday, On a peculiar class of acoustical figures, and on certain forms assumed by groups of particles upon vibrating elastic surfaces, Phil. Trans. Roy. Soc. London 121 (1831) 299–340.
For references see the historical notes in W. Lauterborn and J. Holzfuss, Acoustic chaos, International Journal of Bifurcation and Chaos 1,1 (1991) 13–26.
To give one example, reconstructions of chaotic attractors in hydrodynamics have been reported by Tom Mullin, in: Chaos in physical systems, in: Fractals and Chaos, A. J. Crilly, R. A. Eamshaw, H. Jones (eds.), Springer-Verlag, New York, 1991. Several examples for chaos in mechanical sytems is described in the introductory text F. C. Moon, Chaotic Vibrations, John Wiley & Sons, New York, 1987
A good source of reference and algorithms, including codes in three programming languages, is given in Numerical Recipes, W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Cambridge University Press, 1986.
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Peitgen, HO., Jürgens, H., Saupe, D. (1992). Strange Attractors: The Locus of Chaos. In: Fractals for the Classroom. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-4406-6_6
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