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	Annotated Bibliography Ecological Society of America 2002 Annual Meeting. Tucson, Arizona
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Nonlinear Dynamics.

Textbooks / Reviews.

[as82-85] Abraham, R. H. and C. D. Shaw. 1982-5. Dynamics - The Geometry of Behavior. Parts One-Four: Periodic Behavior. Aerial Press. Santa Cruz, CA.A picture book introduction to nonlinear dynamics: the dynamical essentials without equations. An intellectual and aesthetic tour de force.

[b95] Bélair, J. 1995. Dynamical Disease : Mathematical Analysis of Human Illness. AIP Press. Woodbury, N.Y. In 1977, Michael Mackey and Leon Glass [mg77] conjectured that certain pathologies, e.g., chronic periodic myelogenous leukemia, are characterized by aberrant dynamics which, in turn, reflect abnormal parameter values. The fruits of this conjecture are the subject of [b95].

[bpv86] Bergé, P., Pomeau, Y and C. Vidal. 1986. Order Within Chaos: Towards a Deterministic Approach to Turbulence. J. Wiley. N. Y. Includes both theory and its application to experiments. The prefatory remarks regarding the singular contributions of French mathematics give unique sociological insight.

[cr83] Campbell, D. and H. Rose. 1983. Order in Chaos. North-Holland. Amsterdam, NL. Book version of special issue of Physica D devoted to the Los Alamos symposium of the same name.

[d86] Devaney, R. L. 1986. An Introduction to Chaotic Dynamical Systems. Benjamin Cummings. Menlo Park, CA. Introduction to discrete dynamical systems. Restricted focus (difference equations) allows for rigorous exposition while requiring little mathematical background on the part of the reader. Includes extensive results on the Julia sets of analytic maps. Can be profitably read in conjunction with [gh83].

[g87] Gleick, J. 1987. Chaos: The Making of a New Science. Viking Penguin. New York. Best-selling review of the dynamical renaissance of the 1970's. Recounts the exploits of the so-called "Dynamical Systems Collective" at UC Santa Cruz. One of your faithful authors makes the final chapter.

[gh83] Guckenheimer, J. and P. Holmes. 1983. Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer-Verlag, New York. Rigorous, but readable, first course on mathematical dynamics. In the authors’ opinion, [gh83] is the best all-round introduction to the subject. Requires calculus, matrix algebra and a willingness to wrestle with challenging concepts.

[h84] Hao, B.-L. 1984. Chaos. World Scientific. Singapore. [h90] Hao, B.-L. 1990. Chaos II. World Scientific. Singapore.Landmark papers reprinted with review. The two editions differ slightly with regard to choice of articles.

[h83] Hénon, M. 1983. Numerical explorations of Hamiltonian systems. In, Ioos, G., Helleman, R. H. and R. Stora (eds.) Chaotic Behavior of Deterministic Systems. North-Holland. Amsterdam. Lucid review of Hamiltonian dynamics with emphasis on numerical simulation by one of the pioneers in the field.

[h95] Holland, J. H. 1995. Hidden Order. Addison-Wesley Publishing Company, Inc.

[kg95] Kaplan, D. and Glass, L. 1995. Understanding Nonlinear Dynamics. Springer-Verlag, NY.

[k95] Kuznetsov, Yu. A. 1995. Elements of Applied Bifurcation Theory. Springer-Verlag. New York. Introduction to the bestiary of bifurcations that are the ultimate source of dynamical complexity. The extensive Appendix reviews strategies for numerical identification and continuation of invariant sets and their bifurcations.

[ll92] Lichtenberg, A. J., and M. A. Lieberman. 1992. Regular and Chaotic Dynamics. Springer-Verlag, N.Y. Second edition of the "other" nonlinear dynamics text published in the early 1980's. Emphasis is on conservative systems.

[ny97] Nusse, H. E. and Yorke, J. A. 1997. Dynamics: Numerical Explorations. Second, revised and enlarged edition. Springer-Verlag, NY.

[pjs92] Peitgen, H., Jurgens, H. and Saupe, D. 1992. Chaos and Fractals: New Frontiers of Science. Springer-Verlag, NY. Introduction to nonlinear dynamics from the view point of fractals, the Mandelbrot set, etc. The graphics are absolutely stunning.

[s82] Sparrow, C. S. 1982. The Lorenz Equations: Bifurcations, Chaos and Strange Attractors. Springer-Verlag. New York. A marvelously engaging review of Lorenz’ famous equations. Beautifully written. The dynamical novitiate who masters this book along with [h83] will be well on his way.

[t89] Tabor, M. 1989. Chaos and Integrability in Nonlinear Dynamics: An Introduction. J. Wiley. New York. Introduction to the unexpected complexity that results when one subjects a frictionless pendulum to periodic forcing, couples two or more such pendula, etc. This is the context (three-body problem) in which chaos was first encountered.

[ts02] Thompson, J. M. T and H. B. Stewart. 2002. Nonlinear Dynamics and Chaos. J. Wiley. New York. Second edition of the text first published in 1986. Aimed principally at engineers with experience in linear systems, this book emphasizes the geometric origins of dynamical complexity and sensitive dependence on initial conditions. A good complement to Guckenheimer and Holmes [gh83].

[t90] Tong, H. 1990. Non-Linear Time Series: A Dynamical Systems Approach. Oxford Science Publications, Oxford. Nonlinear dynamics and time series analysis.

[w02] Wolfram, S. 2002. A New Kind of Science. New from the author of Mathematica and numerous papers on cellular automata. Haven't seen it yet.

	Butterfly of chaos. The Lorenz attractor color-coded according to rate of trajectorial evolution. Click image to enlarge. Click here for additional commentary.
	Mixing on the Lorenz attractor. Long-term unpredictability (sensitivity to initial conditions) in chaotic systems is precluded by the fact that nearby trajectories diverge from each other exponentially. Shown here is an ensemble of 100,000 initial points (the yellow dot) on the Lorenz attractor which is shown in gray. As the points evolve, they are spread out over the attractor so that after some time, given any uncertainty as to its initial state, the only thing that could have been predicted are the probabilities that a given trajectory will be in different neighborhoods on the attractor. In other words, only G-d can predict the long-term time evolution of a chaotic system, because only G-d can specify the initial conditions to arbitrary precision. Click the image to start the animation (high speed connection and Netscape recommended). Click here for additional commentary and here for mixing in other chaotic attractors.

[b-g90] Barrow-Green, J. 1996. Poincaré and the Three-Body Problem. Amer. Math. Soc. Providence, RI. Historical review of Poincarè’s discovery of complex dynamics and the work of subsequent investigators. Can be read with profit and pleasure by specialists and non-mathematicians alike.

[l44] Landau, L. D. 1944. On the problem of turbulence. C. R. Acad. Sci. USSR. 44: 311. Transition to turbulence via quasiperiodic motion on tori of increasing dimension.

[l63] Lorenz, E. N. 1963. Nonperiodic deterministic flow. J. Atm. Sci. 357: 282-291. Discovery of chaos in a dissipative dynamical system. The decade-delayed inspiration of [ly75] and all that followed.

[l64] Lorenz, E. N. 1964. The problem of deducing the climate from the governing equations. Tellus. 16: 1-11. Meteorological uncertainty as a consequence of sensitivity to initial conditions.

[p92-94] Poincaré, H. 1892. 1893. 1894. Les Methodes Nouvelles de la Mècanique Celeste. Vols. 1-3 Gauthier-Villars. Paris. Reprinted by Dover, 1957. (Translation: New Methods of Celestial Mechanics. NASA, 1967.) In principio, . . .

[rt71] Ruelle, D. and F. Takens. 1971. On the nature of turbulence. Commun. Math. Phys. 20: 167-192; [nrt78] Newhouse, S. Ruelle, D. and F. Takens. 1978. Occurrence of strange axaxiom attractors near quasi-periodic flows on T^m, m > 3. Commun. Math. Phys. 64: 35-40. Transition to chaos via low-dimensional "strange" attractors. An addendum to [nrt78] acacknowledgeshe fact that most of the results had already been discovered Russian mathematicians whose work was largely unknown in the West.

[s62] Saltzman, B. 1962. Finite amplitude free convection as an initial value problem. J. Atmos. Sci. 19: 329-341. Where the Lorenz equations came from in the first place.

[s64] Sarkovskii, A. N. 1964. Coexistence of cycles of a continuous map of a line into itself. Ukr. Mat. Z. 16: 61-71. "Period Three Implies Chaos" ten years prior to the event.

[s65] Shil’nikov, L. P. 1965. A case of the existence of a countable number of periodic motions. Sov. Math. Dokl. 6: 163-166. First report of what is now called "Shil’nikov chaos." Relevant to the dynamics of continuous chaotic attractors (Rössler [r76], Gilpin [g79] etc.) organized about saddle foci in three dimensions.

[s63] Smale, S. 1963. Diffeomorphisms with many periodic points. Pp. 63-80. In, Cairns, S. S. (ed.) Differential and Combinatorial Topology. Princeton University Press; [s67] 1967. Differentiable dynamical systems. Bull. Amer. Math. Soc. 73: 747-817. Construction of the horseshoe, its properties and the overthrow of the conjecture that Peixoto's theorem [gh83] applies to three- and higher dimensional systems. To the contrary, Smale's work probes that structurally stable systems are not generic in Rⁿ , n > 3.

	Hamiltonian dynamics in the Ikeda map. To enlarge and for commentary, click the figure.

Arguably the centerpiece of theoretical dynamics, the celebrated "KAM" (Kolmogorov-Arnol’d-Moser) theorem asserts that, for certain conservative systems, the phase space can be partitioned into regions of "regular," i.e., periodic and quasiperiodic, and chaotic dynamics. In our opinion, the best review of this topic is [h83]. Lichtenberg and Lieberman [ll92] and Tabor [t89] provide additional detail, but require more effort of the reader. The original literature (below) is difficult.

[a63] Arnol’d, V. I. 1963. Small denominators and problems of stability of motion in classical and celestial mechanics. Russian Math. Surveys. 18: 85-189; [a77] 1977. Loss of stability of self-oscillations close to resonance and versal deformations of equivariant vector fields. Funct. Anal. Appl. 11:1-10; [a83] 1983. Geometrical Methods in the Theory of Ordinary Differential Equations. Springer-Verlag. New York.

	Chaotic attractor in the Mackey-Glass equation, a delay differential equation model of physiological function. Modulo intervals of periodicity, the dimension of Mackey-Glass attractors increases with the length of the time delay [f82]. Click the image to enlarge.

[bbj84] Bak, P., Bohr, T. and M. H. Jensen. 1984. Mode-locking and the transition to chaos in dissipative systems. Phys. Sci. T9: 50-58. Quasiperiodic transition to chaos.

[bb84] Bier, M. and C. Bountis. 1984. Remerging Feigenbaum trees in dynamical systems. Phys. Lett. 104A: 239-244. Not all period-doubling cascades accumulate. Takes the "universal" out of universality theory.

[bsswfj83] Brandstäter, A., Swift, J., Swinney, H. L., Wolf, A., Farmer, J. D. and E. Jen. 1983. Low dimensional chaos in a system with Avogadro number of degrees of freedom. Phys. Rev. Lett. 51: 1442-1445. Experimental evidence for low-dimensional chaos in fluids.

[bk86] Broomhead, D. S. and G. P. King. 1986. Extracting qualitative dynamics from experimental data. Physica 20D: 217-236. One of the first attempts to deal with the problem of "messy" data. Proposed the use of singular value decomposition as a means of distinguishing determinism from noise. By way of introduction, the authors review the essential content of Takens' [t81] reconstruction theorem with exceptional lucidity.

[ce80] Collet, P. and J.-P. Eckmann. 1980. Iterated Maps on the Interval as Dynamical Systems. Birkhäuser, Boston. An early and influential compendium of important results for the logistic map

[cfh82] Crutchfield, J. P., Farmer, J. D. and B. A. Huberman. 1982. Fluctuations and simple chaotic dynamics. Phys. Rev. 92: 45-82. One of the first attempts to assess the effects of extrinsic noise on the dynamics of deterministic systems. Some of the comnclusions since superceded.

[e81] Eckmann, J.-P. 1981. Roads to turbulence in dissipative dynamical systems. Rev. Mod. Phys. 53: 643-653. Routes to chaos: period-doubling, intermittency, etc.

[f82] Farmer, J. D. 1982. Chaotic attractors of an infinite-dimensional dynamical system. Physica 4D: 366-393. Numerical investigations of the Mackey-Glass equation. Documents increasing dimension of the motion in response to cranking up the time delay.

Logistic map bifurcation diagrams. Top. Asymptotic dynamics (yellow) on for the parameter interval, r = [2.95, 4.0]. Bottom. A few of the countably infinite periodic orbits (blue) superposed. Even though there aee no attractors past r = 4.0, the periodic orbits created on r = [0, 4], persist to infinity. The title, "Period-3 Implies Chaos" of [ly75] refers to the fact that for parameter values in beyond the period-3 window (actually the value at which the 3-cycle is born), there exist cycles of all possible period. This result is a direct consequence of Sarkovskii's [s64] theorem of which Li and Yorke were not at the time aware. (Click the diagrams to enlarge.)

[f78] Feigenbaum, M. J. 1978. Qualitative universality for a class of nonlinear transformations. J. Statist. Phys. 19: 25-52. [f79]1979. The universal metric properties of nonlinear transformations. J. Statist. Phys. 21: 669-706. Period-doubling, renormalization and Feigenbaum’a famous ratios: "Yes, Virgina. There is order in chaos."

[gs75] Gollub, J. P. and H. L. Swinney. 1975. Onset of turbulence in a rotating fluid. Phys. Rev. Lett. 35: 927-930. Experimental confirmation of low-dimensional (Ruelle-Takens scenario) transition to turbulence.

[gp83a] Grassberger, P. and I. Procaccia. 1983a. Characterization of strange attractors. Phys. Rev. Lett. 50: 346-349; [gp83b] 1983b. Measuring the strangeness of strange attractors. Physica 9D: 189-208. Introduction of the "correlation dimension" as an estimate of the fractal dimension of a time series. The most widely implemented algorithm for estimating experimental fractal dimensions.

[goy83] Grebogi, C., Ott, E. and J. A. Yorke. 1983. Crises, sudden changes in chaotic attractors and transient chaos. Physica 7D: 181-200; [goy86] 1986. Metamorphoses of basin boundaries in nonlinear dynamical systems. Phys. Rev. Lett. 56: 1011-1014. Fractal basin boundaries.

[h80a] Helleman, R. H. G. 1980a. Nonlinear dynamics. Ann. N. Y. Acad. Sci. 357:; [h80b] 1980b. Self-generated chaotic behavior in nonlinear systems. Pp 165-233. In, Cohen, E. G. D. Fundamental Problems in Statistical Mechanics. North-Holland, Amsterdam. Routes to chaos in conservative and dissipative dynamical systems compared.

[h76] Hénon, M. 1976. A two-dimensional mapping with a strange attractor. Commun. Math. Phys. 50: 69-78. Dissipative chaos in the now famous 2-dimensional discrete dynamical system.

[j81] Jakobson, M. V. 1981. Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Commun. Math. Phys. 81: 39-88. Chaotic parameter values for the logistic map have positive measure but do not form intervals.

[ly75] Li, T. Y. and J. A. Yorke. 1975. Period three implies chaos. Amer. Math. Monthly. 82: 983-992. Periodicity and chaos in one-dimensional maps. Motivated by Lorenz’ observation that strongly contracting flows can be modeled by "1-D" maps. Principal result anticipated by Sarkovskii’s paper [s64] which was then unknown. Along with May’s paper [m76] in Nature, [ly75] got the "chaos revolution" rolling.

[lfl83] Libchaber, A., Fauve, S. and C. Laroche.1983. Two-parameter study of the routes to chaos. Physica D. 7: 73-84. Quasiperidic transition to chaos.

[l80] Lorenz, E. N. 1980. Noisy periodicity and reverse bifurcation. Ann N. Y. Acad. Sci. 357: 130-141. Remerging of semi-periodic attractors.

[mg77] Mackey, M. C. and L. Glass. 1977. Oscillations and chaos in physiological control systems. Science. 197: 287-289. Complex dynamics in delay differential equation models of physiological function. The antecedent of subsequent investigations of so-called "dynamical diseases" [b95].

[m76] May, R. M. 1976. Simple mathematical models with very complicated dynamics. Nature. 262: 459-467. Period-doubling and periodic windows in single species difference equations, i.e., one-dimensional maps. Ecology’s premier contribution to what the British Ecological Society once fatuously referred to as "knowledge at large."

[mss73] Metropolis, M., Stein, M. L. and P. R. Stein. 1973. On finite limit sets for transformations on the unit interval. J. Comb. Theor. A5:25-44. Classification and sequence of periodic orbits in one-dimensional maps.

	Phase space geometry from a time series [t81] as summarized schematically by Broomhead and King [bk86]. To enlarge and for commentary, click the figure.

[pcfs80] Packard, N. H., Crutchfield, J. P., Farmer, J. D. and R. S. Shaw. 1980. Geometry from a time series. Phys. Rev. Lett. 45: 712-716; [t81] Takens, F. 1981. Detecting strange attractors in turbulence. Pp. 366-381. In, Rand, D. A. and L. S. Young (eds.). Dynamical systems and Turbulence. Springer-Verlag. Berlin. With the discovery that phase space dynamics can be reconstructed from univariate time series, experimentalists were given access to the stage on which dynamical dramas are enacted. The importance of this advance, allowing, as it does, direct comparison of theory and observation, cannot be overstated. Packard et al. [pcfs80] proposed the use of derivatives of increasing order, dx/dt, d²x/dt², dx³/dt³, … , as surrogate phase space coordinates. Takens [t81] proved theorems justifying the use of derivatives, as well as time-delayed coordinates, x(t), x(t-T), x(t-2T), … , where T is a delay or "lag." Of the two articles, the second article is the more widely cited.

[r76] Rössler, O. E. 1976. An equation for continuous chaos. Phys. Lett. 35a: 397-398; [r77] 1977. Contrinuous chaos. Pp. 184-187. In, Haken, H. (ed.) Synergetics: A Workhop. Springer-Verlag. Berlin; [r79] 1979. Continuous chaos - four prototype equations. Pp. 376-392. In, Grure, O. and O. E. Rössler (eds.) Bifurcation Theory and Applications in Scientific Disciplines. N. Y. Acad. Sci. New York; [rkppr86] Rössler, O. E., Kahlert, C., Parisi, J., Peinke, J. and B. Röhricht. 1986. Hyperchaos and Julia sets. Z. f. Natürforsch. 41a: 819-822. Rössler's "band" and other simple systems evidencing continuous chaos.

[rss93] Roux, J.-C., Turner, J. S. and H. L. Swinney. 1983. Observation of a strange attractor. Physica D. 8: 257-266. Chaos in the B-Z reaction. The first application of Takens’(1981) embedding theorems to an experimental system. Will one day gain its authors a Nobel Prize (IOHO).

[r79] Ruelle, D. 1979. Sensitive dependence on initial conditions and turbulent behavior. Ann. N. Y. Acad. Sci. 316: 408-416. Chaos implies unpredictability.

[s81] Shaw, R. 1981. Strange attractors, chaotic behavior and information flow. Z. f. Natürforsch. 36a: 80-112. Topology of chaotic behavior; extraction of "1-D" maps from strongly contracting flows; chaos and information theory.

[s77] Stefan, O. 1977. A theorem of Sarkovskii on the existence of periodic orbits of continuous endomorphisms of the real line. Commun. Math. Phys. 54: 237-248. Sarkovskii rediscovered.

[w77] Williams, R. F. 1977. The structure of Lorenz attractors. In, Bernard, P. (Ed.) Lecture Notes in Mathematics. 615: 93-113. Springer-Verlag. New York; [w79] 1979. The bifurcation space of the Lorenz attractor. Ann. N.Y. Acad. Sci. 316: 393-399; [w80] 1980. Structure of Lorenz attractors. Publ. Math. I.H.E.S. 50: 59-72. Complex dynamics in the Lorenz equations; branched manifolds.

	Experimental set-up of Roux et al. [rss83] for studying the BZ reaction under spatially homogeneous conditions. To enlarge and for commentary, click the image.

Among the various sub-disciplines that contributed to the dynamical rennaissance of the late 1970s, none was more important than the study of oscillating chemical and biochemical reactions. The most famous of these (right) was discovered by the Russian worker Belousov [b59] whose findings were later confirmed by Zhabotinskii [z64] - whence the name, the "Belousov-Zhabotinskii" or, more simply, "BZ" reaction. In fact, the first demonstration [od77] of chaos in a chemical system was for the peroxidase-oxidase (PO) reaction. This is a biochemical reaction whereby a hydrogen donor, such as NADH, is oxidized enzymatically. The schematic below shows a proposed [bfso95] mechanism. Clicking on it, allows you to watch the dynamics evolve. Despite the compelling evidence for complex dynamics in biochemical systems such as the PO reaction in vitro and in vivo observations of oscillatory behavior in glycolysis and other systems, the functional significance of biochemical oscillations in living cells remains uncertain. For more on the PO reaction, including additional animations of different dynamical regimes, go here. For the nonlinear dynamics of glycolysis go here. For a introduction to chemical / biochemical oscillators generally, go here.

[bk91] Baier, C. and M. Klein. (eds.) 1991. A Chaotic Hierarchy. World Scientific. Singapore.

[b59] Belousov, B. P. 1959. A periodic chemical reaction and its mechanism. Sb. Ref. Radiats. Med. Medgiz: Moscow, 145-147.

[b95] Berridge, M. J. (ed.) 1995. Ca²⁺ Waves, Gradients and Oscillations. CIBA Found. Symp. J. Wiley. Chichester, UK.

[bfso95] Bronnikova, T. V., Fed'kina, V. R., Schaffer, W. M. and L. F. Olsen. 1995. Period-doubling bifurcations in a detailed model of the peroxidase-oxidase reaction. J. Phys. Chem. 99: 9309-9312.

[bf85] Burger, M. and R. Field. (eds.) 1985. Oscillations and Travelling Waves in Chemical Systems. Wiley-Interscience. NY.

Coupled redox loops in the peroxidase-oxidase reaction. Two cycles share successive reductions of the enzyme intermediates coI à coII à Per3+. Thereafter, Per3+ is re-oxidized directly (peroxidase cycle) to coI or indirectly (oxidase cycle) via the formation of Per2+ and coIII. To watch the reaction evolve, click the figure. The simulation is for a quasiperiodic state near heteroclinicity, i.e., the hole in the "doughnut" is close to disappearing. As it proceeds, colors of the boxes (reactant concentrations) and arrows (reaction rates) will change from dark blue to white and then back as the magnitudes of these quantities wax and wane. The scrolling time series (upper right) is [coI]/[ET], where [ET] is the total enzyme concentration which is fixed. Axes of the phase portrait projection (lower right) are [coI]/[ET] and [coIII]/[ET]. Width of the time series window is 1120 simulated seconds. For additional information, click here.

[ck99] Carafoli, E. and C. Klee. 1999. Calcium as a Cellular Regulator. Oxford Univ. Press. NY and Oxford, UK.

[cpgh73] Chance, B., Pye, E. K., Ghosh, A. K., and B. Hess. 1973. Biological and Biochemical Oscillators. AP. New York.

[cms86] Coffman, A. K, McCormick, W. D. and H. L. Swinney. 1986. Multiplicity in a chemical reaction with one-dimensional dynamics. Phys. Rev. Lett. 56: 999-102.

[ew98] Eichwald, C. and J. Walleczek. 1998. Magnetic field perturbations as a tool for controlling enzyme-regulated and oscillatory biochemical reactions. Biophysical Chemistry. 74: 209-224.

[ep98] Epstein, I. R. and J. A. Pojman. 1998. An Introduction to Nonlinear Chemical Dynamics. Oxford Univ. Press. New York.

[es96] Epstein, I. and K. Showalter. 1996. Nonlinear chemical dynamics: oscillations, patterns and chaos. J. Phys. Chem. 100: 13132-13147.

[fab84] Fed'kina, V. R., Ataullakhanov, F. I., and T. V. Bronnikova. 1984. Computer simulation of sustained oscillations in peroxidase-oxidase reaction. Biophys. Chem. 19: 259-264.

[fb85] Field, R. J. and M. Burger. (eds.) 1985. Oscillations and Travelling Waves in Chemical Systems. J. Wiley. N.Y.

[fg93] Field, R. J. and L. Györgyi (eds.) 1993. Chaos in Chemistry and Biochemistry. World Scientific. Singapore.

[g89] Goldbeter, A. (ed.) 1989. Cell to Cell Signalling: From Experiments to Theoretical Models. AP. London.

[g96] Goldbeter, A. 1996. Biochemical oscillations and Cellular Rhythms. Cambridge Univ. Press. Cambridge, UK.

[gs90] Gray, P. and S. K. Scott. 1990. Chemical Oscillations and Instabilities. Clarendon Press. Oxford, UK.

[h97] Hess, B. 1997. Periodic patterns in biochemical reactions. Quart. Rev. Biophys. 30: 121-17.

[h86] Holden, A. 1986 (ed.) Chaos. Manchester Univ. Press. Manchester, UK.

[m79] Mackey, M. C. 1979. Periodic auto-immune hemolytic anemia: an induced dynamical disease. Bull. Math. Biol. 41: 829-834.

[m82] Meinhardt, H. 1982. Models for Biological Pattern Formation. AP. London.

[mc84] Moore-Ede, M. C. and C. A. Czeisler. (eds.) 1984. Mathematical Models of the Circadian Sleep / Wake Cycle. Raven Press. New York.

	Interior fat torus dynamics in a model [bfso95] of the peroxidase-oxidase reaction. Click image to enlarge. For additional commentary, click here.

[mrd83] Morris, H. C., Ryan, E. E. and R. K. Dodd. 1983. Periodic solutions and chaos in a delay-differential equation modeling haematopoesis. Nonlin. Anal. Theor. Meth. Appl. 7: 623-660.

[od77] Olsen L. F. and H. Degn. 1977. Chaos in an enzyme reaction. Nature. 267: 177-178

[od85] Olsen, L. F. and H. Degn. 1985. Chaos in biological systems. Q. Rev. Biol. 18: 165-225.

[pv84] Pacault, A. and C. Vidal. 1984. Nonequilibrium Dynamics of Chemical systems. Springer-Verlag. Berlin.

[r79] Rapp, P. E. 1979. An atlas of cellular oscillators. J. Exp. Biol. 81: 281-306.

[rhm87] Rensing, L., an der Heiden, U., and M. C. Mackey. 1987. Temporal Disorder in Human Oscillatory Systems. Springer-Verlag. Berlin.

[rc-b84] Ricard, J. and A. Cornish-Bowden. (eds.) 1984. Dynamics of Biochemical Systems. Plenum Press. NY.

[s91] Scott, S. K. 1991. Chemical Chaos. Clarendon Press. Oxford, UK.

[s94] Scott, S. K. 1994. Oscillations, Waves, and Chaos in Chemical Kinetics. Oxford University Press. Oxford. UK.

[s80] Segel, L. A. 1980. Mathematical Models in Molecular and Cellular Biology. Cambridge Univ. Press. Cambridge, UK.

[sws82] Simoyi, R. H., Wolf, A. and H. L. Swinney. 1982. One-dimensional dynamics in a multi-component chemical reaction. Phys. Rev. Lett. 49: 245-248.

[vt98] Verkhatsky, A. and E. C. Toescu. 1998. Integrative Aspects of Calcium Signalling. Plenum Press. NY and London.

[w00] Walleczek, J. (ed.) 2000. Self-Organized Biological Dynamics and Nonlinear Control. Cambridge University Press. Cambridge, UK.

[z64a, z64b] Zhabotinskii, A. M. 1964a. Periodic process of the oxidation of malonic acid in solution. Biofizika. 9: 306-311; 1964b. Periodic liqu id-phase oxidation reactions. Dokl. Akad. Nauk USSR. 157: 392-395.

Complex dynamics in the B-Z reaction. Left. Chaos in a spatially homogeneous experiment [rss83]. Shown is a two-dimensional phase portrait "reconstruction" computed by plotting concentrations of bromide ion, [Br-](t+T) vs. [Br-](t) where [Br-](t) is the experimentally observed concentration at time t, and T is a time delay. Center. Return map specifying temporal sequence of points on Poincaré section indicated by dashed line at the left. Right. Spiral waves of oxidation in a spatially extensive medium. Image courtesy of A. W. Winfree

	Estimating the Lyapunov characteristic exponents (LCEs) of nonlinear dynamical systems. An n-dimensional system has n such quantities which measure rates of expansion and contraction much as the eigenvalues of a linear system characterize the tendency to diverge from or tend toward equilibrium. Top. In a 1-D map, x(i+1) = f[x(i)], the single LCE is determined by the long-term product of the absolute values of the derivatives, (df/dx), at points visited by the evolving itinerary. Bottom. In higher order continuous systems, one montiors the time evolution of a set of basis vectors in the tangent space. Click the pictures to enlarge. (Reproduced from [ll92]).

[bggs80] Bennetin, G., Galgani, L. Giogilli, A. and J.-M. Strelcyn. 1980. Lyapunov characteristic exponents for smooth dynamical systems: a method for computing all of them. Part 1: Theory. Meccanica. 15: 9-20.

[bsswfj83] Brandstäter, A., Swift, J., Swinney, H. L., Wolf, A., Farmer, J. D. and E. Jen. 1983. Low dimensional chaos in a system with Avogadro’s number of degrees of freedom. Phys. Rev. Lett. 51: 1442-1445.

[er85] Eckmann, J.-P. and D. Ruelle. 1985. Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57: 617-656.

[fcfps80] Farmer, D., Crutchfield, J., Froehling, H., Packard, N. and R. Shaw. 1980. Power spectra and mixing properties of chaotic attractors. Ann. N. Y. Acad. Sci. 357: 453-472.

[k58] Kolmogorov, A. N. 1958. A new invariant for transitive dynamical systems. Dokl. Akad. Nauk. USSR. 119: 861-864.

[lk89] Lathrop, D. P., and E. J. Kostelich. 1989. Characterization of an experimental strange attractor by periodic orbits, Phys. Rev. A, 40, 4028-403.

[l81] Ledrappier, F. 1981. Some relations between dimension and Lyapunov exponent. Commun. Math. Phys. 81: 229-238.

[o58] Oseledec, V. I. 1968. A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19: 197.

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Ecological Dynamics:

Childhood Diseases.

	Chaotic SEIR (measles) attractor (in section) color-coded by predictability according to the visible spectrum (red = high; violet = low). For additional commentary, click here.

[am82] Anderson, R. M. and R. M. May. 1982. Directly transmitted infectious diseases: control by vaccination. Science. 215: 1053-1060; [am83] 1983. Vaccination against rubella and measles: Quantitative investigations of different policies. J. Hyg. Camb. 90: 259-325. [ma79] May, R. M. and R. M. Anderson. 1979. Population biology of infectious diseases: II. Nature. 280: 455-461. Influential reviews applying mathematical theory to epidemiology.

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[s85a] Schaffer, W. M. 1985a. Can nonlinear dynamics help us infer mechanisms in ecology and epidemiology? IMA J. Math. Appl. Med. Biol. 2: 221-252; [sk85a] Schaffer, W. M. and M. Kot. 1985a. Nearly one dimensional dynamics in an epidemic. J. Theor. Biol. 112: 403-427. Reconstructed phase portraits and return maps for historical time series of chickenpox, measles and mumps in first world cities. Real world epidemics and SEIR simulations compared. First evidence of complex dynamics in childhood diseases.

[sotf90] Schaffer, W. M., Olsen, L. F., Truty, G. L. and S. L. Fulmer. 1990. The case for chaos in childhood epidemics. Pp. 139-167. In, Krasner, S. (ed.). The Ubiquity of Chaos. AAAS. Washington, D.C. [os90] Olsen, L. F. and W. M. Schaffer. 1990. Chaos vs. noisy periodicity: Alternative hypotheses for childhood epidemics. Science. 249:499-504. The case for complex dynamics in childhood diseases.

[ss83] Schwartz, I. B. and H. L. Smith. 1983. Infinite subharmonic bifurcations in an SEIR epidemic model. J. Math. Biol. 18: 233-253. Period-3, -5, etc., cycles that coexist with those produced by the "main" period-doubling sequence described by [as84]. Unlike the latter orbits, subharmonic cycles do not evidence seasonality. As a consequence, SEIR chaos, which is typically organized about both types of cycles, can evidence alternating periods of seasonal and aseasonal dynamics.

[tos93] Tidd, C. W., Olsen, L. F. and W. M. Schaffer. 1993. The case for chaos in childhood diseases. II. Predicting historical epidemics with mathematical models. Proc. Biol. Sci. Roy. Soc. London. 254: 257-273.

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[achm80] Aronson, D. G., Chory, M. A., Hall, A. and R. P. McGeehee. 1980. A discrete dynamical system with subtly wild behavior. Pp. 339-359. In, Holmes, P. (ed.) New Approaches to Nonlinear Problems in Dynamics. SIAM. Philadelphia, PA; [acm81] 1981. Bifurcations from an invariant circle for two-parameter families of maps of the plane: a computer assisted study. Commun. Math. Phys. 83: 303-351.

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[bst96] Begon, M., Sait, S. M., and D. J. Thompson. 1996. Predator-prey cycles with period shifts between two- and three-species systems. Nature. 381: 311-315.

[bbsfst98] Bjornstead, O. N., Begon, M., Stenseth, N., Flack, W., Sait, S. M. and Thompson, D. J. 1998. Population dynamics of the Indian meal moth: demographic stochasticity and delayed regulatory mechanisms. Journal of Animal Ecology. 67: 110-126.

[cdcd97] Costantino R. F., Desharnais, R. A., Cushing J. M. and B. Dennis. 1997. Chaotic dynamics in an insect population, Science. 275: 389-391.

[ccddh02] Cushing, J. M., Costantino, R. F., Dennis, B., Desharnais, R. A., and S. M. Henson. 2002. Chaos in Ecology: Experimental Population Dynamics. Academic Press. New York. About to be released monograph highlighting the "Beetle Team's" famous "route-to-chaos" experiments.

[cddc96] Cushing, J. M., Dennis, B., Desharnais, R. A., and R. F. Costantino. 1996. An interdisciplinary approach to understanding ecological dynamics, Ecological Modelling. 92: 111-119.

[ccddh98] Cushing, J. M., Costantino, R. F., Dennis, B., Desharnais, R. A. and S. M. Henson. 1998. Nonlinear population dynamics: Models, experiments and data, J. Theor. Biol. 194: 1-9.

[ddchc01] Dennis, B., Desharnais, R. A., Cushing, J. M., Henson, S. M., and R. F. Costantino. 2001. Estimating chaos and complex dynamics in an insect population. Ecol. Monog. 71:277-303. Getting the "lies" out of "Lies, Damned Lies and Statistics."

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[hccddk01] Henson, S. M., Costantino, R. F., Cushing, J. M., Desharnais, R. A., Dennis, B., and King, A. A. 2001. Lattice effects observed in chaotic dynamics of experimental populations, Science. 294: 602-605. Dynamical consequences of the fact that most animals (and many plants) come in whole numbers.

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[e91] Ellner, S. 1991. Detecting low-dimensional chaos in population dynamics data: a critical review. Pp. 63-90, In, Logan, J. A., and Hain, F. P., (eds.), Chaos and Insect Ecology, Virginia Exptl. Station Information Series, 91-3, Blacksburg, VA.

[et91] Ellner, S., and Turchin, P. 1995. Chaos in a noisy world: new methods and evidence from time-series analysis, American Naturalist, 145, 343-375.

[ebbggn98] Ellner, S., Bailey, B. A., Bobashev, G. V., Gallant, A. R., Grenfell, B. T., and Nychka, D. W. 1998. Noise and nonlinearity in measles epidemics: Combining mechanistic and statistical approaches to population modeling, American Naturalist, 151, 425-440.

[hlm76] Hassell, M. P., Lawton, J. H. and R. M. May. 1976. Patterns of dynamical behaviour in single-species populations. J. Anim. Ecol. 48: 471-486. Aborted the nonlinear revolution in ecology before it got started. Source of the claim that ecological chaos is a mathematical curiosity unobservable in nature because it just ain't there.

[h99] Higgins, K. 1999. The impact of age-, space- and stochastic-structure on the extinction probability of a chinook salmon metapopulation. Natural Resource Modeling. 12: 65-108.

[hhsr97] Higgins, K., Hastings, A., Sarvela, J. N. and L. Botsford. 1997. Stochastic dynamics and deterministic skeletons: Population behavior of dungeness crab. Science. 276: 1431-1435.

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[m90] Morris, W. F. 1990. Problems in detecting chaotic behavior in natural populations by fitting simple discrete models. Ecology. 71: 1849-1862. Why [hlm76] may have been wrong.

[pswm00] Perry, J. N., Smith, R. H., Woiwood and D. R. Morse. 2000. Chaos in Real Data. The Analysis of Nonlinear Dynamics from Short Ecological Time Series. Kluwer Academic Publ. Dordrecht. The Netherlands.

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[tt92] Turchin, P. and A. D. Taylor. 1992. Complex dynamics in ecological time series. Ecology. 73: 289-305.

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[ks99] King, A. A. and W. M. Schaffer. 1999. The rainbow bridge: Hamiltonian limits and harmonic resonance in predator-prey dynamics. J. Math. Biol. 39: 439-469.

[b98] Bazikin, A. D. 1998. Nonlinear Dynamics of Interacting Populations. World Scientific. Singapore.

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[c00] Caswell, H. 2000. Matrix Population Models. Sinauer Associates, Inc. Publishers.

[cd91] Costantino, R. F. and Desharnais, R. A. 1991. Population Dynamics and the Tribolium Model: Genetics and Demography. Springer-Verlag, NY.

[ccdds02] Cushing, J. M., Costantino, R. F., Dennis, B., Desharnais, R. A. and S. Henson. 2002. Chaos in Ecology. Elsevier (In press)

[ksotj94] Kendall, B., Schaffer, W. M., Olsen, L. F., Tidd, C. W. and B. L. Jorgensen. 1994. Using chaos to understand biological dynamics. Pp. 184-203. In, Grassman, J. and G. van Straten (eds.) Predictability and Nonlinear Modelling in Natural Sciences and Economics. Kluwer Academic Publ. Dordrecht. [kst97] Kendall, B. E., Schaffer, W. M. and C. W. Tidd. 1997. The impact of chaos on biology. Pp. 190-218. In, Grebogi, C. and J. A. Yorke (eds.) The Impact of Chaos on Science and Society. United Nations Press. Tokyo.

[hhetg93] Hastings, A., Hom, C. L., Ellner, S., Turchin, P., and H. C. H. Godfray. 1993. Chaos in ecology: Is Mother Nature a strange attractor? Ann. Rev. Ecol. Syst. 24:1-33.

[l99] Levin, S. 1999. Fragile Dominion: Complexity and the Commons. Helix Books/Perseus Publishing.

[mj00] Mueller, L. and Joshi, A. 2000. Stability in Model Populations. Princeton Monographs in Population Biology, Princeton University Press.

[rs01] Rai, V. and W. M. Schaffer. 2001. Preface to the Special Edition on Chaos in Ecology. Chaos, Solitons and Fractals. 12: 197-203.

[s85b] Schaffer, W. M. 1985b. Order and chaos in ecological systems. Ecology. 66:93-106. [sk86a] Schaffer, W. M. and M. Kot. 1985. Do strange attractors govern ecological systems? Bioscience. 36: 342-350; 1986a. Differential systems in ecology and epidemiology. Pp. 158-178. In, Holden, A. V. (ed.). Chaos: An Introduction. Univ. Manchester Press, UK; [sk86b] 1986b. The coals that Newcastle forgot: Chaos in ecological systems. Trends in Ecology and Evolution. 1: 58-63. By the mid-1980s, ecological chaos was widely regarded as a mathematical curiosity. This was largely in response to [hlm76] who argued that rates of reproduction and survival observable in natural populations were incompatible with dynamical complexity. No doubt, it also reflected the long-standing aversion of most work-a-day ecologists to mathematical modelling except possibly in the context of specific systems. The papers listed here attempted to reverse that pre-conception. Among the then-novel (for ecology) points they advanced were the following:

1. Complex dynamics and chaos are readily obserevd in continuous models provided one has at least three species (state variables).

2. In some of these cases, the sequence of orbital excursions is predictable by 1-D difference equations, i.e., the situation first observed by Lorenz [l63] and extensively commented upon by Shaw [s81].

3. Surrogate phase portraits can be reconstructed from univariate time series using the method of Takens [t81].

4. In the case of lynx fur returns, the reconstructions suggest period-doubled orbits, i.e., a 20 year cycle, as opposed to a simple 10 year oscillation.

5. In the case of human childhood diseases (microparasitic infections), the reconstructions suggest dynamics ranging from noisy limit cycles (chickenpox) to low dimensional chaos (measles). In some cases (mumps) both patterns were observed, depending on locale.

6. Differential equation (SEIR) models of childhood infections predict time series and phase portrait reconstructions which bear an uncanny resemblance to those observed in historical notifications.

An important consequence of #2 was that the results first reported (in an ecological context) by May [m74, m76] are not restricted to populations with discrete, non-overlapping generations, but instead potentially carry over to the overwhelming majority of populations in nature. In such cases, the parameters determining system dynamics reflect the totality of interactions among system elements - which is to say, ecological chaos does not require biologically unrealistic per capita rates of increase. The final point - the congeruence of prediction and observation in childhood epidemics - emphasized the importance of mathematical modelling as an integral part of assessing the nature of ecological fluctuations in nature, and was therefore anticipatory to the work of Costantino, Cushing, etc., on laboratory populations of flour beetles. Put another way, ecological time series being what they are - short and messy - it is unrealistic to imagine that data analysis by itself will yield unambiguous conclusions of the sort deducible under more controlled circumstances, for example, in the B-Z reaction [rss83].

[ts92] Tong, H. and R. L. Smith. (eds.) 1992. Royal Statistical Society Meeting on Chaos. Journal Royal Statistical Society B. Volume 54.

[t02]Turchin, P. 2002. Complex Population Dynamics: A Theoretical/Empirical Synthesis. Princeton University Press, Princeton, NJ. (In press).

	Multifactorial snowshoe hare experiments real and imagined. According to the model, hare densities plateau (but do not decline!) as the animals reduce their food supply and then crash in response to predator population build-up. (Reproduced from [ks01]. For a copy, write Aaron King.)

[bhs99] Blasius, B. Hupert, A. and L. Stone. 1999. Complex dynamics and phase synchronization in spatially extended ecological systems. Nature. 399: 354-359. Phase coherence [fcfps80] in Rössler-like attractors rediscovered, this time in predator-prey models. The authors conjecture that the (more or less) spatially synchronized oscillations observed in nature in the case of the lynx-hare cycle result therefrom.

[gs00] Gamarra, J. G. P., and Sole, R. V., Bifurcations and chaos in ecology: Lynx returns revisited, Ecol. Lett. 3: 114.

[ks01] King, A. A. and W. M. Schaffer. 2001. The geometry of a population cycle: A mechanistic model of snowshoe hare demography Ecology 82: 814-830.

[s84] Schaffer, W. M. 1984. Stretching and folding in lynx fur returns: evidence for a strange attractor in nature? Am. Nat. 124: 798-820.

[s88] Schaffer, W. M. 1988. Perceiving order in the chaos of nature. Pp. 313-350. In, Boyce, M. (ed.) Life Histories: Theory and Patterns from Mammals. Yale University Press. New Haven.