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Deterministic Chaos Theory and Its Applications to Materials Science

Published online by Cambridge University Press:  29 November 2013

Alan J. Markworth ,
John Stringer  and
Roger W. Rollins
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Over the past 30 years or so, some fascinating new developments have taken place in a field that is now called “nonlinear science.” These developments have been of such magnitude that historian Alan Beyerchen concluded “There is every reason to believe that the westernized world is in the early stages of an intellectual transformation of major proportions, perhaps as significant as the emergence of the modern worldview in the fifteenth through seventeenth centuries.”

These new developments are rooted in work conducted by mathematician Jules Henri Poincaré around the turn of the century. He carried out some calculations regarding planetary orbits and demonstrated the possibility of erratic or “chaotic” dynamical behavior. What is particularly interesting about such behavior is that it is exhibited by deterministic systems, that is, systems that have no stochastic (noisy) character of any kind. Little was done with Poincaré's findings until the 1960s when new work in nonlinear science, based on numerical models, led to the discovery of amazingly complex behavior that can be exhibited even by very simple deterministic systems. Since that time, the discipline has grown enormously and has been applied in practically every area of science and engineering, as well as in other diverse areas such as the financial market, political science, economics, and even Adlerian psychology.

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Type
Technical Features
Information
MRS Bulletin , Volume 20 , Issue 7 , July 1995 , pp. 20 - 28
Copyright
Copyright © Materials Research Society 1995

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References

1.Beyerchen, A.D., “Nonlinear Science and the Unfolding of a New Intellectual Vision,” Papers in Contemporary Studies 6 (1989) p. 25.Google Scholar
2.Kawczyriski, A.L., Raczyński, W., and Baranowski, B., “Analysis of Chaotic Oscillations in a Simple Electrochemical System,” Z. Phys. Chemie Leipzig 269 (1988) p. 596.CrossRefGoogle Scholar
3.Wisdom, J., “Is the Solar System Stable? and Can We Use Chaos to Make Measurements?”, in CHAOS/XAOC — Soviet-American Perspectives on Nonlinear Science, edited by Campbell, D.K. (American Institute of Physics, New York, 1990) p. 275.Google Scholar
4.Gleick, J., Chaos-Making a New Science (Viking, New York, 1987).Google Scholar
5.Thompson, J.M.T. and Stewart, H.B., Nonlinear Dynamics and Chaos-Geometrical Methods for Engineers and Scientists (Wiley, Chichester, U.K., 1986).Google Scholar
6.Moon, F.C., Chaotic Vibrations—An Introduction for Applied Scientists and Engineers (Wiley, New York, 1987).Google Scholar
7.Moon, F.C., Chaotic and Fractal Dynamics—An Introduction for Applied Scientists and Engineers (Wiley, New York, 1992).CrossRefGoogle Scholar
8.Kim, J.H. and Stringer, J., eds., Applied Chaos (Wiley, New York, 1992).Google Scholar
9.Thompson, J.M.T. and Stewart, H.B., Nonlinear Dynamics and Chaos — Geometrical Methods for Engineers and Scientists (Wiley, Chichester, U.K., 1986) p. 3.Google Scholar
10.Lorenz, E.N., “Deterministic Nonperiodic Flow,” J. Atmos. Sciences 20 (1963) p. 130.2.0.CO;2>CrossRefGoogle Scholar
11.Kirkaldy, J.S., “Deterministic Chaos and Eutectoid Phase Transformations,” Scr. Metall. Mater. 24 (1990) p. 179.CrossRefGoogle Scholar
12.Akuezue, H.C. and Stringer, J. (unpublished manuscript).Google Scholar
13.Frank, G.W., Lookman, T., and Nerenberg, M.A.H., “Recovering the Attractor: A Review of Time-Series Analysis,” Can. J. Phys. 68 (1990) p. 711.CrossRefGoogle Scholar
14.Brawley, G.H., Markworth, A.J., and Parmananda, P., “Use of Neural Networks to Predict the Short-Term Behavior of Chaotic Time Series, Including Effects of Superimposed Noise,” in Proc. 26th Southeastern Symp. on System Theory (IEEE Computer Society Press, Los Alamitos, CA 1994) p. 643.CrossRefGoogle Scholar
15.McCoy, J.K., Parmananda, P., Rollins, R.W., and Markworth, A.J., “Chaotic Dynamics in a Model of Metal Passivation,” J. Mater. Res. 8 (1993) p. 1,858.Google Scholar
16.Parmananda, P., Dewald, H.D., and Rollins, R.W., “Mixed-Mode Oscillations in the Electrodissolution of Copper in Acetic Acid/Acetate Buffer,” Electrochim. Acta 39 (1994) p. 917.CrossRefGoogle Scholar
17.Ott, E., Grebogi, C., and Yorke, J.A., “Controlling Chaos,” Phys. Rev. Lett. 64 (1990) p. 1,196.CrossRefGoogle ScholarPubMed
18.Chen, G., “Control and Synchronization of Chaotic Systems (A Bibliography),” Electrical Engineering Department, University of Houston, Texas; available by http://uhoop.egr.uh.edu.pub.TeX.chaos.tex (login name and password: both “anonymous”).Google Scholar
19.Abarbanel, H.D.I., Brown, R., Sidorowich, J.J., and Tsimring, L.Sh., “The Analysis of Observed Chaotic Data in Physical Systems,” Rev. Mod. Phys. 65 (1993) p. 1,331.CrossRefGoogle Scholar
20.Dewald, H.D., Parmananda, P., and Rollins, R.W., “Periodic Current Oscillations in the Anodic Dissolution of Copper in Acetate Buffer,” J. Electroanal. Chem. 306 (1991) p. 297.CrossRefGoogle Scholar
21.Dewald, H.D., Parmananda, P., and Rollins, R.W., “Periodic and Chaotic Current Oscillations at a Copper Electrode in an Acetate Electrolyte,” J. Electrochem. Soc. 140 (1993) p. 1,969.CrossRefGoogle Scholar
22.Rollins, R.W., Parmananda, P., and Sherard, P., “Controlling Chaos in Highly Dissipative Systems: A Simple Recursive Algorithm,” Phys. Rev. E 47 (1993) p. R780.Google ScholarPubMed
23.Parmananda, P., Sherard, P., and Rollins, R.W., “Control of Chaos in an Electrochemical Cell,” Phys Rev. E 47 (1993) p. R3,003.Google Scholar
24.Rhode, M.A., Rollins, R.W., and Vassiliadis, C.A., in “Adaptive Learning to Control Chaos,” Proc. 26th Southeastern Symp. on System Theory (IEEE Computer Society Press, Los Alamitos, CA, 1994) p. 638.CrossRefGoogle Scholar
25.Parmananda, P., Rhode, M.A., Johnson, G.A., Rollins, R.W., Dewald, H.D., and Markworth, A.J., “Stabilization of Unstable Steady States in an Electrochemical System Using Derivative Control,” Phys. Rev. E 49 (1994) p. 5,007.Google Scholar
26.Rhode, M.A., Rollins, R.W., Markworth, A.J., Edwards, K.D., Nguyen, K., Daw, O.S., and Thomas, J.F., J. Appl. Phys. in press.Google Scholar
27.Wilson, E. Bright, “One Hundred Years of Physical Chemistry,” Am. Scientist 74 (1986) p. 70.Google Scholar