Studies in Nonlinear Dynamics

Reviews in Computational Chemistry, Volume 10 Kenny B. Lipkowitz and Donald B. Boyd, Editors VCH Publishers, Inc. New York, 1997 

The properties of atoms and the subatomic particles which comprise them, particularly the electrons, can explain the transformation of matter through chemical reaction. 

The addition of details (such as the elearon shell structure, various bonding theories, etc.) has rendered this view enormously successful in explaining many of the phenomena of interest to chemists. 

The approach advocated in nonlinear dynamics has been developed as a response to the failure of the strictly reductionist approach in elucidating many phenomena of interest above the molecular level. Philip Anderson, a physicist, wrote more than 20 years ago on the necessity of the paradigm shift that has led to this area of science  

The reductionist hypothesis does not by any means imply a constructionist one the ability to reduce everydiing to simple fundamentd laws does not imply the ability to start from those laws and reconstruct the universe. 

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Raima Larter and Kenneth Showalter, Computational Studies in Nonlinear Dynamics. 

Renal Blood Flow Regulation and Arterial Pressure Fluctuations a Case Study in Nonlinear Dynamics, Physiol. Rev. 74, 637-681 (1994). 

Studies in nonlinear dynamics developed rapidly in the last quarter of the twentieth century, and we now have quite a lot of knowledge about the dynamics of systems with a few degrees of freedom. 

R. C. Hilborn, Chaos and Nonlinear Dynamics An Introduction for Scientists and Engineers, Oxford University Press, New York, 1994. See also, R. Latter and K. Showalter, this volume. Computational Studies in Nonlinear Dynamics. 

Sometimes the ODEs that arise in studies in nonlinear dynamics can be solved using explicit methods (such as the forward Euler) which require less computations per step and are thus cheaper and ter to implement. The Runge-Kutta femily of algorithms are a popular implementation of the explicit methods. Runge—Kutta methods begin with a Taylor series expansion the order of the particular Runge-Kutta method used is simply the highest order term retained in the Taylor series. 

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