Dynamic Systems Theory

Arnold, V. L, Afraimovich, V. S., Ilyashenko, Y. S. and Shilnikov, L. P. [1994] Bifurcation Theory Dynamical Systems V. Encyclopaedia of Mathematical Sciences (Springer-Verlag New York). 

The classical microscopic description of molecular processes leads to a mathematical model in terms of Hamiltonian differential equations. In principle, the discretization of such systems permits a simulation of the dynamics. However, as will be worked out below in Section 2, both forward and backward numerical analysis restrict such simulations to only short time spans and to comparatively small discretization steps. Fortunately, most questions of chemical relevance just require the computation of averages of physical observables, of stable conformations or of conformational changes. The computation of averages is usually performed on a statistical physics basis. In the subsequent Section 3 we advocate a new computational approach on the basis of the mathematical theory of dynamical systems we directly solve a... 

Takens, F. Motion under the influence of a strong constraining force. In Global Theory of Dynamical Systems, Evanston 1979 (Z. Nitecki and C. Robinson, eds.). Springer-Verlag, Berlin, Heidelberg, New York (1980)... 

Chapter 4 covers much of the same ground as chapter 3 but from a more formal dynamical systems theory approach. The discrete CA world is examined in the context of what is known about the behavior of continuous dynamical systems, and a number of important methodological tools developed by dynamical systems theory (i.e. Lyapunov exponents, invariant measures, and various measures of entropy and... 

The decay of the spatial block entropy, which gives the amount of information contained in a block of N contiguous site values ai,...,aN needed to predict the value (Jn+i is considerably slower than [block-length), and is therefore indicative of very long and complex correlations we will come back to this point later in chapter 4, following our discussion of dynamical system theory. 

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