Dynamical systems theory

Without immersing ourselves in mathematical rigor, we define a dynamical system - from a physicist s perspective - as being any physical system that evolves in time according to some well-defined rule. To be somewhat more specific, a general continuous dynamical system satisfies the following two properties  

Its state is completely defined at all times by the values of N variables, a i(t), 

where Xi t) represent any physical quantity of interest (position, 

Its temporal evolution is specified by an autonomous system of N, possibly coupled, ordinary first-order differential equations  

Once the initial state x(f = 0) of the system is specified, future states, x(t), are uniquely defined for all times t . Moreover, the uniqueness theorem of the solutions of ordinary differential equations guarantees that trajectories originating from different initial points never intersect. 

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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. 

The exponent /3 decreases from 1.6 to 1.0 as the system size increases from N = 2 to N = 100. Kaneko suggests that this may result from an effective increase in the number cf possible pathways for the zigzag collapse, and thus that the change of 0 with size may be regarded as a path from the dynamical systems theory to statistical mechanical phase transition problems [kaneko89a]. 

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