Global Gheory of a Second Order: Linear Ordinary by Author Unknown

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83, xn oscillates between three values, and then 6 values, 12 values etc. For r > 4, xn diverges. 22 CHAPTER 1. WHY A SCIENCE OF NETWORKS? in ecology by the biologist Robert May [101]. Its meaning refers to cases in which a population (represented by xn , with n = 1, 2, . . indicating the generation of the species) evolves according to two opposite causal effects: there is an increasing rate proportional to the current population (reproduction) in the presence of a limiting factor (starvation) that would decrease the population at a rate proportional to the complementary value of the current population measured in proportion to the capacity of the system.

1 Qualitative and quantitative One of the pioneers of quantum mechanics, Ernest Rutherford, once observed that physical theories are, and should be, quantitative in the following way: All science is either physics or stamp collecting. Qualitative is nothing but poor quantitative. This view of the foundation of science is shared by a majority of scientists and from our experience it reflects the feelings of the scientific community as a whole. The mathematician Rene Thom [173] elaborated on this perspective by pointing out that at the end of the seventeenth century there were two main groups in science, those that followed the dictates of Descartes and those that accepted and practiced the physics of Newton: Descartes, with his vortices, his hooked atoms, and the like, explained everything and calculated nothing.

Chaos is an implicit property of nonlinear dynamical networks, whereas noise is a property of the environment in contact with the network of interest. Chaos can therefore be controlled and predicted over short time intervals, whereas noise can neither be predicted nor controlled, except perhaps through the way it interacts with the network. 4 Simple examples are found in the pseudo-random generators in computers whose simplest form is given by the linear congruential generators implemented in the ANSI C rand() command: Ij+1 = aIj + c (mod m) with a = 1103515245, c = 12345, and m = 232 .

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