By John S. Nicolis

The major goal of those lectures is to tri gger the curiosity of the stressed below graduate pupil of actual, mathematical, engineering, or organic sciences within the new and interesting multidisciplinary sector of the evolution of "large-scale" dynamical structures. this article grew out of a synthesis of relatively heterogeneous mate rial that I provided on numerous events and in several contexts. for instance, from lectures given given that 1972 to first- and final-year undergraduate and primary yr graduate scholars on the institution of Engineering of the college of Patras and from casual seminars provided to a global workforce of graduate and publish doctoral scholars and school individuals on the college of Stuttgart within the aca demic yr 1982-1983. those that look for rigor or perhaps formality during this publication are absolute to be relatively disenchanted. My goal is to begin from "scratch" if attainable, conserving the rea soning heuristic and tied as heavily as attainable to actual instinct; i suppose as must haves simply simple wisdom of (classical) physics (at the extent of the Berkeley sequence or the Feynman lectures), calculus, and a few components of probabil ity idea. this doesn't suggest that I meant to write down a simple publication, yet relatively to dispose of any hassle for an keen reader who, despite incomplete for malistic education, wish to turn into accustomed to the actual rules and con cepts underlying the evolution and dynamics of complicated systems.

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**Example text**

Bifurcation diagram of the rotating nonlinear pendulum oscillator, where ~ =n 2L/g So we see that beyond a certain critical value of the angular velocity n of the frame, the traditional stable steady state 8 =0 becomes unstable, and the new (symmetrical) stable steady-state solutions are given by 8 = arc cos~-l. Of course, we may as well reduce the preceding analysis to the motion of a heavy particle in a potential well. What is the potential function now? The external force acting on the system is F =w6 sin (~COS8 -1).

This is not always the case, however. In order to pass judgement we have to exami ne carefully the "cartography" of the state space around each such new state. , more accessible) candidates than others for receiving the system which, after being expelled from the former state, requires a "landing". In short, we have to examine the "stream1ines" around the available steady states in state space, in the hope of predicting the 1i ke 1i hood of the next state. 13 -perpendicular to the paper, and then looking "down" to appreciate the road system of the landscape.

E(I, ... , Il), ~i=l Ni =N. The number of microscopic arrangements or compZexions responsible for the same macrostate of the system is, of course, W= N! 15) i' If all complexions are a priori equiprobable, then the probability of having one of them responsible for the given macrostate is P =W- 1. Question: What is the uncertainty-and the information to be deduced there from-about which complexion is responsible for the observed macros tate? Think of the system under consideration as divided into two parts 1 and 2.