Learning objectives
<br />Basic notions will be introduced in order to characterize qualitatively the motion in topological and probability spaces, stressing in particular on the concepts of stability and chaoticity, on symbolical dynamics, on the idea of dynamical entropy, on the complexity measures. Examples will refer to problems in Hamiltonian dynamics (short account on the KAM Theory) and, for the discrete case, to problems in statistical mechanics represented by Cellular Automata (Ising, sandpiles). Some methods in numerical simulations and data analysis will be presented.
Prerequisites
<br />Mathematical background of the first level degree in mathematical or physical sciences. Basic notions of classical mechanics and statistics.
Course unit content
<br /><br />Preliminaries in topology and measure theory.<br />One parameter groups, fluxes, maps. <br />Elementary examples of one dimensional maps (circle, quadratic map, ecc.) <br />Shift spaces and symbolical dynamics. <br />Topological conjugacy. Structural stability.<br />Metrical dynamical systems. <br />Invariant measures. <br />Birkhoff Theorem..<br />Ergodicity and Mixing <br />Metrical isomorphism. <br />Shannon entropy and Khinchin Theorem.<br />Kolmogorov Entropy.<br />Bernoulli Shifts.<br />Lyapunov exponents.<br />Predictability and chaos. <br />Time series analysis, Wiener-Khinchin theorem, 1/f noise.<br />Cellular automata and examples (Ising, sandpiles.)
Full programme
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Bibliography
<br />R.L.Devaney: Chaotic Dynamical Systems (Benjamin 1985);<br />A.J. Lichtenberg and M.A. Liebermann: Regular and Stochastic Motion (Springer 1983);<br />V.I.Arnold and A. Avez: Ergodic Problems of Classical Mechanics (Benjamin 1968)<br />D. Ruelle: Chaotic Evolution and Strange Attractors (Cambridge UP 1989) <br />R. Badii and A. Politi: Complexity (Cambridge UP 1997)<br />T. Toffoli and N. Margulis: Cellular Automata Machines (Mit Press 1987)
Teaching methods
<br />Frontal lectures and exercises<br />Final oral examination<br />
Assessment methods and criteria
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Other information
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