SISTEMI DINAMICI, CAOS E COMPLESSITA'

Provide students with a gradual introduction to the science of complex systems through a path that, starting from dynamical systems - both dissipative and conservative - with a few degrees of freedom, already capable of manifesting chaotic behaviors, then moves to the study of systems with many degrees of freedom, to be addressed by means of a statistical approach, with particular attention to non-equilibrium phenomena, systems with long-range interactions and those at the edge of chaos.

Frontal lectures - Audiovisual material - Use of the software NetLogo for the development of agent-based simulations

Introduction to the new science of complexity. Non-linear systems, self-organized criticality, complex networks, emergent phenomena, cellular automata, synchronization, sociophysics and econophysics,

Systems with few degrees of freedom. Dissipative dynamical systems, both continuous (flows) and discrete (maps), with one and two dimensions. Fixed point and limit cycle attractors. Bifurcations. Three-dimensional flows. Routes to chaos. Lyapunov exponents. Fractal size. Hamiltonian systems in one and two dimensions. The KAM theorem.

Systems with many degrees of freedom. Review of thermodynamics. Clausius entropy and time arrow. Order and disorder in the universe. Introduction to classical statistical equilibrium mechanics. Boltzmann Entropy and H Theorem. Gibbs "ensembles" theory. Fine-tuning and the problem of low entropy initial conditions. Introduction to non-extensive statistical mechanics. Tsallis entropy and the generalized Central Limit heorem. Long-range Hamiltonian systems. Complex systems at the Edge of Chaos.

1) Robert C. Hilborn, “Chaos and nonlinear dynamics”, Oxford University Press, 2nd Ed. 2000

2) Steven Strogatz, “Nonlinear dynamics and chaos”, Westview Press 2001

3) K. Huang, “Meccanica Statistica”, Zanichelli 1997

4) A.Pluchino, "La firma della complessità. Una passeggiata al margine del caos", Malcor D' Edizione 2015

5) C.Tsallis, "Introduction to nonextensive statistical mechanics: approaching a complex world", Springer 2008