Gruppo di fisica dei sistemi complessi di Bologna Gruppo di fisica dei sistemi complessi di Bologna
Bologna University Physics Department
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Dynamical Systems

During the last decades non linear dynamics has faced relevant developments and the range of its applications has extended. In the XIX century linear analysis and linear models have been successfully applied in physics and engineering: among non linear systems only the integrable or quasi integrable o ones had been investigated and used. They provided a good description of planetary motions, but were totally inadequate to deal with the turbulent motion of a fluid or a plasma. The theory of dynamical systems, which combines the deterministic evolution with a statistical analysis of motion, via an invariant probability measure, was founded by Birkhoff and Poincare', starting from Boltzmann's statistical mechanics. Integrable systems present regular quasi-periodic motions, whereas the uniformly chaotic ones, though deterministic, are similar to stochastic processes, exhibit a strong dependence on initial conditions, but have simple statistical properties. Between these two extremes there exists a large spectrum of systems with intermediate behavior, where order and chaos interlace. Non conventional geometric structures like fractals, are observed, whose scale invariance properties makes them similar to an endless game of Chinese boxes. When dissipative and time dependent external forces are introduced the asymptotic motion develops on attractors, with fractal like structures, signature of a restored energy balance. The dynamical behavior is examined from a statistical viewpoint, introducing suitable indicators such as the Lyapounov exponents, which measure the divergence rate of nearby orbits and/or the recurrence times spectra, which provide the statistics of time intervals after which the orbit returns close to the initial point. The invariant measure is investigated by decomposing the attractor into subsets with the same scaling law and computing the corresponding dimension. The topology changes when a parameter of the system is varied: these bifurcations provide scenarios for the transition from order to chaos and vice versa. For the quasi integrable Hamiltonian systems the theory of normal forms, allows to conjugate the orbits with a given topology with their homologues having an exact continuous or discrete symmetry, by using a perturbative construction. A distinction of orbits into non resonant, resonant and chaotic is achieved numerically by the Fourier analysis of the orbits, which associates to any of them the corresponding frequency (when it exists). The time evolution of a set of points with assigned initial density, may be followed numerically or by integrating the continuity equation. The same analysis applies when a fluctuating field is present. In this case to a single initial condition correspond many orbits, one for each realization of the noise. The average properties of time evolution are specified by a probability density function, which satisfies the Fokker-Planck diffusion equation, or its generalizations. Such a description is often used to isolate a few significant degrees of freedom in a system, treating the remaining ones as an external system, which produce a fluctuating field. A relevant case is Langevin equation which describes the erratic motion of particles in a fluid where the collisions with molecules are described by a white (non correlated) noise and the medium resistance by a damping term. The equilibrium solution for the probability density function is Boltzmann's distribution where the temperature is given by as the ratio between the square of the noise amplitude and the damping coefficient. The activity of our group on dynamical systems is the oldest and concerned many of the outlined topics. Significant contributions were given to the theory of normal forms for symplectic maps, the corrections of scaling laws in multifractal analysis, the recurrence time spectra, the theory of diffusion for Hamiltonian systems with weak noise and the theory anomalous diffusion.

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