Modelling of Chaotic Processes with Caputo Fractional Order Derivative.

Chaotic dynamical systems are studied in this paper. In the models, integer order time derivatives are replaced with the Caputo fractional order counterparts. A Chebyshev spectral method is presented for the numerical approximation. In each of the systems considered, linear stability analysis is established. A range of chaotic behaviours are obtained at the instances of fractional power which show the evolution of the species in time and space.

Thermal conductivity, relaxation and low-frequency vibrational mode anomalies in glasses: a model using the Fermi-Pasta-Ulam nonlinear Hamiltonian.

We present a nonlinear model that allows exploration of the relationship between energy relaxation, thermal conductivity and the excess of low-frequency vibrational modes (LFVMs) that are present in glasses. The model is a chain of the Fermi-Pasta-Ulam (FPU) type, with nonlinear second neighbour springs added at random. We show that the time for relaxation is increased as LFVMs are removed, while the thermal conductivity diminishes. These results are important in order to understand the role of the cooling speed and thermal conductivity during glass transition. Also, the model provides evidence for the fundamental importance of LFVMs in the FPU problem.