Studies on nonlinear dynamics of driven vibrating systems with dissipation

Nonlinear dissipative continuous and discrete systems at different levels of complexity are considered. Firstly, a general theory for the dynamics of one-dimensional continua with elastically or rigidly mounted rigid parts extending the traditional Sturm-Liouville theory is developed. The solution, taking into account nonlinearities and dissipation as forcing terms, is formed in terms of proper eigenmodes with the aid of the associated orthogonality relations. As an application the dynamics of an arctic offshore structure is studied. The modal analysis with proper eigenmodes is performed. The finite element method-solution is also developed. Secondly, a particle in a symmetric potential well with quartic nonlinearity is studied. The exclusion rule for inversion symmetric attractors is derived. The chaotic motion and its universal features are considered. The effect of a small asymmetry is studied. Finally, the dynamics of an impacting particle with piece-wise analytic solution is considered. The rich chaotic behaviour, the Farey organisation of the solutions, a peculiar group of devil's attractors and the phase locking phenomenon are found. The fractal basin boundaries of multiple coexisting attractors are examined.