DYNAMIC-RESPONSE OF NONLINEAR-SYSTEMS TO RENEWAL IMPULSES BY PATH INTEGRATION

The cell-to-cell mapping (path integration) technique has been devised for non-linear and non-hysteretic systems subjected to random trains of impulses driven by an ordinary renewal point process with gamma-dis tributed integer parameter inter-arrival times (an Erlang process). Si nce the renewal point process does not have independent increments the state vector of the system, consisting of the generalized displacemen ts and velocities, is not a Markov process. Initially, it is shown how the indicated system can be converted to an equivalent Poisson driven system at the expense of introducing additional discrete-valued state variables for which the stochastic equations are also formulated. The reby the original non-Markov system is converted into an equivalent sy stem which does possess the Markov property. The continuous non-diffus ive Markov process is converted, by discretization in time and in phas e space, to a Markov chain. The transition probability matrix is next constructed for a transition time interval short enough to neglect the probability of occurrence of more than one impulse in this interval. The technique of convection and lumping of the probability mass is ess entially based on the technique devised by Koyluoglu and the authors [ 1] for Poisson driven systems. In the present problem this technique h as been suitably modified with due account taken of the artificial, au xiliary, state variables introduced in the description. The method has been applied to the Duffing oscillator, and the results for the stati onary probability density function have been compared with those obtai ned from extensive Monte Carlo simulations. (C) 1996 Academic Press Li mited