Dynamical systems subjected to correlated noise is in most engineering applications the rule rather than the exception (in contrast to white-noise). While for white-noise excitation the input spectrum is essentially flat and there are probabilistic tools to analyze the response, correlated noise is characterized in most cases by broadband spectra with non-uniform distribution of energy across harmonics and for this case the determination of the response statistics is not straightforward.
An important class of engineering systems modeled as dynamical systems under correlated noise is ships and ocean structures subjected to water waves, e.g. ship rolling motion. The random character of the environment that these systems operate in makes it essential to characterize their dynamics in a probabilistic framework (i.e. determine statistics for their motion rather than individual trajectories).
While for dynamical systems subjected under white-noise excitation there is Fokker-Planck-Kolmogorov (FPK) equation that allows for the determination of the full probabilistic structure of the response (probability density function), for nonlinear systems subjected to correlated excitation there are no mathematical tools that can provide us with the probability density function of the response. Such information is really important in order to quantify extreme events such as the extreme rolling event shown above.
The scope of this work is the development of such probabilistic analysis tools and their application to engineering systems that operate in environments with randomness. Below we present a very simple example of a singe degree of freedom dynamical system acting under the influence of a fixed quartic, double-well potential and two forms of external excitation: i) white-noise and ii) colored-noise.
While in the first case of white noise excitation there are two well separated regions where the system "lives" this not the case for the colored noise excitation where we have frequent transitions from one equilibrium point to another. For an engineering system such transitions are crucial and therefore we need tools to quantify them.
The joint response-excitation probability density function theory
The joint response-excitation (RE) pdf theory is a new set of evolution equations that characterize the probability density function for dynamical systems subjected to correlated excitation. Using tools of functional differential equations and suitable finite-dimensional projection operators [read more here...] we derive this set of equations, which are not approximate, but an exact reformulation of the dynamical problem in probabilistic terms (in the same spirit as FPK equation for white noise excitation). For special forms of excitations (such as white-noise) RE equations directly reproduce the existing ones (such as the FPK equation) and to this end they can be viewed as a unified approach for stochastic systems. Current efforts from our group and collaborators focus on the numerical and analytical treatment of these equations.
M. Mohamad, T. Sapsis, Probabilistic description of extreme events in intermittently unstable systems excited by correlated stochastic processes, SIAM J. of Uncertainty Quantification, Submitted (2014). [pdf]
D. Venturi, T. Sapsis, H. Cho, and G. E. Karniadakis, A computable evolution equation for the joint response-excitation probability density function of stochastic dynamical systems, Proceedings of the Royal Society A, 468 (2012) 759. [pdf]
T. Sapsis & G. Athanassoulis, New partial differential equations governing the response-excitation joint probability distributions of nonlinear systems under general stochastic excitation, Probabilistic Engineering Mechanics, 23 (2008) 289. [pdf]