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Response of nonlinear systems to stochastic excitation


Payne, Harold James (1967) Response of nonlinear systems to stochastic excitation. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/H3WE-RD54.


The response of a dynamical system modelled by differential equations with white noise as the forcing term may be represented by a Markov process with incremental moments simply related to the differential equation. The structure of such Markov processes is completely characterized by a transition probability density function which satisfies a partial differential equation known as the Fokker-Planck equation. Sufficient conditions for the uniqueness and convergence of the transition probability density function to the steady-state are established.

Exact solutions for the transition probability density function are known only for linear stochastic differential equations and certain special first order nonlinear systems. Exact solutions for the steady-state density are known for special nonlinear systems. Eigenfunction expansions are shown to provide a convenient vehicle for obtaining approximate solutions for first order systems and for self-excited oscillators. The first term in an asymptotic expansion of the transition probability density function is found for self-excited oscillators.

A class of first passage problems for oscillators, which includes the zero crossing problem, is formulated.

Item Type:Thesis (Dissertation (Ph.D.))
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Caughey, Thomas Kirk
Thesis Committee:
  • Unknown, Unknown
Defense Date:3 May 1967
Record Number:CaltechETD:etd-10022002-114156
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:3864
Deposited By: Imported from ETD-db
Deposited On:03 Oct 2002
Last Modified:21 Dec 2019 02:03

Thesis Files

PDF (Payne_hj_1967.pdf) - Final Version
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