Lyapunov exponents play an important role in the modern theory of non-linear dynamical systems. They characterize the rate of exponential changes of the states or responses of the systems. A systematic investigation is carried out in this research of Lyapunov exponents along with their application to various problems in structural dynamics.

The formulation and various methods of evaluation of Lyapunov exponents for different classes of dynamical systems are studied from both analytical and numerical simulation viewpoints.

Chaotic motion in the transverse vibration of a buckled column under axial periodic load or end displacement, which is modelled as a single degree-of-freedom parametrically excited system, is investigated first. The Lyapunov exponents of the system are evaluated to ascertain the onset of chaotic motion. Other characteristics such as Poincare map, Lyapunov dimension, and power spectrum are also studied to confirm the results obtained.

As an application to discrete stochastic systems, the localization phenomenon in randomly disordered periodic engineering structures is investigated. The localization factor, which characterizes the exponential rate of decay of amplitudes of waves propagating in the disordered periodic structure, is related to the largest Lyapunov exponent of the structure. Furstenberg's theorem on the asymptotic behaviour of a product of random matrices is applied to evaluate the largest Lyapunov exponents of certain disordered periodic structures having engineering applications. The results obtained are also applicable to other mono-coupled randomly disordered periodic structures.

Stochastic bifurcational behaviour of engineering structures exhibiting pitch-fork and Hopf bifurcations is examined by determining the largest Lyapunov exponent of continuous single degree-of-freedom systems. Several asymptotic methods for evaluating the largest Lyapunov exponent such as series expansion, asymptotic expansion of integrals, linear transformation and stochastic averaging are presented. The validity of the approximate results is checked using numerical techniques.

The almost sure stability of multi-degrees-of-freedom systems frequently encountered in problems of elastic stability of structures is investigated. Asymptotic expressions for the largest Lyapunov exponent are obtained under various forms of stochastic loading. These are compared with numerical results obtained by digital simulation.