Largest Lyapunov exponent - Moment Lyapunov exponents

This code contains two functions which assess the largest Lyapunov exponents and the moment Lyapunov exponent of a linear stochastic system.

The stability of dynamic systems driven by random noise is not an easy task, especially when the studied physical problems lead to complicated mathematical formulations that are difficult to handle with analytical approaches. This often happens in practical engineering problems where the dynamic systems are usually characterized by many degrees of freedom and driven by real noise.

This code contains two functions (LE_XIE2006.m and MLE_XIE2009.m) which numerically assess the sample and p-th moment stability of a linear stochastic system driven by an ergodic process (), according to the largest Lyapunov exponent (LE) and moment Lyapunov exponents (MLE), respectively. Both codes are inspired by the seminal works of Prof. Xie Wei-Chau [Xie (2006) and Xie and Huang (2009)].

Dynamic system equation in the state-spece:

Largest Lyapunov exponent:

Moments Lyapunov exponents:

Both codes require the time realization of the state-matrix of the dynamic system (one sample for the largest Lyapunov exponent, multiple samples for the moments).

The following code was developed to evaluate the random flutter sample and p-th moment stability of a suspension bridge parametrically excited by a turbulent wind field [see Barni, N., Bartoli, G., and Mannini, C. (2024)].

To clarify how to use the codes, a simple example of a linear pendulum driven by stochastic damping defined as an Ornstein-Uhlenbeck process is provided (Example_Stochastic_pendulum.m).

List of codes:

• LE_XIE2006.m (function)
• MLE_XIE2009.m (function)
• Example_Stochastic_pendolum.m (script)
• central_difference.m (function)

N.B. According to the LE and MLE definition [see e.g. Xie (2006)], both the observation time T and the number of samples (only for the MLE) have to be large enough to ensure estimators converge numerically (possible check: the slope of MLE at should be as similar as possible to the LE).

References:

• Xie, W.C.: Dynamic Stability of Structures. Dynamic Stability of Structures. Cambridge University Press, New York (2006). https://books.google.it/books?id=Ibe8GeL 0QsC
• Xie, W.C., Huang, Q.: Simulation of Moment Lyapunov Exponents for Linear Homogeneous Stochastic Systems. Journal of Applied Mechanics 76(3), 031001 (2009) https://doi.org/10.1115/1.3063629
• Barni, N., Bartoli, G., and Mannini, C. Lyapunov stability of suspension bridges in turbulent flow. (2024) https://doi.org/10.1007/s11071-024-09931-y