08 December 2008

STAB2004_Use of Lyapunov Exponents to Predict Chaotic Vessel Motions

McCue, L.S. and Troesch, A.W., "Use of Lyapunov exponents to predict chaotic vessel motions," 7th International Ship Stability Workshop, Shanghai, China, November, 2004.


Use of Lyapunov Exponents to Predict Chaotic Vessel Motions
Leigh McCue, Armin Troesch, Department of Naval Architecture and Marine Engineering, University of Michigan

ABSTRACT
It is the aim of this paper to further the use of Lyapunov and local Lyapunov exponent methods for analyzing phenomena involving nonlinear vessel dynamics. Lyapunov exponents represent a means to measure the rate of convergence or divergence of nearby trajectories thus denoting chaos and possibly leading to the onset of conditions that produce capsize. The work developed here makes use of Lyapunov exponent methodologies to study capsize and chaotic behavior in vessels both experimentally and numerically using a multi-degree of freedom computational model. Since, the Lyapunov exponent is defined in the limit as time approaches infinity, one encounters fundamental difficulties using Lyapunov exponents on the capsize problem, which is inherently limited to a finite time. This work also incorporates the use of local Lyapunov exponents, which do not require an infinite time series, to demonstrate their usefulness in analyzing finite time chaotic vessel phenomena. The objective is to demonstrate the value of the Lyapunov exponent and local Lyapunov exponent as a predictive tool with which to indicate regions with crucial sensitivity to initial conditions. Through the intelligent use of Lyapunov exponents in vessel analysis to indicate specific regions of questionable stability, one may significantly reduce the volume of costly simulation and experimentation.

KEYWORDs
capsize, chaos, Lyapunov exponent, local Lyapunov exponent, nonlinear vessel motions

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