TY - JOUR

T1 - The real Schur decomposition estimates Lyapunov characteristic exponents with multiplicity greater than one

AU - Masarati, Pierangelo

AU - Tamer, Aykut

PY - 2016/12/1

Y1 - 2016/12/1

N2 - Lyapunov characteristic exponents are indicators of the nature and of the stability properties of solutions of differential equations. The estimation of Lyapunov exponents of algebraic multiplicity greater than 1 is troublesome. In this work, we intuitively derive an interpretation of higher multiplicity Lyapunov exponents in forms that occur in simple linear time invariant problems of engineering relevance. We propose a method to determine them from the real Schur decomposition of the state transition matrix of the linear, nonautonomous problem associated with the fiducial trajectory. So far, no practical way has been found to formulate the method as an algorithm capable of mitigating over- or underflow in the numerical computation of the state transition matrix. However, this interesting approach in some practical cases is shown to provide quicker convergence than usual methods like the discrete QR and the continuous QR and Singular Value Decomposition (SVD)methods when Lyapunov exponents with multiplicity greater than one are present.

AB - Lyapunov characteristic exponents are indicators of the nature and of the stability properties of solutions of differential equations. The estimation of Lyapunov exponents of algebraic multiplicity greater than 1 is troublesome. In this work, we intuitively derive an interpretation of higher multiplicity Lyapunov exponents in forms that occur in simple linear time invariant problems of engineering relevance. We propose a method to determine them from the real Schur decomposition of the state transition matrix of the linear, nonautonomous problem associated with the fiducial trajectory. So far, no practical way has been found to formulate the method as an algorithm capable of mitigating over- or underflow in the numerical computation of the state transition matrix. However, this interesting approach in some practical cases is shown to provide quicker convergence than usual methods like the discrete QR and the continuous QR and Singular Value Decomposition (SVD)methods when Lyapunov exponents with multiplicity greater than one are present.

U2 - 10.1177/1464419316637275

DO - 10.1177/1464419316637275

M3 - Article

SN - 1464-4193

VL - 230

SP - 568

EP - 578

JO - Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics

JF - Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics

IS - 4

ER -