Estimating covariant Lyapunov vectors from data

07/16/2021
by   Christoph Martin, et al.
0

Covariant Lyapunov vectors (CLVs) characterize the directions along which perturbations in dynamical systems grow. They have also been studied as potential predictors of critical transitions and extreme events. For many applications, it is, however, necessary to estimate the vectors from data since model equations are unknown for many interesting phenomena. We propose a novel method for estimating CLVs based on data records without knowing the underlying equations of the system which is suitable also for high-dimensional data and computationally inexpensive. We demonstrate that this purely data-driven approach can accurately estimate CLVs from data records generated by chaotic dynamical systems of dimension 128 and multiple lower-dimensional systems and thus provides the foundation for numerous future applications in data-analysis and data-based predictions.

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Acknowledgments

The authors of this study are grateful to the BMBF for financial support within the project DADLN (01S19079) and to the Landesforschungsförderung Hamburg for financial support within the project LD-SODA (LFF-FV90).

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I Computing Lyapunov exponents, finite-time Lyapunov exponents and covariant Lyapunov vectors

The LEs are computed using Benettin algorithm [1]. The integration step for computing Lyapunov vectors is for all the models, i.e., twice the time step of the simulations. For the Lorenz attractor and the Josephson junction, we reorthogonalise the vectors every 10 steps, i.e., the reorthogonalisation interval is 0.01. For the Lorenz 96 system, the reorthogonalisation of the vectors is done every 100 steps (). During the computation of the CLVs using Ginelli’s method, we need to use two transients, one in the past and one in the future. The length of both transients for our low dimensional systems is 40. In the case of the Lorenz 96, this length is increased to 100. For the NFM method, the same transients are used, however they are both in the past. We then use a short interval of the future for backwards iteration of the vectors. In the case of Lorenz 96 we use the future interval of 1 and for the lower dimensional systems we use the interval of 0.1.

Ii Results for the Lorenz 96 model with dimension n=32

In Fig. S1 we present the comparison of data-based and equation-based CLVs, LEs, and FTLEs for a Lorenz 96 model of dimension .

Figure S1: CLVs can also be estimated from time series of high-dimensional chaotic systems. Presented here is a Lorenz 96 system with and . (a) Trajectory. (b) LEs from model equations compared to LEs from estimated Jacobians. (c) Time-series of differences between FTLEs from model equations and the FTLEs from estimated Jacobians. (d) Absolute value of the cosine of the angle between the CLVs from model equations and the CLVs based on estimated Jacobians. Ginelli’s method has been used to compute both sets of the CLVs. (e) LEs compared to the average growth rate of the CLVs estimated from the near future. (f) Similar to (d) with both sets of CLVs estimated from the near future.

Iii Results for the stochastic Lorenz attractor with low and intermediate noise

In Figs. S2 and S3, data-based and equation-based quantities are compared for stochastic Lorenz models with the standard deviations of the added white noise and .

Figure S2: CLVs of a stochastic Lorenz attractor: estimated and equation-based results are in very good agreement. (a) Trajectory of a Lorenz system with noise, . (b) LEs from model equations compared to LEs from estimated Jacobians. (c), (d), (e), FTLEs from model equations compared to the FTLEs from estimated Jacobians. (f) Absolute value of the cosine of the angle between the CLVs from model equations and the CLVs based on estimated Jacobians. Ginelli’s method has been used to compute both sets of the CLVs. (g) Similar to (f) with both sets of CLVs estimated from the near future.
Figure S3: CLVs of a stochastic Lorenz attractor: estimated and equation-based results are in very good agreement. (a) Trajectory of a Lorenz system with noise, . (b) LEs from model equations compared to LEs from estimated Jacobians. (c), (d), (e) FTLEs from model equations compared to the FTLEs from estimated Jacobians. (f) Absolute value of the cosine of the angle between the CLVs from model equations and the CLVs based on estimated Jacobians. Ginelli’s method has been used to compute both sets of the CLVs. (g) Similar to (f) with both sets of CLVs estimated from the near future.

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