A novel quantity for identifying the repelling structures of continuous dynamical systems

Guoqiao You; Guoqiao You

doi:10.3934/math.2021202

AIMS Mathematics

2021, Volume 6, Issue 4: 3378-3392. doi: 10.3934/math.2021202

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Research article

A novel quantity for identifying the repelling structures of continuous dynamical systems

Guoqiao You ^,

School of Statistics and Mathematics, Nanjing Audit University, Nanjing 211815, China

Received: 11 August 2020 Accepted: 14 January 2021 Published: 19 January 2021
MSC : 37A25, 37M25, 76M27

We propose a new quantity to study complicated dynamical systems based on the repelling behaviors of particle trajectories throughout the whole time interval under consideration. Since this proposed quantity measures the averaged repelling rate along each particle trajectory against nearby trajectories, we name the quantity the Lagrangian Averaged Repelling Rate (LARR). The LARR is shown to be objective, i.e. unchanged under time-dependent rotations and translations of the coordinate frame. We also compare the proposed LARR with the commonly used concept called the finite time Lyapunov exponent (FTLE), the latter also measures the separation behaviors of particles but only cares about the initial and terminal states of them. An efficient Eulerian algorithm is also proposed to compute the LARR. Numerical examples illustrate the effectiveness of the LARR in measuring the repelling properties of particle trajectories and also the difference between the proposed LARR and the traditional FTLE.
- dynamical systems,
- flow visualization,
- level set method,
- coherent structure,
- finite time Lyapunov exponent
Citation: Guoqiao You. A novel quantity for identifying the repelling structures of continuous dynamical systems[J]. AIMS Mathematics, 2021, 6(4): 3378-3392. doi: 10.3934/math.2021202

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Abstract

We propose a new quantity to study complicated dynamical systems based on the repelling behaviors of particle trajectories throughout the whole time interval under consideration. Since this proposed quantity measures the averaged repelling rate along each particle trajectory against nearby trajectories, we name the quantity the Lagrangian Averaged Repelling Rate (LARR). The LARR is shown to be objective, i.e. unchanged under time-dependent rotations and translations of the coordinate frame. We also compare the proposed LARR with the commonly used concept called the finite time Lyapunov exponent (FTLE), the latter also measures the separation behaviors of particles but only cares about the initial and terminal states of them. An efficient Eulerian algorithm is also proposed to compute the LARR. Numerical examples illustrate the effectiveness of the LARR in measuring the repelling properties of particle trajectories and also the difference between the proposed LARR and the traditional FTLE.

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