Quantifying chaos using Lagrangian descriptors

Abstract

We present and validate simple and efficient methods to estimate the chaoticity of orbits in low dimensional dynamical systems from computations of Lagrangian descriptors (LDs) on short time scales. Two quantities are proposed for determining the chaotic or regular nature of orbits in a system's phase space, which are based on the values of the LDs of these orbits and of nearby ones: The difference (DNLD) and ratio (RNLD) of neighboring orbits' LDs. We find that, typically, these indicators are able to correctly characterize the chaotic or regular nature of orbits to better than 90 % agreement with results obtained by implementing the Smaller Alignment Index (SALI) method, which is a well established chaos detection technique. Further investigating the performance of the two introduced quantities we discuss the effects of the total integration time and of the spacing between the used neighboring orbits on the accuracy of the methods, finding that even typical short time, coarse grid LD computations are sufficient to provide a reliable quantification of a system's chaotic component, using less CPU time than the SALI. In addition to quantifying chaos, the introduced indicators have the ability to reveal details about the systems' local and global chaotic phase space structure. Our findings clearly suggest that LDs can also be used to quantify and investigate chaos in continuous and discrete low dimensional dynamical systems.

References in corpus (7)

• A Theoretical Framework for Lagrangian Descriptors
• Lagrangian Descriptors for Two Dimensional, Area Preserving, Autonomous and Nonautonomous Maps
• A comparison of different indicators of chaos based on the deviation vectors. Application to symplectic mappings
• Lagrangian Descriptors for Stochastic Differential Equations: A Tool for Revealing the Phase Portrait of Stochastic Dynamical Systems
• Global dynamics visualisation from Lagrangian Descriptors. Applications to discrete and continuous systems
• Lagrangian descriptors for open maps
• Wave-packet spreading in disordered soft architected structures

Cited by in corpus (5)

• Performance of chaos diagnostics based on Lagrangian descriptors. Application to the 4D standard map
• Improved detection of chaos with Lagrangian descriptors using differential algebra
• Efficient detection of chaos through the computation of the Generalized Alignment Index (GALI) by the multi-particle method
• Phase space integrity in neural network models of Hamiltonian dynamics: A Lagrangian descriptor approach
• Chaotic resetting: A resetting strategy for deterministic chaotic systems