DynamicalSystems.jl: A Julia software library for chaos and nonlinear dynamics

Datseris, George

doi:10.21105/joss.00598

Cited by 91 publications

(64 citation statements)

References 9 publications

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“…These two attractors will, in general, exhibit different properties, like, e.g., different maximum Lyapunov exponent. The maximum Lyapunov exponents computed with the method of [67] (implementation of DynamicalSystems.jl [68]) are λ (cold max ≈ 1.04 for the cold and λ (warm max ≈ 2.60 for the warm state. The larger Lyapunov exponent of the warm state shows that, as expected, the L96 model is more chaotic for larger val-ues of the forcing.…”

Section: Predicting the Modelmentioning

confidence: 99%

Analysis of a bistable climate toy model with physics-based machine learning methods

Gelbrecht

Lucarini

Boers

et al. 2021

Eur. Phys. J. Spec. Top.

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We propose a comprehensive framework able to address both the predictability of the first and of the second kind for high-dimensional chaotic models. For this purpose, we analyse the properties of a newly introduced multistable climate toy model constructed by coupling the Lorenz ’96 model with a zero-dimensional energy balance model. First, the attractors of the system are identified with Monte Carlo Basin Bifurcation Analysis. Additionally, we are able to detect the Melancholia state separating the two attractors. Then, Neural Ordinary Differential Equations are applied to predict the future state of the system in both of the identified attractors.

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Section: Predicting the Modelmentioning

confidence: 99%

Analysis of a bistable climate toy model with physics-based machine learning methods

Gelbrecht

Lucarini

Boers

et al. 2021

Eur. Phys. J. Spec. Top.

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“…[52,53,54], review articles, e.g. [15,55,56], and software packages such as AUTO [57], DSTOOL [58], PDECont [59], MatCont [60], LOCA [61], JuliaDynamics [62], pde2path [63] have been dedicated to targeting simple or complicated bifurcation-theoretic objects for general dynamical systems. However, such algorithms usually require the solution of linear systems or matrix diagonalization.…”

Section: Looking Aheadmentioning

confidence: 99%

Computational Challenges of Nonlinear Systems

Tuckerman

2020

Emerging Frontiers in Nonlinear Science

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We survey some of the major types of dynamical-systems computations that can be carried out for two or three-dimensional systems of partial differential equations. In order of increasing complexity, we describe methods for calculating steady states and bifurcation diagrams, linear stability and Floquet analysis, and heteroclinic orbits. These are illustrated by computations for Rayleigh-Bénard convection in a cylindrical geometry, the Faraday instability of a fluid layer, the flow past a cylinder and over a square cavity, flow in a cylindrical container with counter-rotating lids, and Bose-Einstein condensation. We discuss some mathematical questions raised by these computations and the need for improved numerical tools.

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“…As these two conditions are very restrictive, a more general way to analyse the stability of the aforementioned linear system is to estimate its maximal Lyapunov exponent (MLE) (Zounes and Rand, 1998), for which the MLE being positive means instability. 3 The MLE has been evaluated using the method described in Benettin et al (1976), whose implementation is available in (Datseris, 2018).…”

Section: Two Triads Coupled By One Modementioning

confidence: 99%

Nonlinear interaction of gravity and acoustic waves

Teruya

Raupp

2020

Tellus A: Dynamic Meteorology and Oceanography

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Here we have investigated the possibility of an inertio-acoustic wave-mode to be unstable with regards to gravity mode perturbations through non-linear triad interactions in the context of a shallow non-hydrostatic model. We have considered highly truncated Galerkin expansions of the perturbations around a resting, hydrostatic and isothermal background state in terms of the eigensolutions of the linear problem. For a single interacting wave triplet, we have shown that an acoustic mode cannot amplify a pair of inertio-gravity perturbations due to the high mismatch among the eigenfrequencies of the three interacting wave-modes, which requires an unrealistically high amplitude of the acoustic mode in order for pump wave instability to occur. In contrast, it has been demonstrated by analysing the dynamics of two triads coupled by a single mode that a non-hydrostatic gravity wave-mode participating in a nearly resonant interaction with two acoustic modes can be unstable to small amplitude perturbations associated with a pair of two hydrostatically balanced inertio-gravity wave-modes. This linear instability yields significant inter-triad energy exchanges if the nonlinearity associated with the second triplet containing the two hydrostatically balanced inertio-gravity modes is restored. Therefore, this inter-triad energy exchanges lead the acoustic modes to yield significant energy modulations in hydrostatic inertio-gravity wave modes. Consequently, our theory suggests that acoustic waves might play an important role in the transient phase of the three-dimensional adjustment process of the atmosphere to both hydrostatic and geostrophic balances.

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DynamicalSystems.jl: A Julia software library for chaos and nonlinear dynamics

Cited by 91 publications

References 9 publications

Analysis of a bistable climate toy model with physics-based machine learning methods

Analysis of a bistable climate toy model with physics-based machine learning methods

Computational Challenges of Nonlinear Systems

Nonlinear interaction of gravity and acoustic waves

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