scholarly journals Transition pathways for a class of high dimensional stochastic dynamical systems with Lévy noise

2021 ◽  
Vol 31 (6) ◽  
pp. 063138
Author(s):  
Jianyu Hu ◽  
Jianyu Chen
Keyword(s):  
Dynamical Systems ◽  
Lévy Noise ◽  
High Dimensional ◽  
Stochastic Dynamical Systems ◽  
Transition Pathways ◽  
Levy Noise

Slow manifolds for dynamical systems with non-Gaussian stable Lévy noise

2019 ◽  
Vol 17 (03) ◽  
pp. 477-511 ◽  
Author(s):  
Shenglan Yuan ◽  
Jianyu Hu ◽  
Xianming Liu ◽  
Jinqiao Duan
Keyword(s):  
Biological Sciences ◽  
Dynamical Systems ◽  
Scale Parameter ◽  
Lévy Noise ◽  
Stochastic Dynamical Systems ◽  
Time Dynamics ◽  
Slow Manifolds ◽  
Long Time ◽  
Non Gaussian ◽  
Levy Noise

This work is concerned with the dynamics of a class of slow–fast stochastic dynamical systems driven by non-Gaussian stable Lévy noise with a scale parameter. Slow manifolds with exponentially tracking property are constructed, and then we eliminate the fast variables to reduce the dimensions of these stochastic dynamical systems. It is shown that as the scale parameter tends to zero, the slow manifolds converge to critical manifolds in distribution, which helps to investigate long time dynamics. The approximations of slow manifolds with error estimate in distribution are also established. Furthermore, we corroborate these results by three examples from biological sciences.

scholarly journals A numerical framework to understand transitions in high-dimensional stochastic dynamical systems

2016 ◽  
Vol 1 (1) ◽  
Author(s):  
Henk A Dijkstra ◽  
Alexis Tantet ◽  
Jan Viebahn ◽  
Erik Mulder ◽  
Mariët Hebbink ◽  
...  
Keyword(s):  
Dynamical Systems ◽  
High Dimensional ◽  
Stochastic Dynamical Systems

scholarly journals Metastability for discontinuous dynamical systems under Lévy noise: Case study on Amazonian Vegetation

2017 ◽  
Vol 7 (1) ◽  
Author(s):  
Larissa Serdukova ◽  
Yayun Zheng ◽  
Jinqiao Duan ◽  
Jürgen Kurths
Keyword(s):  
Dynamical Systems ◽  
Lévy Noise ◽  
Levy Noise

scholarly journals A Gaussian Mixture Model Smoother for Continuous Nonlinear Stochastic Dynamical Systems: Applications

2017 ◽  
Vol 145 (7) ◽  
pp. 2763-2790 ◽  
Author(s):  
Tapovan Lolla ◽  
Pierre F. J. Lermusiaux
Keyword(s):  
Dynamical Systems ◽  
Gaussian Mixture Model ◽  
Mixture Model ◽  
High Dimension ◽  
Gaussian Mixture ◽  
Sudden Expansion ◽  
High Dimensional ◽  
Diffusion Experiment ◽  
Stochastic Dynamical Systems ◽  
Observation Times

The nonlinear Gaussian Mixture Model Dynamically Orthogonal (GMM–DO) smoother for high-dimensional stochastic fields is exemplified and contrasted with other smoothers by applications to three dynamical systems, all of which admit far-from-Gaussian distributions. The capabilities of the smoother are first illustrated using a double-well stochastic diffusion experiment. Comparisons with the original and improved versions of the ensemble Kalman smoother explain the detailed mechanics of GMM–DO smoothing and show that its accuracy arises from the joint GMM distributions across successive observation times. Next, the smoother is validated using the advection of a passive stochastic tracer by a reversible shear flow. This example admits an exact smoothed solution, whose derivation is also provided. Results show that the GMM–DO smoother accurately captures the full smoothed distributions and not just the mean states. The final example showcases the smoother in more complex nonlinear fluid dynamics caused by a barotropic jet flowing through a sudden expansion and leading to variable jets and eddies. The accuracy of the GMM–DO smoother is compared to that of the Error Subspace Statistical Estimation smoother. It is shown that even when the dynamics result in only slightly multimodal joint distributions, Gaussian smoothing can lead to a severe loss of information. The three examples show that the backward inferences of the GMM–DO smoother are skillful and efficient. Accurate evaluation of Bayesian smoothers for nonlinear high-dimensional dynamical systems is challenging in itself. The present three examples—stochastic low dimension, reversible high dimension, and irreversible high dimension—provide complementary and effective benchmarks for such evaluation.

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