Dual canonical systems and dual matrix string equations

2012 ◽  
pp. 113-149
Author(s):  
Lev A. Sakhnovich
Keyword(s):  
Canonical Systems

scholarly journals Discovering governing equations from data by sparse identification of nonlinear dynamical systems

2016 ◽  
Vol 113 (15) ◽  
pp. 3932-3937 ◽  
Author(s):  
Steven L. Brunton ◽  
Joshua L. Proctor ◽  
J. Nathan Kutz
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Measurement Data ◽  
Climate Science ◽  
Model Complexity ◽  
Governing Equations ◽  
Canonical Systems ◽  
Nonlinear Dynamical ◽  
Balance Accuracy ◽  
Wide Range

Extracting governing equations from data is a central challenge in many diverse areas of science and engineering. Data are abundant whereas models often remain elusive, as in climate science, neuroscience, ecology, finance, and epidemiology, to name only a few examples. In this work, we combine sparsity-promoting techniques and machine learning with nonlinear dynamical systems to discover governing equations from noisy measurement data. The only assumption about the structure of the model is that there are only a few important terms that govern the dynamics, so that the equations are sparse in the space of possible functions; this assumption holds for many physical systems in an appropriate basis. In particular, we use sparse regression to determine the fewest terms in the dynamic governing equations required to accurately represent the data. This results in parsimonious models that balance accuracy with model complexity to avoid overfitting. We demonstrate the algorithm on a wide range of problems, from simple canonical systems, including linear and nonlinear oscillators and the chaotic Lorenz system, to the fluid vortex shedding behind an obstacle. The fluid example illustrates the ability of this method to discover the underlying dynamics of a system that took experts in the community nearly 30 years to resolve. We also show that this method generalizes to parameterized systems and systems that are time-varying or have external forcing.

scholarly journals Canonical systems and their limits on stable curves

2014 ◽  
Vol 399 ◽  
pp. 634-656 ◽  
Author(s):  
Ziv Ran
Keyword(s):  
Canonical Systems ◽  
Stable Curves

On the Integrals of Canonical Systems

1941 ◽  
Vol 42 (3) ◽  
pp. 806 ◽  
Author(s):  
Carl Ludwig Siegel
Keyword(s):  
Canonical Systems

Local Structure of Amorphous Canonical Systems

1999 ◽  
Vol 2-6 ◽  
pp. 393-398 ◽  
Author(s):  
P.K. Hung ◽  
D.K. Belashchenko ◽  
V.M. Chieu ◽  
N.T. Duong ◽  
Vo Van Hoang ◽  
...  
Keyword(s):  
Local Structure ◽  
Canonical Systems

Regular Canonical Systems

1989 ◽  
pp. 180-216
Author(s):  
J. Richard Büchi ◽  
Dirk Siefkes
Keyword(s):  
Canonical Systems

scholarly journals Phase Transition in Modified Newtonian Dynamics (MONDian) Self-Gravitating Systems

Entropy ◽  
2021 ◽  
Vol 23 (9) ◽  
pp. 1158
Author(s):  
Mohammad Hossein Zhoolideh Zhoolideh Haghighi ◽  
Sohrab Rahvar ◽  
Mohammad Reza Rahimi Rahimi Tabar
Keyword(s):  
Phase Transition ◽  
Binary Systems ◽  
Dimensional Space ◽  
Gravitational Interaction ◽  
Three Dimensional ◽  
Physical Parameters ◽  
Canonical Systems ◽  
Modified Newtonian Dynamics ◽  
Newtonian Dynamics ◽  
Gravitating Systems

We study the statistical mechanics of binary systems under the gravitational interaction of the Modified Newtonian Dynamics (MOND) in three-dimensional space. Considering the binary systems in the microcanonical and canonical ensembles, we show that in the microcanonical systems, unlike the Newtonian gravity, there is a sharp phase transition, with a high-temperature homogeneous phase and a low-temperature clumped binary one. Defining an order parameter in the canonical systems, we find a smoother phase transition and identify the corresponding critical temperature in terms of the physical parameters of the binary system.

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