Differentiability of the stable foliation for the model Lorenz equations

1981 ◽  
pp. 302-315 ◽  
Author(s):  
Clark Robinson
Keyword(s):  
Lorenz Equations ◽  
Stable Foliation

Nonlinear dynamos: A complex generalization of the Lorenz equations

1985 ◽  
Vol 14 (2) ◽  
pp. 161-176 ◽  
Author(s):  
C.A. Jones ◽  
N.O. Weiss ◽  
F. Cattaneo
Keyword(s):  
Lorenz Equations

On the integrability of intermediate distributions for Anosov diffeomorphisms

1995 ◽  
Vol 15 (2) ◽  
pp. 317-331 ◽  
Author(s):  
M. Jiang ◽  
Ya B. Pesin ◽  
R. de la Llave
Keyword(s):  
Finite Number ◽  
Lyapunov Exponents ◽  
Periodic Orbits ◽  
Three Dimensional ◽  
Dimensional Torus ◽  
Anosov Diffeomorphism ◽  
Anosov Diffeomorphisms ◽  
Stable Foliation

AbstractWe study the integrability of intermediate distributions for Anosov diffeomorphisms and provide an example of a C∞-Anosov diffeomorphism on a three-dimensional torus whose intermediate stable foliation has leaves that admit only a finite number of derivatives. We also show that this phenomenon is quite abundant. In dimension four or higher this can happen even if the Lyapunov exponents at periodic orbits are constant.

Periodic and chaotic solutions of the generalized lorenz equations

1996 ◽  
Vol 54 (2) ◽  
pp. 140-142 ◽  
Author(s):  
M Y Yu ◽  
B Yang
Keyword(s):  
Lorenz Equations

scholarly journals A shooting approach to the Lorenz equations

1992 ◽  
Vol 27 (2) ◽  
pp. 298-304 ◽  
Author(s):  
S. P. Hastings ◽  
W. C. Troy
Keyword(s):  
Lorenz Equations

On the Chaotic Dynamics Associated with the Center Manifold Equations of Double-Diffusive Convection Near a Codimension-Four Bifurcation Point at Moderate Thermal Rayleigh Number

2018 ◽  
Vol 28 (08) ◽  
pp. 1850094 ◽  
Author(s):  
Justin Eilertsen ◽  
Jerry Magnan
Keyword(s):  
Bifurcation Point ◽  
Center Manifold ◽  
Poincaré Map ◽  
Three Dimensional ◽  
Thermosolutal Convection ◽  
Poincare Map ◽  
Lorenz Equations ◽  
Frequency Locking ◽  
Diffusive Convection ◽  
Double Diffusive

We analyze the dynamics of the Poincaré map associated with the center manifold equations of double-diffusive thermosolutal convection near a codimension-four bifurcation point when the values of the thermal and solute Rayleigh numbers, [Formula: see text] and [Formula: see text], are comparable. We find that the bifurcation sequence of the Poincaré map is analogous to that of the (continuous) Lorenz equations. Chaotic solutions are found, and the emergence of strange attractors is shown to occur via three different routes: (1) a discrete Lorenz-like attractor of the three-dimensional Poincaré map of the four-dimensional center manifold equations that forms as the result of a quasi-periodic homoclinic explosion; (2) chaos that follows quasi-periodic intermittency occurring near saddle-node bifurcations of tori; and, (3) chaos that emerges from the destruction of a 2-torus, preceded by frequency locking.

Machine Learning for Chaotic Rayleigh Transitions

2021 ◽  
Author(s):  
Ben Tribelhorn ◽  
H. E. Dillon
Keyword(s):  
Machine Learning ◽  
Cluster Analysis ◽  
Natural Convection ◽  
Prandtl Number ◽  
Learning System ◽  
Lorenz Equations ◽  
Time Step ◽  
Learning Methods ◽  
Unsupervised Machine Learning ◽  
Machine Learning Methods

Abstract This paper is a preliminary report on work done to explore the use of unsupervised machine learning methods to predict the onset of turbulent transitions in natural convection systems. The Lorenz system was chosen to test the machine learning methods due to the relative simplicity of the dynamic system. We developed a robust numerical solution to the Lorenz equations using a fourth order Runge-Kutta method with a time step of 0.001 seconds. We solved the Lorenz equations for a large range of Raleigh ratios from 1–1000 while keeping the geometry and Prandtl number constant. We calculated the spectral density, various descriptive statistics, and a cluster analysis using unsupervised machine learning. We examined the performance of the machine learning system for different Raleigh ratio ranges. We found that the automated cluster analysis aligns well with well known key transition regions of the convection system. We determined that considering smaller ranges of Raleigh ratios may improve the performance of the machine learning tools. We also identified possible additional behaviors not shown in z-axis bifurcation plots. This unsupervised learning approach can be leveraged on other systems where numerical analysis is computationally intractable or more difficult. The results are interesting and provide a foundation for expanding the study for Prandtl number and geometry variations. Future work will focus on applying the methods to more complex natural convection systems, including the development of new methods for Nusselt correlations.

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