Nonlinear dynamos: A complex generalization of the Lorenz equations

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.

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Machine Learning for Chaotic Rayleigh Transitions

ASME 2021 Heat Transfer Summer Conference ◽

10.1115/ht2021-61158 ◽

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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Computer program for Lorenz equations

Fractals and Chaos ◽

10.1887/0750304006/b384b1 ◽

2004 ◽

Keyword(s):

Computer Program ◽

Lorenz Equations

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Map Dynamics Study of the Lorenz Equations

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127497000248 ◽

1997 ◽

Vol 07 (02) ◽

pp. 373-382 ◽

Cited By ~ 8

Author(s):

Olivier Michielin ◽

Paul E. Phillipson

Keyword(s):

Periodic Orbits ◽

Symbolic Dynamics ◽

Strange Attractor ◽

Good Accuracy ◽

Simple Solution ◽

Lorenz Equations ◽

Computer Solution ◽

One Dimensional

The Lorenz equations [Lorenz, 1963], in addition to a strange attractor, display sequences of periodic and aperiodic orbits. Approximate one-dimensional map solutions are heuristically constructed, supplementing previous symbolic dynamics studies, which closely reproduce these sequences. A relatively simple solution reproduces the sequence topology to good accuracy. A second more refined solution reproduces to higher accuracy both the topology and scale of the attractor. The second solution is sufficiently accurate to predict periodic orbits not previously observed and difficult to extract directly from computer solution of the Lorenz equations.

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