Parameter estimation for a chaotic dynamical system with partial observations

Parameter estimation in chaotic dynamical systems is an important and practical issue. Nevertheless, the high-dimensionality and the sensitive dependence on initial conditions typically makes the problem difficult to solve. In this paper, we propose an innovative parameter estimation approach, utilizing numerical differentiation for observation data preprocessing. Given plenty of noisy observations on a portion of dependent variables, numerical differentiation allows them and their derivatives to be accurately approximated. Substituting those approximations into the original system can effectively simplify the parameter estimation problem. As an example, we consider parameter estimation in the well-known Lorenz model given partial noisy observations. According to the Lorenz equations, the estimated parameters can be simply given by least squares regression using the approximated functions provided by data preprocessing. Numerical examples show the effectiveness and accuracy of our method. We also prove the uniqueness and stability of the solution.

Machine Learning for Chaotic Rayleigh Transitions

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.

Conversations with Materials and Diagrams about some of the Intricacies of Oscillatory Motion

This article relates a case study on how a conversation with materials and diagrams – the actual use of materials and diagrams to think, imagine, explain, collaborate, design and build – featured a certain kind of interplay between material and digital components. The physical components present in this setting included a water wheel, which is a wheel driven by flow of water whose rotational motion is a classic example of chaotic dynamics regulated by Lorenz equations. Digital components allowed for real-time graphical displays corresponding to the turning of the water wheel. We selected for this article a sequence of episodes from an interview with Jake, an undergraduate student majoring in engineering. Through a micro-ethnographic analysis, we reflect on how Jake combined the responsiveness of the digital displays with the tangibility of the water wheel to gain insight into some of the intricacies of oscillatory motion.

A completely integrable case in the complex Lorenz equations

By application of the Lie theory of extended groups and for the parameter values σ=1/2, b=1, r1= e^2/2, r2=e/2, e arbitrary we prove that the system of the complex Lorenz equations is algebraically completely integrable. The respective general exact solution i$ expressed by means of Jacobian elliptic functions

New Nonlinear Active Element Dedicated to Modeling Chaotic Dynamics with Complex Polynomial Vector Fields

This paper describes evolution of new active element that is able to significantly simplify the design process of lumped chaotic oscillator, especially if the concept of analog computer or state space description is adopted. The major advantage of the proposed active device lies in the incorporation of two fundamental mathematical operations into a single five-port voltage-input current-output element: namely, differentiation and multiplication. The developed active device is verified inside three different synthesis scenarios: circuitry realization of a third-order cyclically symmetrical vector field, hyperchaotic system based on the Lorenz equations and fourth- and fifth-order hyperjerk function. Mentioned cases represent complicated vector fields that cannot be implemented without the necessity of utilizing many active elements. The captured oscilloscope screenshots are compared with numerically integrated trajectories to demonstrate good agreement between theory and measurement.