Lorenz Equations Research Articles

New Knots in the Lorenz Equations

New Knots in the Lorenz Equations

Representing turbulent statistics with partitions of state space. Part 1. Theory and methodology

This is the first of a two-part paper. We formulate a data-driven method for constructing finite-volume discretizations of an arbitrary dynamical system's underlying Liouville/Fokker–Planck equation. A method is employed that allows for flexibility in partitioning state space, generalizes to function spaces, applies to arbitrarily long sequences of time-series data, is robust to noise and quantifies uncertainty with respect to finite sample effects. After applying the method, one is left with Markov states (cell centres) and a random matrix approximation to the generator. When used in tandem, they emulate the statistics of the underlying system. We illustrate the method on the Lorenz equations (a three-dimensional ordinary differential equation) saving a fluid dynamical application for Part 2 (Souza, J. Fluid Mech., vol. 997, 2024, A2).

A Broken Duet: Multistable Dynamics in Dyadic Interactions.

Misunderstandings in dyadic interactions often persist despite our best efforts, particularly between native and non-native speakers, resembling a broken duet that refuses to harmonise. This paper delves into the computational mechanisms underpinning these misunderstandings through the lens of the broken Lorenz system-a continuous dynamical model. By manipulating a specific parameter regime, we induce bistability within the Lorenz equations, thereby confining trajectories to distinct attractors based on initial conditions. This mirrors the persistence of divergent interpretations that often result in misunderstandings. Our simulations reveal that differing prior beliefs between interlocutors result in misaligned generative models, leading to stable yet divergent states of understanding when exposed to the same percept. Specifically, native speakers equipped with precise (i.e., overconfident) priors expect inputs to align closely with their internal models, thus struggling with unexpected variations. Conversely, non-native speakers with imprecise (i.e., less confident) priors exhibit a greater capacity to adjust and accommodate unforeseen inputs. Our results underscore the important role of generative models in facilitating mutual understanding (i.e., establishing a shared narrative) and highlight the necessity of accounting for multistable dynamics in dyadic interactions.

Rotated Lorenz Curves of Biological Size Distributions Follow Two Performance Equations

The Lorenz curve is used to describe the relationship between the cumulative proportion of household income and the number of households of an economy. The extent to which the Lorenz curve deviates from the line of equality (i.e., y = x) is quantified by the Gini coefficient. Prior models are based on the simulated and empirical data of income distributions. In biology, the Lorenz curves of cell or organ size distributions tend to have similar shapes. When the Lorenz curve is rotated by 135 degrees counterclockwise and shifted to the right by a distance of 2, a three-parameter performance equation (PE), and its generalized version with five parameters (GPE), accurately describe this rotated and right-shifted curve. However, in prior studies, PE and GPE were not compared with the other Lorenz equations, and little is known about whether the skewness of the distribution could influence the validity of these equations. To address these two issues, simulation data from the beta distributions with different skewness values and six empirical datasets of plant (organ) size distributions were used to compare PE and GPE with three other Lorenz equations in describing the rotated and right-shifted plant (organ) size distributions. The root-mean-square error and Akaike information criterion were used to assess the validity of the two performance equations and the three other Lorenz equations. PE and GPE were both validated in describing the rotated and right-shifted simulation and empirical data of plant (organ) distributions. Nevertheless, GPE worked better than PE and the three other Lorenz equations from the perspectives of the goodness of fit, and the trade-off between the goodness of fit and the model structural complexity. Analyses indicate that GPE provides a powerful tool for quantifying size distributions across a broad spectrum of organic entities and can be used in a variety of ecological and evolutionary applications. Even for the simulation data from hypothetical extreme skewed distribution curves, GPE still worked well.

The evolving butterfly: Statistics in a changing attractor

The Earth system is often modeled as a dynamical system in what has come to be known as Earth System Models. When used to study anthropogenically forced climate change, these models are forced in such a way that they are not in a statistically stationary state. Yet, statistical statements are still made about the Earth climate system using only a single trajectory by taking temporal averages. At each moment in time, one draws a sample from a different distribution, raising questions about the utility of temporal averages, in stark contrast to the utility of temporal averages in ergodic systems. This work follows in the tradition of using a toy model to examine properties present in the Earth climate system. We aim to examine how we can make meaningful statistical statements in non-stationary systems when only dealing with a single trajectory. We use the Lorenz equations with a time-varying parameter as a starting point for comparing ensemble averages to temporal averages. We find that, in so far as the control parameter induces a slow and smooth change in the dynamics, the resulting statistics of ensemble averages compare well to those of temporal averages.

Comparison of five equations in describing the variation of leaf area distributions of Alangium chinense (Lour.) Harms.

Previous studies have validated a performance equation (PE) and its generalized version (GPE) in describing the rotated and right-shifted Lorenz curves of organ size (e.g., leaf area and fruit volume) distributions of herbaceous plants. Nevertheless, there are still two questions that have not been adequately addressed by prior work: (i) whether the PE and GPE apply to woody plant species and (ii) how do the PE and GPE perform in comparison with other Lorenz equations when fitting data. To address these deficiencies, we measured the lamina length and width of each leaf on 60 Alangium chinense saplings to compare the performance of the PE and GPE with three other Lorenz equations in quantifying the inequality of leaf area distributions across individual trees. Leaf area is shown to be the product of a proportionality coefficient (k) and leaf length and width. To determine the numerical value of k, we scanned 540 leaves to obtain the leaf area empirically. Using the estimated k, the leaf areas of 60A. chinense saplings were calculated. Using these data, the two performance equations and three other Lorenz equations were then compared and assessed using the root-mean-square error (RMSE) and Akaike information criterion (AIC). The PE and GPE were found to be valid in describing the rotated and right-shifted Lorenz curves of the A. chinense leaf area distributions, and GPE has the lowest RMSE and AIC values. This work validates the GPE as the best model in gauging variations in leaf area of the woody species.

Infinite-memory classical wave-particle entities, attractor-driven active particles, and the diffusionless Lorenz equations.

A classical wave-particle entity (WPE) can materialize as a millimeter-sized droplet walking horizontally on the free surface of a vertically vibrating liquid bath. This WPE comprises a particle (droplet) that shapes its environment by locally exciting decaying standing waves, which, in turn, guides the particle motion. At high amplitude of bath vibrations, the particle-generated waves decay very slowly in time and the particle motion is influenced by the history of waves along its trajectory. In this high-memory regime, WPEs exhibit hydrodynamic quantum analogs where quantum-like statistics arise from underlying chaotic dynamics. Exploration of WPE dynamics in the very high-memory regime requires solving an integrodifferential equation of motion. By using an idealized one-dimensional WPE model where the particle generates sinusoidal waves, we show that in the limit of infinite memory, the system dynamics reduce to a 3D nonlinear system of ordinary differential equations (ODEs) known as the diffusionless Lorenz equations (DLEs). We use our algebraically simple ODE system to explore in detail, theoretically and numerically, the rich set of periodic and chaotic dynamical behaviors exhibited by the WPE in the parameter space. Specifically, we link the geometry and dynamics in the phase-space of the DLE system to the dynamical and statistical features of WPE motion, paving a way to understand hydrodynamic quantum analogs using phase-space attractors. Our system also provides an alternate interpretation of an attractor-driven particle, i.e., an active particle driven by internal state-space variables of the DLE system. Hence, our results might also provide new insights into modeling active particle locomotion.

Entropy structure informed learning for solving inverse problems of differential equations

Entropy, since its first discovery by Ludwig Boltzmann in 1877, has been widely applied in diverse disciplines, including thermodynamics, continuum mechanics, mathematical analysis, machine learning, etc. In this paper, we propose a new method for solving the inverse XDE (ODE, PDE, SDE) problems by utilizing the entropy balance equation instead of the original differential equations. This distinguishing feature constitutes a major difference between our current method and other previous classical methods (e.g. SINDy). Despite concerns about the potential information loss during the compression procedure from the original XDEs to single entropy balance equation, various examples from MM reactions, Schlögl model and chemical Lorenz equations in the form of ODEs to nonlinear porous medium equation and Fokker–Planck equation with a double-well potential in the PDE form all well confirm the accuracy, robustness and reliability of our method, as well as its comparable performance with respect to SINDy.

Shape index, Brouwer degree and Poincaré–Hopf theorem

In this paper, we study the relationship of the Brouwer degree of a vector field with the dynamics of the induced flow. Analogous relations are studied for the index of a vector field. We obtain new forms of the Poincar é–Hopf theorem and of the Borsuk and Hirsch antipodal theorems. As an application, we calculate the Brouwer degree of the vector field of the Lorenz equations in isolating blocks of the Lorenz strange set.

Brinkman–Bénard Convection with Rough Boundaries and Third-Type Thermal Boundary Conditions

The Brinkman–Bénard convection problem is chosen for investigation, along with very general boundary conditions. Using the Maclaurin series, in this paper, we show that it is possible to perform a relatively exact linear stability analysis, as well as a weakly nonlinear stability analysis, as normally performed in the case of a classical free isothermal/free isothermal boundary combination. Starting from a classical linear stability analysis, we ultimately study the chaos in such systems, all conducted with great accuracy. The principle of exchange of stabilities is proven, and the critical Rayleigh number, Rac, and the wave number, ac, are obtained in closed form. An asymptotic analysis is performed, to obtain Rac for the case of adiabatic boundaries, for which ac≃0. A seemingly minimal representation yields a generalized Lorenz model for the general boundary condition used. The symmetry in the three Lorenz equations, their dissipative nature, energy-conserving nature, and bounded solution are observed for the considered general boundary condition. Thus, one may infer that, to obtain the results of various related problems, they can be handled in an integrated manner, and results can be obtained with great accuracy. The effect of increasing the values of the Biot numbers and/or slip Darcy numbers is to increase, not only the value of the critical Rayleigh number, but also the critical wave number. Extreme values of zero and infinity, when assigned to the Biot number, yield the results of an adiabatic and an isothermal boundary, respectively. Likewise, these extreme values assigned to the slip Darcy number yield the effects of free and rigid boundary conditions, respectively. Intermediate values of the Biot and slip Darcy numbers bridge the gap between the extreme cases. The effects of the Biot and slip Darcy numbers on the Hopf–Rayleigh number are, however, opposite to each other. In view of the known analogy between Bénard convection and Taylor–Couette flow in the linear regime, it is imperative that the results of the latter problem, viz., Brinkman–Taylor–Couette flow, become as well known.

Analytical study of the Lorenz system: Existence of infinitely many periodic orbits and their topological characterization

We consider the Lorenz equations, a system of three-dimensional ordinary differential equations modeling atmospheric convection. These equations are chaotic and hard to study even numerically, and so a simpler "geometric model" has been introduced in the seventies. One of the classical problems in dynamical systems is to relate the original equations to the geometric model. This has been achieved numerically by Tucker for the classical parameter values and remains open for general values. In this paper, we establish analytically a relation to the geometric model for a different set of parameter values that we prove must exist. This is facilitated by finding a way to apply topological tools developed for the study of surface dynamics to the more intricate case of three-dimensional flows.

Time Averages and Periodic Attractors at High Rayleigh Number for Lorenz-like Models

Revisiting the Lorenz ’63 equations in the regime of large of Rayleigh number, we study the occurrence of periodic solutions and quantify corresponding time averages of selected quantities. Perturbing from the integrable limit of infinite ρ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho $$\\end{document}, we provide a full proof of existence and stability of symmetric periodic orbits, which confirms previous partial results. Based on this, we expand time averages in terms of elliptic integrals with focus on the much studied average ‘transport,’ which is the mode reduced excess heat transport of the convection problem that gave rise to the Lorenz equations. We find a hysteresis loop between the periodic attractors and the nonzero equilibria of the Lorenz equations. These have been proven to maximize transport, and we show that the transport takes arbitrarily small values in the family of periodic attractors. In particular, when the nonzero equilibria are unstable, we quantify the difference between maximal and typically realized values of transport. We illustrate these results by numerical simulations and show how they transfer to various extended Lorenz models.

A Dynamic Hybrid Cryptosystem Using Chaos and Diffie–Hellman Protocol: An Image Encryption Application

Images with sensitive content require encryption for storage and transmission. Symmetric schemes can cipher them, while an asymmetric cryptosystem can distribute the secret key safely. For this reason, we propose a dynamic hybrid cryptosystem, which ciphers images and transfers its private keys. It has a symmetric algorithm that applies the Lorenz equations for generating different boxes and permutations in every encryption process and round. Since the secret key concatenates two private numbers, an asymmetric algorithm is included for its key distribution. The proposal uses the Diffie–Hellman protocol with ElGamal for obtaining a seed and building 128 strings. Then, the SHA-512 is applied in each of them a number of times associated with the secret key value in its blockchain representation. The resultant strings are concatenated to conform to the public key. Finally, the tests indicate that the cryptosystem resists differential, linear, algebraic, and brute-force attacks. Its cipher quality is high according to the entropy, correlation, DFT, NPCR, UACI, AC, texture analysis, and goodness of fit test. Additionally, occlusion, additive, multiplicative, and the proposed χ2 noise attacks are simulated on encrypted images. Finally, the sharpness loss is measured with the Similarity Parameter and improved with a filter 5 × 5.

A kind of Lagrangian chaotic property of the Arnold–Beltrami–Childress flow

Three-dimensional steady-state Arnold–Beltrami–Childress (ABC) flow has a chaotic Lagrangian structure, and also satisfies the Navier–Stokes (NS) equations with an external force per unit mass. It is well known that, although trajectories of a chaotic system have sensitive dependence on initial conditions, i.e. the famous ‘butterfly effect’, their statistical properties are often insensitive to small disturbances. This kind of chaos (such as governed by the Lorenz equations) is called normal-chaos. However, a new concept, i.e. ultra-chaos, has been reported recently, whose statistics are unstable to tiny disturbances. Thus, ultra-chaos represents higher disorder than normal chaos. In this paper, we illustrate that ultra-chaos widely exists in Lagrangian trajectories of fluid particles in steady-state ABC flow. Moreover, solving the NS equation when $Re=50$ with the ABC flow plus a very small disturbance as the initial condition, it is found that trajectories of nearly all fluid particles become ultra-chaotic when the transition from laminar to turbulence occurs. These numerical experiments and facts highly suggest that ultra-chaos should have a relationship with turbulence. This paper identifies differences between ultra-chaos and sensitivity of statistics to parameters. Possible relationships between ultra-chaos and the Poincaré section, ultra-chaos and ergodicity/non-ergodicity, etc., are discussed. The concept of ultra-chaos opens a new perspective of chaos, the Poincaré section, ergodicity/non-ergodicity, turbulence and their inter-relationships.

Generation of Boxes and Permutations Using a Bijective Function and the Lorenz Equations: An Application to Color Image Encryption

Some images that contain sensitive information and travel through the network require security. Therefore, a symmetric cryptosystem that encrypts images and resists known attacks is developed. Subsequently, in this work, an encryption algorithm known as Image Cipher utilizing Lorenz equation and a Bijective Function—ICLEBF are proposed. In the proposal, the Lorenz equations and the Bijective function are used to generate boxes, the permutation, and schedule keys, considering that all these elements are different in each encryption process. The encryption procedure consists of 14 rounds, where a different box is applied in each round. In this type of algorithm, the impact of quantum computers will be less forceful and can be useful for that epoch. On the other hand, the quality of the encrypted images and the loss of sharpness in decoded images with damage are measured. In addition, an attack from five types of noise (one of which is a developed proposal) is carried out by applying it to encrypted images. Finally, the results of the proposed ICLEBF are compared with other recent image encryption algorithms, including the Advanced Encryption Standard. As a result, this proposal resists known attacks and others that the current standard does not support.

The Extended Non-Elementary Amplitude Functions as Solutions to the Damped Pendulum Equation, the Van der Pol Equation, the Damped Duffing Equation, the Lienard Equation and the Lorenz Equations

The Extended Non-Elementary Amplitude Functions as Solutions to the Damped Pendulum Equation, the Van der Pol Equation, the Damped Duffing Equation, the Lienard Equation and the Lorenz Equations

Existence of Approximate Solutions to Nonlinear Lorenz System under Caputo-Fabrizio Derivative

In this article, we developed sufficient conditions for the existence and uniqueness of an approximate solution to a nonlinear system of Lorenz equations under Caputo-Fabrizio fractional order derivative (CFFD). The required results about the existence and uniqueness of a solution are derived via the fixed point approach due to Banach and Krassnoselskii. Also, we enriched our work by establishing a stable result based on the Ulam-Hyers (U-H) concept. Also, the approximate solution is computed by using a hybrid method due to the Laplace transform and the Adomian decomposition method. We computed a few terms of the required solution through the mentioned method and presented some graphical presentation of the considered problem corresponding to various fractional orders. The results of the existence and uniqueness tests for the Lorenz system under CFFD have not been studied earlier. Also, the suggested method results for the proposed system under the mentioned derivative are new. Furthermore, the adopted technique has some useful features, such as the lack of prior discrimination required by wavelet methods. our proposed method does not depend on auxiliary parameters like the homotopy method, which controls the method. Our proposed method is rapidly convergent and, in most cases, it has been used as a powerful technique to compute approximate solutions for various nonlinear problems.

Mapping the Chaotic Transitions of the Lorenz Equations With Unsupervised Machine Learning

Abstract Prior work has explored utilizing machine learning for the Lorenz system in the time domain. In this work, we have focused on applications of machine learning for predicting the onset of chaotic transitions in the Lorenz system. Our methods included the development of a robust numerical solution to the Lorenz equations using a fourth order Runge–Kutta method. We solved the Lorenz equations for a large range of Raleigh ratios from 1 to 1000. We calculated the power spectral density, various descriptive statistics, and a cluster analysis using unsupervised machine learning. To identify behaviors and regions in the data, we utilize unsupervised learning as it is designed to assist in recognizing patterns without being told or trained by prior knowledge. We confirmed the performance of the machine learning system's ability to identify chaotic transitions independent of expert selection of Raleigh ratio ranges. The system correctly identifies the transitional behaviors described in prior mathematical work. The results indicate that the power spectral density is very important for the clustering. We also found that examining machine learning clusters by dimension (x, y, and z) was important to understand many of the facets of the chaotic transitions. The results provide a visual mapping of the regions where chaotic transitions may occur based on variations in the Prandtl number and geometry constant. Unsupervised machine learning may be used as a tool to characterize the transition regions for these geometries, providing new lenses for the heat transfer community.

Solving Lorenz System by Using Lower Order Symmetrized Runge-Kutta Methods

Runge-Kutta is a widely used numerical method for solving the non-linear Lorenz system. This study focuses on solving the Lorenz equations with the classical parameter values by using the lower order symmetrized Runge-Kutta methods, Implicit Midpoint Rule (IMR), and Implicit Trapezoidal Rule (ITR). We show the construction of the symmetrical method and present the numerical experiments based on the two methods without symmetrization, with one- and two-step active symmetrization in a constant step size setting. For our numerical experiments, we use MATLAB software to solve and plot the graphical solutions of the Lorenz system. We compare the oscillatory behaviour of the solutions and it appears that IMR and two-step active IMR turn out to be chaotic while the rest turn out to be non-chaotic. We also compare the accuracy and efficiency of the methods and the result shows that IMR performs better than the symmetrizers, while two-step active ITR performs better than ITR and one-step active ITR. Based on the results, we conclude that different implicit numerical methods with different steps of active symmetrization can significantly impact the solutions of the non-linear Lorenz system. Since most study on solving the Lorenz system is based on explicit time schemes, we hope this study can motivate other researchers to analyze the Lorenz equations further by using Runge-Kutta methods based on implicit time schemes.

Investigation of optical properties and radioactive attenuation parameters of doped tungsten oxide soda lime silica SLS waste glass

The optical properties and radioactive attenuation parameters of doped tungsten oxide soda-lime-silica (SLS) waste glass were investigated in the current work. A novel series of lead-free transparent glass samples were manufactured with the possibility of developing radiation-absorbing glass shielding materials. The glass composition is WO3–Bi2O3–ZnO–B2O3-SLS. A new approach was employed in the utilization of soda-lime-silica (SLS) glass as an alternative for pure silicon dioxide having 74.1% of SiO2 and other elements in it. The optical properties were investigated with the help of a UV–Visible Spectrophotometer. The results for the optical properties showed that when WO3 content is increased in the glass structure, the Urbach energy ΔE and refractive index n, increase from 0.337 to 0.431 and 2.84 to 2.94 respectively, the optical basicity Λ also increase with the increase in WO3 while the energy optical band gap Eopt registered a decrease. The metallization criterion (M) confirmed that the current glass samples are nonmetallic (insulator) because Rm/Vm < 1 which is confirmed in the condensed matter theory in the Lorentz–Lorenz equations. Moreover, the theoretical radiation shielding parameter results indicate that the linear attenuation coefficient μ and Zeff were increased with an increase in WO3. It can be seen also that the relationship between the μ and HVL is an inverse one; larger values of μ result in lesser thickness of the HVL. Whereas, as the WO3 concentration increases, the half-value layer HVL decreases. The results of this study demonstrate that the glass samples are harmless, transparent to visible light, and effective radiation shielding materials.