Search Results (64)

This paper presents an analytical exploration of how diverse dynamical systems, arising from different scientific domains, can be reformulated (under specific approximations and assumptions) into a common set of equations formally equivalent to the Lorenz system originally derived to model atmospheric convection. Unlike previous studies that focus on analyzing or applying the Lorenz equations, our objective is to show how these equations emerge from distinct models, emphasizing the underlying structural and dynamical similarities. The mathematical steps involved in these reformulations are included. The systems examined include Lorenz’s original atmospheric convection model, the chaotic water wheel, the Maxwell–Bloch equations for lasers, mechanical gyrostat, solar dynamo model, mesoscale reaction dynamics, an interest rate economic model, and a socioeconomic control system. This work includes a discussion of the unifying features that lead to similar qualitative behaviors across seemingly unrelated systems. By highlighting the Lorenz system as a paradigmatic limit of a broad class of nonlinear models, we underscore its relevance as a unifying framework in the study of complex dynamics. Full article

The Gini coefficient, while widely used to quantify inequality in biological size distributions, lacks the capacity to resolve directional asymmetry inherent in Lorenz curves, a critical limitation for understanding skewed resource allocation strategies. To address this, we extend our prior geometric framework of the rotated and right-shifted Lorenz curve (RRLC) by introducing two original asymmetry metrics: the positional shift ratio (P_L, defined as $x_c/\sqrt{2}$, where $x_c$ is the x-coordinate of the RRLC’s maximum value point) and the area ratio (P_A, defined as A_L/(A_L + A_R), where A_L and A_R denote the areas under the left and right segments of the RRLC). These indices uniquely dissect contributions of dominant versus small individuals to overall inequality, with P_L reflecting the peak position of the RRLC and P_A quantifying the area dominance of its left segment. Theoretically, P_L directly links to the classical Lorenz asymmetry coefficient S (defined as $S=x_c^{\prime }+y_c^{\prime }$, where $\left(x_c^{\prime },y_c^{\prime }\right)$ is the tangent point on the original Lorenz curve with a 45° slope) through S = 2 − 2P_L, bridging geometric transformation and parametric asymmetry analysis. Applied to 480 Shibataea chinensis Nakai shoots, our analysis revealed that over 99% exhibited pronounced left-skewed distributions, where abundant large leaves drove the majority of leaf area inequality, challenging assumptions of symmetry in plant canopy resource allocation. The framework’s robustness was further validated by the strong correlation between P_A and P_L. By transforming abstract Lorenz curves into interpretable bell-shaped performance curves, this work provides a novel toolkit for analyzing asymmetric size distributions in ecology. The proposed metrics can be applied to refine light-use models, monitor phenotypic plasticity under environmental stress, and scale trait variations across biological hierarchies, thereby advancing both theoretical and applied research in plant ecology. Full article

Chaotic systems can exhibit completely different behaviors given only slightly different initial conditions, yet it is possible to synchronize them through appropriate coupling. A wide variety of behaviors—complete chaos, complete synchronization, phase synchronization, etc.—across a variety of systems have been identified but rely on systems’ phase space trajectories, which suppress important distinctions between very different behaviors and require access to the differential equations. In this paper, we introduce the Difference Time Series Peak Complexity (DTSPC) algorithm, a technique using entropy as a tool to quantitatively measure synchronization. Specifically, this uses peak pattern complexity created from sampled time series, focusing on the behavior of ringing patterns in the difference time series to distinguish a variety of synchronization behaviors based on the entropic complexity of the populations of various patterns. We present results from the paradigmatic case of coupled Lorenz systems, both identical and non-identical, and across a range of parameters and show that this technique captures the diversity of possible synchronization, including non-monotonicity as a function of parameter as well as complicated boundaries between different regimes. Thus, this peak pattern entropic analysis algorithm reveals and quantifies the complexity of chaos synchronization dynamics, and in particular captures transitional behaviors between different regimes. Full article

In recent years, precipitation concentration indices have become popular, and the daily precipitation concentration index has been widely used worldwide. This index is based on the Lorenz curve fitting. Recently, some biases in the fitting process have been observed in some research. Therefore, this research’s objective consisted of testing the performance of one alternative equation for fitting the Lorenz curve through the analysis of the daily precipitation concentration in Brazil. Daily precipitation data from 735 time series were used to fit the Lorenz curve and calculate the concentration index. Therefore, the goodness of fit was evaluated to determine which equation better describes the empirical data. Results show that the mean value for the concentration index based on Equation (1) was 0.650 ± 0.079, while the mean value based on Equation (2) was 0.624 ± 0.070. The results of the fitting performance show a better fitting with Equation (2) compared to Equation (1) as indicated by R², RSS, and RMSE values, R² = 0.9959 for Equation (1) versus 0.9996 for Equation (2), RSS = 252.78 versus 22.66, and RMSE = 1.5092 versus 0.0501. Thus, Equation (2) can be considered an alternative to improve the calculation of the concentration index. Full article

This work focused on demonstrating the use of dynamic time warping (DTW) as a metric for the elementary effects computation in Morris-based global sensitivity analysis (GSA) of model parameters in multivariate dynamical systems. One of the challenges of GSA on multivariate time-dependent dynamics is the modeling of parameter perturbation effects propagated to all model outputs while capturing time-dependent patterns. The study establishes and demonstrates the use of DTW as a metric of elementary effects across the time domain and the multivariate output domain, which are all aggregated together via the DTW cost function into a single metric value. Unlike the commonly studied coefficient-based functional approximation and covariance decomposition methods, this new DTW-based Morris GSA algorithm implements curve alignment via dynamic programing for cost computation in every parameter perturbation trajectory, which captures the essence of “elementary effect” in the original Morris formulation. This new algorithm eliminates approximations and assumptions about the model outputs while achieving the objective of capturing perturbations across time and the array of model outputs. The technique was demonstrated using an ordinary differential equation (ODE) system of mixed-order adsorption kinetics, Monod-type microbial kinetics, and the Lorenz attractor for chaotic solutions. DTW as a Morris-based GSA metric enables the modeling of parameter sensitivity effects on the entire array of model output variables evolving in the time domain, resulting in parameter rankings attributed to the entire model dynamics. Full article

We develop a hybrid classical–quantum method for solving the Lorenz system. We use the forward Euler method to discretize the system in time, transforming it into a system of equations. This set of equations is solved by using the Variational Quantum Linear Solver (VQLS) algorithm. We present numerical results comparing the hybrid method with the classical approach for solving the Lorenz system. The simulation results demonstrate that the VQLS method can effectively compute solutions comparable to classical methods. The method is easily extended to solving similar nonlinear differential equations. Full article

The fractional differential quadrature method (FDQM) with generalized Caputo derivatives is used in this paper to show a new numerical way to solve fractional Riccati equations and fractional Lorenz systems. Unlike previous FDQM applications that have primarily focused on linear problems, our work pioneers the use of this method for nonlinear fractional initial value problems. By combining Lagrange interpolation polynomials and discrete singular convolution (DSC) shape functions with the generalized Caputo operator, we effectively transform nonlinear fractional equations into algebraic systems. An iterative method is then utilized to address the nonlinearity. Our numerical results, obtained using MATLAB, demonstrate the exceptional accuracy and efficiency of this approach, with convergence rates reaching 10⁻⁸. Comparative analysis with existing methods highlights the superior performance of the DSC shape function in terms of accuracy, convergence speed, and reliability. Our results highlight the versatility of our approach in tackling a wider variety of intricate nonlinear fractional differential equations. Full article

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. Full article

Estimating and controlling dynamical systems from observable time-series data are essential for understanding and manipulating nonlinear dynamics. This paper proposes a probabilistic framework for simultaneously estimating and controlling nonlinear dynamics under noisy observation conditions. Our proposed method utilizes the particle filter not only as a state estimator and a prior estimator for the dynamics but also as a controller. This approach allows us to handle the nonlinearity of the dynamics and uncertainty of the latent state. We apply two distinct dynamics to verify the effectiveness of our proposed framework: a chaotic system defined by the Lorenz equation and a nonlinear neuronal system defined by the Morris–Lecar neuron model. The results indicate that our proposed framework can simultaneously estimate and control complex nonlinear dynamical systems. Full article

The 1960s was an exciting era for atmospheric predictability research: a finite predictability of the atmosphere was uncovered using Lorenz’s models and the well-acknowledged predictability limit of two weeks was estimated using a general circulation model (GCM). Here, we delve into details regarding how a correlation between the two-week predictability limit and a doubling time of five days was established, recognize Lorenz’s pioneering work, and suggest non-impossibility for predictability beyond two weeks. We reevaluate the outcomes of three different approaches—dynamical, empirical, and dynamical-empirical—presented in Lorenz’s and Charney et al.’s papers from the 1960s. Using the intrinsic characteristics of the irregular solutions found in Lorenz’s studies and the dynamical approach, a doubling time of five days was estimated using the Mintz–Arakawa model and extrapolated to propose a predictability limit of approximately two weeks. This limit is now termed “Predictability Limit Hypothesis”, drawing a parallel to Moore’s Law, to recognize the combined direct and indirect influences of Lorenz, Mintz, and Arakawa under Charney’s leadership. The concept serves as a bridge between the hypothetical predictability limit and practical model capabilities, suggesting that long-range simulations are not entirely constrained by the two-week predictability hypothesis. These clarifications provide further support to the exploration of extended-range predictions using both partial differential equation (PDE)-physics-based and Artificial Intelligence (AI)—powered approaches. Full article

This entry examines Lorenz’s error growth models with quadratic and cubic hypotheses, highlighting their mathematical connections to the non-dissipative Lorenz 1963 model. The quadratic error growth model is the logistic ordinary differential equation (ODE) with a quadratic nonlinear term, while the cubic model is derived by replacing the quadratic term with a cubic one. A variable transformation shows that the cubic model can be converted to the same form as the logistic ODE. The relationship between the continuous logistic ODE and its discrete version, the logistic map, illustrates chaotic behaviors, demonstrating computational chaos with large time steps. A variant of the logistic ODE is proposed to show how finite predictability horizons can be determined, emphasizing the continuous dependence on initial conditions (CDIC) related to stable and unstable asymptotic values. This review also presents the mathematical relationship between the logistic ODE and the non-dissipative Lorenz 1963 model. Full article

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 $\sqrt{2}$, 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. Full article

This article explores the evolution of Lorenz trajectories within attractors. Specifically, based on the characteristics of the tangents to trajectories, we derive quantitative standards for determining the spatial position of trajectory lines. The Lorenz trajectory is decomposed into four parts. This standard is objective and quantitative and is independent of the initial field of the Lorenz equation and the calculation scheme; importantly, it is designed based on the inherent dynamic characteristics of the Lorenz equation. Linear fitting of the trajectories in the left and right equilibrium point regions shows that the trajectories lie on planes, indicating the existence of linear features in the nonlinear system. This study identifies the fundamental causes of chaos in the Lorenz equation using the microscopic evolution and local characteristics of the trajectories, and indicating that the spatial position of the initial field is important for their predictability. We theoretically demonstrate that mutation is essentially self-regulation within chaotic systems. This scheme is designed according to the evolution characteristics of Lorenz trajectories, and thus has certain methodological limitations that mean it may not be applicable to other chaotic systems. However, it does depict the causes of chaos and elucidates the sensitivity of differential equations to initial values in terms of trajectory evolution. Full article

Lorenz’ seminal work on global atmospheric energetics improved our understanding of the general circulation. With the advent of Regional Climate Models (RCMs), it is important to have a limited-area energetic budget available that is applicable for both weather and climate, analogous to Lorenz’ global atmospheric energetics. A regional-scale energetic budget is obtained in this study by applying Reynolds decomposition rules to quadratic forms of the kinetic energy K and the available enthalpy A, to obtain time mean and time deviation contributions. According to the employed definition, the time mean energy contributions are decomposed in a component associated with the time-averaged atmospheric state and a component due to the time-averaged statistics of transient eddies; these contributions are suitable for the study of the climate over a region of interest. Energy fluctuations (the deviations of instantaneous energies from their climate value) that are appropriate for weather studies are split into quadratic and linear contributions. The sum of all the contributions returns exactly to the total primitive kinetic energy and available enthalpy equations. Full article

In this paper will be considered a three-dimensional autonomous quadratic polynomial system of first-order differential equations with three real parameters, the so-called T-system. This system is symmetric relative to the $Oz$-axis and represents a special type of the generalized Lorenz system. The approach of this work will consist of the study of the nonlinear dynamics of this system through the Kosambi–Cartan–Chern (KCC) geometric theory. More exactly, we will focus on the associated system of second-order differential equations (SODE) from the point of view of Jacobi stability by determining the five invariants of the KCC theory. These invariants determine the internal geometrical characteristics of the system, and particularly, the deviation curvature tensor is decisive for Jacobi stability. Furthermore, we will look for necessary and sufficient conditions that the system parameters must satisfy in order to have Jacobi stability for every equilibrium point. Full article