Lorenz Model Research Articles

On bifurcations of Lorenz attractors in the Lyubimov–Zaks model

We provide numerical evidence for the existence of the Lorenz and the Rovella (contracting Lorenz) attractors in the generalization of the Lorenz model proposed by Lyubimov and Zaks. The Lorenz attractor is robustly chaotic (pseudohyperbolic) in contrast to the Rovella attractor, which is only measure-persistent (it exists for a set of parameter values, which is nowhere dense but has a positive Lebesgue measure). It is well known that in this model, for certain values of parameters, there exists a homoclinic butterfly (a pair of homoclinic loops) to the symmetric saddle equilibrium, which is neutral, i.e., its eigenvalues λ2<λ1<0<γ are such that the saddle index ν=−λ1/γ is equal to ∼1. The birth of the Lorenz attractor at this codimension-two bifurcation is established by means of numerical verification of the Shilnikov criterion. For the birth of the Rovella attractor, we propose a new criterion, which is also verified numerically.

Sparse nonlinear models of chaotic electroconvection.

Convection is a fundamental fluid transport phenomenon, where the large-scale motion of a fluid is driven, for example, by a thermal gradient or an electric potential. Modelling convection has given rise to the development of chaos theory and the reduced-order modelling of multiphysics systems; however, these models have been limited to relatively simple thermal convection phenomena. In this work, we develop a reduced-order model for chaotic electroconvection at high electric Rayleigh number. The chaos in this system is related to the standard Lorenz model obtained from Rayleigh–Benard convection, although our system is driven by a more complex three-way coupling between the fluid, the charge density, and the electric field. Coherent structures are extracted from temporally and spatially resolved charge density fields via proper orthogonal decomposition (POD). A nonlinear model is then developed for the chaotic time evolution of these coherent structures using the sparse identification of nonlinear dynamics (SINDy) algorithm, constrained to preserve the symmetries observed in the original system. The resulting model exhibits the dominant chaotic dynamics of the original high-dimensional system, capturing the essential nonlinear interactions with a simple reduced-order model.

The onset of Rayleigh–Bénard convection and heat transfer under two‐frequency rotation modulation

AbstractThe impact of 16 combinations of sinusoidal (sine) and nonsinusoidal (square, triangular, and sawtooth) time‐periodic Coriolis force (rotation modulation) on Rayleigh–Bénard convection in a Newtonian liquid is studied in this paper. This consideration is made to capture the possible effects of two‐frequency rotation modulation on stability, that is, the onset of convection and the simultaneous amount of heat transfer in the system. The Venezian approach has been asserted on the linearized Lorenz model to derive the correction Rayleigh number as a function of the two frequencies. The Lorenz model with nonlinearities is evaluated numerically to assess the quantity of heat transfer in the system. The present study states that in comparison with existing studies of no‐modulation and single‐frequency rotation modulation, two‐frequency rotation modulation yields higher stability bounds and thus diminishes the heat transfer. Heat transfer is found to be enhanced by a pair of coprime integers associated with the harmonics in the system.

Parameter estimation for a chaotic dynamical system with partial observations

Abstract 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.

Individual effects of four types of rotation modulation on Rayleigh–Bénard convection in a ferromagnetic fluid with couple stress

AbstractThe effect of trigonometric sine, square, triangular, and sawtooth wave types of rotation modulation on Rayleigh–Bénard convection in a ferromagnetic fluid with couple stress is investigated in this paper using linear and nonlinear analyses. The expression for the critical Rayleigh number and the correction Rayleigh number is deduced from the three‐mode linearized Lorenz model using the Venezian approach. The effect of rotation modulation on heat transport is studied using the generalized fifth‐order Lorenz model. The study reveals that the Taylor number stabilizes the no‐modulation system and decreases the heat transport, and this situation remains so in the presence of rotation modulation. It is found that the effect of all four types of modulation is to stabilize the system and diminish heat transport. It is also observed that the sawtooth wave type of modulation has the least diminishing effect on heat transport and the square wave type of modulation diminishes the most.

Lorenz attractors and the modular surface * *TP was supported by the Israel Science Foundation (Grant No. 1504/18).

We define an extension of the geometric Lorenz model, defined on the three sphere. This geometric model has an invariant one dimensional trefoil knot, a union of invariant manifolds of the singularities. It is similar to the invariant trefoil knot arising in the classical Lorenz flow near the classical parameters. We prove that this geometric model is topologically equivalent to the geodesic flow on the modular surface, once compactifying the latter.

A kernel principal component analysis of coexisting attractors within a generalized Lorenz model

Based on recent studies that reveal the coexistence of chaotic and non-chaotic solutions using a generalized Lorenz model (GLM), a revised view on the dual nature of weather has been proposed by Shen et al. [41,42], as follows: the entirety of weather is a superset consisting of both chaotic and non-chaotic processes. Since better predictability for non-chaotic processes can be expected, an effective detection of regular or chaotic solutions can improve our confidence in numerical weather and climate predictions. In this study, by performing a kernel principal component analysis of coexisting attractors obtained from the GLM, we illustrate that the time evolution of the first eigenvector of the kernel matrix, referred to as the first kernel principal component (K-PC), is effective for the classification of chaotic and non-chaotic orbits. The spatial distribution of the first K-PC within a two-dimensional phase space can depict the shape of a decision boundary that separates the chaotic and non-chaotic orbits. We additionally present how a large number (e.g., 128 or 256) of K-PCs can be used for the reconstruction of data in order to illustrate the different portions of the phase space occupied by chaotic and non-chaotic orbits, respectively.

Исследование взаимосвязи продовольственных, энергетических и водных ресурсов с помощью трехсекторальной модели Лоренца

Работа посвящена проблеме создания новых методов для комплексного моделирования и управления риском, которые позволят исследовать синергетические взаимодействия между источниками рисков различного происхождения в условиях неопределенности. Предложен подход к исследованию взаимосвязи продовольственных, водных и энергетических ресурсов с помощью трехсекторальной модели Лоренца, которая объединяет в единой структуре однотипным образом описанные сектора экономики, каждый из которых рассматривается в сроках уровня производительности, количества рабочих мест. и уровня структурных нарушений В результате математического моделирования определены условия возникновения детерминированного хаоса в минимальной модели экономического развития и выявлены возможные причины возрастающей уязвимости глобальной экономики к малым изменениям параметров управления. Рассмотрена задача определения эффективных управлений с целью минимизации суммарных структурных нарушений за выбранный интервал времени. В результате модельных экспериментов обнаружены траектории изменения параметров управления, позволяющие уменьшить число структурных нарушений. Это достигается за счет изменений соотношения уровней пропозиции и спроса продукции, спроса и предложения на создание рабочих мест. Рассмотрено влияние случайных возмущений на стохастическую деформацию детерминированных аттракторов модели Лоренца. Показано, что при случайных возбуждениях траектории стохастической системы покидают детерминированный аттрактор и образуют вокруг него некоторый пучок с соответствующим вероятностным распределением. Рассмотрена возможность дальнейшего усложнения модели за счет учета других секторов экономики с помощью модели Лоренца в комплексной форме. Задача исследования n секторов экономик в этом случае сводится к рассмотрению поведения ансамбля n связанных осцилляторов, генерирующих колебания с частотами ωn соответственно. Коллективная синхронизация данных осцилляторов может быть исследована с помощью модели Курамото. Задача управления социально-экономическим развитием в условиях возникновения хаотических режимов сводится для комплексной модели к управлению частотой поля с ненулевым средним, генерируемым связанными осцилляторами.

Linear and nonlinear triple diffusive convection in the presence of sinusoidal/non-sinusoidal gravity modulation: A comparative study

The non-uniform vertical vibrations (gravity modulation) or g-jitter or time-periodic body force, can be realized by oscillating the system vertically. The time periodic variations are considered to vary sinusoidally (trigonometric cosine) or non-sinusoidally (square, sawthooth and triangular) to study three-component convection in a Newtonian fluid using linear and non-linear analyses. In the linear theory, the expressions for the Rayleigh number and the correction Rayleigh number are obtained by using a regular perturbation method. The eigen value is obtained by adopting the classical Venezian approach. The generalized Lorenz model is derived using a Fourier series with additional modes and under the assumption of the Boussinesq approximation and small-scale convection motion. The resulting non-autonomous Lorenz model is solved numerically to quantify the heat and mass transports. The results are considered against the background of the results of no modulation. Study reveals that all the four types of gravity modulation delay the onset of convection and thereby lead to the diminished heat transfer situation. It is also found that in the case of trigonometric type of gravity modulation heat and mass transports are between those of the cases of triangular and square types. The results in respect of saw-tooth and triangular wave modulations are identical.

On a problem of linearized stability for fractional difference equations

This paper discusses the problem of linearized stability for nonlinear fractional difference equations. Computational methods based on appropriate linearization theorem are standardly applied in bifurcation analysis of dynamical systems. However, in the case of fractional discrete systems, a theoretical background justifying its use is still missing. Therefore, the main goal of this paper is to fill in the gap. We consider a general autonomous system of fractional difference equations involving the backward Caputo fractional difference operator and prove that any equilibrium of this system is asymptotically stable if the zero solution of the corresponding linearized system is asymptotically stable. Moreover, these asymptotic stability conditions for equilibria of the system are described via location of all the characteristic roots in a specific area of the complex plane. In the planar case, these conditions are given even explicitly in terms of trace and determinant of the appropriate Jacobi matrix. The results are applied to a fractional predator-prey model and the fractional Lorenz model. Related experiments are supported by a numerical code that is appended as well

A study of Darcy–Bénard regular and chaotic convection using a new local thermal non-equilibrium formulation

The onset of Darcy–Bénard regular and chaotic convection in a porous medium is studied by considering phase-lag effects that naturally arise in the thermal non-equilibrium heat transfer problem between the fluid and solid phases. A new type of heat equation is derived for both the phases. Using a double Fourier series and a novel decomposition, an extended Vadasz–Lorenz model with three phase-lag effects is derived. New parameters arise due to the phase-lag effects between local acceleration, convective acceleration, and thermal diffusion. The principle of exchange of stabilities is found to be valid and the subcritical instability is discounted. The new perspective supports the finding of an analytical expression for the critical Darcy–Rayleigh numbers representing, respectively, the onset of regular and chaotic convection. The understanding of the transition from the local thermal non-equilibrium situation to the local thermal equilibrium one is also best explained through the new perspective. In its present elegant form, the extended Vadasz–Lorenz system with three phase-lag effects is analyzed using the largest Lyapunov exponent and the bifurcation diagram. It is found that the lag effects not only give rise to a quantitative difference in the above two metrics concerning chaos, but also present a qualitative difference as well in the form of the very nature of chaos.

On the Localization in Strongly Coupled Ensemble Data Assimilation Using a Two‐Scale Lorenz Model

AbstractFor coupled numerical models with different components (domains), there are two kinds of assimilation strategies applied for producing ocean analysis and initial condition of predictions: the strongly coupled data assimilation (SCDA) and weakly coupled data assimilation (WCDA). The former needs to accurately estimate cross‐component error covariances, which is much challenging, especially when a small ensemble size's Kalman filter‐based algorithm is used and a coupled model has the components of different spatiotemporal scales. In this study, we propose a new scheme for the ensemble adjustment Kalman filter (EAKF) to address cross‐component localization, a critical issue in estimating the cross‐component error covariance in SCDA, based on a two‐scale Lorenz ’96 coupled mode with different temporal and spatial scales. Emphasis places on designing the cross‐component localization factors in the framework of multiple spatial scales. The result shows that the SCDA can provide much more accurate estimations of the states than the WCDA when the new proposed cross‐component localization is used. A further analysis reveals that the advantage of the SCDA over the WCDA is attributed to the assimilation of observations from the small‐scale model in the coupled system, whereas the contribution of the assimilation of observations from the large‐scale model is not obvious. This study offers a useful technique to develop SCDA system in operational prediction models, which is being pursued in the prediction community.

Combining data assimilation and machine learning to infer unresolved scale parametrization.

In recent years, machine learning (ML) has been proposed to devise data-driven parametrizations of unresolved processes in dynamical numerical models. In most cases, the ML training leverages high-resolution simulations to provide a dense, noiseless target state. Our goal is to go beyond the use of high-resolution simulations and train ML-based parametrization using direct data, in the realistic scenario of noisy and sparse observations. The algorithm proposed in this work is a two-step process. First, data assimilation (DA) techniques are applied to estimate the full state of the system from a truncated model. The unresolved part of the truncated model is viewed as a model error in the DA system. In a second step, ML is used to emulate the unresolved part, a predictor of model error given the state of the system. Finally, the ML-based parametrization model is added to the physical core truncated model to produce a hybrid model. The DA component of the proposed method relies on an ensemble Kalman filter while the ML parametrization is represented by a neural network. The approach is applied to the two-scale Lorenz model and to MAOOAM, a reduced-order coupled ocean-atmosphere model. We show that in both cases, the hybrid model yields forecasts with better skill than the truncated model. Moreover, the attractor of the system is significantly better represented by the hybrid model than by the truncated model. This article is part of the theme issue 'Machine learning for weather and climate modelling'.

Modulated gravity effects on nonlinear convection in viscoelastic ferromagnetic fluids between two horizontal parallel plates

AbstractIn this study, the analysis of nonlinear stability with viscoelastic ferromagnetic fluids as working media is performed by finite‐amplitude perturbations. The solution of the resulting nonautonomous system of the Lorenz model (generalized Khayat–Lorenz model of four modes) is obtained numerically to measure the amount of heat transport. The effects of elastic parameters, Prandtl number, modulation parameters, buoyancy magnetic parameter, and nonbuoyancy magnetic parameter on heat transport are studied. Heat transport is quantified through the average Nusselt number, which is determined by solving the scaled Lorenz model. As limiting cases of the study, the results of Newtonian, Maxwell, Rivlin–Ericksen fluids are determined. The results obtained have been presented graphically.

Non-linear Rayleigh-Benard Magnetoconvection in Temperature-sensitive Newtonian Liquids with Variable Heat Source

The present paper aims at weak non-linear stability analysis followed by linear analysis of nite-amplitude Rayleigh-Benard magneto convection problem in an electrically conducting Newtonian liquid with heat source/sink. It is shown that the internal Rayleigh number, ther- morheological parameter, and the Chandrasekhar number in uence the onset of convection. The generalized Lorenz model derived for the prob- lem is essentially the classical Lorenz model but with some coecient depending on the variable heat source (sink), viscosity, and the applied magnetic eld. The result of the parameters' in uence on the critical Rayleigh number explains their in uence on the Nusselt number. It is found that an increasing strength of the magnetic eld is to stabilize the system and diminishes heat transport whereas the heat source and variable viscosity in-tandem to work system unstable and enhances heat transfer.

Study of Brinkman–Bènard nanofluid convection with idealistic and realistic boundary conditions and by considering the effects of shape of nanoparticles

AbstractThis study deals with linear and weakly non‐linear stability analyses of Brinkman–Bènard convection in nanoliquid‐saturated porous enclosures. Water with a dilute concentration of molybdenum disulfide nanoparticles with 0.06 volume fraction and 30% glass fiber‐reinforced polycarbonate as a porous medium with porosity 0.88 are considered to be a working medium. The analytical solution is obtained in the present study for idealistic and realistic boundary conditions, and their results are compared. An analytically intractable Lorenz model with quadratic nonlinearities is reduced to a tractable Ginzburg–Landau amplitude equation with cubic nonlinearity using the multiscale method. Nanoparticles with different shapes are considered in the study, and their effects on the onset and heat transfer are discussed in great detail graphically in the presence of other parameters arising in the problem.

Nonlinear analysis of the effect of viscoelasticity on ferroconvection

AbstractThis paper concerns a nonlinear analysis of the effects of viscoelasticity on convection in ferroliquids. We consider the Oldroyd model for the constitutive equation of the liquid. The linear stability analysis yields the critical value of the Rayleigh number for the onset of oscillatory convection in Maxwell and Jeffrey ferroliquids. The use of a minimal mode double Fourier series in the nonlinear perturbation equations yields a Khayat–Lorenz model for the ferromagnetic liquid, and that is scaled further to get the classical Lorenz model as a limiting case. The scaled Khayat–Lorenz model thus obtained is solved numerically and the solution is used to compute the time‐dependent Nusselt number, which quantifies the heat transport. The results are analyzed for the dependence of the time‐averaged Nusselt number on different parameters.

Eigenvector-spatial localisation

We present a new multiscale covariance localisation method for ensemble data assimilation that is based on the estimation of eigenvectors and subsequent projections, together with traditional spatial localisation applied with a range of localisation lengths. In short, we estimate the leading, large-scale eigenvectors from the sample covariance matrix obtained by spatially smoothing the ensemble (treating small scales as noise) and then localise the resulting sample covariances with a large length scale. After removing the projection of each ensemble member onto the leading eigenvectors, the process may be repeated using less smoothing and tighter localizations or, in a final step, using the resulting, residual ensemble and tight localisation to represent covariances in the remaining subspace. We illustrate the use of the new multiscale localisation method in simple numerical examples and in cycling data assimilation experiments with the Lorenz Model III. We also compare the proposed new method to existing multiscale localisation and to single-scale localisation.

Online learning of both state and dynamics using ensemble Kalman filters

<p style='text-indent:20px;'>The reconstruction of the dynamics of an observed physical system as a surrogate model has been brought to the fore by recent advances in machine learning. To deal with partial and noisy observations in that endeavor, machine learning representations of the surrogate model can be used within a Bayesian data assimilation framework. However, these approaches require to consider long time series of observational data, meant to be assimilated all together. This paper investigates the possibility to learn both the dynamics and the state online, i.e. to update their estimates at any time, in particular when new observations are acquired. The estimation is based on the ensemble Kalman filter (EnKF) family of algorithms using a rather simple representation for the surrogate model and state augmentation. We consider the implication of learning dynamics online through (ⅰ) a global EnKF, (ⅰ) a local EnKF and (ⅲ) an iterative EnKF and we discuss in each case issues and algorithmic solutions. We then demonstrate numerically the efficiency and assess the accuracy of these methods using one-dimensional, one-scale and two-scale chaotic Lorenz models.</p>

A Novel of New 7D Hyperchaotic System with Self-Excited Attractors and Its Hybrid Synchronization.

In this study, a novel 7D hyperchaotic model is constructed from the 6D Lorenz model via the nonlinear feedback control technique. The proposed model has an only unstable origin point. Thus, it is categorized as a model with self-excited attractors. And it has seven equations which include 19 terms, four of which are quadratic nonlinearities. Various important features of the novel model are analyzed, including equilibria points, stability, and Lyapunov exponents. The numerical simulation shows that the new class exhibits dynamical behaviors such as chaotic and hyperchaotic. This paper also presents the hybrid synchronization for a novel model via Lyapunov stability theory.