Lorenz Equations Research Articles

Maximal transport in the Lorenz equations

Abstract We derive rigorous upper bounds on the transport 〈 X Y 〉 where 〈 ⋅ 〉 indicates time average, for solutions of the Lorenz equations without assuming statistical stationarity. The bounds are saturated by nontrivial steady (albeit often unstable) states, and hence they are sharp. Moreover, using an optimal control formulation we prove that no other flow protocol of the same strength, i.e., no other function of time X ( t ) driving the Y ( t ) and Z ( t ) variables while satisfying the basic balance 〈 X 2 〉 = 〈 X Y 〉 , produces higher transport.

A New Piecewise Linear Hyperchaotic Circuit

When the polarity information in diffusionless Lorenz equations is preserved or removed, a new piecewise linear hyperchaotic system results with only signum and absolute-value nonlinearities. Dynamical equations have seven terms without any quadratic or higher order polynomials and, to our knowledge, are the simplest hyperchaotic system. Therefore, a relatively simple hyperchaotic circuit using diodes is constructed. The circuit requires no multipliers or inductors, as are present in other hyperchaotic circuits, and it has not been previously reported.

Statistical Properties of Lorenz-like Flows, Recent Developments and Perspectives

We comment on the mathematical results about the statistical behavior of Lorenz equations and its attractor, and more generally on the class of singular hyperbolic systems. The mathematical theory of such kind of systems turned out to be surprisingly difficult. It is remarkable that a rigorous proof of the existence of the Lorenz attractor was presented only around the year 2000 with a computer-assisted proof together with an extension of the hyperbolic theory developed to encompass attractors robustly containing equilibria. We present some of the main results on the statistical behavior of such systems. We show that for attractors of three-dimensional flows, robust chaotic behavior is equivalent to the existence of certain hyperbolic structures, known as singular-hyperbolicity. These structures, in turn, are associated with the existence of physical measures: in low dimensions, robust chaotic behavior for flows ensures the existence of a physical measure. We then give more details on recent results on the dynamics of singular-hyperbolic (Lorenz-like) attractors: (1) there exists an invariant foliation whose leaves are forward contracted by the flow (and further properties which are useful to understand the statistical properties of the dynamics); (2) there exists a positive Lyapunov exponent at every orbit; (3) there is a unique physical measure whose support is the whole attractor and which is the equilibrium state with respect to the center-unstable Jacobian; (4) this measure is exact dimensional; (5) the induced measure on a suitable family of cross-sections has exponential decay of correlations for Lipschitz observables with respect to a suitable Poincaré return time map; (6) the hitting time associated to Lorenz-like attractors satisfy a logarithm law; (7) the geometric Lorenz flow satisfies the Almost Sure Invariance Principle (ASIP) and the Central Limit Theorem (CLT); (8) the rate of decay of large deviations for the volume measure on the ergodic basin of a geometric Lorenz attractor is exponential; (9) a class of geometric Lorenz flows exhibits robust exponential decay of correlations; (10) all geometric Lorenz flows are rapidly mixing and their time-1 map satisfies both ASIP and CLT.

Clean numerical simulation for some chaotic systems using the parallel multiple-precision Taylor scheme

An improved parallel multiple-precision Taylor (PMT) scheme is developed to obtain clean numerical simulation (CNS) solutions of chaotic ordinary differential equations (ODEs). The new version program is about 500 times faster than the reported solvers developed in the MATHEMATICA, and also 2–3 times faster than the older version (PMT-1.0) of the scheme. This solver has the ability to yield longer solutions of Lorenz equations [up to 5000 TU (time unit)]. The PMT-1.1 scheme is applied to a selection of chaotic systems including the Chen, Rossler, coupled Lorenz and Lu systems. The T c -M and T c -K diagrams for these chaotic systems are presented and used to analyze the computation parameters for long-term solutions. The reliable computation times of these chaotic equations are obtained for single- and double-precision computation.

New parametric approach for the general Lorenz system

We are concerned with the study of the system of coupled of motion for a system of two-level atoms interacting with a single-mode field in the laser cavity on resonance. The passage from the equation of motion to the Lorenz equations is established. Some theoretical aspects of the slowly time-varying phenomena in cavities are discussed and the linear stability analysis is presented. A new approach to solve these nonlinear coupled differential equations is explored and its generalization to the case of variable coefficients is performed.

Computer Assisted Proof of Transverse Saddle-to-Saddle Connecting Orbits for First Order Vector Fields

In this paper we introduce a computational method for proving the existence of generic saddle-to-saddle connections between equilibria of first order vector fields. The first step consists of rigorously computing high order parametrizations of the local stable and unstable manifolds. If the local manifolds intersect, the Newton–Kantorovich theorem is applied to validate the existence of a so-called short connecting orbit. If the local manifolds do not intersect, a boundary value problem with boundary values in the local manifolds is rigorously solved by a contraction mapping argument on a ball centered at the numerical solution, yielding the existence of a so-called long connecting orbit. In both cases our argument yields transversality of the corresponding intersection of the manifolds. The method is applied to the Lorenz equations, where a study of a pitchfork bifurcation with saddle-to-saddle stability is done and where several proofs of existence of short and long connections are obtained.

Chaos in hydrodynamic BL Herculis models

We present non-linear, convective, BL Her-type hydrodynamic models that show complex variability characteristic for deterministic chaos. The bifurcation diagram reveals a rich structure, with many phenomena detected for the first time in hydrodynamic models of pulsating stars. The phenomena include not only period doubling cascades en route to chaos (detected in earlier studies) but also periodic windows within chaotic band, type-I and type-III intermittent behaviour, interior crisis bifurcation and others. Such phenomena are known in many textbook chaotic systems, from the simplest discrete logistic map, to more complex systems like Lorenz equations. We discuss the physical relevance of our models. Although except of period doubling such phenomena were not detected in any BL Her star, chaotic variability was claimed in several higher luminosity siblings of BL Her stars - RV Tau variables, and also in longer-period, luminous irregular pulsators. Our models may help to understand these poorly studied stars. Particularly interesting are periodic windows which are intrinsic property of chaotic systems and are not necessarily caused by resonances between pulsation modes, as sometimes claimed in the literature.

Computional Dynamics for Diffusionless Lorenz Equations with Periodic Parametric Perturbation

The dynamics of diffusionless Lorenz equations (DLE) with periodic parametric perturbation is studied through numerical and experimental investigations in this paper. A method for calculating Lyapunov exponents (LEs), Lyapunov dimension (LD) from time series is presented. Furthermore, bifurcation and some complex dynamic behaviors such as periodic, quasi-periodic motion and chaos which occurred in the system are analyzed. And an algorithm for detecting unstable periodic orbits (UPOs) is presented. Also, give some numerical simulation studies of the system in order to verify the analytic results.

Bifurcation Structure of Augmented Lorenz Equations and Synchronizability of Coupled Augmented Lorenz Oscillators

Bifurcation Structure of Augmented Lorenz Equations and Synchronizability of Coupled Augmented Lorenz Oscillators

Estimating uncertainties in statistics computed from direct numerical simulation

Rigorous assessment of uncertainty is crucial to the utility of direct numerical simulation (DNS) results. Uncertainties in the computed statistics arise from two sources: finite statistical sampling and the discretization of the Navier–Stokes equations. Due to the presence of non-trivial sampling error, standard techniques for estimating discretization error (such as Richardson extrapolation) fail or are unreliable. This work provides a systematic and unified approach for estimating these errors. First, a sampling error estimator that accounts for correlation in the input data is developed. Then, this sampling error estimate is used as part of a Bayesian extension of Richardson extrapolation in order to characterize the discretization error. These methods are tested using the Lorenz equations and are shown to perform well. These techniques are then used to investigate the sampling and discretization errors in the DNS of a wall-bounded turbulent flow at Reτ ≈ 180. Both small (Lx/δ × Lz/δ = 4π × 2π) and large (Lx/δ × Lz/δ = 12π × 4π) domain sizes are investigated. For each case, a sequence of meshes was generated by first designing a “nominal” mesh using standard heuristics for wall-bounded simulations. These nominal meshes were then coarsened to generate a sequence of grid resolutions appropriate for the Bayesian Richardson extrapolation method. In addition, the small box case is computationally inexpensive enough to allow simulation on a finer mesh, enabling the results of the extrapolation to be validated in a weak sense. For both cases, it is found that while the sampling uncertainty is large enough to make the order of accuracy difficult to determine, the estimated discretization errors are quite small. This indicates that the commonly used heuristics provide adequate resolution for this class of problems. However, it is also found that, for some quantities, the discretization error is not small relative to sampling error, indicating that the conventional wisdom that sampling error dominates discretization error for this class of simulations needs to be reevaluated.

A new finding of the existence of hidden hyperchaotic attractors with no equilibria

The paper presents a new four-dimensional hyperchaotic system developed by extension of the generalized diffusionless Lorenz equations. The model is shown to not be equivalent to any hyperchaotic system that the authors know of. In particular, the model does not display any equilibria, but can exhibit two-scroll hyperchaos as well as chaotic, quasiperiodic and periodic dynamics. For certain parameter values, coexisting attractors can be observed, e.g. hyperchaotic and periodic attractors. Investigation of the proposed system is performed through a combination of numerical simulation and mathematical analysis in order to obtain time plots, phase portraits, Lyapunov exponents, and Poincaré sections.

The theoretical study of the turning period in numerical weather prediction models based on the Lorenz equations

Based on the Lorenz equations, the dynamics of the weather turning period is studied about numerical weather prediction. Through the analysis of the stability of equilibrium points of the Lorenz equations, we get the surfaces which separate the quasi-stable region and quasi-unstable region. In the quasi-stable region, the path curve of the Lorenz equations can remain relatively stable around the equilibrium points, however in the quasi-unstable region, the path curve of the Lorenz equations can spring from this equilibrium point to another one. This is one of the important dynamic characteristics of the Lorenz system, and the paper give new method and theory for the detection of the abrupt change of climate.

Rigorous Numerics for Nonlinear Differential Equations Using Chebyshev Series

A computational method based on Chebyshev series to rigorously compute solutions of initial and boundary value problems of analytic nonlinear vector fields is proposed. The idea is to recast solutions as fixed points of an operator defined on a Banach space of rapidly decaying Chebyshev coefficients and to use the so-called radii polynomials to show the existence of a unique fixed point near an approximate solution. As applications, solutions of initial value problems in the Lorenz equations and symmetric connecting orbits in the Gray--Scott equation are rigorously computed. The symmetric connecting orbits are obtained by solving a boundary value problem with one of the boundary values in the stable manifold.

Mathematical modeling is more than fitting equations.

Comments on the comments made by Brown et al. (see record 2013-24609-001). The article by Brown et al. regarding the Fredrickson and Losada (see record 2005-11834-001) article discussed the use of differential equations in science and repeated our earlier observation (Luoma, Hämäläinen, & Saarinen, 2008) that there is lack of justification for the use of the Lorenz equations in the latter article. In this comment we want to point out that Brown et al. presented a very narrow view on mathematical modeling in behavioral research. We describe how the conceptual use of mathematical models is essential in many fields.

자기장 및 열하중을 받는 복합재료 원통셸의 진동 및 안정성해석

In this paper vibration and stability analysis of laminated composite shells based on the first order shear deformation theory(FSDT) for two different boundary conditions(clamped-clamped, simply supported) are performed. Structural model of cross-ply symmetric laminated composite cylindrical shells subjected to a combination of magnetic and thermal fields is developed via Hamilton's variational principle. These coupled equations of motion are based on the electromagnetic equations(Faraday, Ampere, Ohm, and Lorenz equations)and thermal equations which are involved in constitutive equations. Extended Galerkin method is adopted to obtain the discretized equations of motion. Variations of dynamic characteristics of composite shells with applied magnetic field, temperature gradient, laminate thickness-ratio and radius ratio for two boundary conditions are investigated and pertinent conclusions are derived.

SOME ATTRACTORS IN THE EXTENDED COMPLEX LORENZ MODEL

We address the question of finding the attractors of the extended complex Lorenz model (ℂLM), which is obtained by extending the space from ℝ3 to ℂ3, and defining the model by the same equations as the classical Lorenz model (LM). We have numerical evidence of two strong attractors unrelated to the Lorenz attractor. We calculate its Lyapunov exponents and show that two of them are 0, and the other four are double and negative. Hence the attractors are nonchaotic. We show that they have a quasi-periodic nature. To decipher the structure of these attractors, we introduce the imaginary Lorenz model (𝕀LM), which is defined in the same space ℂ3 by multiplying with [Formula: see text] the Lorenz equations. Both models locally commute, and with its help we account for the double Lyapunov exponent 0 and show evidence that the basin of attraction of each attractor is a big open set of ℂ3. The chaotic limit set Lℂ ⊂ ℂ3 obtained from the classical Lorenz attractor L0 of (LM) by moving it with the (𝕀LM) has two positive Lyapunov exponents, but only captures a set of 6D-volume 0 in its basin of attraction. Hence this attractor may be hyperchaotic in ℝ5.

Comment on “A constructive proof on the existence of globally exponentially attractive set and positive invariant set of general Lorenz family”, P. Yu, X.X. Liao, S.L. Xie, Y.L. Fu [Commun Nonlinear Sci Numer Simulat 14 (2009) 2886–2896

In the commented paper the authors study some aspects of boundedness in the general Lorenz family, ẋ=σ(y-x),ẏ=ρx-γy-xz,ż=-βz+xy, considering that it contains four independent parameters. However, as we show here by means of a linear scaling in time and coordinates, they are dealing with a system homothetically equivalent to the Lorenz system. Consequently, the novel and interesting results they provide for the general Lorenz family can be obtained working directly with the Lorenz equations, that is, dealing only with three independent parameters.

Sequences of gluing bifurcations in an analog electronic circuit

We report on the experimental investigation of gluing bifurcations in the analog electronic circuit which models a dynamical system of the third order: Lorenz equations with an additional quadratic nonlinearity. Variation of one of the resistances in the circuit changes the coefficient at this nonlinearity and enables transition from the Lorenz route to chaos to a different scenario which leads, through the sequence of homoclinic bifurcations, from periodic oscillations of the voltage to the chaotic state. A single bifurcation "glues" in the phase space two stable periodic orbits and creates a new one, with the doubled length: a bifurcation sequence results in the birth of the chaotic attractor.

Probabilistic methods for physics

We present an asymptotic method giving a probability of presence of the iterated spots of Rd by a polynomial function f. We use the well-known Perron Frobenius operator (PF) that lets certain sets and measure invariant by f. Probabilistic solutions can exist for the deterministic iteration. If the theoretical result is already known, here we quantify these probabilities. This approach seems interesting to use for computing situations when the deterministic methods don't run. Among the examined applications, are asymptotic solutions of Lorenz, Navier-Stokes or Hamilton's equations. In this approach, linearity induces many difficult problems, all of whom we have not yet resolved.

A New Method for Parameter Sensitivity Analysis of Lorenz Equations

A new method for parameter sensitivity analysis of Lorenz equations is presented. The sensitivity equations are derived based on the staggered methods. Experimental results indicate that it is possible to determine effects of parameters on model variables so that we can eliminate the less effective ones. Robustness can also be verified in some confidence intervals by simply looking at the corresponding phase portraits. This enables us to control the system. Although the stability properties of the Lorenz equations are studied extensively, to the best knowledge of the authors, the PSA of Lorenz equations has not been considered which is the main goal of this paper.