Energy cycle for the Lorenz attractor

Pelino, Vinicio; Maimone, Filippo; Pasini, Antonello

doi:10.1016/j.chaos.2013.09.005

Cited by 39 publications

(19 citation statements)

References 31 publications

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“…These systems were constructed via mathematics and simulations whereas the Lorenz system [6] was modeled on a physical process. A few studies have investigated the application of the Lorenz system in meteorology [7] and mechanics [8]. The main research focus of these numerical systems as well as the Lorenz system has been on their dynamic analysis.…”

Section: Introductionmentioning

confidence: 99%

“…The recurrence of dynamics [17] and energy cycling [8] for the Lorenz system were also investigated with the understanding obtained from the Kolmogorov system.…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, the Casimir function, like enstrophy or potential vorticity in a fluid dynamic context, is very useful in analyzing stability conditions and global description of a dynamical system. It represents a constant of the motion of the Hamiltonian system; moreover it defines a foliation of the phase space [8,19]. The energetics of the Lorenz system using the Casimir function has already been studied [20].…”

Section: Introductionmentioning

confidence: 99%

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Energy cycle and bound of Qi chaotic system

Zhang

2017

Chaos, Solitons & Fractals

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The Qi chaotic system is transformed into a Kolmogorov-type system, thereby facilitating the analysis of energy exchange in its different forms. Regarding four forms of energy, the vector field of this chaotic system is decomposed into four forms of torque: inertial, internal, dissipative, and external. The rate of change of the Casimir function is equal to the exchange power between the dissipative energy and the supplied energy. The exchange power governs the orbital behavior and the cycling of energy. With the rate of change of Casimir function, a general bound and least upper bound of the Qi chaotic attractor are proposed. A detailed analysis with illustrations is conducted to uncover insights, in particular, cycling among the different types of energy for this chaotic attractor and key factors producing the different types of dynamic modes.

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Section: Introductionmentioning

confidence: 99%

“…The recurrence of dynamics [17] and energy cycling [8] for the Lorenz system were also investigated with the understanding obtained from the Kolmogorov system.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Energy cycle and bound of Qi chaotic system

Zhang

2017

Chaos, Solitons & Fractals

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“…Névir and Blender [22] developed a Nambu representation of the nondissipative parts of the classical Lorenz-63 [12] model for X 1 , Y 2 , and Z 1 (see also [1,24,3] and [25] for a classification). With the conservation laws for H and C the Lorenz equations are in terms of x = (X 1 , Y 2 , Z 1 )…”

Section: Nambu Structurementioning

confidence: 99%

Nambu representation of an extended Lorenz model with viscous heating

Blender¹,

Lucarini²

2013

Physica D: Nonlinear Phenomena

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We consider the Nambu and Hamiltonian representations of RayleighBénard convection with a nonlinear thermal heating effect proportional to the Eckert number (Ec). The model we use is an extension of the classical Lorenz-63 model with 4 kinematic and 6 thermal degrees of freedom. The conservative parts of the dynamical equations which include all nonlinearities satisfy Liouville's theorem and permit a conserved Hamiltonian H for arbitrary Ec. For Ec = 0 two independent conserved Casimir functions exist, one of these is associated with unavailable potential energy and is also present in the Lorenz-63 truncation. This Casimir C is used to construct a Nambu representation of the conserved part of the dynamical system. The thermal heating effect can be represented either by a second canonical Hamiltonian or as a gradient (metric) system using the time derivativeĊ of the Casimir. The results demonstrate the impact of viscous heating in the total energy budget and in the Lorenz energy cycle for kinetic and available potential energy.

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“…Casimir energy plays a vital role in the energy analysis of rigid body systems. By the Casimir energy method of geometric mechanics, Pelino and coworkers [19] presented the energy cycle of the Lorenz system employing the Kolmogorov model. Qi and Liang [20,21] studied the mechanics of the Qi fourwing chaotic system and the Qi chaotic system to perform a forcing analysis, where the variables of the systems are interpreted as angular momenta of a forced-dissipative rigid body.…”

Section: Introductionmentioning

confidence: 99%

Modeling of a Chaotic Gyrostat System and Mechanism Analysis of Dynamics Using Force and Energy

Yang

2019

Complexity

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Incorporating both viscous friction torque and external torque, a mathematical model of a gyrostat system consisting of three rotors and a fixed outer frame is configured. This model is transformed into a Kolmogorov-type system for force analysis. The force field in the gyrostat system includes four different torques—inertial, internal, dissipative, and external. Correspondingly, four different energies are identified and the interconversion of energies is analyzed. The Casimir power being equivalent to the error power between the supplied power and the dissipative power is found and used to analyze the mechanism underlying the different dynamical behaviors. A four-wing chaotic attractor is found when the gyrostat is subject to a combination of inertial, internal, and dissipative torques as well as the addition of an external torque. The bifurcation of the Casimir power and leaps in Casimir energy level are used as indicators of the different definitive dynamics, which is demonstrated to be identical to that appearing in the state bifurcation and the Lyapunov exponent spectrum.

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Energy cycle for the Lorenz attractor

Cited by 39 publications

References 31 publications

Energy cycle and bound of Qi chaotic system

Energy cycle and bound of Qi chaotic system

Nambu representation of an extended Lorenz model with viscous heating

Modeling of a Chaotic Gyrostat System and Mechanism Analysis of Dynamics Using Force and Energy

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