Experiments with a Malkus–Lorenz water wheel: Chaos and Synchronization

Illing, Lucas; Fordyce, Rachel; Saunders, A. M.; Ormond, R.

doi:10.1119/1.3680533

Cited by 17 publications

(17 citation statements)

References 35 publications

(39 reference statements)

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“…Indeed, if both y(t) and z(t) andỹ(t) andz(t) satisfy (22) with the same x(t) albeit with different initial conditions, then the differences Δy = y −ỹ and Δz = z −z satisfy the homogeneous linear system d dt…”

Section: Analysis and Bounds On Transport By Arbitrary Stirringmentioning

confidence: 99%

“…where η(t) and ζ(t) are Lagrange multipliers enforcing (22) and the real variable μ is the Lagrange multiplier enforcing the intensity constraint. Ignoring initial (and final) conditions for the moment, the Euler-Lagrange equations obtained by setting δF/δy, δF/δz, and ∂F/∂μ to zero are, respectively,…”

Section: Background Analysismentioning

confidence: 99%

“…The optimal control problem is a constrained optimization problem: we seek to determine the maximum possible value, over all admissible functions x(t), of the long-time average xy where y(t) and z(t) solve (22).…”

Section: Background Analysismentioning

confidence: 99%

“…We numerically constructed some time-periodic solutions of the optimal control problem defined by (22), (24), and (25) by computationally continuing analytical time-periodic solutions from the linearized problem at small Pe into the strongly nonlinear regime. In the limit Pe → 0 the linearized Euler-Lagrange equations arė…”

Section: Time Periodic Stirring Protocolsmentioning

confidence: 99%

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Maximal transport in the Lorenz equations

Souza

Doering

2015

Physics Letters A

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Section: Analysis and Bounds On Transport By Arbitrary Stirringmentioning

confidence: 99%

Section: Background Analysismentioning

confidence: 99%

Section: Background Analysismentioning

confidence: 99%

Section: Time Periodic Stirring Protocolsmentioning

confidence: 99%

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Maximal transport in the Lorenz equations

Souza

Doering

2015

Physics Letters A

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“…También son numerosas las observaciones experimentales, como las de Weiss en láseres [22,23] u otras como las de Illing en norias de agua [24].…”

Section: C Versatilidad Del Modelo De Lorenzunclassified

Didactic application of numerical analysis in nonlinear dynamics: Lorenz model study

García-Ferrer¹,

Roldán²,

Silva³

et al. 2017

Opt. Pura Apl.

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ABSTRACT:We describe a practice designed for the numerical study of the Lorenz model that is a central model in the physics of lasers. The didactical objectives pursued in this practice have a dual nature, considering both the introduction to the knowledge of a physical paradigm of deterministic chaos as the training for the use of certain computational tools for its characterization. The method used to achieve programming is finding solutions of the Lorenz model and systematically studying of their temporal evolution using a Mathematica program. In the academic context, the practice is designed to be included in the curriculum of the degree in physics and to facilitate adaptation to other matters in this area, such as quantum optics, fluids, mechanical vibrations, etc. We first study the steady states, and their linear stability, of the Lorenz model equations and then numerically study the different types of dynamic behavior. We pay special attention to the deterministic chaotic behavior and to the sequence of bifurcations leading from periodic to chaotic behavior (routes to chaos).Key words: Lorenz model, chaos, nonlinear dynamics, laser, didactic, Mathematica. RESUMEN:Describimos una práctica diseñada para el estudio numérico de un modelo central en la física del láser, el modelo de Lorenz. Los objetivos didácticos que se persiguen en la práctica tienen una naturaleza doble, por un parte, la introducción al conocimiento de un paradigma físico del caos determinista y, por otra, la capacitación para la utilización de ciertas herramientas computacionales que permiten su caracterización. El método utilizado para alcanzarlos es la búsqueda de las soluciones del modelo de Lorenz y el estudio sistemático de su evolución temporal mediante el programa Mathematica. En el contexto académico, la práctica está diseñada para ser incluida en el currículum del grado en Física y para facilitar su adaptación a otras materias dentro de este ámbito, como puede ser óptica cuántica, fluidos, oscilaciones mecánicas, etc. Partiendo de las ecuaciones del modelo de Lorenz, estudiamos sus estados estacionarios y su estabilidad lineal, para después estudiar numéricamente los distintos tipos de comportamiento dinámico. Prestamos especial atención al comportamiento caótico determinista y a la secuencia de bifurcaciones que lleva de los comportamientos periódicos a los caóticos (rutas al caos).Palabras clave: Modelo de Lorenz, caos, dinámica no lineal, láser, didáctica, Mathematica. REFERENCES AND LINKS / REFERENCIAS Y ENLACES[1] E.N. Lorenz, "Deterministic nonperiodic flow", J. Atm. Sci. 20, 130-141 (1963).

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Abarbanel

2013

Understanding Complex Systems

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Experiments with a Malkus–Lorenz water wheel: Chaos and Synchronization

Cited by 17 publications

References 35 publications

Maximal transport in the Lorenz equations

Maximal transport in the Lorenz equations

Didactic application of numerical analysis in nonlinear dynamics: Lorenz model study

Examples as a Guide to the Issues

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