Chaos in the Lorenz equations: a computer-assisted proof

Mischaikow, Konstantin; Mrożek, Marian

doi:10.1090/s0273-0979-1995-00558-6

Cited by 209 publications

(143 citation statements)

References 7 publications

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“…Oddly enough, the original equations introduced by Lorenz have remained a puzzle. A few computer-assisted proofs, however, have quite recently been announced, see [3], [6], [12]. These papers deal with subsets of A which are not attracting, and therefore only concern a set of trajectories having measure zero.…”

Section: Background To the Problemmentioning

confidence: 99%

A Rigorous ODE Solver and Smale’s 14th Problem

Tucker

2002

Found. Comput. Math.

399

288

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Abstract. We present an algorithm for computing rigorous solutions to a large class of ordinary differential equations. The main algorithm is based on a partitioning process and the use of interval arithmetic with directed rounding. As an application, we prove that the Lorenz equations support a strange attractor, as conjectured by Edward Lorenz in 1963. This conjecture was recently listed by Steven Smale as one of several challenging problems for the twenty-first century. We also prove that the attractor is robust, i.e., it persists under small perturbations of the coefficients in the underlying differential equations. Furthermore, the flow of the equations admits a unique SRB measure, whose support coincides with the attractor. The proof is based on a combination of normal form theory and rigorous computations.

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Section: Background To the Problemmentioning

confidence: 99%

A Rigorous ODE Solver and Smale’s 14th Problem

Tucker

2002

Found. Comput. Math.

399

288

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“…Thus the above is indeed the solution of system (9). The next step is to show the uniqueness of the solution.…”

Section: Theoremmentioning

confidence: 84%

“…The next step is to show the uniqueness of the solution. Let assume that we can find a different function F 1 ptq satisfying system (9). Then:…”

Section: Theoremmentioning

confidence: 99%

“…Within the class of mathematical systems of equations describing some chaotic system, we can say that the well-known Lorenz model is perhaps the most classical and exemplary problem since it is, as far we are aware, the primitive model of chaotic behavior [7][8][9][10]. This model is a modified and simplified form of the former complex model of Saltzman to portray buoyancy-driven convection patterns in the classical rectangular Rayleigh-Bernard problem used in thermal convection between two plates perpendicular to the direction of the Earth's gravitational force.…”

Section: Introductionmentioning

confidence: 99%

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Chaos on the Vallis Model for El Niño with Fractional Operators

Alkahtani

Atangana

2016

Entropy

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Abstract:The Vallis model for El Niño is an important model describing a very interesting physical problem. The aim of this paper is to investigate and compare the models using both integer and non-integer order derivatives. We first studied the model with the local derivative by presenting for the first time the exact solution for equilibrium points, and then we presented the exact solutions with the numerical simulations. We further examined the model within the scope of fractional order derivatives. The fractional derivatives used here are the Caputo derivative and Caputo-Fabrizio type. Within the scope of fractional derivatives, we presented the existence and unique solutions of the model. We derive special solutions of both models with Caputo and Caputo-Fabrizio derivatives. Some numerical simulations are presented to compare the models. We obtained more chaotic behavior from the model with Caputo-Fabrizio derivative than other one with local and Caputo derivative. When compare the three models, we realized that, the Caputo derivative plays a role of low band filter when the Caputo-Fabrizio presents more information that were not revealed in the model with local derivative.

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“…M. Eidenschink and Mischaikow [3] are currently developing numerical algorithms for computing connection matrices for systems of ordinary differential equations. This, combined with the work of M. Mrozek and Mischaikow [22,23], provides a technique for rigorously determining the connection matrices for fixed choices of θ. In the language of travelling waves, the connections found via the connection matrix correspond most closely to Fischer waves.…”

Section: Let η Denote the Number Of Negative Eigenvalues Ofmentioning

confidence: 99%

Algebraic transition matrices in the Conley index theory

Franzosa

Mischaikow

1998

Trans. Amer. Math. Soc.

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Abstract. We introduce the concept of an algebraic transition matrix. These are degree zero isomorphisms which are upper triangular with respect to a partial order. It is shown that all connection matrices of a Morse decomposition for which the partial order is a series-parallel admissible order are related via a conjugation with one of these transition matrices. This result is then restated in the form of an existence theorem for global bifurcations. Simple examples of how these results can be applied are also presented.

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Chaos in the Lorenz equations: a computer-assisted proof

Cited by 209 publications

References 7 publications

A Rigorous ODE Solver and Smale’s 14th Problem

A Rigorous ODE Solver and Smale’s 14th Problem

Chaos on the Vallis Model for El Niño with Fractional Operators

Algebraic transition matrices in the Conley index theory

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