Unstable periodic orbit detection for ODEs with periodic forcing

Deane, Jonathan H. B.; Marsh, L.

doi:10.1016/j.physleta.2006.06.088

Cited by 10 publications

(3 citation statements)

References 10 publications

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“…The location of UPOs has been an important and a well studied problem by physicists [11][12][13][14][15] and mathematicians using a vast number of numerical algorithms. Obtaining accurate information of UPOs is thus a very interesting task.…”

Section: Introductionmentioning

confidence: 99%

A database of rigorous and high-precision periodic orbits of the Lorenz model

Barrio

Dena

Tucker

2015

Computer Physics Communications

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Section: Introductionmentioning

confidence: 99%

A database of rigorous and high-precision periodic orbits of the Lorenz model

Barrio

Dena

Tucker

2015

Computer Physics Communications

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“…Therefore, it is of great significance to develop efficient numerical methods and techniques for locating periodic orbits. In literature, various numerical algorithms of identifying periodic orbits have been proposed so far [7][8][9][10][11][12][13][14] , among which the best known is probably the Newton-Raphson method.…”

Section: Introductionmentioning

confidence: 99%

A reduced variational approach for searching cycles in high-dimensional systems

Wang¹,

Lan²

2022

Preprint

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Searching recurrent patterns in complex systems with high-dimensional phase spaces is an important task in diverse fields. In the current work, an improved scheme is proposed to accelerate the recently designed variational approach for finding periodic orbits in systems with chaotic dynamics based on the existence of inertial manifold widely observed in various spatially extended systems, especially those with high dimensions. On the premise of keeping exponential convergence of the variational method, an effective loop evolution equation is derived to greatly reduce the storage and computing time. With repeated modification of local coordinates and evolution of the guess loop being carried out alternately, the rapid convergence and the stability of the reduction scheme are effectively achieved. The dimension of local coordiante subspaces is generally larger than the number of nonnegative Lyapunov exponents to ensure the exponential convergence. The proposed scheme is successfully demonstrated on several well-known examples and expected to supply a powerful tool in the exploration of high-dimensional nonlinear systems.

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“…There are several works developing different numerical algorithms to compute periodic orbits [11][12][13][14][15][16][17][18][19] , but most of them do not work with high-precision. This is a serious difficulty if our problem requires obtaining periodic orbits with many precision digits.…”

Section: Introductionmentioning

confidence: 99%

High‐Precision Continuation of Periodic Orbits

Dena

Rodríguez

Serrano

et al. 2012

Abstract and Applied Analysis

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Obtaining periodic orbits of dynamical systems is the main source of information about how the orbits, in general, are organized. In this paper, we extend classical continuation algorithms in order to be able to obtain families of periodic orbits with high-precision. These periodic orbits can be corrected to get them with arbitrary precision. We illustrate the method with two important classical Hamiltonian problems.

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Unstable periodic orbit detection for ODEs with periodic forcing

Cited by 10 publications

References 10 publications

A database of rigorous and high-precision periodic orbits of the Lorenz model

A database of rigorous and high-precision periodic orbits of the Lorenz model

A reduced variational approach for searching cycles in high-dimensional systems

High‐Precision Continuation of Periodic Orbits

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