Continuation of connecting orbits in 3d-ODEs. (ii) cycle-to-cycle connections.

E.J. Doedel, B.W. Kooi, G.A.K. van Voorn, Y.A. Kuznetzov

Research output: Contribution to JournalArticleAcademicpeer-review

Abstract

In Part I of this paper we have discussed new methods for the numerical continuation of point-to-cycle connecting orbits in three-dimensional autonomous ODE's using projection boundary conditions. In this second part we extend the method to the numerical continuation of cycle-to-cycle connecting orbits. In our approach, the projection boundary conditions near the cycles are formulated using eigenfunctions of the associated adjoint variational equations, avoiding costly and numerically unstable computations of the monodromy matrices. The equations for the eigenfunctions are included in the defining boundary-value problem, allowing a straightforward implementation in AUTO, in which only the standard features of the software are employed. Homotopy methods to find the connecting orbits are discussed in general and illustrated with an example from population dynamics. Complete AUTO demos, which can be easily adapted to any autonomous three-dimensional ODE system, are freely available. © 2009 World Scientific Publishing Company.
Original languageEnglish
Pages (from-to)159-169
JournalInternational Journal of Bifurcation and Chaos
Volume19
DOIs
Publication statusPublished - 2009

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Connecting Orbits
Continuation
Orbits
Numerical Continuation
Cycle
Eigenvalues and eigenfunctions
Boundary conditions
Eigenfunctions
Population dynamics
Projection
Homotopy Method
Three-dimensional
Boundary value problems
Variational Equation
Adjoint Equation
Monodromy
Population Dynamics
Unstable
Boundary Value Problem
Software

Cite this

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title = "Continuation of connecting orbits in 3d-ODEs. (ii) cycle-to-cycle connections.",
abstract = "In Part I of this paper we have discussed new methods for the numerical continuation of point-to-cycle connecting orbits in three-dimensional autonomous ODE's using projection boundary conditions. In this second part we extend the method to the numerical continuation of cycle-to-cycle connecting orbits. In our approach, the projection boundary conditions near the cycles are formulated using eigenfunctions of the associated adjoint variational equations, avoiding costly and numerically unstable computations of the monodromy matrices. The equations for the eigenfunctions are included in the defining boundary-value problem, allowing a straightforward implementation in AUTO, in which only the standard features of the software are employed. Homotopy methods to find the connecting orbits are discussed in general and illustrated with an example from population dynamics. Complete AUTO demos, which can be easily adapted to any autonomous three-dimensional ODE system, are freely available. {\circledC} 2009 World Scientific Publishing Company.",
author = "E.J. Doedel and B.W. Kooi and {van Voorn}, G.A.K. and Y.A. Kuznetzov",
year = "2009",
doi = "10.1142/S0218127409022804",
language = "English",
volume = "19",
pages = "159--169",
journal = "International Journal of Bifurcation and Chaos",
issn = "0218-1274",
publisher = "World Scientific Publishing Co. Pte Ltd",

}

Continuation of connecting orbits in 3d-ODEs. (ii) cycle-to-cycle connections. / Doedel, E.J.; Kooi, B.W.; van Voorn, G.A.K.; Kuznetzov, Y.A.

In: International Journal of Bifurcation and Chaos, Vol. 19, 2009, p. 159-169.

Research output: Contribution to JournalArticleAcademicpeer-review

TY - JOUR

T1 - Continuation of connecting orbits in 3d-ODEs. (ii) cycle-to-cycle connections.

AU - Doedel, E.J.

AU - Kooi, B.W.

AU - van Voorn, G.A.K.

AU - Kuznetzov, Y.A.

PY - 2009

Y1 - 2009

N2 - In Part I of this paper we have discussed new methods for the numerical continuation of point-to-cycle connecting orbits in three-dimensional autonomous ODE's using projection boundary conditions. In this second part we extend the method to the numerical continuation of cycle-to-cycle connecting orbits. In our approach, the projection boundary conditions near the cycles are formulated using eigenfunctions of the associated adjoint variational equations, avoiding costly and numerically unstable computations of the monodromy matrices. The equations for the eigenfunctions are included in the defining boundary-value problem, allowing a straightforward implementation in AUTO, in which only the standard features of the software are employed. Homotopy methods to find the connecting orbits are discussed in general and illustrated with an example from population dynamics. Complete AUTO demos, which can be easily adapted to any autonomous three-dimensional ODE system, are freely available. © 2009 World Scientific Publishing Company.

AB - In Part I of this paper we have discussed new methods for the numerical continuation of point-to-cycle connecting orbits in three-dimensional autonomous ODE's using projection boundary conditions. In this second part we extend the method to the numerical continuation of cycle-to-cycle connecting orbits. In our approach, the projection boundary conditions near the cycles are formulated using eigenfunctions of the associated adjoint variational equations, avoiding costly and numerically unstable computations of the monodromy matrices. The equations for the eigenfunctions are included in the defining boundary-value problem, allowing a straightforward implementation in AUTO, in which only the standard features of the software are employed. Homotopy methods to find the connecting orbits are discussed in general and illustrated with an example from population dynamics. Complete AUTO demos, which can be easily adapted to any autonomous three-dimensional ODE system, are freely available. © 2009 World Scientific Publishing Company.

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DO - 10.1142/S0218127409022804

M3 - Article

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SP - 159

EP - 169

JO - International Journal of Bifurcation and Chaos

JF - International Journal of Bifurcation and Chaos

SN - 0218-1274

ER -