---
_id: '16557'
abstract:
- lang: eng
  text: <jats:p> We combine the techniques of almost invariant sets (using tree structured
    box elimination and graph partitioning algorithms) with invariant manifold and
    lobe dynamics techniques. The result is a new computational technique for computing
    key dynamical features, including almost invariant sets, resonance regions as
    well as transport rates and bottlenecks between regions in dynamical systems.
    This methodology can be applied to a variety of multibody problems, including
    those in molecular modeling, chemical reaction rates and dynamical astronomy.
    In this paper we focus on problems in dynamical astronomy to illustrate the power
    of the combination of these different numerical tools and their applicability.
    In particular, we compute transport rates between two resonance regions for the
    three-body system consisting of the Sun, Jupiter and a third body (such as an
    asteroid). These resonance regions are appropriate for certain comets and asteroids.
    </jats:p>
author:
- first_name: Michael
  full_name: Dellnitz, Michael
  last_name: Dellnitz
- first_name: Oliver
  full_name: Junge, Oliver
  last_name: Junge
- first_name: Wang Sang
  full_name: Koon, Wang Sang
  last_name: Koon
- first_name: Francois
  full_name: Lekien, Francois
  last_name: Lekien
- first_name: Martin W.
  full_name: Lo, Martin W.
  last_name: Lo
- first_name: Jerrold E.
  full_name: Marsden, Jerrold E.
  last_name: Marsden
- first_name: Kathrin
  full_name: Padberg, Kathrin
  last_name: Padberg
- first_name: Robert
  full_name: Preis, Robert
  last_name: Preis
- first_name: Shane D.
  full_name: Ross, Shane D.
  last_name: Ross
- first_name: Bianca
  full_name: Thiere, Bianca
  last_name: Thiere
citation:
  ama: Dellnitz M, Junge O, Koon WS, et al. Transport in Dynamical Astronomy and Multibody
    Problems. <i>International Journal of Bifurcation and Chaos</i>. 2005:699-727.
    doi:<a href="https://doi.org/10.1142/s0218127405012545">10.1142/s0218127405012545</a>
  apa: Dellnitz, M., Junge, O., Koon, W. S., Lekien, F., Lo, M. W., Marsden, J. E.,
    … Thiere, B. (2005). Transport in Dynamical Astronomy and Multibody Problems.
    <i>International Journal of Bifurcation and Chaos</i>, 699–727. <a href="https://doi.org/10.1142/s0218127405012545">https://doi.org/10.1142/s0218127405012545</a>
  bibtex: '@article{Dellnitz_Junge_Koon_Lekien_Lo_Marsden_Padberg_Preis_Ross_Thiere_2005,
    title={Transport in Dynamical Astronomy and Multibody Problems}, DOI={<a href="https://doi.org/10.1142/s0218127405012545">10.1142/s0218127405012545</a>},
    journal={International Journal of Bifurcation and Chaos}, author={Dellnitz, Michael
    and Junge, Oliver and Koon, Wang Sang and Lekien, Francois and Lo, Martin W. and
    Marsden, Jerrold E. and Padberg, Kathrin and Preis, Robert and Ross, Shane D.
    and Thiere, Bianca}, year={2005}, pages={699–727} }'
  chicago: Dellnitz, Michael, Oliver Junge, Wang Sang Koon, Francois Lekien, Martin
    W. Lo, Jerrold E. Marsden, Kathrin Padberg, Robert Preis, Shane D. Ross, and Bianca
    Thiere. “Transport in Dynamical Astronomy and Multibody Problems.” <i>International
    Journal of Bifurcation and Chaos</i>, 2005, 699–727. <a href="https://doi.org/10.1142/s0218127405012545">https://doi.org/10.1142/s0218127405012545</a>.
  ieee: M. Dellnitz <i>et al.</i>, “Transport in Dynamical Astronomy and Multibody
    Problems,” <i>International Journal of Bifurcation and Chaos</i>, pp. 699–727,
    2005.
  mla: Dellnitz, Michael, et al. “Transport in Dynamical Astronomy and Multibody Problems.”
    <i>International Journal of Bifurcation and Chaos</i>, 2005, pp. 699–727, doi:<a
    href="https://doi.org/10.1142/s0218127405012545">10.1142/s0218127405012545</a>.
  short: M. Dellnitz, O. Junge, W.S. Koon, F. Lekien, M.W. Lo, J.E. Marsden, K. Padberg,
    R. Preis, S.D. Ross, B. Thiere, International Journal of Bifurcation and Chaos
    (2005) 699–727.
date_created: 2020-04-15T09:15:58Z
date_updated: 2022-01-06T06:52:52Z
department:
- _id: '101'
doi: 10.1142/s0218127405012545
language:
- iso: eng
page: 699-727
publication: International Journal of Bifurcation and Chaos
publication_identifier:
  issn:
  - 0218-1274
  - 1793-6551
publication_status: published
status: public
title: Transport in Dynamical Astronomy and Multibody Problems
type: journal_article
user_id: '15701'
year: '2005'
...
---
_id: '16627'
abstract:
- lang: eng
  text: <jats:p> The computation of global invariant manifolds has seen renewed interest
    in recent years. We survey different approaches for computing a global stable
    or unstable manifold of a vector field, where we concentrate on the case of a
    two-dimensional manifold. All methods are illustrated with the same example —
    the two-dimensional stable manifold of the origin in the Lorenz system. </jats:p>
author:
- first_name: B.
  full_name: Krauskopf, B.
  last_name: Krauskopf
- first_name: H. M.
  full_name: Osinga, H. M.
  last_name: Osinga
- first_name: E. J.
  full_name: Doedel, E. J.
  last_name: Doedel
- first_name: M. E.
  full_name: Henderson, M. E.
  last_name: Henderson
- first_name: J.
  full_name: Guckenheimer, J.
  last_name: Guckenheimer
- first_name: A.
  full_name: Vladimirsky, A.
  last_name: Vladimirsky
- first_name: M.
  full_name: Dellnitz, M.
  last_name: Dellnitz
- first_name: O.
  full_name: Junge, O.
  last_name: Junge
citation:
  ama: Krauskopf B, Osinga HM, Doedel EJ, et al. A Survey of Methods for Computing
    (un)stable Manifolds of Vector Fields. <i>International Journal of Bifurcation
    and Chaos</i>. 2005:763-791. doi:<a href="https://doi.org/10.1142/s0218127405012533">10.1142/s0218127405012533</a>
  apa: Krauskopf, B., Osinga, H. M., Doedel, E. J., Henderson, M. E., Guckenheimer,
    J., Vladimirsky, A., … Junge, O. (2005). A Survey of Methods for Computing (un)stable
    Manifolds of Vector Fields. <i>International Journal of Bifurcation and Chaos</i>,
    763–791. <a href="https://doi.org/10.1142/s0218127405012533">https://doi.org/10.1142/s0218127405012533</a>
  bibtex: '@article{Krauskopf_Osinga_Doedel_Henderson_Guckenheimer_Vladimirsky_Dellnitz_Junge_2005,
    title={A Survey of Methods for Computing (un)stable Manifolds of Vector Fields},
    DOI={<a href="https://doi.org/10.1142/s0218127405012533">10.1142/s0218127405012533</a>},
    journal={International Journal of Bifurcation and Chaos}, author={Krauskopf, B.
    and Osinga, H. M. and Doedel, E. J. and Henderson, M. E. and Guckenheimer, J.
    and Vladimirsky, A. and Dellnitz, M. and Junge, O.}, year={2005}, pages={763–791}
    }'
  chicago: Krauskopf, B., H. M. Osinga, E. J. Doedel, M. E. Henderson, J. Guckenheimer,
    A. Vladimirsky, M. Dellnitz, and O. Junge. “A Survey of Methods for Computing
    (Un)Stable Manifolds of Vector Fields.” <i>International Journal of Bifurcation
    and Chaos</i>, 2005, 763–91. <a href="https://doi.org/10.1142/s0218127405012533">https://doi.org/10.1142/s0218127405012533</a>.
  ieee: B. Krauskopf <i>et al.</i>, “A Survey of Methods for Computing (un)stable
    Manifolds of Vector Fields,” <i>International Journal of Bifurcation and Chaos</i>,
    pp. 763–791, 2005.
  mla: Krauskopf, B., et al. “A Survey of Methods for Computing (Un)Stable Manifolds
    of Vector Fields.” <i>International Journal of Bifurcation and Chaos</i>, 2005,
    pp. 763–91, doi:<a href="https://doi.org/10.1142/s0218127405012533">10.1142/s0218127405012533</a>.
  short: B. Krauskopf, H.M. Osinga, E.J. Doedel, M.E. Henderson, J. Guckenheimer,
    A. Vladimirsky, M. Dellnitz, O. Junge, International Journal of Bifurcation and
    Chaos (2005) 763–791.
date_created: 2020-04-16T08:21:52Z
date_updated: 2022-01-06T06:52:53Z
department:
- _id: '101'
doi: 10.1142/s0218127405012533
language:
- iso: eng
page: 763-791
publication: International Journal of Bifurcation and Chaos
publication_identifier:
  issn:
  - 0218-1274
  - 1793-6551
publication_status: published
status: public
title: A Survey of Methods for Computing (un)stable Manifolds of Vector Fields
type: journal_article
user_id: '15701'
year: '2005'
...
---
_id: '16535'
abstract:
- lang: eng
  text: <jats:p> Recently multilevel subdivision techniques have been introduced in
    the numerical investigation of complicated dynamical behavior. We illustrate the
    applicability and efficiency of these methods by a detailed numerical study of
    Chua's circuit. In particular we will show that there exist two regions in phase
    space which are almost invariant in the sense that typical trajectories stay inside
    each of these sets on average for quite a long time. </jats:p>
author:
- first_name: Michael
  full_name: Dellnitz, Michael
  last_name: Dellnitz
- first_name: Oliver
  full_name: Junge, Oliver
  last_name: Junge
citation:
  ama: Dellnitz M, Junge O. Almost Invariant Sets in Chua’s Circuit. <i>International
    Journal of Bifurcation and Chaos</i>. 1997:2475-2485. doi:<a href="https://doi.org/10.1142/s0218127497001655">10.1142/s0218127497001655</a>
  apa: Dellnitz, M., &#38; Junge, O. (1997). Almost Invariant Sets in Chua’s Circuit.
    <i>International Journal of Bifurcation and Chaos</i>, 2475–2485. <a href="https://doi.org/10.1142/s0218127497001655">https://doi.org/10.1142/s0218127497001655</a>
  bibtex: '@article{Dellnitz_Junge_1997, title={Almost Invariant Sets in Chua’s Circuit},
    DOI={<a href="https://doi.org/10.1142/s0218127497001655">10.1142/s0218127497001655</a>},
    journal={International Journal of Bifurcation and Chaos}, author={Dellnitz, Michael
    and Junge, Oliver}, year={1997}, pages={2475–2485} }'
  chicago: Dellnitz, Michael, and Oliver Junge. “Almost Invariant Sets in Chua’s Circuit.”
    <i>International Journal of Bifurcation and Chaos</i>, 1997, 2475–85. <a href="https://doi.org/10.1142/s0218127497001655">https://doi.org/10.1142/s0218127497001655</a>.
  ieee: M. Dellnitz and O. Junge, “Almost Invariant Sets in Chua’s Circuit,” <i>International
    Journal of Bifurcation and Chaos</i>, pp. 2475–2485, 1997.
  mla: Dellnitz, Michael, and Oliver Junge. “Almost Invariant Sets in Chua’s Circuit.”
    <i>International Journal of Bifurcation and Chaos</i>, 1997, pp. 2475–85, doi:<a
    href="https://doi.org/10.1142/s0218127497001655">10.1142/s0218127497001655</a>.
  short: M. Dellnitz, O. Junge, International Journal of Bifurcation and Chaos (1997)
    2475–2485.
date_created: 2020-04-15T08:31:50Z
date_updated: 2022-01-06T06:52:52Z
department:
- _id: '101'
doi: 10.1142/s0218127497001655
language:
- iso: eng
page: 2475-2485
publication: International Journal of Bifurcation and Chaos
publication_identifier:
  issn:
  - 0218-1274
  - 1793-6551
publication_status: published
status: public
title: Almost Invariant Sets in Chua's Circuit
type: journal_article
user_id: '15701'
year: '1997'
...
---
_id: '16510'
abstract:
- lang: eng
  text: <jats:p> In an array of coupled oscillators, synchronous chaos may occur in
    the sense that all the oscillators behave identically although the corresponding
    motion is chaotic. When a parameter is varied this fully symmetric dynamical state
    can lose its stability, and the main purpose of this paper is to investigate which
    type of dynamical behavior is expected to be observed once the loss of stability
    has occurred. The essential tool is a classification of Lyapunov exponents based
    on the symmetry of the underlying problem. This classification is crucial in the
    derivation of the analytical results but it also allows an efficient computation
    of the dominant Lyapunov exponent associated with each symmetry type. We show
    how these dominant exponents determine the stability of invariant sets possessing
    various instantaneous symmetries, and this leads to the idea of symmetry breaking
    bifurcations of chaotic attractors. Finally, the results and ideas are illustrated
    for several systems of coupled oscillators. </jats:p>
author:
- first_name: Philip J.
  full_name: Aston, Philip J.
  last_name: Aston
- first_name: Michael
  full_name: Dellnitz, Michael
  last_name: Dellnitz
citation:
  ama: Aston PJ, Dellnitz M. Symmetry Breaking Bifurcations of Chaotic Attractors.
    <i>International Journal of Bifurcation and Chaos</i>. 1995:1643-1676. doi:<a
    href="https://doi.org/10.1142/s021812749500123x">10.1142/s021812749500123x</a>
  apa: Aston, P. J., &#38; Dellnitz, M. (1995). Symmetry Breaking Bifurcations of
    Chaotic Attractors. <i>International Journal of Bifurcation and Chaos</i>, 1643–1676.
    <a href="https://doi.org/10.1142/s021812749500123x">https://doi.org/10.1142/s021812749500123x</a>
  bibtex: '@article{Aston_Dellnitz_1995, title={Symmetry Breaking Bifurcations of
    Chaotic Attractors}, DOI={<a href="https://doi.org/10.1142/s021812749500123x">10.1142/s021812749500123x</a>},
    journal={International Journal of Bifurcation and Chaos}, author={Aston, Philip
    J. and Dellnitz, Michael}, year={1995}, pages={1643–1676} }'
  chicago: Aston, Philip J., and Michael Dellnitz. “Symmetry Breaking Bifurcations
    of Chaotic Attractors.” <i>International Journal of Bifurcation and Chaos</i>,
    1995, 1643–76. <a href="https://doi.org/10.1142/s021812749500123x">https://doi.org/10.1142/s021812749500123x</a>.
  ieee: P. J. Aston and M. Dellnitz, “Symmetry Breaking Bifurcations of Chaotic Attractors,”
    <i>International Journal of Bifurcation and Chaos</i>, pp. 1643–1676, 1995.
  mla: Aston, Philip J., and Michael Dellnitz. “Symmetry Breaking Bifurcations of
    Chaotic Attractors.” <i>International Journal of Bifurcation and Chaos</i>, 1995,
    pp. 1643–76, doi:<a href="https://doi.org/10.1142/s021812749500123x">10.1142/s021812749500123x</a>.
  short: P.J. Aston, M. Dellnitz, International Journal of Bifurcation and Chaos (1995)
    1643–1676.
date_created: 2020-04-15T07:23:50Z
date_updated: 2022-01-06T06:52:52Z
department:
- _id: '101'
doi: 10.1142/s021812749500123x
language:
- iso: eng
page: 1643-1676
publication: International Journal of Bifurcation and Chaos
publication_identifier:
  issn:
  - 0218-1274
  - 1793-6551
publication_status: published
status: public
title: Symmetry Breaking Bifurcations of Chaotic Attractors
type: journal_article
user_id: '15701'
year: '1995'
...
---
_id: '16550'
author:
- first_name: Michael
  full_name: Dellnitz, Michael
  last_name: Dellnitz
- first_name: Michael
  full_name: Field, Michael
  last_name: Field
- first_name: Martin
  full_name: Golubitsky, Martin
  last_name: Golubitsky
- first_name: Jun
  full_name: Ma, Jun
  last_name: Ma
- first_name: Andreas
  full_name: Hohmann, Andreas
  last_name: Hohmann
citation:
  ama: Dellnitz M, Field M, Golubitsky M, Ma J, Hohmann A. Cycling Chaos. <i>International
    Journal of Bifurcation and Chaos</i>. 1995:1243-1247. doi:<a href="https://doi.org/10.1142/s0218127495000909">10.1142/s0218127495000909</a>
  apa: Dellnitz, M., Field, M., Golubitsky, M., Ma, J., &#38; Hohmann, A. (1995).
    Cycling Chaos. <i>International Journal of Bifurcation and Chaos</i>, 1243–1247.
    <a href="https://doi.org/10.1142/s0218127495000909">https://doi.org/10.1142/s0218127495000909</a>
  bibtex: '@article{Dellnitz_Field_Golubitsky_Ma_Hohmann_1995, title={Cycling Chaos},
    DOI={<a href="https://doi.org/10.1142/s0218127495000909">10.1142/s0218127495000909</a>},
    journal={International Journal of Bifurcation and Chaos}, author={Dellnitz, Michael
    and Field, Michael and Golubitsky, Martin and Ma, Jun and Hohmann, Andreas}, year={1995},
    pages={1243–1247} }'
  chicago: Dellnitz, Michael, Michael Field, Martin Golubitsky, Jun Ma, and Andreas
    Hohmann. “Cycling Chaos.” <i>International Journal of Bifurcation and Chaos</i>,
    1995, 1243–47. <a href="https://doi.org/10.1142/s0218127495000909">https://doi.org/10.1142/s0218127495000909</a>.
  ieee: M. Dellnitz, M. Field, M. Golubitsky, J. Ma, and A. Hohmann, “Cycling Chaos,”
    <i>International Journal of Bifurcation and Chaos</i>, pp. 1243–1247, 1995.
  mla: Dellnitz, Michael, et al. “Cycling Chaos.” <i>International Journal of Bifurcation
    and Chaos</i>, 1995, pp. 1243–47, doi:<a href="https://doi.org/10.1142/s0218127495000909">10.1142/s0218127495000909</a>.
  short: M. Dellnitz, M. Field, M. Golubitsky, J. Ma, A. Hohmann, International Journal
    of Bifurcation and Chaos (1995) 1243–1247.
date_created: 2020-04-15T09:04:16Z
date_updated: 2022-01-06T06:52:52Z
department:
- _id: '101'
doi: 10.1142/s0218127495000909
language:
- iso: eng
page: 1243-1247
publication: International Journal of Bifurcation and Chaos
publication_identifier:
  issn:
  - 0218-1274
  - 1793-6551
publication_status: published
status: public
title: Cycling Chaos
type: journal_article
user_id: '15701'
year: '1995'
...
---
_id: '16551'
abstract:
- lang: eng
  text: <jats:p> Spiral patterns have been observed experimentally, numerically, and
    theoretically in a variety of systems. It is often believed that these spiral
    wave patterns can occur only in systems of reaction–diffusion equations. We show,
    both theoretically (using Hopf bifurcation techniques) and numerically (using
    both direct simulation and continuation of rotating waves) that spiral wave patterns
    can appear in a single reaction–diffusion equation [ in u(x, t)] on a disk, if
    one assumes "spiral" boundary conditions (u<jats:sub>r</jats:sub> = mu<jats:sub>θ</jats:sub>).
    Spiral boundary conditions are motivated by assuming that a solution is infinitesimally
    an Archimedian spiral near the boundary. It follows from a bifurcation analysis
    that for this form of spirals there are no singularities in the spiral pattern
    (technically there is no spiral tip) and that at bifurcation there is a steep
    gradient between the "red" and "blue" arms of the spiral. </jats:p>
author:
- first_name: Michael
  full_name: Dellnitz, Michael
  last_name: Dellnitz
- first_name: Martin
  full_name: Golubitsky, Martin
  last_name: Golubitsky
- first_name: Andreas
  full_name: Hohmann, Andreas
  last_name: Hohmann
- first_name: Ian
  full_name: Stewart, Ian
  last_name: Stewart
citation:
  ama: Dellnitz M, Golubitsky M, Hohmann A, Stewart I. Spirals in Scalar Reaction–Diffusion
    Equations. <i>International Journal of Bifurcation and Chaos</i>. 1995:1487-1501.
    doi:<a href="https://doi.org/10.1142/s0218127495001149">10.1142/s0218127495001149</a>
  apa: Dellnitz, M., Golubitsky, M., Hohmann, A., &#38; Stewart, I. (1995). Spirals
    in Scalar Reaction–Diffusion Equations. <i>International Journal of Bifurcation
    and Chaos</i>, 1487–1501. <a href="https://doi.org/10.1142/s0218127495001149">https://doi.org/10.1142/s0218127495001149</a>
  bibtex: '@article{Dellnitz_Golubitsky_Hohmann_Stewart_1995, title={Spirals in Scalar
    Reaction–Diffusion Equations}, DOI={<a href="https://doi.org/10.1142/s0218127495001149">10.1142/s0218127495001149</a>},
    journal={International Journal of Bifurcation and Chaos}, author={Dellnitz, Michael
    and Golubitsky, Martin and Hohmann, Andreas and Stewart, Ian}, year={1995}, pages={1487–1501}
    }'
  chicago: Dellnitz, Michael, Martin Golubitsky, Andreas Hohmann, and Ian Stewart.
    “Spirals in Scalar Reaction–Diffusion Equations.” <i>International Journal of
    Bifurcation and Chaos</i>, 1995, 1487–1501. <a href="https://doi.org/10.1142/s0218127495001149">https://doi.org/10.1142/s0218127495001149</a>.
  ieee: M. Dellnitz, M. Golubitsky, A. Hohmann, and I. Stewart, “Spirals in Scalar
    Reaction–Diffusion Equations,” <i>International Journal of Bifurcation and Chaos</i>,
    pp. 1487–1501, 1995.
  mla: Dellnitz, Michael, et al. “Spirals in Scalar Reaction–Diffusion Equations.”
    <i>International Journal of Bifurcation and Chaos</i>, 1995, pp. 1487–501, doi:<a
    href="https://doi.org/10.1142/s0218127495001149">10.1142/s0218127495001149</a>.
  short: M. Dellnitz, M. Golubitsky, A. Hohmann, I. Stewart, International Journal
    of Bifurcation and Chaos (1995) 1487–1501.
date_created: 2020-04-15T09:05:30Z
date_updated: 2022-01-06T06:52:52Z
department:
- _id: '101'
doi: 10.1142/s0218127495001149
language:
- iso: eng
page: 1487-1501
publication: International Journal of Bifurcation and Chaos
publication_identifier:
  issn:
  - 0218-1274
  - 1793-6551
publication_status: published
status: public
title: Spirals in Scalar Reaction–Diffusion Equations
type: journal_article
user_id: '15701'
year: '1995'
...