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Interactive initialization and continuation of homoclinic and heteroclinic orbis in MATLAB

Virginie De Witte, Willy Govaerts (UGent) , Yuri Kuznetsov and Mark Friedman
(2012) ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE. 38(3).
Author
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Abstract
MATCONT is a MATLAB continuation package for the interactive numerical study of a range of parameterized nonlinear dynamical systems, in particular ODEs, that allows to compute curves of equilibria, limit points, Hopf points, limit cycles, flip, fold and torus bifurcation points of limit cycles. It is now possible to continue homoclinic-to-hyperbolic-saddle and homoclinic-to-saddle-node orbits in MATCONT. The implementation is done using the continuation of invariant subspaces, with the Riccati equations included in the defining system. A key feature is the possibility to initiate both types of homoclinic orbits interactively, starting from an equilibrium point and using a homotopy method. All known codimension-two homoclinic bifurcations are tested for during continuation. The test functions for inclination-flip bifurcations are implemented in a new and more efficient way. Heteroclinic orbits can now also be continued and an analogous homotopy method can be used for the initialization.
Keywords
Global bifurcation, homotopy, numerical continuation, LOCATING CONNECTING ORBITS, INVARIANT SUBSPACES, NUMERICAL COMPUTATION, BIFURCATION-ANALYSIS

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MLA
De Witte, Virginie, et al. “Interactive Initialization and Continuation of Homoclinic and Heteroclinic Orbis in MATLAB.” ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE, vol. 38, no. 3, 2012, doi:10.1145/2168773.2168776.
APA
De Witte, V., Govaerts, W., Kuznetsov, Y., & Friedman, M. (2012). Interactive initialization and continuation of homoclinic and heteroclinic orbis in MATLAB. ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE, 38(3). https://doi.org/10.1145/2168773.2168776
Chicago author-date
De Witte, Virginie, Willy Govaerts, Yuri Kuznetsov, and Mark Friedman. 2012. “Interactive Initialization and Continuation of Homoclinic and Heteroclinic Orbis in MATLAB.” ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE 38 (3). https://doi.org/10.1145/2168773.2168776.
Chicago author-date (all authors)
De Witte, Virginie, Willy Govaerts, Yuri Kuznetsov, and Mark Friedman. 2012. “Interactive Initialization and Continuation of Homoclinic and Heteroclinic Orbis in MATLAB.” ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE 38 (3). doi:10.1145/2168773.2168776.
Vancouver
1.
De Witte V, Govaerts W, Kuznetsov Y, Friedman M. Interactive initialization and continuation of homoclinic and heteroclinic orbis in MATLAB. ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE. 2012;38(3).
IEEE
[1]
V. De Witte, W. Govaerts, Y. Kuznetsov, and M. Friedman, “Interactive initialization and continuation of homoclinic and heteroclinic orbis in MATLAB,” ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE, vol. 38, no. 3, 2012.
@article{2129354,
  abstract     = {{MATCONT is a MATLAB continuation package for the interactive numerical study of a range of parameterized nonlinear dynamical systems, in particular ODEs, that allows to compute curves of equilibria, limit points, Hopf points, limit cycles, flip, fold and torus bifurcation points of limit cycles. It is now possible to continue homoclinic-to-hyperbolic-saddle and homoclinic-to-saddle-node orbits in MATCONT. The implementation is done using the continuation of invariant subspaces, with the Riccati equations included in the defining system. A key feature is the possibility to initiate both types of homoclinic orbits interactively, starting from an equilibrium point and using a homotopy method. All known codimension-two homoclinic bifurcations are tested for during continuation. The test functions for inclination-flip bifurcations are implemented in a new and more efficient way. Heteroclinic orbits can now also be continued and an analogous homotopy method can be used for the initialization.}},
  articleno    = {{18}},
  author       = {{De Witte, Virginie and Govaerts, Willy and Kuznetsov, Yuri and Friedman, Mark}},
  issn         = {{0098-3500}},
  journal      = {{ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE}},
  keywords     = {{Global bifurcation,homotopy,numerical continuation,LOCATING CONNECTING ORBITS,INVARIANT SUBSPACES,NUMERICAL COMPUTATION,BIFURCATION-ANALYSIS}},
  language     = {{eng}},
  number       = {{3}},
  pages        = {{34}},
  title        = {{Interactive initialization and continuation of homoclinic and heteroclinic orbis in MATLAB}},
  url          = {{http://dx.doi.org/10.1145/2168773.2168776}},
  volume       = {{38}},
  year         = {{2012}},
}
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