-
- 1
-
BASHIR AL-HDAIBAT, WILLY GOVAERTS, YURI A. KUZNETSOV AND HIL G.E. MEIJER, Initialization of homoclinic solutions near Bogdanov-Takens points: Lindstedt-Poincaré compared with regular perturbation method, SIAM J. Appl. Dyn. Syst. 15(2). p.952-980 (2016).
- 2
-
E.L. ALLGOWER AND K. GEORG, Numerical Continuation Methods: An introduction, Springer-Verlag, 1990 .
- 3
-
W.J. BEYN, A. CHAMPNEYS, E. DOEDEL, W. GOVAERTS, YU.A. KUZNETSOV, AND
B. SANDSTEDE, Numerical continuation and computation of normal forms. In:
B. Fiedler, G. Iooss, and N. Kopell (eds.) ``Handbook of Dynamical Systems :
Vol 2", Elsevier 2002, pp 149 - 219.
- 4
-
CHAMPNEYS, A.R. AND KUZNETSOV YU.A. 1994. Numerical detection and continuation of
codimension-two homoclinic orbits. Int. J. Bifurcation Chaos, 4(4), 785-822.
- 5
-
CHAMPNEYS, A.R., KUZNETSOV YU.A. AND SANDSTEDE B. 1996. A numerical toolbox for
homoclinic bifurcation analysis. Int. J. Bifurcation Chaos, 6(5), 867-887.
- 6
-
C. DE BOOR AND B. SWARTZ, Collocation at Gaussian points, SIAM Journal on Numerical Analysis 10 (1973), pp. 582-606.
- 7
-
DEMMEL, J.W., DIECI, L. AND FRIEDMAN, M.J. 2001. Computing connecting orbits via an
improved algorithm for continuing invariant subspaces. SIAM J. Sci. Comput.,
22(1), 81-94.
- 8
-
V. DE WITTE, W. GOVAERTS, YU. A. KUZNETSOV AND M. FRIEDMAN, Interactive Initialization and Continuation of Homoclinic and Heteroclinic Orbits in MATLAB, ACM Transactions on Mathematical Software. Volume 38, Issue 3, Article Number: 18, DOI: 10.1145/2168773.2168776 Published: APR 2012
- 9
-
V. DE WITTE, F. DELLA ROSSA, W.GOVAERTS AND YU.A. KUZNETSOV, Numerical Periodic Normalization for Codim2 Bifurcations of Limit Cycles: Computational Formulas, Numerical Implementation, and Examples, SIAM J. Applied Dynamical
Systems 12,2 (2013) 722-788.
DOI: 10.1137/120874904
- 10
-
A. DHOOGE, W. GOVAERTS AND YU. A. KUZNETSOV, MATCONT : A MATLAB package for numerical bifurcation
analysis of ODEs, ACM Transactions on Mathematical Software 29(2) (2003), pp. 141-164.
- 11
-
A. DHOOGE, W.GOVAERTS, YU. A. KUZNETSOV, H. G. E. MEIJER AND B. SAUTOIS: New features of the software MatCont
for bifurcation analysis of dynamical systems, Mathematical and Computer Modelling of Dynamical Systems, Vol. 14(2),
pp. 147-175, Published: 2008.
https://doi.org/10.1080/13873950701742754.
- 12
-
E. DOEDEL AND J KERNÉVEZ, AUTO: Software for continuation problems in ordinary differential equations with applications, California Institute of Technology, Applied Mathematics, 1986.
- 13
-
DOEDEL, E.J. AND FRIEDMAN, M.J.:
Numerical computation of heteroclinic orbits,
J. Comp. Appl. Math. 26 (1989) 155-170.
- 14
-
E.J. DOEDEL, A.R. CHAMPNEYS, T.F. FAIRGRIEVE, YU.A. KUZNETSOV, B. SANDSTEDE AND X.J. WANG,
AUTO97-00 :
Continuation and Bifurcation Software for
Ordinary Differential Equations (with HomCont), User's Guide,
Concordia University, Montreal, Canada (1997-2000).
(http://indy.cs.concordia.ca).
- 15
-
DOEDEL, E.J., GOVAERTS W., KUZNETSOV, YU.A.: Computation of Periodic Solution Bifurcations in ODEs using Bordered Systems, SIAM Journal on Numerical Analysis 41,2(2003) 401-435.
- 16
-
DOEDEL, E.J., GOVAERTS, W., KUZNETSOV, YU.A., DHOOGE, A.: Numerical continuation of branch points of equilibria and periodic orbits, Int. J. Bifurcation and Chaos,
15(3) (2005), 841-860.
- 17
-
ERMENTROUT, B.: Simulating, Analyzing, and Animating Dynamical Systems.
Siam Publications, Philadelphia, 2002.
- 18
-
FREIRE, E., RODRIGUEZ-LUIS, A., GAMERO E. AND PONCE, E., A case study for homoclinic chaos in an autonomous electronic circuit: A trip form Takens-Bogdanov to Hopf- Shilnikov, Physica D 62 (1993) 230-253.
- 19
-
FRIEDMAN, M., GOVAERTS, W., KUZNETSOV, YU.A. AND SAUTOIS, B. 2005.
Continuation of homoclinic orbits in MATLAB. Lecture Notes in Computer Science, 3514, 263-270.
- 20
-
GENESIO, R. AND TESI, A. Harmonic balance methods for the analysis
of chaotic dynamics in nonlinear systems. Automatica 28 (1992), 531-548.
- 21
-
GENESIO, R., TESI, A., AND VILLORESI, F., Models of complex dynamics in
nonlinear systems. Systems Control Lett. 25 (1995), 185-192.
- 22
-
W.J.F. GOVAERTS, Numerical Methods for Bifurcations of Dynamical Equilibria, SIAM, 2000.
- 23
-
GOVAERTS, W. AND SAUTOIS, B.: Phase response curves, delays and
synchronization in MATLAB. Lecture Notes in Computer Science, 3992 (2006), 391-398.
- 24
-
GOVAERTS, W. AND SAUTOIS, B.: Computation of the phase response curve: a
direct numerical approach. Neural Comput. 18(4) (2006), 817-847.
- 25
-
YU. A. KUZNETSOV, Elements of Applied Bifurcation Theory, Springer-Verlag, 1998. (third edition 2004).
- 26
-
YU. A. KUZNETSOV AND V.V. LEVITIN, CONTENT: Integrated Environment for analysis of dynamical systems. CWI, Amsterdam 1997:
ftp://ftp.cwi.nl/pub/CONTENT
- 27
-
MATLAB, The Mathworks Inc.,
http://www.mathworks.com
.
- 28
-
YU. A. KUZNETSOV, W. GOVAERTS, E.J. DOEDEL AND A. DHOOGE, Numerical periodic normalization for codim 1 bifurcations of limit cycles, SIAM J. Numer. Anal. 43 (2005) 1407-1435.
- 29
-
YU.A. KUZNETSOV, H.G.E. MEIJER, W. GOVAERTS AND B. SAUTOIS, Switching to nonhyperbolic cycles from codim 2 bifurcations of equilibria in ODEs, Physica D 237 No. 23 (2008) 3061-3068 (ISSN 0167-2789).
- 30
-
KUZNETSOV, YU.A., MEIJER H.G.E., AL HDAIBAT, B. AND GOVAERTS, W., Improved homoclinic predictor for Bogdanov-Takens Bifurcation, International Journal of Bifurcations and Chaos, 24(4) (2014). Article Number: 1450057.
DOI: 10.1142/S0218127414500576
- 31
-
YU.A. KUZNETSOV, H.G.E. MEIJER, B. AL-HDAIBAT AND W. GOVAERTS, Accurate Approximation of Homoclinic Solutions in Gray-Scott Kinetic Model. International Journal of Bifurcation and Chaos, Volume: 25(9) August 2015.
Article Number: 1550125
DOI: 10.1142/S0218127415501254
- 32
-
W. MESTROM, Continuation of limit cycles in MATLAB, Master Thesis,
Mathematical Institute, Utrecht University, The Netherlands, 2002.
- 33
-
MORRIS, C., LECAR,H., Voltage oscillations in the barnacle giant muscle fiber,Biophys J. 35 (1981) 193-213.
- 34
-
N. NEIRYNCK, Advances in numerical bifurcation software: MatCont. PhD thesis, Ghent University, Belgium 2019.
https://biblio.ugent.be/publication/8615817.
- 35
-
A. RIET, A Continuation Toolbox in MATLAB, Master Thesis, Mathematical
Institute, Utrecht University, The Netherlands, 2000.
- 36
-
C. STÉPHANOS, Sur une extension du calcul des substitutions linéaires, J. Math. Pures
Appl. 6 (1900) 73-128.
- 37
-
TERMAN, D., Chaotic spikes arising from a model of bursting in excitable membranes, Siam J. Appl. Math. 51 (1991) 1418-1450.
- 38
-
TERMAN, D., The transition from bursting to continuous spiking in excitable membrane models, J. Nonlinear Sci. 2, (1992) 135-182.