Analysis of bifurcations of limit cycles with Lyapunov exponents and numerical normal forms

Physica D: Nonlinear Phenomena

Volumn 269, Issue , 2014, Pages 126-141

  De Witte, V ^a   Govaerts, W ^a   Kuznetsov, Yu A ^b,c   Meijer, H G E ^c  

^a GHENT UNIVERSITY   (Belgium)

^b UTRECHT UNIVERSITY   (Netherlands)

^c UNIVERSITY OF TWENTE   (Netherlands)

Author keywords

3 torus; 4 torus; Double Neimark Sacker; Flip Neimark Sacker; Fold Neimark Sacker; Normal form

Indexed keywords

BIFURCATION (MATHEMATICS); BOUNDARY VALUE PROBLEMS; LYAPUNOV FUNCTIONS; PHASE SPACE METHODS; VIBRATIONS (MECHANICAL);

3-TORUS; 4-TORUS; BIFURCATION SCENARIO; LINEAR BOUNDARY VALUE PROBLEM; NEIMARK - SACKERS; NEIMARK-SACKER BIFURCATION; NORMAL FORM; STABLE INVARIANT TORI;

LYAPUNOV METHODS;

EID: 84890911410     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physd.2013.12.002     Document Type: Article

References (32)

1. E.J. Doedel, A.R. Champneys, T.F. Fairgrieve, Yu.A. Kuznetsov, B. Sandstede, X.J. Wang, AUTO97: continuation and bifurcation software for ordinary differential equations (with HomCont), 1997.
2. A. Dhooge, W. Govaerts, and Yu.A. Kuznetsov MATCONT: a MATLAB package for numerical bifurcation analysis of ODEs ACM Trans. Math. Software 29 2 2003 141 164 (Pubitemid 40001448)
3. A. Dhooge, W. Govaerts, Y.A. Kuznetsov, H.G.E. Meijer, and B. Sautois New features of the software MatCont for bifurcation analysis of dynamical systems Math. Comput. Model. Dyn. Syst. 14 2 2008 147 175
4. V.I. Arnol'd Geometrical Methods in the Theory of Ordinary Differential Equations 1983 Springer-Verlag New York
5. G. Iooss Bifurcation of Maps and Applications North-Holland Mathematics Studies vol. 36 1979 North-Holland Pub. Co. Amsterdam
6. J. Guckenheimer, and Ph. Holmes Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields 1983 Springer-Verlag New York
7. Yu.A. Kuznetsov Elements of Applied Bifurcation Theory third ed. 2004 Springer-Verlag New York
8. Yu.A. Kuznetsov, and H.G.E. Meijer Remarks on interacting Neimark-Sacker bifurcations J. Difference Equ. Appl. 12 10 2006 1009 1035 (Pubitemid 44671768)
9. W. Govaerts, R. Khoshsiar Ghaziani, Yu.A. Kuznetsov, and H.G.E. Meijer Numerical methods for two-parameter local bifurcation analysis of maps SIAM J. Sci. Comput. 29 6 2007 2644 2667
10. R. Vitolo, H. Broer, and C. Simó Routes to chaos in the Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms Nonlinearity 23 8 2010 1919 1948
11. C. Simó Analytical and numerical computation of invariant manifolds C. Benest, C. Froeschlé, Modern Methods in Celestial Mechanics 1990 Editions Frontiéres 285 330
12. J. Guckenheimer, and B. Meloon Computing periodic orbits and their bifurcations with automatic differentiation SIAM J. Sci. Comput. 22 3 2000 951 985 (Pubitemid 32548796)
13. Yu.A. Kuznetsov, and H.G.E. Meijer Numerical normal forms for codim 2 bifurcations of fixed points with at most two critical eigenvalues SIAM J. Sci. Comput. 26 6 2005 1932 1954 (Pubitemid 41641357)
14. G. Iooss Global characterization of the normal form for a vector field near a closed orbit J. Differential Equations 76 1 1988 47 76
15. G. Iooss, and M. Adelmeyer Topics in Bifurcation Theory and Applications Advanced Series in Nonlinear Dynamics vol. 3 1992 World Sci. Pub. Co. Inc. River Edge, New York
16. 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 4 2005 1407 1435 (Pubitemid 44231202)
17. V. De Witte, F. Della Rossa, W. Govaerts, and Yu. Kuznetsov Numerical periodic normalization for codim 2 bifurcations of limit cycles - computational formulas, numerical implementation, and examples SIAM J. Appl. Dyn. Syst. 12 2 2013 722 788
18. V. De Witte, W. Govaerts, Y.A. Kuznetsov, H.G.E. Meijer, Numerical periodic normalization for codim 2 bifurcations of limit cycles with center manifold of dimension higher than 3. http://arxiv.org/abs/1210.6205.
19. V. Arnol'd, V. Afrajmovich, Yu. Il'yashenko, and L. Shil'nikov Dynamical Systems. Vol. V Encyclopaedia of Mathematical Sciences vol. 5 1994 Springer-Verlag Berlin Bifurcation Theory and Catastrophe Theory, A Translation of "Current problems in mathematics. Fundamental directions". Vol. 5 (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1986. Translation by N. D. Kazarinoff
20. J. Los Nonnormally hyperbolic invariant curves for maps in R 3 and doubling bifurcation Nonlinearity 2 1 1989 149
21. R. Vitolo, H. Broer, and C. Simó Quasi-periodic bifurcations of invariant circles in low-dimensional dissipative dynamical systems Regul. Chaotic Dyn. 16 1-2 2011 154 184
22. Yu.A. Kuznetsov, and V.V. Levitin CONTENT: A Multiplatform Environment for Analyzing Dynamical Systems 1997 Dynamical Systems Laboratory, CWI Amsterdam
23. C. De Boor, and B. Swartz Collocation at Gaussian points SIAM J. Numer. Anal. 10 4 1973 582 606
24. U.M. Ascher, R.M.M. Mattheij, and R.D. Russell Numerical Solution of Boundary Value Problems for Ordinary Differential Equations Classics in Applied Mathematics vol. 13 1995 Society for Industrial and Applied Mathematics (SIAM) Philadelphia, PA Corrected reprint of the 1988 original
25. S. Wieczorek, and W.W. Chow Self-induced chaos in a single-mode inversionless laser Phys. Rev. Lett. 97 11 2006 113903
26. 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 23 2008 3061 3068
27. M. Rosenzweig Paradox of enrichment: destabilization of exploitation ecosystems in ecological time Science 171 3969 1971 385 387
28. V. Jansen Regulation of predator-prey systems through spatial interactions: a possible solution to the paradox of enrichment Oikos 74 3 1995 384 390 (Pubitemid 26454068)
29. V. Jansen The dynamics of two diffusively coupled predator-prey populations Theor. Popul. Biol. 59 2 2001 119 131 (Pubitemid 34152369)
30. S. Fatimah, and F. Verhulst Suppressing flow-induced vibration by parametric excitation Nonlinear Dynam. 31 2003 275 297
31. Yu.A. Kuznetsov Numerical normalization techniques for all codim 2 bifurcations of equilibria in ODEs SIAM J. Numer. Anal. 36 4 1999 1104 1124 (Pubitemid 129567507)
32. E.J. Doedel, W. Govaerts, and Yu.A. Kuznetsov Computation of periodic solution bifurcations in ODEs using bordered systems SIAM J. Numer. Anal. 41 2 2003 401 435