A planar model system for the saddle-node Hopf bifurcation with global reinjection

Nonlinearity

Volumn 17, Issue 4, 2004, Pages 1119-1151

  Krauskopf, Bernd ^a   Oldeman, Bart E ^a  

^a UNIVERSITY OF BRISTOL   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 3142771892     PISSN: 09517715     EISSN: None     Source Type: Journal    
DOI: 10.1088/0951-7715/17/4/001     Document Type: Article

References (43)

1. Ashwin P, Rucklidge A M and Sturman R 2003 Two-state intermittency near a symmetric interaction of saddle-node and Hopf bifurcations: a case study from dynamo theory Physica D at press (Preprint http://www.maths.leeds.ac.uk/ ~rsturman/ABSTRACTS/ARS_snhopf_abs.html)
2. Back A, Guckenheimer J, Myers M, Wicklin F and Worfolk P 1992 DsTool: computer assisted exploration of dynamical systems Not. Am. Math. Soc. 39 303-9
3. Blows T R and Rousseau C 1993 Bifurcation at infinity in polynomial vector fields J. Diff. Eqns 104 215-42
4. Broer H W and Vegter G 1992 Bifurcational aspects of parametric resonance Dynamics Reported, New Series vol 1 (Berlin: Springer) pp 1-51
5. Champneys A R and Kirk V 2003 The entwined wiggling of homoclinic curves emerging from saddle-node/Hopf instabilities Physica D at press (Preprint 2003.18 University of Bristol http://www.enm.bris.ac.uk/anm/preprints/2003rl8. html)
6. Champneys A R and Kuznetsov Yu A 1994 Numerical detection and continuation of codimension-two homoclinic bifurcations Int. J. Bifurc. Chios 4 785-822
7. Champneys A R, Kuznetsov Yu A and Sandstede B 1996 A numerical toolbox for homoclinic bifurcation analysis Int. J. Bifurc. Chaos 6 867-87
8. Chow S N, Deng B and Fiedler B 1990 Homoclinic bifurcation at resonant eigenvalues J. Dyn. Diff. Eqns 2 177-244
9. Chow S N, Li C and Wang D 1994 Normal Forms and Bifurcation of Planar Vector Fields (Cambridge: Cambridge University Press)
10. Chow S N and Lin X B 1990 Bifurcation of a homoclinic orbit with a saddle-node equilibrium Diff. Integ. Eqns 3 435-66
11. Crooks C 2000 Nonlinear dynamics of an optically injected diode laser Master's Thesis University of Bristol
12. De Jagher P C, Van der Graaf W A and Lenstra D 1996 Relaxation- oscillation phenomena in an injection locked semiconductor laser Quantum Semiclass. Opt. 8 805-22
13. Doedel E J, Paffenroth R C, Champneys A R, Fairgrieve T F, Kuznetsov Yu A, Sandstede B and Wang X 2001 AUTO 2000: Continuation and Bifurcation Software for Ordinary Differential Equations (with HomCont) http://sourceforge.net/ projects/auto2000/
14. Gaspard P 1993 Local birth of homoclinic chaos Physica D 62 94-122
15. Gavrilov N K 1978 On some bifurcations of equilibria with a zero and a pair of purely imaginary eigenvalues Methods of the Qualitative Theory of Differential Equations ed E A Leontovich-Adronova (Gorr'kij: Gor'kov Gos. University) pp 33-40 (in Russian)
16. Gavrilov N K 1985 On some bifurcations of equilibria of codimension two of divergence-free vector fields Methods of the Qualitative Theory of Differential Equations (Gorr'kij: Gor'kov Gos. University) pp 46-55 (in Russian)
17. Gavrilov N K 1987 On bifurcations of equilibria with a zero and a pair of purely imaginary eigenvalues with an additional degeneracy Methods of the Qualitative Theory of Differential Equations (Gorr'kij: Gor'kov Gos. University) pp 43-51 (in Russian)
18. Guckenheimer J 1981 On a codimension two bifurcation Dynamical Systems and Turbulence (Springer Lecture Notes in Mathematics vol 998) ed D A Rand and L S Young (Berlin: Springer) pp 99-142
19. Guckenheimer J and Holmes P 1983 Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (Berlin: Springer)
20. Kirk V 1991 Breaking of symmetry in the saddle-node Hopf bifurcation Phys. Lett. A 154 243-8
21. Knobloch E and Moore D 1990 Minimal model of binary fluid convection Phys. Rev. A 42 4693-709
22. Krauskopf B 1997 Bifurcations at ∞ in a model for 1:4 resonance Ergod. Theory Dynam. Syst. 17 899-931
23. Krauskopf B and Rousseau C 1997 Codimension-three unfoldings of reflectionally symmetric planar vector fields Nonlinearity 10 1115-50
24. Krauskopf B, Tollenaar N and Lenstra D 1998 Tori and their bifurcations in an optically injected semiconductor laser Opt. Commun. 156 158-69
25. Krauskopf B, Van der Graaf W A and Lenstra D 1997 Bifurcations of relaxation oscillations in an optically injected diode laser Quantum Semiclass. Opt. 9 797-809
26. Krauskopf B and Wieczorek S M 2002 Accumulating regions of winding periodic orbits in optically driven lasers Physica D 173 97-113
27. Kuznetsov Yu A 1998 Elements of Applied Bifurcation Theory 2nd edn (Berlin: Springer)
28. Lukyanov V 1982 Bifurcations of dynamical systems with 'saddle-node' separatrix loops Diff. Eqns 18 1049-59
29. Moroz I 1990 Multiple instabilities in rotating convection Geophys. Astrophys. Fluid Dyn. 53 183-204
30. Mullin T, Tavener S J and Cliffe K A 1989 An experimental and numerical study of a codimension-2 bifurcation in a rotating annulus Europhys. Lett. 8 251-6
31. Nozdrachova V 1982 Bifurcation of a noncourse separatrix loop Diff. Eqns 18 1098-104
32. Shil'nikov A, Nicolis G and Nicolis C 1995 Bifurcation and predictability analysis of a low-order atmospheric circulation mode Int. J. Bifurc. Chaos 5 1701-11
33. Simpson T B 2003 Mapping the nonlinear dynamics of a distributed feedback laser subject to external optical injection Opt. Commun. 215 135-51
34. Sneyd J, LeBeau A and Yule D 2000 Travelling waves of calcium in pancreatic acinar cells: model construction and bifurcation analysis Physica D 145 158-79
35. Solari H G and Oppo G L 1994 Laser with injected signal: perturbation of an invariant circle Opt. Commun. 111 173-90
36. Van Tartwijk G H M and Agrawal G P 1998 Laser instabilities: a modern perspective Prog. Quantum Electron. 22 43-122
37. Van Tartwijk G H M and Lenstra D 1995 Semiconductor lasers with optical injection and feedback Quantum Semiclass. Opt. 7 87-143
38. Vitolo R 2003 Bifurcations of attractors in 3D diffeomorphisms: a study in experimental mathematics PhD Thesis University of Groningen
39. Wieczorek S M and Krauskopf B 2003 Bifurcations of n-homoclinic orbits in optically injected lasers Applied Nonlinear Mathematics Preprint 2003.14, University of Bristol http://www.enm.bris.ac.uk/anm/preprints/2003r14.html
40. Wieczorek S M, Krauskopf B and Lenstra D 1999 A unifying view of bifurcations in a semiconductor laser subject to optical injection Opt. Commun. 172 279-95
41. Wieczorek S M, Simpson T B, Krauskopf B and Lenstra D 2003 Bifurcation transitions in an optically injected diode laser: theory and experiment Opt. Commun. 215 125-34
42. Yeung M K S and Strogatz S H 1998 Nonlinear dynamics of a solid-state laser with injection Phys. Rev. E 58 4421-35
43. Zimmermann M, Natiello M and Solari H 1997 Shil'nikov-saddle-node interaction near a codimension 2 bifurcation: laser with injected signal Physica D 109 293-314