Nonlinear numerics

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

Volumn 7, Issue 9, 1997, Pages 2127-2143

  Doedel, Eusebius ^a  

^a Gina Cody School of Engineering and Computer Science   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0039835889     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127497001576     Document Type: Article

References (49)

1. Allgower, E. L. & Georg, K. [1997] "Numerical path following," Handbook of Numerical Analysis, Vol. 5, eds. Ciarlet, P. G. & Lions, J. L. (North Holland Publishing).
2. Aronson, D. G., Doedel, E. J. & Othmer, H. G. [1991] "The dynamics of coupled current-biased Josephson junctions - Part II," Int. J. Bifurcation and Chaos 1(1), 51-66.
3. Aronson, D. G., Doedel, E. J. & Terman, D. H. [1997] "A codimension-two point associated with coupled Josephson junctions," to appear in Nonlinearity.
4. N symmetry," Nonlinearity 4, 861-902.
5. Ascher, U. M., Mattheij, R. M. M. & Russell, R. D. [1988] Numerical Solution of Boundary Value Problems for Ordinary Differential Equations (Prentice-Hall).
6. Ascher, U. M. & Spiteri, R. J. [1995] "Collocation software for boundary value differential-algebraic equations," SIAM J. Sci. Comput. 15(4), 938-952.
7. Back, A., Guckenheimer, J., Myers, M. R., Wicklin, F. J. & Worfolk, P. A. [1992] "DsTool: Computer assisted exploration of dynamical systems," Notices Amer. Math. Soc. 39(4), 303-309.
8. Bank, R. E. [1990] "PLTMG, A software package for solving elliptic partial differential equations," SIAM.
9. Beyn, W.-J. [1990] "The numerical computation of connecting orbits in dynamical systems," IMA J. Numer. Anal. 9, 379-405.
10. Broer, H. W., Osinga, H. M. & Vegter, G. [1996] "On the computation of normally hyperbolic invariant manifolds," in Progress in Nonlinear Differential Equations and Their Applications, Vol. 19, eds. Broer, H. W., Hoveijn, I. & Takens, F. (Birkhäuser Verlag), pp. 423-447.
11. Champneys, A. R. & Kuznetsov, Yu. A. [1994] "Numerical detection and continuation of codimension-two homoclinic bifurcations," Int. J. Bifurcation and Chaos 4, 795-822.
12. Champneys, A. R., Kuznetsov, Yu. A. & Sandstede, B. [1995] "An AUTO86 driver for homoclinic bifurcation analysis," Report AM-R9516, CWI Amsterdam.
13. Chow, S.-N. & Hale, J. K. [1982] Methods of Bifurcation Theory (Springer Verlag).
14. de Boor, C. & Swartz, B. [1973] "Collocation at Gaussian points," SIAM J. Numer. Anal. 10, 582-606.
15. Decroly, O. & Goldbeter, A. [1987] "From simple to complex oscillatory behavior: Analysis of bursting in a multiply regulated biochemical system," J. Theor. Biol. 124, 219-250.
16. Dieci, L. & Lorenz, J. [1997] "Lyapunov-type numbers and torus breakdown: Numerical aspects and a case study," Numerical Algorithms 14, 79-102.
17. Doedel, E. J. [1997] "On the construction of discretizations of elliptic partial differential equations," to appear in J. Diff. Equations Appl.
18. Doedel, E. J., Keller, H. B. & Kernévez, J. P. [1991a] "Numerical analysis and control of bifurcation problems - Part I," Int. J. Bifurcation and Chaos 1(3), 493-520.
19. Doedel, E. J., Keller, H. B. & Kernévez, J. P. [1991b] "Numerical analysis and control of bifurcation problems - Part II," Int. J. Bifurcation and Chaos 1(4), 745-772.
20. Doedel, E. J., Wang, X. J. & Fairgrieve, T. F. [1995] "AUTO 94: Software for continuation and bifurcation problems in ordinary differential equations," Technical Report CRPC-95-2, California Institute of Technology.
21. Ermentrout, B. [1995] "XPPAUT1.61 - The differential equations tool," Department of Mathematics, University of Pittsburgh.
22. Fairgrieve, T. F. & Jepson, A. D. [1991] "O.K. Floquet multipliers," SIAM J. Numer. Anal. 28(5), 1446-1462.
23. Fenichel, N. [1971] "Persistence and smoothness of invariant manifolds for flows," Indiana Univ. Math. J. 21(3), 193-226.
24. Friedman, M. & Doedel, E. J. [1991] "Numerical computation and continuation of invariant manifolds connecting fixed points," SIAM J. Numer. Anal. 28, 789-808.
25. Gatermann, K. & Hohmann, A. [1991] "Symbolic exploitation of symmetry in numerical path-following," Impact Comput. Sci. Eng. 3, 330-365.
26. Giannakopoulos, F. Personal communication, Department of Mathematics, University of Cologne, Germany.
27. Govaerts, W. [1997] "Computation of singularities in large nonlinear systems," SIAM J. Numer. Anal. 34(3), 867-880.
28. Hadley, P., Beasley, M. R. & Wiesenfeld, K. [1988] "Phase locking of Josephson-junction series arrays," Phys. Rev. B38, 8712-8719.
29. Henderson, M. [1993] "Computing implicitly defined surfaces: Two parameter continuation," IBM Research Report RC 18777, IBM Thomas Watson Research Center, Yorktown Heights, New York.
30. Isaacson, E. & Keller, H. B. [1966] Analysis of Numerical Methods (Wiley).
31. Kearfott, R. B. & Xing, Z. [1994] "An interval step control for continuation methods," SIAM J. Numer. Anal. 31(3), 892-914.
32. Keller, H. B. [1977] "Numerical solution of bifurcation and nonlinear eigenvalue problems," in Applications of Bifurcation Theory, ed. Rabinowitz, P. H. (Academic Press), pp. 359-384.
33. Kevrekidis, I. G. & Jolly, M. S. [1987] "On the use of interactive graphics in the numerical study of chemical dynamics," AIChE Annual Meeting, New York, paper 22c.
34. Khibnik, A. I., Kuznetsov, Yu. A., Levitin, V. V. & Nikolaev, E. N. [1993] "Continuation techniques and interactive software for bifurcation analysis of ode's and iterated maps," Physica D62, 360-371.
35. Khibnik, A. I., Roose, D. & Chua, L. O. [1993] "On periodic orbits and homoclinic bifurcations in Chua's circuit with a smooth nonlinearity," Int. J. Bifurcation and Chaos 3(2), 363-384.
36. Kuznetsov, Yu. A. [1995] Elements of Applied Bifurcation Theory (Springer-Verlag).
37. Kuznetsov, Yu. A. & Levitin, V. V. [1997] "CONTENT, a multiplatform continuation environment," CWI Amsterdam, Technical Report.
38. Lentini, M. & Keller, H. B. [1980] "Boundary value problems over semi-infinite intervals and their numerical solution," SIAM J. Numer. Anal. 17, 557-604.
39. Lorenz, J. [1982] "Nonlinear boundary value problems with turning points and properties of difference schemes," Lecture Notes in Mathematics 942 (Springer-Verlag).
40. Nusse, H. E. & Yorke, J. A. [1994] Dynamics: Numerical Explorations (Springer-Verlag).
41. Plant, R. E. [1981] "Bifurcation and resonance in a model for bursting nerve cells," J. Math. Biol. 11, 15-32.
42. Rheinboldt, W. C. [1986] "Numerical analysis of parametrized nonlinear equations," University of Arkansas Lecture Notes in the Mathematical Sciences (Wiley-Interscience).
43. Rinzel, J. [1985] "Bursting oscillations in an excitable membrane model," in Ordinary and Partial Differential Equations, eds. Sleeman, B. D. & Jarvis, R. J. (Springer-Verlag), pp. 304-316.
44. Russell, R. D. & Christiansen, J. [1978] "Adaptive mesh selection strategies for solving boundary value problems," SIAM J. Numer. Anal. 15, 59-80.
45. Seydel, R. [1995] From Equilibrium to Chaos. Practical Bifurcation and Stability Analysis 2nd edition (Else" vier, New York).
46. Stech, H. Personal communication, Department of Mathematics, University of Minnesota, Duluth.
47. Terman, D. [1992] "The transition from bursting to continuous spiking in excitable membrane models," J. Nonlinear Science 2, 135-182.
48. Wang, X. J. & Doedel, E. J. [1995] "AUTO94P: An experimental parallel version of AUTO," Technical Report CRPC-95-3, Center for Research on Parallel Computing, California Institute of Technology, Pasadena CA 91125.
49. Wicklin, F. J. [1995] "Pisces: A platform for implicit surfaces and curves and the exploration of singularities," Technical Report GCG 89, Geometry Center, University of Minnesota, Minneapolis, (http://www.geom.umn.edu/locate/pisces/).