Nonlinear numerics

Journal of the Franklin Institute

Volumn 334, Issue 5-6, 1997, Pages 1049-1073

  Doedel, Eusebius J ^a  

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

Author keywords

[No Author keywords available]

Indexed keywords

ALGORITHMS; BIFURCATION (MATHEMATICS); NUMERICAL ANALYSIS; SOFTWARE ENGINEERING;

NONLINEAR NUMERICAL BIFURCATION ANALYSIS;

NONLINEAR EQUATIONS;

EID: 0031223379     PISSN: 00160032     EISSN: None     Source Type: Journal    
DOI: 10.1016/s0016-0032(97)00027-6     Document Type: Article

References (52)

1. Rheinboldt, W. C., Numerical Analysis of Parametrized Nonlinear Equations. University of Arkansas Lecture Notes in the Mathematical Sciences, Wiley-Interscience, 1986.
2. Seydel, R., From Equilibrium to Chaos. Practical Bifurcation and Stability Analysis, 2nd edn. Springer, New York, 1994.
3. Doedel, E. J., Keller, H. B. and Kernévez J. P., Numerical analysis and control of bifurcation problems: (I) Bifurcation in finite dimensions. Internat. J. Bifur. Chaos, 1991, 1, 493-520.
4. Doedel, E. J., Keller, H. B. and Kernévez J. P., Numerical analysis and control of bifurcation problems: (II) Bifurcation in infinite dimensions. Internat. J. Bifur. Chaos, 1991, 1, 745-772.
5. Allgower, E. L. and Georg, K., Numerical path following. In Handbook of Numerical Analysis, Vol. 5, ed. P. G. Ciarlet and J. L. Lions. North Holland Publishing, 1996.
6. H. B. Keller, Numerical solution of bifurcation and nonlinear eigenvalue problems. In Applications of Bifurcation Theory, ed. P. H. Rabinowitz. Academic Press, 1977, pp. 359-384.
7. Kearfott, R. B., An interval step control for continuation methods. SIAM J. Numer. Anal., 1994, 31(3), 892-914.
8. Chow, S.-N. and Hale, J., Methods of Bifurcation Theory. Springer, 1982.
9. Wittenberg and Holmes, F., Princeton University, 1996.
10. Govaerts, W., Computation of singularities in large nonlinear systems. SIAM J. Numer. Anal., 1997, 34(3), 867-880.
11. Isaacson, E. and Keller, H. B., Analysis of Numerical Methods. Wiley, 1966.
12. Lentini, M. and Keller H. B., Boundary value problems over semi-infinite intervals and their numerical solution. SIAM J. Numer. Anal., 1980, 17, 557-604.
13. Beyn, W.-J., The numerical computation of connecting orbits in dynamical systems. IMA J. Numer. Anal., 1990, 9, 379-405.
14. Friedman, M. J. and Doedel E. J., Numerical computation and continuation of invariant manifolds connecting fixed points. SIAM J. Numer. Anal., 1991, 28, 789-808.
15. Ascher, U. M., Matthei j, R. M. M. and Russell, R. D., Numerical Solution of Boundary Value Problems for Ordinary Differential Equations. Prentice-Hall, 1988.
16. de Boor, C. and Swartz B., Collocation at Gaussian points. SIAM J. Numer. Anal., 1973, 10, 582-606.
17. Fairgrieve, T. F. and Jepson A. D., O.K. Floquet multipliers. SIAM J. Numer. Anal., 1991, 28(5), 1446-1462.
18. Wang, X.-J. and Doedel, E.J., AUTO94P: An experimental parallel version of AUTO. CRPC-95-3, Center for Research on Parallel Computing, California Institute of Technology, Pasadena CA 91125, 1995.
19. Doedel, E. J., On the construction of discretizations of elliptic partial differential equations, J. Difference Equations and Applications, to appear.
20. Lorenz, J., Nonlinear boundary value problems with turning points and properties of difference schemes. In Singular Perturbation Theory and Applications, ed. W. Eckhaus and E. M. de Jager. Lecture Notes in Mathematics Vol. 942, Springer, 1982.
21. Plant, R. E., Bifurcation and resonance in a model for bursting nerve cells. J. Math. Biol., 1981, 11, 15-32.
22. Rinzel, J., Bursting oscillations in an excitable membrane model. In Ordinary and Partial Differential Equations, ed. B. D. Sleeman and R. J. Jarvis. Springer, 1985, pp. 304-316.
23. Terman, D., The transition from bursting to continuous spiking in excitable membrane models. J. Nonlinear Sci., 1992, 2, 135-182.
24. Decroly, O. and Goldbeter A., From simple to complex oscillatory behaviour: Analysis of bursting in a multiply regulated biochemical system. J. Theoret. Biol., 1987, 124, 219-250.
25. Khibnik, A. I., Kuznetsov, Yu. A., Levitin, V. V. and Nikolaev E. N., Continuation techniques and interactive software for bifurcation analysis of ODE's and iterated maps. Phys. D, 1993, 62, 360-371.
26. Hadley, P., Beasley, M. R. and Wiesenfeld K., Phase locking of Josephson-junction series arrays. Phys. Rev. B, 1988, 38, 8712-8719.
27. N symmetry. Nonlinearity, 1991, 4, 861-902.
28. Aronson, D. G., Doedel, E. J. and Terman, D. H., A codimension-two point associated with coupled Josephson junctions, to appear in Nonlinearity.
29. Back, A., Guckenheimer, J., Myers, M. R., Wicklin, F. J. and Worfolk P. A., DsTool: Computer assisted exploration of dynamical systems. Notices Amer. Math. Soc., 1992, 39(4), 303-309.
30. Kuznetsov, Y. A. and Levitin, V. V., CONTENT, a multiplatform continuation environment. Technical report, CWI Amsterdam, 1996.
31. Khibnik, A. I., Roose, D. and Chua L. O., On periodic orbits and homoclinic bifurcations in Chua's circuit with a smooth nonlinearity. Internat. J. Bifur. Chaos, 1993, 3(2), 363-384.
32. Doedel, E. J., Champneys, A. R., Fairgrieve, T. F., Kuznetsov, Y. A., Sandstede, B. and Wang, X.-J., AUTO97: Continuation and bifurcation software for ordinary differential equations. Department of Computer Science, Concordia University, Montreal, Canada, 1997 (Available by FTP from ftp.cs.concordia.ca. in directory pub/doedel/auto).
33. Doedel, E. J., Aronson, D. G. and Othmer H. G., The dynamics of coupled current-biased Josephson junctions II. Internat. J. Bifur. Chaos, 1991, 1(1), 51-66.
34. Gatermann, K. and Hohmann A., Symbolic exploitation of symmetry in numerical pathfollowing. Impact Comput. Sci. Engrg., 1991, 3, 330-365.
35. Kuznetsov, Yuri. A., Elements of Applied Bifurcation Theory. Springer, 1995.
36. Champneys, A. R. and Kuznetsov Yu. A., Numerical detection and continuation of codimension-two homoclinic bifurcations. Internat. J. Bifur. Chaos, 1994, 4, 795-822.
37. Champneys, A. R., Kuznetsov, Yu. A. and Sandstede, B., HomCont: An AUTO86 driver for homoclinic bifurcation analysis, version 2.0. Technical report, CWI, Amsterdam, 1995.
38. Henderson, M., Computing implicitly denned surfaces: two parameter continuation. IBM Research Report RC 18777, IBM Thomas Watson Research Center, Yorktown Heights, New York, 1993.
39. Wicklin, F. J., Pisces: A platform for implicit surfaces and curves and the exploration of singularities. Technical Report GCG89, Geometry Center, University of Minnesota, Minneapolis, 1995.
40. Kevrekidis, I. G. and Jolly, M. S., On the use of interactive graphics in the numerical study of chemical dynamics. In AIChE Annual Meeting, New York, paper 22c, 1987.
41. Nusse, H. E. and Yorke, J. A., Dynamics: Numerical Explorations. Springer, 1994.
42. Fenichel, N., Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J., 1971, 21(3), 193-226.
43. Lorenz, J., Lyapunov-type numbers and torus breakdown: Numerical aspects and a case study. Technical report, Mathematics Department, University of New Mexico, 1996.
44. Broer, H. W., Osinga, H. M. and Vegter, G., On the computation of normally hyperbolic invariant manifolds. In Progress in Nonlinear Differential Equations and Their Applications, Vol. 19, ed. H. W. Broer, I. Hoveijn and F. Takens. Birkhäuser, 1996, pp. 423-447.
45. Ermentrout, B., XPPAUT1.61-the differential equations tool. Technical report, Department of Mathematics, University of Pittsburgh, 1995.
46. Stech, H., A numerical analysis of the structure of periodic orbits in autonomous functional differential equations. In Dynamics of Infinite Dimensional Systems, ed. S.-N. Chow, and J. K. Hale. Springer, Berlin, 1987, pp. 325-337.
47. Doedel, E. J. and Leung P. C., Numerical techniques for bifurcation problems in delay equations. Cong. Num., 1982, 34, 225-237 (Proc. 11th Manitoba Conf. on Num. Math. and Comp., Univ. of Manitoba, Winnipeg, Canada).
48. Cometto, L., Dahmen, M. and Giannakopoulos, F., On the number of p-zeroes of a quasi-polynomial with application to the stability and Hopf-bifurcation of systems of differential equations with delays. Technical report, Department of Mathematics, University of Cologne, 1996.
49. Luzyanina, T., Engelborghs, K., Lust, K. and Roose, D., Computation, continuation and bifurcation analysis of periodic solutions of delay differential equations. Internat. J. Bifur. Chaos, to appear.
50. Ascher, U. M. and Spiteri R. J., Collocation software for boundary value differentialalgebraic equations. SIAM J. Sci. Comput., 1995, 15, 938-952.
51. Bank, R. E., PLTMG, A Software Package for Solving Elliptic Partial Differential Equations. SIAM, 1990.
52. Shroff, G. M. and Keller H. B., Stabilization of unstable procedures: The recursive projection method. SIAM J. Numer. Anal., 1993, 30(4), 1099-1120.