zbMATH — the first resource for mathematics

Continuation techniques and interactive software for bifurcation analysis of ODEs and iterated maps. (English) Zbl 0784.34030
Summary: We present a numerical technique for the analysis of local bifurcations which is based on the continuation of structurally unstable invariant sets in a suitable phase-parameter space. The invariant sets involved in our study are equilibrium points and limit cycles of autonomous ODEs, periodic solutions of time-periodic nonautonomous ODEs, fixed points and periodic orbits of iterated maps. The more general concept of a continuation strategy is also discussed. It allows the analysis of various singularities of generic systems and of their mutual relationships. The approach is extended to codimension three singularities. We introduce several bifurcation functions and show how to use them to construct well-posed continuation problems. The described continuation technique is supported by an interactive graphical program called LOCBIF. We discuss briefly the concepts of the LOCBIF interface and give some examples of typical applications.

34C23 Bifurcation theory for ordinary differential equations
65J99 Numerical analysis in abstract spaces
37G99 Local and nonlocal bifurcation theory for dynamical systems
65D30 Numerical integration
65L99 Numerical methods for ordinary differential equations
68N99 Theory of software
Full Text: DOI
[1] Allgower, E.L.; Georg, K., ()
[2] Afrajmovich, V.S.; Arnold, V.I.; Il’yashenko, Yu.S.; Šil’nikov, L.P., Theory of bifurcations, (), [Russian original (VINITI, Moscow, 1985)]
[3] Arnold, V.I., Geometrical methods in the theory of ordinary differential equations, (1982), Springer New York
[4] Bazykin, A.D., Mathematical biophysics of interacting populations, (1985), Nauka Moscow, (in Russian) · Zbl 0605.92015
[5] Borisyuk, G.; Borisyuk, R.; Khibnik, A., Analysis of oscillatory regimes of a coupled neural oscillator system with application to visual cortex modeling, (), in press
[6] Doedel, E.; Keller, H.B.; Kernevez, J.P.; Doedel, E.; Keller, H.B.; Kernevez, J.P., Numerical analysis and control in bifurcation problems, (II) bifurcations in infinite dimensions, Int. J. bifurcation chaos, Int. J. bifurcations chaos, 2, 745-772, (1992) · Zbl 0876.65060
[7] Doedel, E.; Kernevez, J.P., AUTO: software for continuation and bifurcation problems in ordinary differential equations, (1986), California Institute of Technology Pasadena
[8] Gantmacher, F.R., ()
[9] Gatermann, K.; Hohmann, A., Symbolic exploitation of symmetry in numerical path following, (1990), Kondrad-Zuse-Zentrum für Informationstechnik Berlin, preprint
[10] Guckenheimer, J.; Holmes, P., Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, (1983), Springer New York · Zbl 0515.34001
[11] Jepson, A.D.; Spence, A., Singular points and their computation, (), 195-209 · Zbl 0579.65048
[12] Hairer, E.; Nørsett, S.P.; Wanner, G., Solving ordinary differential equations I. nonstiff problems, () · Zbl 1185.65115
[13] Hairer, E.; Wanner, G., Solving ordinary differential equations II. stiff and differential-algebraic problems, () · Zbl 0729.65051
[14] Hassard, B.; Kazarinoff, N.D.; Wan, Y.-H., Theory and applications of Hopf bifurcations, (1981), Cambridge Univ. Press Cambridge · Zbl 0474.34002
[15] Keller, H.B., Numerical solution of bifurcation and nonlinear eigenvalue problems, (), 359-384 · Zbl 0581.65043
[16] Khibnik, A.I., Study of critical phenomenas in biological kinetic problems, (), unpublished (in Russian) · Zbl 0783.65060
[17] Khibnik, A.I., Numerical methods in bifurcation analysis of dynamical systems: a continuation approach, (), 162-197, (in Russian) · Zbl 0783.65060
[18] Khibnik, A.I., LINLBF: A program for continuation and bifurcation analysis of equilibria up to codimension three, (), 283-296 · Zbl 0705.34001
[19] Khibnik, A.I., Using trax: A tutorial to accompany trax, A program for simulation and analysis of dynamical systems, (1990), Exeter Software New York
[20] Khibnik, A.I.; Borisyuk, R.M.; Roose, D., Numerical bifurcation analysis of a model of coupled neural oscillators, (), 215-228 · Zbl 0755.92003
[21] Khibnik, A.; Kuznetsov, Yu.; Levitin, V.; Nikolaev, E., LOCBIF: interactive local bifurcation analyser, (1990), Research Computing Centre, USSR Academy of Sciences Pushchino, version 1.1, Report
[22] Khibnik, A.I.; Shnol, E.E., Software for qualitative analysis of differential equations, (1982), USSR Academy of Sciences Pushchino, (in Russian)
[23] Kuznetsov, Yu.A., One-dimensional invariant manifolds of ordinary differential equations depending upon parameters, (), (in Russian)
[24] Kuznetsov, Yu.A.; Rinaldi, S., Numerical analysis of the flip bifurcation of maps, Appl. math. comp., 43, 231-236, (1991) · Zbl 0729.65050
[25] Kuznetsov, Yu.A.; Muratory, S.; Rinaldi, S., Bifurcations and chaos in a periodic predator-prey model, Int. J. bifurcation chaos, 2, 117-128, (1992) · Zbl 1126.92316
[26] Levitin, V.V., Trax: simulation and analysis of dynamical systems, (1989), Exeter Software New York
[27] Rheinboldt, W.C., Numerical analysis of parametrized nonlinear equations, (1986), Wiley New York · Zbl 0572.65034
[28] Rheinboldt, W.C.; Roose, D.; Seydel, R., Aspects of continuation software, (), 261-268 · Zbl 0711.65037
[29] Rosendorf, P.; Orsag, J.; Schreiber, I.; Marek, M., Interactive system for studies in nonlinear dynamics, (), 269-282 · Zbl 0711.34057
[30] Seydel, R., From equilibrium to chaos: practical bifurcation and stability analysis, (1988), Elsevier Amsterdam · Zbl 0652.34059
[31] Seydel, R., Tutorial on continuation, Int. J. bifurcation chaos, 1, 3-11, (1991) · Zbl 0760.34014
[32] Wan, Y.-H., Computation of the stability condition for the Hopf bifurcation of diffeomorfisms on \(R\)^{2}, SIAM J. appl. math., 34, 167-175, (1978)
[33] Werner, B.; Janovsky, V., Computation of Hopf branches bifurcating from Takens-bogdanov points for problems with symmetries, (), 377-388
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.