Continuation techniques and interactive software for bifurcation analysis of ODEs and iterated maps. 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.

References in zbMATH (referenced in 65 articles )

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  1. Behrens, D. A.; Feichtinger, G.; Prskawetz, A.: Complex dynamics and control of arms race (1997)
  2. Diekmann, Odo: The many facets of evolutionary dynamics (1997)
  3. Doedel, Eusebius J.: Nonlinear numerics (1997)
  4. Ghezzi, Luca L.; Piccardi, Carlo: PID control of a chaotic system: An application to an epidemiological model (1997)
  5. Govaerts, W.: Computation of singularities in large nonlinear systems (1997)
  6. Govaerts, W.; Guckenheimer, J.; Khibnik, A.: Defining functions for multiple Hopf bifurcations (1997)
  7. Gragnani, Alessandra: Bifurcation analysis of two predator-prey models (1997)
  8. Kooi, B. W.; Boer, M. P.; Kooijman, S. A. L. M.: Complex dynamic behaviour of autonomous microbial food chains (1997)
  9. Kooi, W. B.; Boer, M. P.; Kooijman, S. A. L. M.: Mass balance equation versus logistic equation in food chains (1997)
  10. Dangelmayr, Gerhard; Fiedler, Bernold; Kirchgässner, Klaus; Mielke, Alexander: Dynamics of nonlinear waves in dissipative systems: reduction, bifurcation and stability (1996)
  11. Feichtinger, G.; Forst, C. V.; Piccardi, C.: A nonlinear dynamical model for the dynastic cycle. (1996)
  12. Janovský, Vladimír; Plecháč, Petr: Local numerical analysis of Hopf bifurcation (1996)
  13. Kuznetsov, Yu. A.; Rinaldi, S.: Remarks on food chain dynamics (1996)
  14. Luzyanina, Tatyana; Roose, Dirk: Numerical stability analysis and computation of Hopf bifurcation points for delay differential equations (1996)
  15. Tesi, A.; Abed, E. H.; Genesio, R.; Wang, H. O.: Harmonic balance analysis of period-doubling bifurcations with implications for control of nonlinear dynamics (1996)
  16. Borisyuk, Galina N.; Borisyuk, Roman M.; Khibnik, Alexander I.; Roose, Dirk: Dynamics and bifurcations of two coupled neural oscillators with different connection types (1995)
  17. Jansen, V. A. A.: Effects of dispersal in a tri-trophic metapopulation model (1995)
  18. Roose, D.; Lust, K.; Champneys, A.; Spence, A.: A Newton-Picard shooting method for computing periodic solutions of large-scale dynamical systems. (1995)
  19. Cymbalyuk, G. S.; Nikolaev, E. V.; Borisyuk, R. M.: In-phase and antiphase self-oscillations in a model of two electrically coupled pacemakers (1994)
  20. Feichtinger, Gustav; Hommes, Cars H.; Milik, Alexandra: Complex dynamics in a threshold advertising model (1994)