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. Berns, Daniel W.; Moiola, Jorge L.; Chen, Guanrong: Detecting period-doubling bifurcation: an approximate monodromy matrix approach. (2001)
  2. Edwards, Andrew M.; Bees, Martin A.: Generic dynamics of a simple plankton population model with a non-integer exponent of closure (2001)
  3. Greiner, Alfred; Feichtinger, Gustav; Haunschmied, Josef L.; Kort, Peter M.; Hartl, Richard F.: Optimal periodic development of a pollution generating tourism industry (2001)
  4. Jansen, Vincent A. A.: The dynamics of two diffusively coupled predator--prey populations. (2001)
  5. Zarnitsina, V. I.; Ataullakhanov, F. I.; Lobanov, A. I.; Morozova, O. L.: Dynamics of spatially nonuniform patterning in the model of blood coagulation (2001)
  6. Chen, Guanrong; Moiola, Jorge L.; Wang, Hua O.: Bifurcation control: theories, methods, and applications. (2000)
  7. Gaunersdorfer, A.: Endogenous fluctuations in a simple asset pricing model with heterogeneous agents (2000)
  8. Goeree, J. K.; Hommes, C. H.: Heterogeneous beliefs and the non-linear cobweb model (2000)
  9. Govaerts, Willy J. F.: Numerical methods for bifurcations of dynamical equilibria (2000)
  10. Tuinstra, J.: A discrete and symmetric price adjustment process on the simplex (2000)
  11. Eirola, Timo: Detection of generic one parameter bifurcations of Hamiltonian equilibria (1999)
  12. Moiola, Jorge L.; Chua, Leon O.: Hopf bifurcations and degeneracies in Chua’s circuit. -- A perspective from a frequency domain approach (1999)
  13. Tuinstra, Jan; Weddepohl, Claus: On the equivalence between the overlapping-generations model and cyclical general-equilibrium models (1999)
  14. Boer, M. P.; Kooi, B. W.; Kooijman, S. A. L. M.: Food chain dynamics in the chemostat (1998)
  15. de Roos, André M.; McCauley, Edward; Wilson, William G.: Pattern formation and the spatial scale of interaction between predators and their prey (1998)
  16. De Vries, Gerda; Sherman, Arthur; Zhu, Hsiu-Rong: Diffusively coupled bursters: Effects of cell heterogeneity (1998)
  17. Govaerts, W.; Kuznetsov, Yu. A.; Sijnave, B.: Implementation of Hopf and double-Hopf continuation using bordering methods (1998)
  18. Gragnani, A.; Gatto, M.; Rinaldi, S.: Acidic deposition, plant pests, and the fate of forest ecosystems (1998)
  19. Kooi, B. W.; Boer, M. P.; Kooijman, S. A. L. M.: Consequences of population models for the dynamics of food chains (1998)
  20. Prskawetz, A.: Government debt, budget surplus, and popularity of politicians (1998)