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.

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  1. Guillot, Louis; Cochelin, Bruno; Vergez, Christophe: A Taylor series-based continuation method for solutions of dynamical systems (2019)
  2. Pei, Lijun; Wang, Shuo: Double Hopf bifurcation of differential equation with linearly state-dependent delays via MMS (2019)
  3. Gol’dshtein, V.; Krapivnik, N.; Yablonsky, G.: About bifurcational parametric simplification (2015)
  4. Merrison-Hort, Robert; Yousif, Nada; Njap, Felix; Hofmann, Ulrich G.; Burylko, Oleksandr; Borisyuk, Roman: An interactive channel model of the basal ganglia: bifurcation analysis under healthy and Parkinsonian conditions (2013)
  5. Torresi, A. M.; Calandrini, G. L.; Bonfili, P. A.; Moiola, J. L.: Generalized Hopf bifurcation in a frequency domain formulation (2012)
  6. Diekmann, Odo; Gyllenberg, Mats; Metz, J. A. J.; Nakaoka, Shinji; de Roos, Andre M.: Daphnia revisited: Local stability and bifurcation theory for physiologically structured population models explained by way of an example (2010)
  7. Nandakumar, K.; Chatterjee, Anindya: Continuation of limit cycles near saddle homoclinic points using splines in phase space (2009)
  8. Bolzoni, Luca; De Leo, Giulio A.; Gatto, Marino; Dobson, Andrew P.: Body-size scaling in an SEI model of wildlife diseases (2008)
  9. Dercole, Fabio: \textscBPcont: An \textscautodriver for the continuation of branch points of algebraic and boundary-value problems (2008)
  10. El Abdllaoui, Abderrahim; Auger, Pierre; Kooi, Bob W.; Bravo de la Parra, Rafael; Mchich, Rachid: Effects of density-dependent migrations on stability of a two-patch predator-prey model (2007)
  11. Govaerts, W.; Ghaziani, R. Khoshsiar; Kuznetsov, Yu. A.; Meijer, H. G. E.: Numerical methods for two-parameter local bifurcation analysis of maps (2007)
  12. Auger, Pierre; Kooi, Bob W.; Bravo de la Parra, Rafael; Poggiale, Jean-Christophe: Bifurcation analysis of a predator-prey model with predators using hawk and dove tactics (2006)
  13. Casagrandi, Renato; Bolzoni, Luca; Levin, Simon A.; Andreasen, Viggo: The SIRC model and influenza A (2006)
  14. Itovich, Griselda R.; Moiola, Jorge L.: On period doubling bifurcations of cycles and the harmonic balance method (2006)
  15. Kooi, B. W.; Troost, T. A.: Advantage of storage in a fluctuating environment (2006)
  16. Itovich, Griselda R.; Moiola, Jorge L.: Double Hopf bifurcation analysis using frequency domain methods (2005)
  17. Kooi, B. W.; Kuijper, L. D. J.; Kooijman, S. A. L. M.: Consequences of symbiosis for food web dynamics (2004)
  18. Dhooge, A.; Govaerts, W.; Kuznetsov, Yu. A.: MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs (2003)
  19. Casagrandi, Renato; Gatto, Marino: Habitat destruction, environmental catastrophes, and metapopulation extinction. (2002)
  20. Kooi, B. W.; Kuijper, L. D. J.; Boer, M. P.; Kooijman, S. A. L. M.: Numerical bifurcation analysis of a tri-trophic food web with omnivory (2002)

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