Bifurcations of maps in the software package CONTENT. The qualitative behaviour of iterates of a map can be very complicated. One approach to these phenomena starts with the simplest situation, the case where the map has a fixed point. Under parameter variations, the fixed point typically moves until a bifurcation value is reached and one of three possible more complex phenomena is encountered. These are fold, flip and Neimark - Sacker bifurcations; they are called codimension one phenomena because they generically appear in problems with one free parameter. The software package CONTENT (continuation environment) combines numerical methods (integration, numerical continuation etcetera) with symbolic methods (e.g. symbolic derivatives) and allows (among other things) to numerically continue fixed points and to detect, compute and continue fold points, flip points and Neimark - Sacker points. To the best of our knowledge content is the only softwaxe that allows to detect and compute all codimension two points on such curves, including strong resonances and degenerate Neimark - Sacker bifurcations. The paper provides details on defining systems and test functions implemented in content for these purposes. We show the power of the software by studying the behaviour of an electromechanical device that exhibits a complicated bifurcation behaviour, the so - called Sommerfeld’s efFect. In this example the map is defined by the time integration of a three - dimensional dynamical system over a fixed time interval.

References in zbMATH (referenced in 25 articles , 2 standard articles )

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  1. Net, M.; Sánchez, J.: Continuation of bifurcations of periodic orbits for large-scale systems (2015)
  2. Meng, Xinzhu; Liu, Rui; Zhang, Tonghua: Adaptive dynamics for a non-autonomous Lotka-Volterra model with size-selective disturbance (2014)
  3. Della Rossa, Fabio; Fasani, Stefano; Rinaldi, Sergio: Potential Turing instability and application to plant-insect models (2012)
  4. Páez Chávez, Joseph: Discretizing dynamical systems with generalized Hopf bifurcations (2011)
  5. Barnett, William A.; Duzhak, Evgeniya A.: Empirical assessment of bifurcation regions within New Keynesian models (2010)
  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. Bakri, Taoufik; Meijer, Hil G.E.; Verhulst, Ferdinand: Emergence and bifurcations of Lyapunov manifolds in nonlinear wave equations (2009)
  8. Duan, Lixia; Lu, Qishao; Cheng, Daizhan: Bursting of Morris-Lecar neuronal model with current-feedback control (2009)
  9. Pribylova, Lenka: Bifurcation routes to chaos in an extended Van der Pol’s equation applied to economic models (2009)
  10. Kuznetsov, Yu.A.; Meijer, H.G.E.; Govaerts, W.; Sautois, B.: Switching to nonhyperbolic cycles from codim 2 bifurcations of equilibria in ODEs (2008)
  11. Bakri, T.: Parametric excitation in nonlinear dynamics (2005)
  12. Hüls, Thorsten: Instability helps virtual flies to mate (2005)
  13. Kuznetsov, Yu.A.; Meijer, H.G.E.: Numerical normal forms for codim 2 bifurcations of fixed points with at most two critical eigenvalues (2005)
  14. Bakri, T.; Nabergoj, R.; Tondl, A.; Verhulst, F.: Parametric excitation in non-linear dynamics (2004)
  15. Beyn, Wolf-Jürgen; Hüls, Thorsten: Error estimates for approximating non-hyperbolic heteroclinic orbits of maps (2004)
  16. Dhooge, A.; Govaerts, W.; Kuznetsov, Yu.A.: MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs (2003)
  17. Dercole, Fabio; Rinaldi, Sergio: Evolution of cannibalistic traits: scenarios derived from adaptive dynamics (2002)
  18. Richter, Frank; Sailer, Manfred: Underspecified semantics in HPSG (2001)
  19. Birol, İnanç; Teymour, Fouad: Statics and dynamics of multiple cubic autocatalytic reactions (2000)
  20. Cann, Ronnie; Grover, Claire; Miller, Philip: Grammatical interfaces in HPSG (2000)

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