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 37 articles , 2 standard articles )

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  1. Páez Chávez, Joseph; Zhang, Zhi; Liu, Yang: A numerical approach for the bifurcation analysis of nonsmooth delay equations (2020)
  2. Ye, Min; Zuo, Hongkun: Stability analysis of regular and chaotic (\mathrmCa^2+) oscillations in astrocytes (2020)
  3. Spyrou, Kostas J.; Themelis, Nikos; Kontolefas, Ioannis: Nonlinear surge motions of a ship in bi-chromatic following waves (2018)
  4. Detroux, T.; Renson, L.; Masset, L.; Kerschen, G.: The harmonic balance method for bifurcation analysis of large-scale nonlinear mechanical systems (2015)
  5. Net, M.; Sánchez, J.: Continuation of bifurcations of periodic orbits for large-scale systems (2015)
  6. Meng, Xinzhu; Liu, Rui; Zhang, Tonghua: Adaptive dynamics for a non-autonomous Lotka-Volterra model with size-selective disturbance (2014)
  7. 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)
  8. Della Rossa, Fabio; Fasani, Stefano; Rinaldi, Sergio: Potential Turing instability and application to plant-insect models (2012)
  9. Savin, Dmitry V.; Savin, Alexey V.; Kuznetsov, Alexander P.; Kuznetsov, Sergey P.; Feudel, Ulrike: The self-oscillating system with compensated dissipation -- the dynamics of the approximate discrete map (2012)
  10. Páez Chávez, Joseph: Discretizing dynamical systems with generalized Hopf bifurcations (2011)
  11. Barnett, William A.; Duzhak, Evgeniya A.: Empirical assessment of bifurcation regions within New Keynesian models (2010)
  12. 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)
  13. Klausmeier, C. A.: Successional state dynamics: a novel approach to modeling nonequilibrium foodweb dynamics (2010)
  14. Bakri, Taoufik; Meijer, Hil G. E.; Verhulst, Ferdinand: Emergence and bifurcations of Lyapunov manifolds in nonlinear wave equations (2009)
  15. Duan, Lixia; Lu, Qishao; Cheng, Daizhan: Bursting of Morris-Lecar neuronal model with current-feedback control (2009)
  16. Pribylova, Lenka: Bifurcation routes to chaos in an extended Van der Pol’s equation applied to economic models (2009)
  17. Kuznetsov, Yu. A.; Meijer, H. G. E.; Govaerts, W.; Sautois, B.: Switching to nonhyperbolic cycles from codim 2 bifurcations of equilibria in ODEs (2008)
  18. Su, Rui; He, Daihai: Using CONTENT 1.5 to analyze an SIR model for childhood infectious diseases (2008)
  19. Heider, Pascal: Rating the performance of shooting methods for the computation of periodic orbits (2006)
  20. Bakri, T.: Parametric excitation in nonlinear dynamics (2005)

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