October  1999, 5(4): 741-752. doi: 10.3934/dcds.1999.5.741

Types of change of stability and corresponding types of bifurcations


Universidad Autónoma Metropolitana, Unidad Iztapalapa, Av. Michoacán y la Purísima, Apdo. Postal 55-534, Mexico 09340, D.F., Mexico, Mexico

Received  August 1998 Revised  May 1999 Published  July 1999

The general topic is the connection between a change of stability of an equilibrium point or invariant set $M$ of a (semi-) dynamical system depending on a parameter and a bifurcation of $M$ (generalizing the Hopf bifurcation). In particular, we address the case where $M$ is unstable (for instance a saddle) for a certain value $\lambda_0$ of a parameter $\lambda$, and stable for certain nearby values. Two kinds of bifurcations are considered: "extracritical", i.e. splitting of the set $M$ as $\lambda$ passes the value $\lambda_0$, and "critical" (also called "vertical"), a term which refers to the accumulation of closed invariant set at $M$ for $\lambda=\lambda_0$. Also, two kinds of change of stability are considered, corresponding to the presence or absence of a certain generalized equistability property for $\lambda\ne\lambda_0$. Connections are established between the type of change of stability and the types of bifurcation arising from them.
Citation: L. Aguirre, P. Seibert. Types of change of stability and corresponding types of bifurcations. Discrete & Continuous Dynamical Systems, 1999, 5 (4) : 741-752. doi: 10.3934/dcds.1999.5.741

Anatoly Neishtadt. On stability loss delay for dynamical bifurcations. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 897-909. doi: 10.3934/dcdss.2009.2.897


Mary Ballyk, Ross Staffeldt, Ibrahim Jawarneh. A nutrient-prey-predator model: Stability and bifurcations. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 2975-3004. doi: 10.3934/dcdss.2020192


Sze-Bi Hsu, Ming-Chia Li, Weishi Liu, Mikhail Malkin. Heteroclinic foliation, global oscillations for the Nicholson-Bailey model and delay of stability loss. Discrete & Continuous Dynamical Systems, 2003, 9 (6) : 1465-1492. doi: 10.3934/dcds.2003.9.1465


Xiang-Ping Yan, Wan-Tong Li. Stability and Hopf bifurcations for a delayed diffusion system in population dynamics. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 367-399. doi: 10.3934/dcdsb.2012.17.367


Leonid Braverman, Elena Braverman. Stability analysis and bifurcations in a diffusive predator-prey system. Conference Publications, 2009, 2009 (Special) : 92-100. doi: 10.3934/proc.2009.2009.92


Chun-Hsiung Hsia, Tian Ma, Shouhong Wang. Rotating Boussinesq equations: Dynamic stability and transitions. Discrete & Continuous Dynamical Systems, 2010, 28 (1) : 99-130. doi: 10.3934/dcds.2010.28.99


Tian Ma, Shouhong Wang. Tropical atmospheric circulations: Dynamic stability and transitions. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1399-1417. doi: 10.3934/dcds.2010.26.1399


Alexey G. Mazko. Positivity, robust stability and comparison of dynamic systems. Conference Publications, 2011, 2011 (Special) : 1042-1051. doi: 10.3934/proc.2011.2011.1042


Sung Kyu Choi, Namjip Koo. Stability of linear dynamic equations on time scales. Conference Publications, 2009, 2009 (Special) : 161-170. doi: 10.3934/proc.2009.2009.161


Yiqiu Mao, Dongming Yan, ChunHsien Lu. Dynamic transitions and stability for the acetabularia whorl formation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5989-6004. doi: 10.3934/dcdsb.2019117


K Najarian. On stochastic stability of dynamic neural models in presence of noise. Conference Publications, 2003, 2003 (Special) : 656-663. doi: 10.3934/proc.2003.2003.656


Ricardo J. Alonso, Irene M. Gamba. Gain of integrability for the Boltzmann collisional operator. Kinetic & Related Models, 2011, 4 (1) : 41-51. doi: 10.3934/krm.2011.4.41


Graeme Wake, Anthony Pleasants, Alan Beedle, Peter Gluckman. A model for phenotype change in a stochastic framework. Mathematical Biosciences & Engineering, 2010, 7 (3) : 719-728. doi: 10.3934/mbe.2010.7.719


Diana M. Thomas, Ashley Ciesla, James A. Levine, John G. Stevens, Corby K. Martin. A mathematical model of weight change with adaptation. Mathematical Biosciences & Engineering, 2009, 6 (4) : 873-887. doi: 10.3934/mbe.2009.6.873


Nicolas Fourrier, Irena Lasiecka. Regularity and stability of a wave equation with a strong damping and dynamic boundary conditions. Evolution Equations & Control Theory, 2013, 2 (4) : 631-667. doi: 10.3934/eect.2013.2.631


Yan Zhang, Wanbiao Ma, Hai Yan, Yasuhiro Takeuchi. A dynamic model describing heterotrophic culture of chorella and its stability analysis. Mathematical Biosciences & Engineering, 2011, 8 (4) : 1117-1133. doi: 10.3934/mbe.2011.8.1117


Anna Cima, Armengol Gasull, Víctor Mañosa. Parrondo's dynamic paradox for the stability of non-hyperbolic fixed points. Discrete & Continuous Dynamical Systems, 2018, 38 (2) : 889-904. doi: 10.3934/dcds.2018038


Lan Jia, Liang Li. Stability and dynamic transition of vegetation model for flat arid terrains. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021189


Hayato Chiba, Georgi S. Medvedev. The mean field analysis of the kuramoto model on graphs Ⅱ. asymptotic stability of the incoherent state, center manifold reduction, and bifurcations. Discrete & Continuous Dynamical Systems, 2019, 39 (7) : 3897-3921. doi: 10.3934/dcds.2019157


Jianhe Shen, Shuhui Chen, Kechang Lin. Study on the stability and bifurcations of limit cycles in higher-dimensional nonlinear autonomous systems. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 231-254. doi: 10.3934/dcdsb.2011.15.231

2020 Impact Factor: 1.392


  • PDF downloads (41)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]