• Previous Article
    Bifurcation delay - the case of the sequence: Stable focus - unstable focus - unstable node
  • DCDS-S Home
  • This Issue
  • Next Article
    Survival of subthreshold oscillations: The interplay of noise, bifurcation structure, and return mechanism
December  2009, 2(4): 897-909. doi: 10.3934/dcdss.2009.2.897

On stability loss delay for dynamical bifurcations


Loughborough University, School of Mathematics, Leicestershire LE11 3TU, United Kingdom

Received  September 2008 Revised  February 2009 Published  September 2009

In the classical bifurcation theory, behavior of systems depending on a parameter is considered for values of this parameter close to some critical, bifurcational one. In the theory of dynamical bifurcations a parameter is changing slowly in time and passes through a value that would be bifurcational in the classical static theory. Some arising here phenomena are drastically different from predictions derived by the static approach. Let at a bifurcational value of a parameter an equilibrium or a limit cycle loses its asymptotic linear stability but remains non-degenerate. It turns out that in analytic systems the stability loss delays inevitably: phase points remain near the unstable equilibrium (cycle) for a long time after the bifurcation; during this time the parameter changes by a quantity of order 1. Such delay is not in general found in non-analytic (even infinitely smooth) systems. A survey of some background on stability loss delay phenomenon is presented in this paper.
Citation: 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

Yong Xu, Bin Pei, Rong Guo. Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2257-2267. doi: 10.3934/dcdsb.2015.20.2257


Alexandre Vidal. Periodic orbits of tritrophic slow-fast system and double homoclinic bifurcations. Conference Publications, 2007, 2007 (Special) : 1021-1030. doi: 10.3934/proc.2007.2007.1021


Renato Huzak. Cyclicity of the origin in slow-fast codimension 3 saddle and elliptic bifurcations. Discrete & Continuous Dynamical Systems, 2016, 36 (1) : 171-215. doi: 10.3934/dcds.2016.36.171


Luca Dieci, Cinzia Elia. Smooth to discontinuous systems: A geometric and numerical method for slow-fast dynamics. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2935-2950. doi: 10.3934/dcdsb.2018112


Renato Huzak, P. De Maesschalck, Freddy Dumortier. Primary birth of canard cycles in slow-fast codimension 3 elliptic bifurcations. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2641-2673. doi: 10.3934/cpaa.2014.13.2641


Ilya Schurov. Duck farming on the two-torus: Multiple canard cycles in generic slow-fast systems. Conference Publications, 2011, 2011 (Special) : 1289-1298. doi: 10.3934/proc.2011.2011.1289


Anatoly Neishtadt, Carles Simó, Dmitry Treschev, Alexei Vasiliev. Periodic orbits and stability islands in chaotic seas created by separatrix crossings in slow-fast systems. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 621-650. doi: 10.3934/dcdsb.2008.10.621


Jie Xu, Yu Miao, Jicheng Liu. Strong averaging principle for slow-fast SPDEs with Poisson random measures. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2233-2256. doi: 10.3934/dcdsb.2015.20.2233


Chunhua Shan. Slow-fast dynamics and nonlinear oscillations in transmission of mosquito-borne diseases. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021097


Chungang Shi, Wei Wang, Dafeng Chen. Weak time discretization for slow-fast stochastic reaction-diffusion equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (12) : 6285-6310. doi: 10.3934/dcdsb.2021019


Alexandre Caboussat, Allison Leonard. Numerical solution and fast-slow decomposition of a population of weakly coupled systems. Conference Publications, 2009, 2009 (Special) : 123-132. doi: 10.3934/proc.2009.2009.123


Valery A. Gaiko. The geometry of limit cycle bifurcations in polynomial dynamical systems. Conference Publications, 2011, 2011 (Special) : 447-456. doi: 10.3934/proc.2011.2011.447


Yancong Xu, Deming Zhu, Xingbo Liu. Bifurcations of multiple homoclinics in general dynamical systems. Discrete & Continuous Dynamical Systems, 2011, 30 (3) : 945-963. doi: 10.3934/dcds.2011.30.945


Lana Horvat Dmitrović. Box dimension and bifurcations of one-dimensional discrete dynamical systems. Discrete & Continuous Dynamical Systems, 2012, 32 (4) : 1287-1307. doi: 10.3934/dcds.2012.32.1287


Héctor Barge, José M. R. Sanjurjo. Higher dimensional topology and generalized Hopf bifurcations for discrete dynamical systems. Discrete & Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2021204


C. Connell Mccluskey. Lyapunov functions for tuberculosis models with fast and slow progression. Mathematical Biosciences & Engineering, 2006, 3 (4) : 603-614. doi: 10.3934/mbe.2006.3.603


Yuri Kifer. Another proof of the averaging principle for fully coupled dynamical systems with hyperbolic fast motions. Discrete & Continuous Dynamical Systems, 2005, 13 (5) : 1187-1201. doi: 10.3934/dcds.2005.13.1187


Seung-Yeal Ha, Dohyun Kim, Jinyeong Park. Fast and slow velocity alignments in a Cucker-Smale ensemble with adaptive couplings. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4621-4654. doi: 10.3934/cpaa.2020209


Chunhua Jin. Boundedness and global solvability to a chemotaxis-haptotaxis model with slow and fast diffusion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1675-1688. doi: 10.3934/dcdsb.2018069


Andrea Giorgini. On the Swift-Hohenberg equation with slow and fast dynamics: well-posedness and long-time behavior. Communications on Pure & Applied Analysis, 2016, 15 (1) : 219-241. doi: 10.3934/cpaa.2016.15.219

2020 Impact Factor: 2.425


  • PDF downloads (267)
  • HTML views (0)
  • Cited by (12)

Other articles
by authors

[Back to Top]