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December  2009, 2(4): 897-909. doi: 10.3934/dcdss.2009.2.897

On stability loss delay for dynamical bifurcations

Anatoly Neishtadt 1,

1. 

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.
Keywords: dynamical bifurcations., Slow-fast systems.
Mathematics Subject Classification: Primary: 34E10, 34E13, 34E15; Secondary: 34C26, 34C2.
Citation: Anatoly Neishtadt. On stability loss delay for dynamical bifurcations. Discrete and Continuous Dynamical Systems - S, 2009, 2 (4) : 897-909. doi: 10.3934/dcdss.2009.2.897
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