1997
DOI: 10.1098/rspa.1997.0110
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Dynamic period–doubling bifurcations of a unimodal map

Abstract: Period-doubling bifurcations of a fairly general unimodal map y n+1 = f (y n , x n ) are considered with linearly varying parameter x n+1 = x n ± ε (ε 1). The sweeping delays the apparent bifurcations. There are structurally different types of trajectories sweeping forwards (+ε) and sweeping backwards (−ε). A matched asymptotic approach is used to analyse the system. Following Baesens, adiabatic manifolds are obtained for the period-1 and period-2 regions. An outer expansion that is singular at the bifurcation… Show more

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Cited by 8 publications
(14 citation statements)
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“…It is convenient to assume also that the Schwartzian derivative of f is everywhere negative. In all these respects, the characteristics of the map are those assumed in the previous linear analysis (Davies & Rangavajhula 1997).…”
Section: Analytic Approximations For a Periodically Perturbed Mapmentioning
confidence: 99%
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“…It is convenient to assume also that the Schwartzian derivative of f is everywhere negative. In all these respects, the characteristics of the map are those assumed in the previous linear analysis (Davies & Rangavajhula 1997).…”
Section: Analytic Approximations For a Periodically Perturbed Mapmentioning
confidence: 99%
“…The singularity at this value is not pertinent to our discussion of period doubling, so we assume that x 0 and α are chosen so that x does not reach this value. (Some examples of the saddle-node, transcritical and pitchfork bifurcations to be expected at this value of x if it is reached are discussed in Davies & Rangavajhula (1997).) Of course, equation (3.3) does not necessarily provide a good description of the actual response in the region above the period-doubling bifurcation, where the period-1 response is now unstable.…”
Section: Analytic Approximations For a Periodically Perturbed Mapmentioning
confidence: 99%
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