Periodic solutions for: ẋ(t) = λf(x(t),x(t – 1))

Arino, Ovide; Benkhalti, Rachid

doi:10.1017/s030821050002775x

Cited by 5 publications

(3 citation statements)

References 6 publications

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“…If x n (t) = 0 then x n (t + s) has exactly 2n zeroes for s ∈ [−1, 0]. The branches of periodic orbits that appear at these successive Hopf bifurcations can be extended up to ǫ = 0, the period converges to r = r 0 = 1 and the amplitude converges to a > 0 [2], and the oscillations tend to a square-wave-like shape when ǫ → 0. So, Eq.…”

Section: Metastability and Rapidly Oscillating Periodic Orbitsmentioning

confidence: 99%

“…The Hopf bifurcation theorem of Eichmann has three first order derivatives hypotheses (H 1 ) (H 2 ) and (H 3 ), three second order derivatives hypotheses (H 4 ) (H 5 ) (H 6 ) and three spectral hypothesis (L 1 ), (L 2 ) and (L 3 ). For convenience of the reader we state these hypotheses here before proving that they are satisfied by equations ( 1) and (2).…”

Section: A Proof Of Theoremmentioning

confidence: 99%

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Metastable periodic patterns in singularly perturbed state-dependent delayed equations

Pellegrin

Ragazzo

Malta

et al. 2014

Physica D: Nonlinear Phenomena

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We consider the scalar delayed differential equation ǫẋ(t) = −x(t)+f (x(t−r)), where ǫ > 0, r = r(x, ǫ) and f represents either a positive feedback df /dx > 0 or a negative feedback df /dx < 0. When the delay is a constant, i.e. r(x, ǫ) = 1, this equation admits metastable rapidly oscillating solutions that are transients whose duration is of order exp(c/ǫ), for some c > 0. In this paper we investigate whether this metastable behavior persists when the delay r(x, ǫ) depends non trivially on the state variable x. Our conclusion is that for negative feedback, the persistence of the metastable behavior depends only on the way r(x, ǫ) depends on ǫ and not on the feedback f . In contrast, for positive feedback, for metastable solutions to exist it is further required that the feedback f is an odd function and the delay r(x, ǫ) is an even function. Our analysis hinges upon the introduction of state dependent transtion layer equations that describe the profiles of the transient oscillations. One novel result is that state dependent delays may lead to metastable dynamics in equations that cannot support such regimes when the delay is constant.

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Section: Metastability and Rapidly Oscillating Periodic Orbitsmentioning

confidence: 99%

Section: A Proof Of Theoremmentioning

confidence: 99%

Metastable periodic patterns in singularly perturbed state-dependent delayed equations

Pellegrin

Ragazzo

Malta

et al. 2014

Physica D: Nonlinear Phenomena

View full text Add to dashboard Cite

show abstract

“…However, as the delay is increased, the origin undergoes successive Hopf bifurcations leading to the generation of periodic orbits [12]. Thus, for large delays, there are solutions that do not converge to any equilibria.…”

Section: )mentioning

confidence: 99%