Degenerate Bogdanov–Takens bifurcations in a one-dimensional transport model of a fusion plasma

Blank, de Hj Hugo; Kuznetsov, Yu. А.; Pekker, Mark J.; Veldman, D.W.M.

doi:10.1016/j.physd.2016.05.008

“…The presence of this type of systems has been reported in different applications, see [16,2,24]. On the study of bifurcation phenomena in these systems many contributions have been made.…”

Section: Introductionmentioning

confidence: 99%

“…1 Facultad de Ingeniería y Ciencias, Departamento de Ciencias Naturales y Matemáticas, Pontificia Universidad Javeriana-Cali, Cali, Colombia. 2,3 Departamento de Matemática Aplicada, Escuela Técnica Superior de Ingeniería, Avda. de los Descubrimientos, 41092 Sevilla, Spain.…”

Section: Introductionunclassified

“…Later on, the above work was quoted in [23], where a numerical analysis of the same model was performed. In [10,12,11], it is considered the system ẋ = y, ẏ = µ 1 + µ 2 x − x 3 + y(µ 3 + µ 4 x − x 2 ), (2) and the authors showed that it can be written as a perturbed Hamiltonian system, reporting the maximum number of limit cycles. Later, by taking the parameter µ 4 = 0 in (2), the authors in [4,5,7,6] analyzed the system as a Liénard system, its local bifurcations were characterized, and a numerical study of the global bifurcations was done.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Bifurcation set for a disregarded Bogdanov-Takens unfolding: Application to 3D cubic memristor oscillators

Amador

¹

,

Freire

²

,

Ponce

³

2021

Nonlinear Dyn

2

0

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We derive the bifurcation set for a not previously considered threeparametric Bogdanov-Takens unfolding, showing that it is possible express its vector field as two different perturbed cubic Hamiltonians. By using several firstorder Melnikov functions, we obtain for the first time analytical approximations for the bifurcation curves corresponding to homoclinic and heteroclinic connections, which along with the curves associated to local bifurcations organize the parametric regions with different structures of periodic orbits.As an application of these results, we study a family of 3D memristor oscillators, for which the characteristic function of the memristor is a cubic polynomial. We show that these systems have an infinity number of invariant manifolds, and by adding one parameter that stratifies the 3D dynamics of the family, it is shown that the dynamics in each stratum is topologically equivalent to a representant of the above unfolding. Also, based upon the bifurcation set obtained, we show the existence of closed surfaces in the 3D state space which are foliated by periodic orbits. Finally, we clarify some misconceptions that arise from the numerical simulations of these systems, emphasizing the important role played by the existence of invariant manifolds.

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“…The presence of this type of systems has been reported in different applications, see [16,2,24]. On the study of bifurcation phenomena in these systems many contributions have been made.…”

Section: Introductionmentioning

confidence: 99%

“…Later on, the above work was quoted in [23], where a numerical analysis of the same model was performed. In [10,12,11], it is considered the system ẋ = y, ẏ = µ 1 + µ 2 x − x 3 + y(µ 3 + µ 4 x − x 2 ), (2) and the authors showed that it can be written as a perturbed Hamiltonian system, reporting the maximum number of limit cycles. Later, by taking the parameter µ 4 = 0 in (2), the authors in [4,5,7,6] analyzed the system as a Liénard system, its local bifurcations were characterized, and a numerical study of the global bifurcations was done.…”

Section: Introductionmentioning

confidence: 99%

Bifurcation set for a disregarded Bogdanov-Takens unfolding. Application to 3D cubic memristor oscillators

Amador¹,

Freire²,

Ponce³

2018

Preprint

0

View full text Add to dashboard Cite

We derive the bifurcation set for a not previously considered threeparametric Bogdanov-Takens unfolding, showing that it is possible express its vector field as two different perturbed cubic Hamiltonians. By using several firstorder Melnikov functions, we obtain for the first time analytical approximations for the bifurcation curves corresponding to homoclinic and heteroclinic connections, which along with the curves associated to local bifurcations organize the parametric regions with different structures of periodic orbits.As an application of these results, we study a family of 3D memristor oscillators, for which the characteristic function of the memristor is a cubic polynomial. We show that these systems have an infinity number of invariant manifolds, and by adding one parameter that stratifies the 3D dynamics of the family, it is shown that the dynamics in each stratum is topologically equivalent to a representant of the above unfolding. Also, based upon the bifurcation set obtained, we show the existence of closed surfaces in the 3D state space which are foliated by periodic orbits. Finally, we clarify some misconceptions that arise from the numerical simulations of these systems, emphasizing the important role played by the existence of invariant manifolds.

show abstract