A proof of Perkoʼs conjectures for the Bogdanov–Takens system

Gasull, Armengol; Giacomini, Héctor; Pérez-González, Set; Torregrosa, Joan

doi:10.1016/j.jde.2013.07.006

Cited by 8 publications

(12 citation statements)

References 19 publications

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“…x B is decreasing as b increases. It is the same for (5) because in the transformation (15) x is independent on b, i.e., x A increases and x B decreases as b increases.…”

Section: Figure 1 Equilibria At Infinitymentioning

confidence: 91%

“…We claim that the homoclinic loop given in Proposition 2 is close to the origin E 0 when |a|, |b| are sufficiently small. In fact, system (5) can be equivalently written as (16) by transformation (15). Thus, by Proposition 2 system (16) has a homoclinic loop when b = ϕ 1 (a), which implies that b/a → −12/5 as a → 0.…”

Section: Figure 1 Equilibria At Infinitymentioning

confidence: 92%

“…In [25] Perko gives global phase portraits of (2) for global parameters (µ 1 , µ 2 ) ∈ R 2 and proves that (2) has the same qualitative behavior concerning limit cycles and homoclinic loops for global parameters as that for small parameters. Moreover, Perko states two conjectures about the function corresponding to the saddle-loop bifurcation curve in the parameter space, which are solved recently by Gasull et al in [15]. The global bifurcation diagram and global phase portraits are investigated in [3] for a Plant-Herbivore model and in [27] for a Predator-Prey system.…”

Section: Hebai Chen and Xingwu Chenmentioning

confidence: 99%

See 2 more Smart Citations

Global phase portraits of a degenerate Bogdanov-Takens system with symmetry (Ⅱ)

Chen¹,

Chen²

2018

Discrete &Amp; Continuous Dynamical Systems - B

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The degenerate Bogdanov-Takens systemẋ = y−(a 1 x+a 2 x 3),ẏ = a 3 x + a 4 x 3 has two normal forms, one of which is investigated in [Disc. Cont. Dyn. Syst. B (22)2017, 1273-1293] and global behavior is analyzed for general parameters. To continue this work, in this paper we study the other normal form and perform all global phase portraits on the Poincaré disc. Since the parameters are not restricted to be sufficiently small, some classic bifurcation methods for small parameters, such as the Melnikov method, are no longer valid. We find necessary and sufficient conditions for existences of limit cycles and homoclinic loops respectively by constructing a distance function among orbits on the vertical isocline curve and further give the number of limit cycles for parameters in different regions. Finally we not only give the global bifurcation diagram, where global existences and monotonicities of the homoclinic bifurcation curve and the double limit cycle bifurcation curve are proved, but also classify all global phase portraits.

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“…x B is decreasing as b increases. It is the same for (5) because in the transformation (15) x is independent on b, i.e., x A increases and x B decreases as b increases.…”

Section: Figure 1 Equilibria At Infinitymentioning

confidence: 91%

Section: Figure 1 Equilibria At Infinitymentioning

confidence: 92%

Section: Hebai Chen and Xingwu Chenmentioning

confidence: 99%

See 1 more Smart Citation

Global phase portraits of a degenerate Bogdanov-Takens system with symmetry (Ⅱ)

Chen¹,

Chen²

2018

Discrete &Amp; Continuous Dynamical Systems - B

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“…Since for the most of the cases this manifold is not algebraic, this computation is not easy. In this work we will approach κ by looking for algebraic approximations of S. These approximations will be obtained following similar tools to the ones developed in [7,8,9]. For fixed values of the parameters of the model, an alternative approach for obtaining κ would be to apply numerical methods to compute S. However, in this work, we center our efforts in obtaining analytic results.…”

Section: Introductionmentioning

confidence: 99%

Quantitative analysis of competition models

Chiralt

Ferragut

Gasull

et al. 2017

Nonlinear Analysis: Real World Applications

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Abstract. We study a 2-species Lotka-Volterra type differential system, modeling competition between two species and having a coexistence equilibrium in the first quadrant. In case that this equilibrium is of saddle type, its stable manifold divides the first quadrant into two zones. Then, depending on the zone where the initial condition lies, one of the species will extinct and the other will go to an equilibrium. Using this separatrix we introduce a measure to discern which species has more chance of surviving. This measure is given by a non-negative real number κ, that we will call persistence ratio, that only depends on the parameters of the system. In some cases, we can give simple explicit expressions for κ. When this is not possible, we use several dynamical tools to obtain effective approximations of it.

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“…Recently, in [14], a technique is developed to localize separatrix bifurcations, which is applied in [15] to give fine estimates for the Bogdanov-Takens separatrix cycle. This technique does not apply for the family (X k m ) m .…”

Section: Introductionmentioning

confidence: 99%

Bifurcation of the separatrix skeleton in some 1-parameter families of planar vector fields

Caubergh

2015

Journal of Differential Equations

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Abstract. This article deals with the bifurcation of polycycles and limit cycles within the 1-parameter families of planar vector fields X k m , defined byẋ = y 3 −x 2k+1 ,ẏ = −x+my 4k+1 , where m is a real parameter and k ≥ 1 integer. The bifurcation diagram for the separatrix skeleton of X k m in function of m is determined and the one for the global phase portraits of (X 1 m ) m∈R is completed. Furthermore for arbitrary k ≥ 1 some bifurcation and finiteness problems of periodic orbits are solved. Among others, the number of periodic orbits of X k m is found to be uniformly bounded independent of m ∈ R and the Hilbert number for (X k m ) m∈R , that thus is finite, is found to be at least one.

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A proof of Perkoʼs conjectures for the Bogdanov–Takens system

Cited by 8 publications

References 19 publications

Global phase portraits of a degenerate Bogdanov-Takens system with symmetry (Ⅱ)

Global phase portraits of a degenerate Bogdanov-Takens system with symmetry (Ⅱ)

Quantitative analysis of competition models

Bifurcation of the separatrix skeleton in some 1-parameter families of planar vector fields

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