Dissecting a Resonance Wedge on Heteroclinic Bifurcations

Rodrigues, Alexandre A. P.

doi:10.1007/s10955-021-02811-4

Cited by 3 publications

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“…In case (B) we observe the existence of the two fixed points, suggesting that the chosen parameters are within a resonant tongue. In (C), the map h a is not a diffeomorphism meaning that the invariant curve broke up; it corresponds to the point when the unstable manifold of the saddle (in the "Arnold tongue" [5,38]) turns around. In case (D), Property (H7) holds, meaning that the unstable manifold of a saddle within the "Arnold tongue" crosses each leaf of the stable foliation of the saddles of the torus's ghost.…”

Section: Mechanism To Create Rank-one Attractors: a Geometrical Inter...mentioning

confidence: 99%

"Large" strange attractors in the unfolding of a heteroclinic attractor

Rodrigues¹

2021

Preprint

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In this paper we present a mechanism for the emergence of strange attractors in a one-parameter family of differential equations acting on a 3-dimensional sphere. When the parameter is zero, its flow exhibits an attracting heteroclinic network (Bykov network) made by two 1-dimensional connections and one 2-dimensional separatrix between hyperbolic saddles-foci with different Morse indices. After slightly increasing the parameter, while keeping the 1-dimensional connections unaltered, we concentrate our study in the case where the 2-dimensional invariant manifolds of the equilibria do not intersect. We will show that, for a set of parameters close enough to zero with positive Lebesgue measure, the dynamics exhibits strange attractors winding around an annulus in the phase space, supporting Sinai-Ruelle-Bowen (SRB) measures. We prove the existence of a sequence of parameter values for which the family exhibits a superstable sink. We also characterise the transition from a Bykov network to a strange attractor.

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Section: Mechanism To Create Rank-one Attractors: a Geometrical Inter...mentioning

confidence: 99%

"Large" strange attractors in the unfolding of a heteroclinic attractor

Rodrigues¹

2021

Preprint

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SIR Model with Vaccination: Bifurcation Analysis

Carvalho

Rodrigues

2023

Qual. Theory Dyn. Syst.

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There are few adapted SIR models in the literature that combine vaccination and logistic growth. In this article, we study bifurcations of a SIR model where the class of Susceptible individuals grows logistically and has been subject to constant vaccination. We explicitly prove that the endemic equilibrium is a codimension two singularity in the parameter space $$(\mathcal {R}_0, p)$$ ( R 0 , p ) , where $$\mathcal {R}_0$$ R 0 is the basic reproduction number and p is the proportion of Susceptible individuals successfully vaccinated at birth. We exhibit explicitly the Hopf, transcritical, Belyakov, heteroclinic and saddle-node bifurcation curves unfolding the singularity. The two parameters $$(\mathcal {R}_0, p)$$ ( R 0 , p ) are written in a useful way to evaluate the proportion of vaccinated individuals necessary to eliminate the disease and to conclude how the vaccination may affect the outcome of the epidemic. We also exhibit the region in the parameter space where the disease persists and we illustrate our main result with numerical simulations, emphasizing the role of the parameters.

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"Large" strange attractors in the unfolding of a heteroclinic attractor

Rodrigues¹

2022

DCDS

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<p style='text-indent:20px;'>We present a mechanism for the emergence of strange attractors in a one-parameter family of differential equations defined on a 3-dimensional sphere. When the parameter is zero, its flow exhibits an attracting heteroclinic network (Bykov network) made by two 1-dimensional connections and one 2-dimensional separatrix between two saddles-foci with different Morse indices. After slightly increasing the parameter, while keeping the 1-dimensional connections unaltered, we concentrate our study in the case where the 2-dimensional invariant manifolds of the equilibria do not intersect. We will show that, for a set of parameters close enough to zero with positive Lebesgue measure, the dynamics exhibits strange attractors winding around the "ghost'' of a torus and supporting Sinai-Ruelle-Bowen (SRB) measures. We also prove the existence of a sequence of parameter values for which the family exhibits a superstable sink and describe the transition from a Bykov network to a strange attractor.</p>

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Dissecting a Resonance Wedge on Heteroclinic Bifurcations

Cited by 3 publications

References 49 publications

"Large" strange attractors in the unfolding of a heteroclinic attractor

"Large" strange attractors in the unfolding of a heteroclinic attractor

SIR Model with Vaccination: Bifurcation Analysis

"Large" strange attractors in the unfolding of a heteroclinic attractor

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