Normal Forms and Bifurcation of Planar Vector Fields

Chow, Shui-Nee; Li, Chengzhi; Wang, Duo

doi:10.1017/cbo9780511665639

Cited by 534 publications

(498 citation statements)

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“…Generally, it has been shown that oscillations about the non-trivial equilibrium (P 1 ) can occur when the sign of 3 changes (it should be noted, from Routh Hurwitz criteria, that if 3 = 0, then the polynomial (3.10) has complex conjugate roots). To prove the existence of Hopf bifurcation, it suffices to verify the transversality condition (Chow et al 1994). This is done below.…”

Section: Hopf Bifurcation Analysismentioning

confidence: 99%

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Mathematical assessment of the role of temperature and rainfall on mosquito population dynamics

Abdelrazec

Gumel

2016

J. Math. Biol.

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A new stage-structured model for the population dynamics of the mosquito (a major vector for numerous vector-borne diseases), which takes the form of a deterministic system of non-autonomous nonlinear differential equations, is designed and used to study the effect of variability in temperature and rainfall on mosquito abundance in a community. Two functional forms of eggs oviposition rate, namely the Verhulst-Pearl logistic and Maynard-Smith-Slatkin functions, are used. Rigorous analysis of the autonomous version of the model shows that, for any of the oviposition functions considered, the trivial equilibrium of the model is locally-and globally-asymptotically stable if a certain vectorial threshold quantity is less than unity. Conditions for the existence and global asymptotic stability of the non-trivial equilibrium solutions of the model are also derived. The model is shown to undergo a Hopf bifurcation under certain conditions (and that increased density-dependent competition in larval mortality reduces the likelihood of such bifurcation). The analyses reveal that the Maynard-Smith-Slatkin oviposition function sustains more oscillations than the Verhulst-Pearl logistic function (hence, it is more suited, from ecological viewpoint, for modeling the egg oviposition process). The non-autonomous model is shown to have a globally-asymptotically stable trivial periodic solution, for each of the oviposition functions, when the associated reproduction threshold is less than unity. Furthermore, this model, in the absence of density-dependent mortality rate for larvae, has a unique and globally-asymptotically stable periodic solution under certain conditions. Numerical simulations of the non-autonomous model, using mosquito surveillance and weather data from the Peel region of Ontario, Canada, show a peak mosquito abundance for temperature and rainfall values in the range [20−25] • C and [15-35] mm, respectively. These ranges are recorded in the Peel region between July and August (hence, this study suggests that anti-mosquito control effects should be intensified during this period).

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Section: Hopf Bifurcation Analysismentioning

confidence: 99%

“…The reason is that the transversality condition may fail at some points if only one parameter is used (Chow et al 1994;Yu 2005). The nature of the Hopf bifurcation property of the model (3.1) is investigated numerically.…”

Section: Part (Ii)mentioning

confidence: 99%

Mathematical assessment of the role of temperature and rainfall on mosquito population dynamics

Abdelrazec

Gumel

2016

J. Math. Biol.

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“…The Hopf-saddle-node bifurcation of equilibria of vector fields has been investigated by several authors [8,18,21,25,28,34,36,41,43,44,57]. Let X α be a C ∞ -family of vector fields on R 3 , where α ∈ R p is a multi-parameter.…”

Section: Dynamics Of Hopf-saddle-node Vector Fieldsmentioning

confidence: 99%

“…Notice that G is not in Poincaré normal form, due to the presence of the non-resonant term ε 2 z 4 . By normal form theory [25,55], there is a transformation such that this term is removed in the new coordinates. We write G in the new coordinates, and restrict to terms of order four in (w, z).…”

Section: Analysis Of a Vector Field Approximationmentioning

confidence: 99%

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Hopf saddle-node bifurcation for fixed points of 3D-diffeomorphisms: Analysis of a resonance ‘bubble’

Broer

Simó

Vitolo

2008

Physica D: Nonlinear Phenomena

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The dynamics near a Hopf-saddle-node bifurcation of fixed points of diffeomorphisms is analysed by means of a case study: a two-parameter model map G is constructed, such that at the central bifurcation the derivative has two complex conjugate eigenvalues of modulus one and one real eigenvalue equal to 1. To investigate the effect of resonances, the complex eigenvalues are selected to have a 1:5 resonance. It is shown that, near the origin of the parameter space, the family G has two secondary Hopf-saddle-node bifurcations of period five points. A cone-like structure exists in the neighbourhood, formed by two surfaces of saddle-node and a surface of Hopf bifurcations. Quasi-periodic bifurcations of an invariant circle, forming a frayed boundary, are numerically shown to occur in model G. Along such Cantor-like boundary, an intricate bifurcation structure is detected near a 1:5 resonance gap. Subordinate quasi-periodic bifurcations are found nearby, suggesting the occurrence of a cascade of quasi-periodic bifurcations.

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Bibliography

2015

Introduction to Nonlinear Oscillations

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Normal Forms and Bifurcation of Planar Vector Fields

Cited by 534 publications

References 0 publications

Mathematical assessment of the role of temperature and rainfall on mosquito population dynamics

Mathematical assessment of the role of temperature and rainfall on mosquito population dynamics

Hopf saddle-node bifurcation for fixed points of 3D-diffeomorphisms: Analysis of a resonance ‘bubble’

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