Singularities of reversible vector fields

Teixeira, Marco Antônio

doi:10.1016/s0167-2789(96)00183-2

Cited by 39 publications

(45 citation statements)

References 4 publications

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“…[28,13]. For a = 2 the reversible system (18) is Hamiltonian, and as long as the coefficient a is positive this Hamiltonian nature prevails.…”

Section: Discussionmentioning

confidence: 99%

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Quasi-periodic bifurcations in reversible systems

Hanßmann

2010

Regul. Chaot. Dyn.

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Invariant tori of integrable dynamical systems occur both in the dissipative and in the conservative context, but only in the latter the tori are parametrised by phase space variables. This allows for quasi-periodic bifurcations within a single given system, induced by changes of the normal behaviour of the tori. It turns out that in a non-degenerate reversible system all semi-local bifurcations of co-dimension 1 persist, under small non-integrable perturbations, on large Cantor sets.

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“…[28,13]. For a = 2 the reversible system (18) is Hamiltonian, and as long as the coefficient a is positive this Hamiltonian nature prevails.…”

Section: Discussionmentioning

confidence: 99%

“…Then at y = y 0 a pair of tori (15) bifurcates off from T n × R m × {(0, 0)}, elliptic in the supercritical case a(y 0 ) · b(y 0 ) > 0 and hyperbolic in the subcritical case a(y 0 ) · b(y 0 [18,28].…”

Section: Pitchfork Bifurcationsmentioning

confidence: 99%

Quasi-periodic bifurcations in reversible systems

Hanßmann

2010

Regul. Chaot. Dyn.

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“…Here ξ 2 undergoes a pitchfork bifurcation giving rise to 2 saddles and turning itself into a saddle centre. In this case centre manifold theory allows to follow the evolution of small bifurcating solutions in a family of reversible planar vector fields and it is well-known that such a reversible pitchfork bifurcation generically gives rise to a heteroclinic cycle (see for instance [18]). …”

Section: Heteroclinic Orbits For F − (X µ κ)mentioning

confidence: 99%

Bifurcation of a reversible Hamiltonian system from a fixed point with fourfold eigenvalue zero

Wagenknecht¹

2002

Dynamical Systems

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We study bifurcations from a fixed point with fourfold eigenvalue zero occurring in a two degrees of freedom Hamiltonian system of second order ODEs which is additionally reversible with respect to two different linear involutions. Using techniques from Catastrophe Theory we are led to a codimension 2 problem and obtain two different unfoldings of the singularity related to the hyperbolic and elliptic umbilic, respectively.The analysis of the unfolded systems is essentially concerned with the existence and properties of homoclinic and heteroclinic orbits. Our studies are motivated by a problem from nonlinear optics concerning the existence of solitons in a χ 2 -medium.

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“…See for example in Buzzi et al (2009) andTeixeira (1997). It allows to always fix the involution as the following:…”

Section: Introductionmentioning

confidence: 99%

Reversible-equivariant systems and matricial equations

Teixeira

Martins

2011

An. Acad. Bras. Ciênc.

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This paper uses tools in group theory and symbolic computing to classify the representations of finite groups with order lower than, or equal to 9 that can be derived from the study of local reversible-equivariant vector fields in R 4 . The results are obtained by solving matricial equations. In particular, we exhibit the involutions used in a local study of reversible-equivariant vector fields. Based on such approach we present, for each element in this class, a simplified Belitskii normal form.

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Singularities of reversible vector fields

Cited by 39 publications

References 4 publications

Quasi-periodic bifurcations in reversible systems

Quasi-periodic bifurcations in reversible systems

Bifurcation of a reversible Hamiltonian system from a fixed point with fourfold eigenvalue zero

Reversible-equivariant systems and matricial equations

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