The bifurcation analysis and the Conley index in mechanics

Bolsinov, Alexey V.; Borisov, Adrijan; Mamaev, Ivan S.

doi:10.20537/nd1103017

Cited by 2 publications

(11 citation statements)

References 9 publications

(11 reference statements)

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Considering that N (2) 22 > 0 we conclude that N (2) is positive definite. Therefore, the signature of N is (+ + +−) as stated.…”

Section: 11)mentioning

confidence: 82%

“…In hyperbolic geometry, there are 3 types of non-trivial isometry, known as elliptic, hyperbolic and parabolic transformations and correspondingly 3 types of 1-parameter subgroup of SO (2,1). The elliptic transformations are characterised by having precisely one fixed point in the hyperbolic plane.…”

Section: The Case Of Negative Curvaturementioning

confidence: 99%

“…Moreover, the Casimir function C restricted to the branch of elliptic RE has a maximum at q = q * . 2 This assumption is done without loss of generality since one may eliminate these quantities from the equations of motion (2.8) by rescaling time t → t µ 1 , and the momenta p → p , m → m , where κ = µ 1 Gµ 2 .…”

Section: (I) All Hyperbolic Re Are (Linearly) Unstablementioning

confidence: 99%

“…The Hessian matrix of H −M 2 at such points is given by D 2 H −M 2 (α, q, 0, 0) =: N = N (1) 0 0 N (2) ,…”

Section: (I) All Hyperbolic Re Are (Linearly) Unstablementioning

confidence: 99%

See 3 more Smart Citations

Reduction and relative equilibria for the two-body problem on spaces of constant curvature

Borisov

García-Naranjo

Mamaev

et al. 2018

Celest Mech Dyn Astr

Self Cite

View full text Add to dashboard Cite

We consider the two-body problem on surfaces of constant non-zero curvature and classify the relative equilibria and their stability. On the hyperbolic plane, for each q > 0 we show there are two relative equilibria where the masses are separated by a distance q. One of these is geometrically of elliptic type and the other of hyperbolic type. The hyperbolic ones are always unstable, while the elliptic ones are stable when sufficiently close, but unstable when far apart.On the sphere of positive curvature, if the masses are different, there is a unique relative equilibrium (RE) for every angular separation except π/2. When the angle is acute, the RE is elliptic, and when it is obtuse the RE can be either elliptic or linearly unstable. We show using a KAM argument that the acute ones are almost always nonlinearly stable. If the masses are equal there are two families of relative equilibria: one where the masses are at equal angles with the axis of rotation ('isosceles RE') and the other when the two masses subtend a right angle at the centre of the sphere. The isosceles RE are elliptic if the angle subtended by the particles is acute and is unstable if it is obtuse. At π/2, the two families meet and a pitchfork bifurcation takes place. Right-angled RE are elliptic away from the bifurcation point.In each of the two geometric settings, we use a global reduction to eliminate the group of symmetries and analyse the resulting reduced equations which live on a 5-dimensional phase space and possess one Casimir function.

show abstract

“…Considering that N (2) 22 > 0 we conclude that N (2) is positive definite. Therefore, the signature of N is (+ + +−) as stated.…”

Section: 11)mentioning

confidence: 82%

Section: The Case Of Negative Curvaturementioning

confidence: 99%

Section: (I) All Hyperbolic Re Are (Linearly) Unstablementioning

confidence: 99%

“…The Hessian matrix of H −M 2 at such points is given by D 2 H −M 2 (α, q, 0, 0) =: N = N (1) 0 0 N (2) ,…”

Section: (I) All Hyperbolic Re Are (Linearly) Unstablementioning

confidence: 99%

See 2 more Smart Citations

Reduction and relative equilibria for the two-body problem on spaces of constant curvature

Borisov

García-Naranjo

Mamaev

et al. 2018

Celest Mech Dyn Astr

Self Cite

View full text Add to dashboard Cite

show abstract

“…Количество классических работ по этой тематике весьма велико. В последнее время вопросы об устойчивости таких движений связываются с топологией со-ответствующих интегрируемых систем, отображением момента и бифуркационными диа-граммами, отражающими все особенности слоений фазового пространства (см., например, работы [4,30,29]). В динамике твердого тела особое место занимает класс движений, назы-ваемых равномерными вращениями, в которых вектор угловой скорости тела постоянен в подвижной и неподвижной системах отсчета.…”

Section: исходные соотношения и постановка задачиunclassified

Phase topology of the Kowalevski–Sokolov top

Ryabov¹,

Savushkin²

2015

Nelin. Dinam.

View full text Add to dashboard Cite

The bifurcation analysis and the Conley index in mechanics

Cited by 2 publications

References 9 publications

Reduction and relative equilibria for the two-body problem on spaces of constant curvature

Reduction and relative equilibria for the two-body problem on spaces of constant curvature

Phase topology of the Kowalevski–Sokolov top

Contact Info

Product

Resources

About