A multiplicity of localized buckling modes for twisted rod equations

doi:10.1098/rspa.1996.0132

Cited by 49 publications

(19 citation statements)

References 35 publications

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“…Therefore we consider the next nontrivial normal form for (8), drop the constant terms 7] -152c27]2 -~~~c47]3 and divide by~~. This yields…”

Section: Higher Order Analysismentioning

confidence: 99%

“…This is a family of Hamiltonian systems defined by the Hamiltonian function (8) where (.. 1.. ) denotes the standard inner product and A is an external parameter the perturbing force field depends upon. The small parameter e is introduced by a scaling x f--+ eX, y f--+ ey, H f--+ e 2 H, whence (8) describes the dynamics close to the origin.…”

Section: The 3d Henon-heiles Familymentioning

confidence: 99%

“…The family (8) has been normalized twice in [14], yielding a fourth order polynomial in the Xi and Yi' This normal form Hamiltonian commutes with 5 as well as with the quadratic part…”

Section: The 3d Henon-heiles Familymentioning

confidence: 99%

“…The latter two references actually deal with reversible systems, however it is shown in [5,26] that this problem is equivalent to the Hamiltonian Hopf bifurcation. Examples also appear in buckling problems [8,7]. Examples from celestial mechanics are the restricted problem of three bodies [23], where this problem finds its 1 origin, and, more recently, the 3D Henon-Heiles family [14, 16J related to the movement of galaxies.…”

Section: Introductionmentioning

confidence: 99%

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Algebraic Methods for Determining Hamiltonian Hopf Bifurcations in Three-Degree-of-Freedom Systems

Hanßmann

Meer

2005

J Dyn Diff Equat

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When considering bifurcations, the type of bifurcation is usually classified by comparing to standard situations or normal forms. It is shown how Hamiltonian Hopf bifurcations can be determined in three-degree-of-freedom systems, as is done in this paper for the 3D Henon-Heiles family. After a careful formulation of the local once reduced system in terms of properly chosen invariants the system can be compared to the standard form to determine the presence of non-degenerate Hamiltonian Hopf bifurcations.

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“…Therefore we consider the next nontrivial normal form for (8), drop the constant terms 7] -152c27]2 -~~~c47]3 and divide by~~. This yields…”

Section: Higher Order Analysismentioning

confidence: 99%

Section: The 3d Henon-heiles Familymentioning

confidence: 99%

“…The family (8) has been normalized twice in [14], yielding a fourth order polynomial in the Xi and Yi' This normal form Hamiltonian commutes with 5 as well as with the quadratic part…”

Section: The 3d Henon-heiles Familymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Algebraic Methods for Determining Hamiltonian Hopf Bifurcations in Three-Degree-of-Freedom Systems

Hanßmann

Meer

2005

J Dyn Diff Equat

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“…Branching is today well known in a number of disciplines, amongst others in optics [1], granular matter [2], structural mechanics [3][4] [5][6] [7] [8], and mostly in fluid dynamics [1] [9][10] [11][12] [13], and magnetohydrodynamics [14] [15]. The first studies into the topic have probably emerged in the field of binary-fluid convection, where spatially localised convection rolls have been observed in water-ethanol mixtures [9] or helium [10].…”

Section: Introductionmentioning

confidence: 99%

Snaking bifurcations in a self-excited oscillator chain with cyclic symmetry

Papangelo

Grolet

Salles

et al. 2017

Communications in Nonlinear Science and Numerical Simulation

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Snaking bifurcations in a chain of mechanical oscillators are studied. The individual oscillators are weakly nonlinear and subject to self-excitation and subcritical Hopf-bifurcations with some parameter ranges yielding bistability. When the oscillators are coupled to their neighbours, snaking bifurcations result, corresponding to localised vibration states. The snaking patterns do seem to be more complex than in previously studied continuous systems, comprising a plethora of isolated branches and also a large number of similar but not identical states, originating from the weak coupling of the phases of the individual oscillators.

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Spatial chaos of 3‐D elastica with the Kirchhoff gyrostat analogy using Melnikov integrals

Leung

Kuang

2004

Numerical Meth Engineering

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SUMMARYThe Kirchhoff kinetic analogy, from which the similarity between the governing equations for the static spatial equilibrium of a 3-D elastica and those for the temporal dynamics of a rigid body is constituted, is revisited. The Melnikov integrals for detecting chaos cannot be easily formed for an elastica. We shall modify the previous procedure of using the Melnikov integrals for detecting temporal chaos for a gyrostat to solve the spatial chaos problem of an elastica. One way to find the disturbed Hamiltonian equations of the equilibrium equations of the elastica are by means of Deprit's variables. Using the Melnikov integrals, the spatially chaotic deformation patterns can be resulted from the homoclinic transversal intersections of the stable and unstable manifolds at a saddle point in the Poincare map when the elastica is disturbed by constant stress-resultants. The Melnikov integrals are integrated for detecting homoclinic intersections. The equations governing the evolution of the stress-couples and the stress-resultants are numerically integrated using the fourth Runge-Kutta algorithms to crosscheck the analytical results. The phase portraits of the Poincare sections are created on a number of hyper-planes to demonstrate the chaotic patterns of the stationary spatial deformations along the centreline of the elastica under weightless conditions. The bounded, non-periodic solutions to the equilibrium of the elastica clearly show the existence of spatially chaotic deformation patterns. The findings in this paper are useful in many structures in which the elastica are important. Examples are the mathematical modelling of the DNA super-coiling, submarine cables and the design of the tethered satellites.

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A multiplicity of localized buckling modes for twisted rod equations

Abstract: B y A. R. C h a m p n e y s 1 a n d J. M. T. T h o m p s o n 2

Cited by 49 publications

References 35 publications

Algebraic Methods for Determining Hamiltonian Hopf Bifurcations in Three-Degree-of-Freedom Systems

Algebraic Methods for Determining Hamiltonian Hopf Bifurcations in Three-Degree-of-Freedom Systems

Snaking bifurcations in a self-excited oscillator chain with cyclic symmetry

Spatial chaos of 3‐D elastica with the Kirchhoff gyrostat analogy using Melnikov integrals

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