104 results on '"Holger R. Dullin"'
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2. A New Twisting Somersault: 513XD.
- Author
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William Tong and Holger R. Dullin
- Published
- 2017
- Full Text
- View/download PDF
3. Instability of Equilibria for the Two-Dimensional Euler Equations on the Torus.
- Author
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Holger R. Dullin, Robert Marangell, and Joachim Worthington
- Published
- 2016
- Full Text
- View/download PDF
4. Generating Hyperbolic Singularities in Semitoric Systems Via Hopf Bifurcations.
- Author
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Holger R. Dullin and álvaro Pelayo
- Published
- 2016
- Full Text
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5. Twisting Somersault.
- Author
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Holger R. Dullin and William Tong
- Published
- 2016
- Full Text
- View/download PDF
6. On the C 8/3-regularisation of simultaneous binary collisions in the planar four-body problem
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Holger R. Dullin and Nathan Duignan
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Planar ,Applied Mathematics ,Mathematical analysis ,General Physics and Astronomy ,Binary number ,Statistical and Nonlinear Physics ,Mathematical Physics ,Mathematics - Abstract
The dynamics of the four-body problem allows for two binary collisions to occur simultaneously. It is known that in the collinear four-body problem this simultaneous binary collision (SBC) can be block-regularised, but that the resulting block map is only C 8/3 differentiable. In this paper, it is proved that the C 8/3 differentiability persists for the SBC in the planar four-body problem. The proof uses several geometric tools, namely, blow-up, normal forms, dynamics near normally hyperbolic manifolds of equilibrium points, and Dulac maps.
- Published
- 2021
7. The Equilateral Pentagon at Zero Angular Momentum: Maximal Rotation through Optimal Deformation.
- Author
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William Tong and Holger R. Dullin
- Published
- 2012
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8. Resonances and Twist in Volume-Preserving Mappings.
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Holger R. Dullin and James D. Meiss
- Published
- 2012
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- View/download PDF
9. Quadratic Volume-Preserving Maps: Invariant Circles and Bifurcations.
- Author
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Holger R. Dullin and James D. Meiss
- Published
- 2009
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10. A Poincaré Section for the General Heavy Rigid Body.
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Sven Schmidt, Holger R. Dullin, and Peter H. Richter
- Published
- 2009
- Full Text
- View/download PDF
11. Instability of unidirectional flows for the 2D α-Euler equations
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Shibi Vasudevan, Holger R. Dullin, Joachim Worthington, Robert Marangell, and Yuri Latushkin
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Physics ,Direct sum ,Computer Science::Information Retrieval ,Applied Mathematics ,010102 general mathematics ,Essential spectrum ,Zero (complex analysis) ,Torus ,General Medicine ,Characterization (mathematics) ,01 natural sciences ,Instability ,Euler equations ,010101 applied mathematics ,Combinatorics ,symbols.namesake ,symbols ,0101 mathematics ,Analysis ,Eigenvalues and eigenvectors - Abstract
We study stability of unidirectional flows for the linearized 2D \begin{document}$ \alpha $\end{document} -Euler equations on the torus. The unidirectional flows are steady states whose vorticity is given by Fourier modes corresponding to a vector \begin{document}$ \mathbf p \in \mathbb Z^{2} $\end{document} . We linearize the \begin{document}$ \alpha $\end{document} -Euler equation and write the linearized operator \begin{document}$ L_{B} $\end{document} in \begin{document}$ \ell^{2}(\mathbb Z^{2}) $\end{document} as a direct sum of one-dimensional difference operators \begin{document}$ L_{B,\mathbf q} $\end{document} in \begin{document}$ \ell^{2}(\mathbb Z) $\end{document} parametrized by some vectors \begin{document}$ \mathbf q\in\mathbb Z^2 $\end{document} such that the set \begin{document}$ \{\mathbf q +n \mathbf p:n \in \mathbb Z\} $\end{document} covers the entire grid \begin{document}$ \mathbb Z^{2} $\end{document} . The set \begin{document}$ \{\mathbf q +n \mathbf p:n \in \mathbb Z\} $\end{document} can have zero, one, or two points inside the disk of radius \begin{document}$ \|\mathbf p\| $\end{document} . We consider the case where the set \begin{document}$ \{\mathbf q +n \mathbf p:n \in \mathbb Z\} $\end{document} has exactly one point in the open disc of radius \begin{document}$ \mathbf p $\end{document} . We show that unidirectional flows that satisfy this condition are linearly unstable. Our main result is an instability theorem that provides a necessary and sufficient condition for the existence of a positive eigenvalue to the operator \begin{document}$ L_{B, {\mathbf q}} $\end{document} in terms of equations involving certain continued fractions. Moreover, we are also able to provide a complete characterization of the corresponding eigenvector. The proof is based on the use of continued fractions techniques expanding upon the ideas of Friedlander and Howard.
- Published
- 2020
12. Regularisation for Planar vector fields
- Author
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Holger R. Dullin and Nathan Duignan
- Subjects
Dynamical systems theory ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Degenerate energy levels ,General Physics and Astronomy ,Statistical and Nonlinear Physics ,Dynamical Systems (math.DS) ,Rational function ,37C10, 37C15, 37C25, 70F10, 70F15 ,01 natural sciences ,010101 applied mathematics ,Quadratic equation ,Singularity ,FOS: Mathematics ,Vector field ,Gravitational singularity ,Differentiable function ,Mathematics - Dynamical Systems ,0101 mathematics ,Mathematical Physics ,Mathematics - Abstract
This paper serves as a first foray on regularisation for planar vector fields. Motivated by singularities in celestial mechanics, the block regularisation of a generic class of degenerate singularities is studied. The paper is concerned with asymptotic properties of the transition map between a section before and after the singularity. Block regularisation is reviewed before topological and explicit conditions for the $ C^0 $-regularity of the map are given. Computation of the $ C^1 $-regularisation is reduced to summing residues of a rational function. It is shown that the transition map is in general only finitely differentiable and a method of computing the map is conveyed. In particular, a perturbation of a toy example derived from the 4-body problem is shown to be $ C^{4/3} $. The regularisation of all homogeneous quadratic vector fields is computed., Comment: 30 pages, 7 figures, Preprint
- Published
- 2019
13. Extended Phase Diagram of the Lorenz Model.
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Holger R. Dullin, Sven Schmidt, Peter H. Richter, and S. K. Grossmann
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- 2007
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14. Symbolic Codes for Rotational Orbits.
- Author
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Holger R. Dullin, James D. Meiss, and David G. Sterling
- Published
- 2005
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15. Self-stabilization of light sails by damped internal degrees of freedom
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M. Z. Rafat, Holger R. Dullin, Boris T. Kuhlmey, Alessandro Tuniz, Haoyuan Luo, Dibyendu Roy, Sean Skinner, Tristram J. Alexander, Michael S. Wheatland, and C. Martijn de Sterke
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Physics::Space Physics ,Classical Physics (physics.class-ph) ,FOS: Physical sciences ,General Physics and Astronomy ,Physics - Classical Physics ,Optics (physics.optics) ,Physics - Optics - Abstract
We consider the motion of a light sail that is accelerated by a powerful laser beam. We derive the equations of motion for two proof-of-concept sail designs with damped internal degrees of freedom. Using linear stability analysis we show that perturbations of the sail movement in all lateral degrees of freedom can be damped passively. This analysis also shows complicated behaviour akin to that associated with exceptional points in PT-symmetric systems in optics and quantum mechanics. The excess heat that is produced by the damping mechanism is likely to be substantially smaller than the expected heating due to the partial absorption of the incident laser beam by the sail., 7 figures
- Published
- 2021
16. Taylor series and twisting-index invariants of coupled spin–oscillators
- Author
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Holger R. Dullin, Sonja Hohloch, and Jaume Alonso
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Pure mathematics ,FOS: Physical sciences ,General Physics and Astronomy ,Zero-point energy ,Dynamical Systems (math.DS) ,01 natural sciences ,symbols.namesake ,0103 physical sciences ,FOS: Mathematics ,Taylor series ,Mathematics - Dynamical Systems ,0101 mathematics ,Invariant (mathematics) ,Mathematical Physics ,Mathematics ,Nonlinear Sciences - Exactly Solvable and Integrable Systems ,Physics ,010102 general mathematics ,37J05, 37J15, 37J35, 53D20, 70H05, 70H06 ,Mathematics - Symplectic Geometry ,symbols ,Symplectic Geometry (math.SG) ,Gravitational singularity ,010307 mathematical physics ,Geometry and Topology ,Exactly Solvable and Integrable Systems (nlin.SI) - Abstract
About six years ago, semitoric systems on 4-dimensional manifolds were classified by Pelayo & Vu Ngoc by means of five invariants. A standard example of such a system is the coupled spin-oscillator on $\mathbb{S}^2 \times \mathbb{R}^2$. Calculations of three of the five semitoric invariants of this system (namely the number of focus-focus singularities, the generalised semitoric polygon, and the height invariant) already appeared in the literature, but the so-called twisting index was not yet computed and, of the so-called Taylor series invariant, only the linear terms were known. In the present paper, we complete the list of invariants for the coupled spin-oscillator by calculating higher order terms of the Taylor series invariant and by computing the twisting index. Moreover, we prove that the Taylor series invariant has certain symmetry properties that make the even powers in one of the variables vanish and allow us to show superintegrability of the coupled spin-oscillator on the zero energy level., 38 pages, 6 figures. Minor details of the exposition made clearer and small sign mistake in intermediate steps fixed
- Published
- 2019
17. Relative equilibria of the 3-body problem in $\mathbb{R}^4$
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Alain Albouy, Holger R. Dullin, Institut de Mécanique Céleste et de Calcul des Ephémérides (IMCCE), Institut national des sciences de l'Univers (INSU - CNRS)-Observatoire de Paris, and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Université de Lille-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Lyapunov function ,Angular momentum ,Control and Optimization ,Rank (linear algebra) ,Motion (geometry) ,FOS: Physical sciences ,Dynamical Systems (math.DS) ,Physics - Classical Physics ,01 natural sciences ,symbols.namesake ,Dimension (vector space) ,[MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph] ,FOS: Mathematics ,0101 mathematics ,Mathematics - Dynamical Systems ,Mathematical Physics ,Mathematics ,Lyapunov stability ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Classical Physics (physics.class-ph) ,Mathematical Physics (math-ph) ,Three-body problem ,010101 applied mathematics ,37N05, 70F10, 70F15, 70H33, 53D20 ,Mechanics of Materials ,Bounded function ,symbols ,Geometry and Topology - Abstract
The classical equations of the Newtonian 3-body problem do not only define the familiar 3-dimensional motions. The dimension of the motion may also be 4, and cannot be higher. We prove that in dimension 4, for three arbitrary positive masses, and for an arbitrary value (of rank 4) of the angular momentum, the energy possesses a minimum, which corresponds to a motion of relative equilibrium which is Lyapunov stable when considered as an equilibrium of the reduced problem. The nearby motions are nonsingular and bounded for all time. We also describe the full family of relative equilibria, and show that its image by the energy-momentum map presents cusps and other interesting features., 18 pages, 6 figures
- Published
- 2020
18. Monodromy in Prolate Spheroidal Harmonics
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Diana M. H. Nguyen, Holger R. Dullin, and Sean R. Dawson
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Physics ,Integrable system ,Applied Mathematics ,010102 general mathematics ,Spectrum (functional analysis) ,37J15, 37J35, 53D20, 53D22, 81Q20 ,FOS: Physical sciences ,Mathematical Physics (math-ph) ,Dynamical Systems (math.DS) ,Eigenfunction ,Mathematics::Spectral Theory ,Quantum number ,01 natural sciences ,010305 fluids & plasmas ,Monodromy ,Special functions ,0103 physical sciences ,Quantum system ,FOS: Mathematics ,0101 mathematics ,Mathematics - Dynamical Systems ,Wave function ,Mathematical Physics ,Mathematical physics - Abstract
We show that spheroidal wave functions viewed as the essential part of the joint eigenfunction of two commuting operators of $L_2(S^2)$ has a defect in the joint spectrum that makes a global labelling of the joint eigenfunctions by quantum numbers impossible. To our knowledge this is the first explicit demonstration that quantum monodromy exists in a class of classically known special functions. Using an analogue of the Laplace-Runge-Lenz vector we show that the corresponding classical Liouville integrable system is symplectically equivalent to the C. Neumann system. To prove the existence of this defect we construct a classical integrable system that is the semi-classical limit of the quantum integrable system of commuting operators. We show that this is a semi-toric system with a non-degenerate focus-focus point, such that there is monodromy in the classical and the quantum system., 26 pages, 11 figures
- Published
- 2020
19. Symplectic classification of coupled angular momenta
- Author
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Sonja Hohloch, Jaume Alonso, and Holger R. Dullin
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Integrable system ,FOS: Physical sciences ,General Physics and Astronomy ,Dynamical Systems (math.DS) ,01 natural sciences ,Hamiltonian system ,symbols.namesake ,FOS: Mathematics ,Taylor series ,Mathematics - Dynamical Systems ,0101 mathematics ,Invariant (mathematics) ,Mathematical Physics ,Mathematical physics ,Mathematics ,Hopf bifurcation ,Nonlinear Sciences - Exactly Solvable and Integrable Systems ,Applied Mathematics ,Physics ,010102 general mathematics ,Degenerate energy levels ,37J05, 37J15, 37J35, 53D20, 70H05, 70H06 ,Statistical and Nonlinear Physics ,010101 applied mathematics ,Mathematics - Symplectic Geometry ,Phase space ,symbols ,Symplectic Geometry (math.SG) ,Exactly Solvable and Integrable Systems (nlin.SI) ,Symplectic geometry - Abstract
The coupled angular momenta are a family of completely integrable systems that depend on three parameters and have a compact phase space. They correspond to the classical version of the coupling of two quantum angular momenta and they constitute one of the fundamental examples of so-called semitoric systems. Pelayo & Vu Ngoc have given a classification of semitoric systems in terms of five symplectic invariants. Three of these invariants have already been partially calculated in the literature for a certain parameter range, together with the linear terms of the so-called Taylor series invariant for a fixed choice of parameter values. In the present paper we complete the classification by calculating the polygon invariant, the height invariant, the twisting-index invariant, and the higher-order terms of the Taylor series invariant for the whole family of systems. We also analyse the explicit dependence of the coefficients of the Taylor series with respect to the three parameters of the system, in particular near the Hopf bifurcation where the focus-focus point becomes degenerate., 59 pages, 16 figures
- Published
- 2020
20. Symmetry reduction of the 3-body problem in <tex-math id='M1'>\begin{document}$ \mathbb{R}^4 $\end{document}</tex-math>
- Author
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Jürgen Scheurle and Holger R. Dullin
- Subjects
Physics ,Lyapunov stability ,Angular momentum ,Control and Optimization ,Computer Science::Information Retrieval ,Applied Mathematics ,Symmetry reduction ,Three-body problem ,Combinatorics ,Maxima and minima ,symbols.namesake ,Mechanics of Materials ,Phase space ,symbols ,Geometry and Topology ,Hamiltonian (quantum mechanics) ,Symplectic geometry - Abstract
The 3-body problem in \begin{document}$ \mathbb{R}^4 $\end{document} has 24 dimensions and is invariant under translations and rotations. We do the full symplectic symmetry reduction and obtain a reduced Hamiltonian in local symplectic coordinates on a reduced phase space with 8 dimensions. The Hamiltonian depends on two parameters \begin{document}$ \mu_1 > \mu_2 \ge 0 $\end{document} , related to the conserved angular momentum. The limit \begin{document}$ \mu_2 \to 0 $\end{document} corresponds to the 3-dimensional limit. We show that the reduced Hamiltonian has three relative equilibria that are local minima and hence Lyapunov stable when \begin{document}$ \mu_2 $\end{document} is sufficiently small. This proves the existence of balls of initial conditions of full dimension that do not contain any orbits that are unbounded.
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- 2020
21. A Lagrangian Fibration of the Isotropic 3-Dimensional Harmonic Oscillator with Monodromy
- Author
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Holger Waalkens, Holger R. Dullin, Konstantinos Efstathiou, Irina Chiscop, and Dynamical Systems, Geometry & Mathematical Physics
- Subjects
Coordinate system ,QUANTUM PHASE-TRANSITIONS ,HYDROGEN-ATOM ,01 natural sciences ,CLASSIFICATION ,Hamiltonian system ,HAMILTONIAN-SYSTEMS ,NORMALIZATION ,symbols.namesake ,0103 physical sciences ,0101 mathematics ,Mathematics::Symplectic Geometry ,Mathematical Physics ,Harmonic oscillator ,Mathematical physics ,Physics ,010102 general mathematics ,Fibration ,Statistical and Nonlinear Physics ,Prolate spheroidal coordinates ,RESONANCE ,Quantum number ,PERTURBATIONS ,Monodromy ,symbols ,010307 mathematical physics ,Hamiltonian (quantum mechanics) - Abstract
The isotropic harmonic oscillator in dimension 3 separates in several different coordinate systems. Separating in a particular coordinate system defines a system of three Poisson commuting integrals and, correspondingly, three commuting operators, one of which is the Hamiltonian. We show that the Lagrangian fibration defined by the Hamiltonian, the z component of the angular momentum, and a quartic integral obtained from separation in prolate spheroidal coordinates has a non-degenerate focus-focus point, and hence, non-trivial Hamiltonian monodromy for sufficiently large energies. The joint spectrum defined by the corresponding commuting quantum operators has non-trivial quantum monodromy implying that one cannot globally assign quantum numbers to the joint spectrum. Published under license by AIP Publishing.
- Published
- 2019
22. Poisson Structure of the Three-Dimensional Euler Equations in Fourier Space
- Author
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Holger R. Dullin, Joachim Worthington, and James D. Meiss
- Subjects
Statistics and Probability ,General Physics and Astronomy ,FOS: Physical sciences ,Dynamical Systems (math.DS) ,Poisson distribution ,01 natural sciences ,010305 fluids & plasmas ,symbols.namesake ,Simple (abstract algebra) ,Poisson manifold ,Stability theory ,0103 physical sciences ,FOS: Mathematics ,0101 mathematics ,Mathematics - Dynamical Systems ,Mathematical Physics ,Mathematics ,Hamiltonian mechanics ,010102 general mathematics ,Mathematical analysis ,Statistical and Nonlinear Physics ,Mathematical Physics (math-ph) ,Euler equations ,Fourier transform ,Modeling and Simulation ,symbols ,Hamiltonian (control theory) - Abstract
We derive a simple Poisson structure in the space of Fourier modes for the vorticity formulation of the Euler equations on a three-dimensional periodic domain. This allows us to analyse the structure of the Euler equations using a Hamiltonian framework. The Poisson structure is valid on the divergence free subspace only, and we show that using a projection operator it can be extended to be valid in the full space. We then restrict the simple Poisson structure to the divergence-free subspace on which the dynamics of the Euler equations take place, reducing the size of the system of ODEs by a third. The projected and the restricted Poisson structures are shown to have the helicity as a Casimir invariant. We conclude by showing that periodic shear flows in three dimensions are equilibria that correspond to singular points of the projected Poisson structure, and hence that the usual approach to study their nonlinear stability through the Energy-Casimir method fails.
- Published
- 2018
23. A New Twisting Somersault
- Author
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William Tong and Holger R. Dullin
- Published
- 2018
24. Scattering invariants in Euler's two-center problem
- Author
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Holger R. Dullin, Holger Waalkens, Konstantinos Efstathiou, N. Martynchuk, and Dynamical Systems, Geometry & Mathematical Physics
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Integrable system ,General Physics and Astronomy ,FOS: Physical sciences ,01 natural sciences ,CLASSIFICATION ,Hamiltonian system ,Action-angle coordinates ,INTEGRABLE HAMILTONIAN-SYSTEMS ,symbols.namesake ,scattering map ,0101 mathematics ,MONODROMY ,Mathematical Physics ,NEIGHBORHOODS ,Mathematical physics ,Mathematics ,Liouville integrability ,Scattering ,Applied Mathematics ,010102 general mathematics ,37J35, 34L25, 57R22, 70F99, 70H05 ,Statistical and Nonlinear Physics ,Mathematical Physics (math-ph) ,FREEDOM ,Celestial mechanics ,scattering monodromy ,010101 applied mathematics ,Monodromy ,Euler's formula ,symbols ,Scattering theory ,FOCUS-FOCUS - Abstract
The problem of two fixed centers was introduced by Euler as early as in 1760. It plays an important role both in celestial mechanics and in the microscopic world. In the present paper we study the spatial problem in the case of arbitrary (both positive and negative) strengths of the centers. Combining techniques from scattering theory and Liouville integrability, we show that this spatial problem has topologically non-trivial scattering dynamics, which we identify as scattering monodromy. The approach that we introduce in this paper applies more generally to scattering systems that are integrable in the Liouville sense.
- Published
- 2018
- Full Text
- View/download PDF
25. Using the Geometric Phase to Optimise Planar Somersaults
- Author
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William Tong and Holger R. Dullin
- Subjects
Angular momentum ,Computer science ,Applied Mathematics ,Mathematical analysis ,70E55, 70E15, 74A99, 93B99, 53Z05 ,Phase (waves) ,Process (computing) ,Equations of motion ,Classical Physics (physics.class-ph) ,FOS: Physical sciences ,Physics - Classical Physics ,Dynamical Systems (math.DS) ,01 natural sciences ,010305 fluids & plasmas ,010101 applied mathematics ,Planar ,Geometric phase ,0103 physical sciences ,FOS: Mathematics ,Reversing ,Mathematics - Dynamical Systems ,0101 mathematics ,Rotation (mathematics) - Abstract
We derive the equations of motion for the planar somersault, which consist of two additive terms. The first is the dynamic phase that is proportional to the angular momentum, and the second is the geometric phase that is independent of angular momentum and depends solely on the details of the shape change. Next, we import digitised footage of an elite athlete performing 3.5 forward somersaults off the 3m springboard, and use the data to validate our model. We show that reversing and reordering certain sections of the digitised dive can maximise the geometric phase without affecting the dynamic phase, thereby increasing the overall rotation achieved. Finally, we propose a theoretical planar somersault consisting of four shape changing states, where the optimisation lies in finding the shape change strategy that maximises the overall rotation of the dive. This is achieved by balancing the rotational contributions from the dynamic and geometric phases, in which we show the geometric phase plays a small but important role in the optimisation process., Comment: 21 pages, 14 figures, work from PhD thesis of William Tong
- Published
- 2018
- Full Text
- View/download PDF
26. Semi-global symplectic invariants of the spherical pendulum
- Author
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Holger R. Dullin
- Subjects
Pure mathematics ,Nonlinear Sciences - Exactly Solvable and Integrable Systems ,Applied Mathematics ,Spherical pendulum ,Mathematical analysis ,Classical Physics (physics.class-ph) ,FOS: Physical sciences ,Theta function ,Physics - Classical Physics ,Dynamical Systems (math.DS) ,37J35, 37J15, 37J40, 70H06, 70H08, 37G20 ,Action (physics) ,Mathematics - Symplectic Geometry ,FOS: Mathematics ,Vanishing cycle ,Symplectic Geometry (math.SG) ,Elliptic integral ,Point (geometry) ,Mathematics - Dynamical Systems ,Exactly Solvable and Integrable Systems (nlin.SI) ,Analysis ,Hamiltonian (control theory) ,Mathematics ,Symplectic geometry - Abstract
We explicitly compute the semi-global symplectic invariants near the focus-focus point of the spherical pendulum. A modified Birkhoff normal form procedure is presented to compute the expansion of the Hamiltonian near the unstable equilibrium point in Eliasson-variables. Combining this with explicit formulas for the action we find the semi-global symplectic invariants near the focus-focus point introduced by Vu Ngoc 2003. We also show that the Birkhoff normal form is the inverse of a complete elliptic integral over a vanishing cycle. To our knowledge this is the first time that semi-global symplectic invariants near a focus-focus point have been computed explicitly. We close with some remarks about the pendulum, for which the invariants can be related to theta functions in a beautiful way., Comment: 27 pages, 2 figures
- Published
- 2013
27. Semi-global symplectic invariants of the Euler top
- Author
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George Papadopoulos and Holger R. Dullin
- Subjects
Pure mathematics ,Mathematics::Dynamical Systems ,Control and Optimization ,Applied Mathematics ,Regular solution ,FOS: Physical sciences ,Inverse ,Mathematical Physics (math-ph) ,Picard–Fuchs equation ,Semi global ,symbols.namesake ,Mathematics - Symplectic Geometry ,Mechanics of Materials ,FOS: Mathematics ,Euler's formula ,symbols ,Symplectic Geometry (math.SG) ,Geometry and Topology ,Invariant (mathematics) ,70E17, 70E15 ,Mathematics::Symplectic Geometry ,Mathematical Physics ,Symplectic geometry ,Mathematics ,Hyperbolic equilibrium point - Abstract
We compute the semi-global symplectic invariants near the hyperbolic equilibrium points of the Euler top. The Birkhoff normal form at the hyperbolic point is computed using Lie series. The actions near the hyperbolic point are found using Frobenius expansion of its Picard-Fuchs equation. We show that the Birkhoff normal form can also be found by inverting the regular solution of the Picard-Fuchs equation. Composition of the singular action integral with the Birkhoff normal form gives the semi-global symplectic invariant. Finally, we discuss the convergence of these invariants and show that in a neighbourhood of the separatrix the pendulum is not symplectically equivalent to any Euler top., 18 pages, 4 figures, PDFLaTeX, submitted to The Journal of Geometric Mechanics (JGM)
- Published
- 2013
28. Stability Results for Idealised Shear Flows on a Rectangular Periodic Domain
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Holger R. Dullin and Joachim Worthington
- Subjects
Physics ,Applied Mathematics ,010102 general mathematics ,Torus ,Dynamical Systems (math.DS) ,Stability result ,Condensed Matter Physics ,01 natural sciences ,Omega ,Euler equations ,010101 applied mathematics ,Computational Mathematics ,symbols.namesake ,symbols ,Euler's formula ,Stationary flow ,FOS: Mathematics ,Physics::Atomic Physics ,0101 mathematics ,Mathematics - Dynamical Systems ,Hamiltonian (quantum mechanics) ,Mathematical Physics ,Mathematical physics ,Linear stability - Abstract
We present a new linearly stable solution of the Euler fluid flow on a torus. On a two-dimensional rectangular periodic domain $$[0,2\pi )\times [0,2\pi / \kappa )$$ for $$\kappa \in \mathbb {R}^+$$ , the Euler equations admit a family of stationary solutions given by the vorticity profiles $$\Omega ^*(\mathbf {x})= \Gamma \cos (p_1x_1+ \kappa p_2x_2)$$ . We show linear stability for such flows when $$p_2=0$$ and $$\kappa \ge |p_1|$$ (equivalently $$p_1=0$$ and $$\kappa {|p_2|}\le {1}$$ ). The classical result due to Arnold is that for $$p_1 = 1, p_2 = 0$$ and $$\kappa \ge 1$$ the stationary flow is nonlinearly stable via the energy-Casimir method. We show that for $$\kappa \ge |p_1| \ge 2, p_2 = 0$$ the flow is linearly stable, but one cannot expect a similar nonlinear stability result. Finally we prove nonlinear instability for all steady states satisfying $$p_1^2+\kappa ^2{p_2^2}>\frac{{3(\kappa ^2+1)}}{4(7-4\sqrt{3})}$$ . The modification and application of a structure-preserving Hamiltonian truncation is discussed for the anisotropic case $$\kappa \ne 1$$ . This leads to an explicit Lie-Poisson integrator for the approximate system, which is used to illustrate our analytical results.
- Published
- 2016
29. A New Twisting Somersault - 513XD
- Author
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Holger R. Dullin and William Tong
- Subjects
Sequence ,Computer science ,Applied Mathematics ,70E55, 70E15, 74A99, 93B99, 53Z05 ,General Engineering ,Motion (geometry) ,Equations of motion ,Classical Physics (physics.class-ph) ,FOS: Physical sciences ,Physics - Classical Physics ,Dynamical Systems (math.DS) ,Space (mathematics) ,01 natural sciences ,010305 fluids & plasmas ,symbols.namesake ,Classical mechanics ,Modeling and Simulation ,0103 physical sciences ,Euler's formula ,symbols ,FOS: Mathematics ,Mathematics - Dynamical Systems ,010306 general physics - Abstract
We present the mathematical framework of an athlete modelled as a system of coupled rigid bodies to simulate platform and springboard diving. Euler's equations of motion are generalised to non-rigid bodies, and are then used to innovate a new dive sequence that in principle can be performed by real world athletes. We begin by assuming shape changes are instantaneous so that the equations of motion simplify enough to be solved analytically, and then use this insight to present a new dive (513XD) consisting of 1.5 somersaults and 5 twists using realistic shape changes. Finally, we demonstrate the phenomenon of converting pure somersaulting motion into pure twisting motion by using a sequence of impulsive shape changes, which may have applications in other fields such as space aeronautics., Comment: 26 pages, 17 figures, work from PhD thesis of William Tong
- Published
- 2016
- Full Text
- View/download PDF
30. Defect in the Joint Spectrum of Hydrogen due to Monodromy
- Author
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Holger R. Dullin, Holger Waalkens, and Dynamical Systems, Geometry & Mathematical Physics
- Subjects
Quantum Physics ,Nonlinear Sciences - Exactly Solvable and Integrable Systems ,010102 general mathematics ,General Physics and Astronomy ,FOS: Physical sciences ,Mathematical Physics (math-ph) ,Prolate spheroidal coordinates ,Quantum number ,Magnetic quantum number ,01 natural sciences ,37J35, 81Q20 ,Azimuthal quantum number ,Quantization (physics) ,Monodromy ,Quantum mechanics ,Quantum process ,0103 physical sciences ,Principal quantum number ,0101 mathematics ,Exactly Solvable and Integrable Systems (nlin.SI) ,010306 general physics ,Quantum Physics (quant-ph) ,Mathematical Physics ,Mathematics - Abstract
In addition to the well known case of spherical coordinates the hydrogen atom separates in three further coordinate systems. Separating in a particular coordinate system defines a system of three commuting operators. We show that the joint spectrum of the Hamilton operator, and the $z$-components of the angular momentum and quantum Laplace-Runge-Lenz vectors obtained from separation in prolate spheroidal coordinates has quantum monodromy for energies sufficiently close to the ionization threshold. This means that one cannot globally assign quantum numbers to the joint spectrum. Whereas the principal quantum number $n$ and the magnetic quantum number $m$ correspond to the Bohr-Sommerfeld quantization of globally defined classical actions a third quantum number cannot be globally defined because the third action is globally multi valued., Comment: 5 pages, 5 figures
- Published
- 2016
- Full Text
- View/download PDF
31. Symmetry reduction by lifting for maps
- Author
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Héctor E. Lomelí, James D. Meiss, and Holger R. Dullin
- Subjects
Pure mathematics ,FOS: Physical sciences ,General Physics and Astronomy ,01 natural sciences ,010305 fluids & plasmas ,symbols.namesake ,0103 physical sciences ,0101 mathematics ,Invariant (mathematics) ,Mathematical Physics ,Poincaré map ,Mathematics ,Nonlinear Sciences - Exactly Solvable and Integrable Systems ,Applied Mathematics ,010102 general mathematics ,Statistical and Nonlinear Physics ,Symmetry reduction ,Nonlinear Sciences - Chaotic Dynamics ,Homogeneous space ,symbols ,37C80 ,Chaotic Dynamics (nlin.CD) ,Exactly Solvable and Integrable Systems (nlin.SI) ,Noether's theorem ,Curse of dimensionality ,Symplectic geometry - Abstract
We study diffeomorphisms that have one-parameter families of continuous symmetries. For general maps, in contrast to the symplectic case, existence of a symmetry no longer implies existence of an invariant. Conversely, a map with an invariant need not have a symmetry. We show that when a symmetry flow has a global Poincar\'{e} section there are coordinates in which the map takes a reduced, skew-product form, and hence allows for reduction of dimensionality. We show that the reduction of a volume-preserving map again is volume preserving. Finally we sharpen the Noether theorem for symplectic maps. A number of illustrative examples are discussed and the method is compared with traditional reduction techniques., Comment: laTeX, 31 pages, 5 figures
- Published
- 2012
32. The 1:±2 resonance
- Author
-
Richard Cushman, Sven Schmidt, Heinz Hanßmann, and Holger R. Dullin
- Subjects
Surface (mathematics) ,Physics ,Equilibrium point ,symbols.namesake ,Mathematics (miscellaneous) ,Monodromy ,Critical point (thermodynamics) ,Mathematical analysis ,symbols ,Torus ,Hamiltonian (quantum mechanics) ,Harmonic oscillator ,Hamiltonian system - Abstract
On the linear level elliptic equilibria of Hamiltonian systems are mere superpositions of harmonic oscillators. Non-linear terms can produce instability, if the ratio of frequencies is rational and the Hamiltonian is indefinite. In this paper we study the frequency ratio ±1/2 and its unfolding. In particular we show that for the indefinite case (1:−2) the frequency ratio map in a neighborhood of the origin has a critical point, i.e. the twist condition is violated for one torus on every energy surface near the energy of the equilibrium. In contrast, we show that the frequency map itself is non-degenerate (i.e. the Kolmogorov non-degeneracy condition holds) for every torus in a neighborhood of the equilibrium point. As a by product of our analysis of the frequency map we obtain another proof of fractional monodromy in the 1:−2 resonance.
- Published
- 2007
33. Geodesic flow on three-dimensional ellipsoids with equal semi-axes
- Author
-
Holger R. Dullin and Chris M. Davison
- Subjects
37J15, 37J35, 53D25, 70H06, 70H33 ,Surface (mathematics) ,Integrable system ,Image (category theory) ,Mathematical analysis ,Degenerate energy levels ,FOS: Physical sciences ,Mathematical Physics (math-ph) ,Ellipsoid ,Action (physics) ,Mathematics (miscellaneous) ,Convex polytope ,Symmetry (geometry) ,Mathematical Physics ,Mathematics - Abstract
Following on from our previous study of the geodesic flow on three dimensional ellipsoid with equal middle semi-axes, here we study the remaining cases: Ellipsoids with two sets of equal semi-axes with $SO(2) \times SO(2)$ symmetry, ellipsoids with equal larger or smaller semi-axes with SO(2) symmetry, and ellipsoids with three semi-axes coinciding with SO(3) symmetry. All of these cases are Liouville-integrable, and reduction of the symmetry leads to singular reduced systems on lower-dimensional ellipsoids. The critical values of the energy-momentum maps and their singular fibers are completely classified. In the cases with SO(2) symmetry there are corank 1 degenerate critical points; all other critical points are non-degenreate. We show that in the case with $SO(2) \times SO(2)$ symmetry three global action variables exist and the image of the energy surface under the energy-momentum map is a convex polyhedron. The case with SO(3) symmetry is non-commutatively integrable, and we show that the fibers over regular points of the energy-casimir map are $T^2$ bundles over $S^2$., 34 pages, 10 figures
- Published
- 2007
34. The Diver with a Rotor
- Author
-
Sudarsh Bharadwaj, William Tong, Nathan Duignan, Karen Leung, and Holger R. Dullin
- Subjects
General Mathematics ,FOS: Physical sciences ,Dynamical Systems (math.DS) ,Physics - Classical Physics ,01 natural sciences ,010305 fluids & plasmas ,law.invention ,Simple (abstract algebra) ,law ,0103 physical sciences ,FOS: Mathematics ,Elliptic integral ,0101 mathematics ,Special case ,Mathematics - Dynamical Systems ,Computer Science::Databases ,Mathematics ,Nonlinear Sciences - Exactly Solvable and Integrable Systems ,Rotor (electric) ,010102 general mathematics ,Mathematical analysis ,70E55, 70E15, 74A99, 93B99, 53Z05 ,Classical Physics (physics.class-ph) ,Rigid body ,Rigid body dynamics ,Geometric phase ,Exactly Solvable and Integrable Systems (nlin.SI) ,Rotation (mathematics) - Abstract
We present and analyse a simple model for the twisting somersault. The model is a rigid body with a rotor attached which can be switched on and off. This makes it simple enough to devise explicit analytical formulas whilst still maintaining sufficient complexity to preserve the shape-changing dynamics essential for twisting somersaults in springboard and platform diving. With `rotor on' and with `rotor off' the corresponding Euler-type equations can be solved, and the essential quantities characterising the dynamics, such as the periods and rotation numbers, can be computed in terms of complete elliptic integrals. Thus we arrive at explicit formulas for how to achieve a dive with m somersaults and n twists in a given total time. This can be thought of as a special case of a geometric phase formula due to Cabrera 2007., 15 pages, 6 figures
- Published
- 2015
35. Syzygies in the two center problem
- Author
-
Holger R. Dullin and Richard Montgomery
- Subjects
Pure mathematics ,Integrable system ,Symbolic dynamics ,General Physics and Astronomy ,Motion (geometry) ,FOS: Physical sciences ,Center (group theory) ,Dynamical Systems (math.DS) ,01 natural sciences ,010305 fluids & plasmas ,symbols.namesake ,Computer Science::Discrete Mathematics ,0103 physical sciences ,FOS: Mathematics ,Mathematics - Dynamical Systems ,0101 mathematics ,Mathematical Physics ,Mathematics ,Nonlinear Sciences - Exactly Solvable and Integrable Systems ,Applied Mathematics ,010102 general mathematics ,Statistical and Nonlinear Physics ,Torus ,16. Peace & justice ,Collision ,Bijection ,Euler's formula ,symbols ,37B10, 37J35, 70H06, 34C25 ,Exactly Solvable and Integrable Systems (nlin.SI) ,Computer Science::Formal Languages and Automata Theory - Abstract
We give a complete symbolic dynamics description of the dynamics of Euler's problem of two fixed centers. By analogy with the 3-body problem we use the collinearities (or syzygies) of the three bodies as symbols. We show that motion without collision on regular tori of the regularised integrable system are given by so called Sturmian sequences. Sturmian sequences were introduced by Morse and Hedlund in 1940. Our main theorem is that the periodic Sturmian sequences are in one to one correspondence with the periodic orbits of the two center problem. Similarly, finite Sturmian sequences correspond to collision-collision orbits., Comment: 28 pages, 16 figures
- Published
- 2015
- Full Text
- View/download PDF
36. Twisting Somersault
- Author
-
Holger R. Dullin and William Tong
- Subjects
Angular momentum ,Motion (geometry) ,FOS: Physical sciences ,Physics - Classical Physics ,Dynamical Systems (math.DS) ,Rotation ,01 natural sciences ,03 medical and health sciences ,symbols.namesake ,0302 clinical medicine ,0103 physical sciences ,FOS: Mathematics ,Mathematics - Dynamical Systems ,010306 general physics ,Rotation number ,Mathematics ,Mathematical analysis ,70E55, 70E15, 74A99, 93B99, 53Z05 ,Classical Physics (physics.class-ph) ,030229 sport sciences ,Rigid body ,Euler equations ,Geometric phase ,Modeling and Simulation ,symbols ,Euler's formula ,Analysis - Abstract
A complete description of twisting somersaults is given using a reduction to a time-dependent Euler equation for non-rigid body dynamics. The central idea is that after reduction the twisting motion is apparent in a body frame, while the somersaulting (rotation about the fixed angular momentum vector in space) is recovered by a combination of dynamic and geometric phase. In the simplest "kick-model" the number of somersaults $m$ and the number of twists $n$ are obtained through a rational rotation number $W = m/n$ of a (rigid) Euler top. This rotation number is obtained by a slight modification of Montgomery's formula [9] for how much the rigid body has rotated. Using the full model with shape changes that take a realistic time we then derive the master twisting-somersault formula: An exact formula that relates the airborne time of the diver, the time spent in various stages of the dive, the numbers $m$ and $n$, the energy in the stages, and the angular momentum by extending a geometric phase formula due to Cabrera [3]. Numerical simulations for various dives agree perfectly with this formula where realistic parameters are taken from actual observations., Comment: 16 pages, 6 figures, work from PhD thesis of William Tong
- Published
- 2015
- Full Text
- View/download PDF
37. Spectra of Sol-Manifolds: Arithmetic and Quantum Monodromy
- Author
-
Alexander P. Veselov, Holger R. Dullin, and Alexey V. Bolsinov
- Subjects
Pure mathematics ,FOS: Physical sciences ,58J50 ,Mathematics - Spectral Theory ,symbols.namesake ,Integer ,35P20 ,FOS: Mathematics ,Number Theory (math.NT) ,Spectral Theory (math.SP) ,Mathematics::Symplectic Geometry ,Mathematical Physics ,Eigenvalues and eigenvectors ,Mathematics ,Conjecture ,Mathematics - Number Theory ,Nonlinear Sciences - Exactly Solvable and Integrable Systems ,Statistical and Nonlinear Physics ,Torus ,Mathematical Physics (math-ph) ,Mathematics::Spectral Theory ,Mathematics::Geometric Topology ,Mathieu function ,Unimodular matrix ,Monodromy ,symbols ,Binary quadratic form ,Exactly Solvable and Integrable Systems (nlin.SI) - Abstract
The spectral problem of three-dimensional manifolds M_A admitting Sol-geometry in Thurston's sense is investigated. Topologically M_A are torus bundles over a circle with a unimodular hyperbolic gluing map A. The eigenfunctions of the corresponding Laplace-Beltrami operators are described in terms of the modified Mathieu functions. It is shown that the multiplicities of the eigenvalues are the same for generic values of the parameters in the metric and are directly related to the number of representations of an integer by a given indefinite binary quadratic form. As a result the spectral statistics is shown to disagree with the Berry-Tabor conjecture. The topological nature of the monodromy for both classical and quantum systems on Sol-manifolds is demonstrated., Comment: 28 pages, 8 figures
- Published
- 2006
38. Normal forms for 4D symplectic maps with twist singularities
- Author
-
Holger R. Dullin, Alexey V. Ivanov, and James D. Meiss
- Subjects
Cusp (singularity) ,Hamiltonian mechanics ,010102 general mathematics ,Invariant manifold ,Mathematical analysis ,Statistical and Nonlinear Physics ,Condensed Matter Physics ,01 natural sciences ,010305 fluids & plasmas ,symbols.namesake ,Singularity ,0103 physical sciences ,symbols ,Gravitational singularity ,0101 mathematics ,Hamiltonian (quantum mechanics) ,Symplectomorphism ,Mathematics ,Symplectic geometry - Abstract
We derive a normal form for a near-integrable, four-dimensional (4D) symplectic map with a fold or cusp singularity in its frequency mapping. The normal form is obtained for when the frequency is near a resonance and the mapping is approximately given by the time-T mapping of a two-degree-of-freedom Hamiltonian flow. Consequently, there is an energy-like invariant. The fold Hamiltonian is similar to the well-studied one-degree-of-freedom case, but is essentially non-integrable when the direction of the singular curve in action does not coincide with curves of the resonance module. We show that many familiar features, such as multiple island chains and reconnecting invariant manifolds, are retained even in this case. The cusp Hamiltonian has an essential coupling between its two degrees of freedom even when the singular set is aligned with the resonance module. Using averaging, we approximately reduce this case to one degree of freedom as well. The resulting Hamiltonian and its perturbation with small cusp-angle is analyzed in detail.
- Published
- 2006
39. Another look at the saddle-centre bifurcation: Vanishing twist
- Author
-
Holger R. Dullin and Alexey V. Ivanov
- Subjects
Transcritical bifurcation ,Mathematical analysis ,Statistical and Nonlinear Physics ,Saddle-node bifurcation ,Torus ,Invariant (mathematics) ,Condensed Matter Physics ,Bifurcation diagram ,Rotation number ,Bifurcation ,Saddle ,Mathematical physics ,Mathematics - Abstract
In the saddle-centre bifurcation a pair of periodic orbits is created “out of nothing” in a Hamiltonian system with two degrees of freedom. It is the generic bifurcation with multiplier one. We show that “out of nothing” should be replaced by “out of a twistless torus”. More precisely, we show that invariant tori of the normal form have vanishing twist right before the appearance of the new orbits. Vanishing twist means that the derivative of the rotation number with respect to the action for constant energy vanishes. We explicitly derive the position of the twistless torus in phase and in parameter space near the saddle-centre bifurcation. The theory is applied to the area preserving Henon map.
- Published
- 2005
40. Twistless Tori Near Low-Order Resonances
- Author
-
Holger R. Dullin and Alexey V. Ivanov
- Subjects
Statistics and Probability ,Period-doubling bifurcation ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Saddle-node bifurcation ,Bifurcation diagram ,Transcritical bifurcation ,Pitchfork bifurcation ,Homoclinic bifurcation ,Infinite-period bifurcation ,Nonlinear Sciences::Pattern Formation and Solitons ,Blue sky catastrophe ,Mathematics - Abstract
In this paper, we investigate the behavior of the twist near low-order resonances of a periodic orbit or an equilibrium of a Hamiltonian system with two degrees of freedom. Namely, we analyze the case where the Hamiltonian has multiple eigenvalues (the Hamiltonian Hopf bifurcation) or a zero eigenvalue near the equilibrium and the case where the system has a periodic orbit whose multipliers are equal to 1 (the saddle-center bifurcation) or −1 (the period-doubling bifurcation). We show that the twist does not vanish at least in a small neighborhood of the period-doubling bifurcation. For the saddle-center bifurcation and the resonances of the equilibrium under consideration, we prove the existence of a “twistless” torus for sufficiently small values of the bifurcation parameter. An explicit dependence of the energy corresponding to the twistless torus on the bifurcation parameter is derived. Bibliography: 6 titles.
- Published
- 2005
41. A new integrable system on the sphere
- Author
-
Holger R. Dullin and Vladimir S. Matveev
- Subjects
Mathematics - Differential Geometry ,Nonlinear Sciences - Exactly Solvable and Integrable Systems ,Integrable system ,General Mathematics ,FOS: Physical sciences ,Dynamical Systems (math.DS) ,Mathematical Physics (math-ph) ,37J35, 58F07, 58F17, 70H06, 70E40 ,Mathematical research ,Algebra ,Differential Geometry (math.DG) ,Mathematics - Symplectic Geometry ,FOS: Mathematics ,Symplectic Geometry (math.SG) ,Exactly Solvable and Integrable Systems (nlin.SI) ,Mathematics - Dynamical Systems ,Mathematics::Symplectic Geometry ,Mathematical Physics ,Mathematics - Abstract
We present a new Liouville-integrable natural Hamiltonian system on the (cotangent bundle of the) two-dimensional sphere. The second integral is cubic in the momenta., LaTeX, 15 pages
- Published
- 2004
42. Quantum monodromy in the two-centre problem
- Author
-
A. Junge, Holger Waalkens, and Holger R. Dullin
- Subjects
Integrable system ,Mathematical analysis ,Monodromy theorem ,General Physics and Astronomy ,Statistical and Nonlinear Physics ,Quantum number ,Hamiltonian system ,Singularity ,Monodromy ,Invariant (mathematics) ,Mathematics::Symplectic Geometry ,Quantum ,Mathematical Physics ,Mathematics ,Mathematical physics - Abstract
Using modern tools from the geometric theory of Hamiltonian systems it is shown that electronic excitations in diatoms which can be modelled by the two-centre problem exhibit a complicated case of classical and quantum monodromy. This means that there is an obstruction to the existence of global quantum numbers in these classically integrable systems. The symmetric case of H+2 and the asymmetric case of H He++ are explicitly worked out. The asymmetric case has a non-local singularity causing monodromy. It coexists with a second singularity which is also present in the symmetric case. An interpretation of monodromy is given in terms of the caustics of invariant tori.
- Published
- 2003
43. Twist singularities for symplectic maps
- Author
-
James D. Meiss and Holger R. Dullin
- Subjects
Cusp (singularity) ,Applied Mathematics ,Mathematical analysis ,General Physics and Astronomy ,Statistical and Nonlinear Physics ,symbols.namesake ,Matrix (mathematics) ,Jacobian matrix and determinant ,symbols ,Twist ,Symplectomorphism ,Moment map ,Mathematical Physics ,Rotation number ,Symplectic geometry ,Mathematics - Abstract
Near a nonresonant, elliptic fixed point, a symplectic map can be transformed into Birkhoff normal form. In these coordinates, the dynamics is represented entirely by the Lagrangian “frequency map” that gives the rotation number as a function of the action. The twist matrix, given by the Jacobian of the rotation number, describes the anharmonicity in the system. When the twist is singular the frequency map need not be locally one-to-one. Here we investigate the occurrence of fold and cusp singularities in the frequency map. We show that folds necessarily occur near third order resonances. We illustrate the results by numerical computations of frequency maps for a quadratic, symplectic map.
- Published
- 2003
44. About ergodicity in the family of limaçon billiards
- Author
-
Arnd Bäcker and Holger R. Dullin
- Subjects
Mathematics::Dynamical Systems ,Limaçon ,Applied Mathematics ,Ergodicity ,Mathematical analysis ,FOS: Physical sciences ,General Physics and Astronomy ,Statistical and Nonlinear Physics ,Saddle-node bifurcation ,Nonlinear Sciences - Chaotic Dynamics ,Nonlinear Sciences::Chaotic Dynamics ,Continuation ,Cardioid ,Limit (mathematics) ,Chaotic Dynamics (nlin.CD) ,Dynamical billiards ,Mathematical Physics ,Bifurcation ,Mathematics - Abstract
By continuation from the hyperbolic limit of the cardioid billiard we show that there is an abundance of bifurcations in the family of limacon billiards. The statistics of these bifurcation shows that the size of the stable intervals decreases with approximately the same rate as their number increases with the period. In particular, we give numerical evidence that arbitrarily close to the cardioid there are elliptic islands due to orbits created in saddle node bifurcations. This shows explicitly that if in this one parameter family of maps ergodicity occurs for more than one parameter the set of these parameter values has a complicated structure., 17 pages, 9 figures
- Published
- 2001
45. Actions of the Neumann systems via Picard–Fuchs equations
- Author
-
Alexander P. Veselov, Holger Waalkens, Peter H. Richter, and Holger R. Dullin
- Subjects
Differential equation ,Mathematical analysis ,Von Neumann stability analysis ,Statistical and Nonlinear Physics ,Condensed Matter Physics ,Picard–Fuchs equation ,Action (physics) ,Neumann series ,symbols.namesake ,Von Neumann algebra ,symbols ,Neumann boundary condition ,Abelian von Neumann algebra ,Mathematics - Abstract
The Neumann system describing the motion of a particle on an n-dimensional sphere with an anisotropic harmonic potential has been celebrated as one of the best understood integrable systems of classical mechanics. The present paper adds a detailed discussion and the determination of its action integrals, using differential equations rather than standard integral formulas. We show that the actions of the Neumann system satisfy a Picard–Fuchs equation which in suitable coordinates has a rather simple form for arbitrary n. We also present an explicit form of the related Gaus–Manin equations. These formulas are used for the numerical calculation of the actions of the Neumann system.
- Published
- 2001
46. Self-rotation number using the turning angle
- Author
-
Holger R. Dullin, James D. Meiss, and David Sterling
- Subjects
Hénon map ,Sequence ,Orientation (geometry) ,Mathematical analysis ,Orbit (dynamics) ,Statistical and Nonlinear Physics ,Geometry ,Limit (mathematics) ,Condensed Matter Physics ,Rotation (mathematics) ,SIMPLE algorithm ,Rotation number ,Mathematics - Abstract
The self-rotation number, as defined by Peckham, is the rotation rate of the image of a point about itself. Here we use the notion of “turning angle” to give a simplified algorithm to compute the self-rotation number for maps that “avoid an angle”. We show that the orientation preserving Henon map does avoid an angle. Moreover, the self-rotation number for orbits of the Henon map can be computed once and for all at the anti-integrable limit by a simple algorithm depending upon the symbol sequence for the orbit.
- Published
- 2000
47. Generalized Hénon maps: the cubic diffeomorphisms of the plane
- Author
-
James D. Meiss and Holger R. Dullin
- Subjects
Hénon map ,Polynomial ,Quadratic equation ,Bounded set ,Plane (geometry) ,Bounded function ,Mathematical analysis ,Statistical and Nonlinear Physics ,Limit (mathematics) ,Diffeomorphism ,Condensed Matter Physics ,Mathematics - Abstract
In general a polynomial diffeomorphism of the plane can be transformed into a composition of generalized Henon maps. These maps exhibit some of the familiar properties of the quadratic Henon map, including a bounded set of bounded orbits and an anti-integrable limit. We investigate in particular the cubic, area-preserving case, which reduces to two, two-parameter families of maps. The bifurcations of low period orbits of these maps are discussed in detail.
- Published
- 2000
48. Stability of Levitrons
- Author
-
Holger R. Dullin and Robert W. Easton
- Subjects
Applied Mathematics ,Computational Mechanics ,Exact theory ,Statistical and Nonlinear Physics ,Constant field ,Condensed Matter Physics ,Levitron ,Adiabatic theorem ,symbols.namesake ,Classical mechanics ,Magnet ,Levitation ,symbols ,Six degrees of freedom ,Hamiltonian (quantum mechanics) ,Spinning ,Mathematics - Abstract
The Levitron is a magnetic spinning top which can levitate in the constant field of a repelling base magnet. An explanation for the stability of the Levitron using an adiabatic approximation has been given by Berry. In experiments the top eventually loses stability at a critical spin rate which cannot be predicted by Berry’s approach. The present work develops an exact theory of the Levitron with six degrees of freedom which allows for the calculation of critical spin rates. The main result is a complete classification of possible Levitrons that allow for an interval of stable spin rates. Stability of the relative equilibrium is lost in Hamiltonian Hopf bifurcations if either the spin rate is too large or too small.
- Published
- 1999
49. Symbolic dynamics and the discrete variational principle
- Author
-
Holger R. Dullin
- Subjects
Pure mathematics ,Mathematical analysis ,Symbolic dynamics ,General Physics and Astronomy ,Statistical and Nonlinear Physics ,Constructive ,Nonlinear Sciences::Chaotic Dynamics ,Cardioid ,Variational principle ,Piecewise ,Dynamical billiards ,Convex function ,Mathematical Physics ,Symplectic geometry ,Mathematics - Abstract
We show how to construct symbolic dynamics for the class of 2d-dimensional twist mappings generated by piecewise strictly convex/concave generating functions. The method is constructive and gives an efficient way to find all periodic orbits of these high-dimensional symplectic mappings. It is illustrated with the cardioid and the stadium billiard.
- Published
- 1998
50. Stability of minimal periodic orbits
- Author
-
James D. Meiss and Holger R. Dullin
- Subjects
Physics ,Maxima and minima ,Pure mathematics ,Variational principle ,General Physics and Astronomy ,Positive-definite matrix ,Twist ,Composition (combinatorics) ,Stability (probability) ,Action (physics) ,Symplectic geometry - Abstract
Symplectic twist maps are obtained from a Lagrangian variational principle. It is well known that nondegenerate minima of the action correspond to hyperbolic orbits of the map when the twist is negative definite and the map is two-dimensional. We show that for more than two dimensions, periodic orbits with minimal action in symplectic twist maps with negative definite twist are not necessarily hyperbolic. In the proof we show that in the neighborhood of a minimal periodic orbit of period n , the n th iterate of the map is again a twist map. This is true even though in general the composition of twist maps is not a twist map.
- Published
- 1998
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