Symplectic integration of Hamiltonian systems using polynomial maps

Rangarajan, Govindan

doi:10.1016/s0375-9601(01)00409-1

Cited by 3 publications

(2 citation statements)

References 21 publications

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“…Since the action of the linear partM on phase space variables is well known and is already a polynomial action, we only refactorize the nonlinear part of the map using N polynomial maps. 32 This is done as follows:…”

Section: Symplectic Integration Using Polynomial Mapsmentioning

confidence: 99%

Polynomial Map Factorization of Symplectic Maps

Rangarajan

2003

Int. J. Mod. Phys. C

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Long-term stability studies of nonlinear Hamiltonian systems require symplectic integration algorithms which are both fast and accurate. In this paper, we study a symplectic integration method wherein the symplectic map representing the Hamiltonian system is refactorized using polynomial symplectic maps. This method is analyzed for the three degree of freedom case. Finally, we apply this algorithm to study a large particle storage ring.

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Section: Symplectic Integration Using Polynomial Mapsmentioning

confidence: 99%

Polynomial Map Factorization of Symplectic Maps

Rangarajan

2003

Int. J. Mod. Phys. C

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“…Maps of type (1) also appear in symplectic discretizations of natural Lagrangian systems (see e.g. [4,5,30] and formulae (7), (8) below). My interest in Hénon-like maps is due to the fact that they appear as rescaled first-return maps at homoclinic bifurcations, near a homoclinic tangency in particular (see [6,7,11,8,9] and section 2).…”

Section: Polynomial Approximations By Hénon-like Mapsmentioning

confidence: 99%

Polynomial approximations of symplectic dynamics and richness of chaos in non-hyperbolic area-preserving maps

Turaev

2002

Nonlinearity

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It is shown that every symplectic diffeomorphism of R 2n can be approximated, in the C ∞ -topology, on any compact set, by some iteration of some map of the form (x, y) → (y + η, −x + ∇V (y)) where x ∈ R n , y ∈ R n , and V is a polynomial R n → R and η ∈ R n is a constant vector. For the case of area-preserving maps (i.e. n = 1), it is shown how this result can be applied to prove that C r -universal maps (a map is universal if its iterations approximate dynamics of all C r -smooth area-preserving maps altogether) are dense in the C r -topology in the Newhouse regions.

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A Symplectic Algorithm for the Stability Analysis of Nonlinear Parametric Excited Systems

Ying

Chen

2009

Int. J. Str. Stab. Dyn.

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A symplectic numerical approach to the stability analysis of nonlinear parametric excited systems with multi-degree-of-freedom is developed and the stability of a nonlinear vibration system depends on the dynamic behavior of its perturbation. The nonlinear parameter-varying di®erential equations for the perturbation motion are expressed in the form of Hamiltonian equations with time-varying Hamiltonian, and are converted further into the classical Hamiltonian equations with extended time-invariant Hamiltonian by augmenting the state variables. The solution to the augmented Hamiltonian equations has the symplectic structure in terms of the symplectic transformation. Then the di®erence equations of the symplectic Runge-Kutta algorithm with a su±cient condition are constructed, which is proved to preserve the intrinsic symplectic structure of the original solution. In particular, the symplectic Gauss-Runge-Kutta algorithm with stage 2 and order 4 is proposed and applied to the stability analysis of a nonlinear system. Unstable regions based on the nonlinear periodic-parameter perturbation equation are obtained by using the symplectic Gauss-Runge-Kutta algorithm, analytical solution method and non-symplectic conventional Runge-Kutta algorithm to verify the higher accuracy of the proposed algorithm. Unstable regions based on the nonlinear perturbation are given to illustrate the improvement over those based on the linear perturbation. The developed symplectic approach to the stability analysis can preserve the symplectic structure of the original system and is applicable to nonlinear parametric excited systems with multi-degree-of-freedom.

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Symplectic integration of Hamiltonian systems using polynomial maps

Cited by 3 publications

References 21 publications

Polynomial Map Factorization of Symplectic Maps

Polynomial Map Factorization of Symplectic Maps

Polynomial approximations of symplectic dynamics and richness of chaos in non-hyperbolic area-preserving maps

A Symplectic Algorithm for the Stability Analysis of Nonlinear Parametric Excited Systems

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