Workshop on KAM Theory and Geometric Integration ， Banff, Canada (Organizers: Walter Craig, Erwan Faou, Benoit Grebert) 保结构算法构造及应用会议 （组织者：孙雅娟，王雨顺）. Numerical Stability of Hamiltonian Systems by Symplectic Integration Zaijiu Shang AMSS, CAS, Beijing June 9, 2011, Banff 2011 年 12 月 7 日，北京.
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Workshop on KAM Theory and Geometric Integration，Banff, Canada (Organizers: Walter Craig, Erwan Faou, Benoit Grebert)保结构算法构造及应用会议（组织者：孙雅娟，王雨顺）
Numerical Stability of Hamiltonian Systems by Symplectic IntegrationZaijiu ShangAMSS, CAS, Beijing June 9, 2011, Banff
2011年12月7日，北京
------ K. Feng, in the Proc. of the Conf. on Applied and Industry Mathematics, the Netherlands: Kluwer Academic Publisher, 1991. Ed. by R. Spigler.
(Channel and Scovel 1990, J. M. Sanz-serna 1994, …)
Skeleton of dynamics
(Steady states in time-invariant sense):
Theme of the dynamical system:
Harmonic oscillator systems in the real plane
where skis the k–th truncation of the formal solution of the Hamilton-Jacobian equation of type (P,q) and is initial value.
---------Non-analytic symmetric methods
(proved by McLachlan, Sun,etc without any quotation
(Casas, Murua,…)
---------- contradict the numerical observations!
if suitably choose the indecies and
e.g.,
for most dionphantine frequency vectors, which was already checked numerically by Faou etc.
(Oberwolfach workshop “Geometric numerical integration” ,
March 19-25, 2006, Numer. Math. 2008)
well defined and in involution on the union of the numerical invariant tori, in the Whitney’s sense, with large Lebesgue measure in the phase space. The numerical tori are the level sets of these first integrals.
is the backward Hamiltonian of the symplectic integrator.
The results can be extended to nearly integrable Hamiltonian systems and systems with Ruessman’s nondegeneracy condition (Ding & -S, 2011).
--- in the exponentially close sense, numerical invariant tori exist and approximate the invariant tori of the discretized nearly integrable systems
---Any numerical solution is still “very stable” even after “very long” time steps.
(dissipative invariant tori: Stoffer (1998), Hairer & Lubich (1999))
Nonlinear stability analysis in the KAM setting is numerically not useful!
(E. Faou etc. and the French groups).
--- beyond my knowledge and understanding.
size such that the time- map of the formal phase flow
of this Hamiltonian system is identity to the symplectic integrator with time step size .
If the formal Hamiltonian is convergent, then the symplectic integrator preserves this Hamiltonian function.
In general, however, the formal Hamiltonian is divergent!
Numerical KAM convergent in a large set of the phase space!
(BIT, 2004):
The level set of ( in stead, some finite truncation) is compact if original energy surface is compact.
(Example: nonlinear oscillation)
--- it is still difficult to give out an effective bound of time step size guaranteeing the numerical stability!
--- Model example
Theorem (L. Song’s thesis, 2009). Assume that a Hamiltonian system of one degrees of freedom has a homoclinic trajectory. When a symplectic integrator applies to this system, then the homoclinic trajectory of the system will be split into stable and unstable trajectories of the integrator which intersect transversally in general and span a chaotic web with exponentially small maximal width with respect to the p-th power of time-step size of the integrator, where p is the accuracy order of the integrator.
Thank you for your attention!