scholarly journals TRANSPORT IN DYNAMICAL ASTRONOMY AND MULTIBODY PROBLEMS

2005 ◽  
Vol 15 (03) ◽  
pp. 699-727 ◽  
Author(s):  
MICHAEL DELLNITZ ◽  
OLIVER JUNGE ◽  
WANG SANG KOON ◽  
FRANCOIS LEKIEN ◽  
MARTIN W. LO ◽  
...  
Keyword(s):  
Reaction Rates ◽  
Invariant Sets ◽  
Computational Technique ◽  
Dynamical Astronomy ◽  
Lobe Dynamics ◽  
Numerical Tools ◽  
Dynamical Features ◽  
Three Body ◽  
Multibody Problems ◽  
Almost Invariant Sets

We combine the techniques of almost invariant sets (using tree structured box elimination and graph partitioning algorithms) with invariant manifold and lobe dynamics techniques. The result is a new computational technique for computing key dynamical features, including almost invariant sets, resonance regions as well as transport rates and bottlenecks between regions in dynamical systems. This methodology can be applied to a variety of multibody problems, including those in molecular modeling, chemical reaction rates and dynamical astronomy. In this paper we focus on problems in dynamical astronomy to illustrate the power of the combination of these different numerical tools and their applicability. In particular, we compute transport rates between two resonance regions for the three-body system consisting of the Sun, Jupiter and a third body (such as an asteroid). These resonance regions are appropriate for certain comets and asteroids.

scholarly journals TRANSPORT IN DYNAMICAL ASTRONOMY AND MULTIBODY PROBLEMS

2006 ◽  
pp. 3-31
Author(s):  
MICHAEL DELLNITZ ◽  
OLIVER JUNGE ◽  
WANG SANG KOON ◽  
FRANCOIS LEKIEN ◽  
MARTIN W. LO ◽  
...  
Keyword(s):  
Dynamical Astronomy ◽  
Multibody Problems

scholarly journals Instability of the isolated spectrum for W-shaped maps

2012 ◽  
Vol 33 (4) ◽  
pp. 1052-1059
Author(s):  
ZHENYANG LI ◽  
PAWEŁ GÓRA
Keyword(s):  
The Other ◽  
Invariant Sets ◽  
Second Eigenvalue ◽  
Almost Invariant Sets

AbstractIn this note we consider the W-shaped map $W_0=W_{s_1,s_2}$ with ${1}/{s_1}+{1}/{s_2}=1$ and show that the eigenvalue $1$ is not stable. We do this in a constructive way. For each perturbing map $W_a$ we show the existence of a ‘second’ eigenvalue $\lambda _a$, such that $\lambda _a\to 1$ as $a\to 0$, which proves instability of the isolated spectrum of $W_0$. At the same time, the existence of second eigenvalues close to 1 causes the maps $W_a$to behave in a metastable way. There are two almost-invariant sets, and the system spends long periods of consecutive iterations in each of them, with infrequent jumps from one to the other.

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