scholarly journals Using Invariant Manifolds to Construct Symbolic Dynamics for Three-Dimensional Volume-Preserving Maps

2017 ◽  
Vol 16 (1) ◽  
pp. 729-769 ◽  
Author(s):  
Bryan Maelfeyt ◽  
Spencer A. Smith ◽  
Kevin A. Mitchell
Keyword(s):  
Symbolic Dynamics ◽  
Invariant Manifolds ◽  
Three Dimensional ◽  
Volume Preserving

scholarly journals GEOMETRY OF HOMOCLINIC CONNECTIONS IN A PLANAR CIRCULAR RESTRICTED THREE-BODY PROBLEM

2007 ◽  
Vol 17 (04) ◽  
pp. 1151-1169 ◽  
Author(s):  
MARIAN GIDEA ◽  
JOSEP J. MASDEMONT
Keyword(s):  
Symbolic Dynamics ◽  
Body Problem ◽  
Invariant Manifolds ◽  
Libration Point ◽  
Homoclinic Orbits ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Lyapunov Orbits ◽  
Homoclinic Connections ◽  
Three Body

The stable and unstable invariant manifolds associated with Lyapunov orbits about the libration point L1between the primaries in the planar circular restricted three-body problem with equal masses are considered. The behavior of the intersections of these invariant manifolds for values of the energy between that of L1and the other collinear libration points L2, L3is studied using symbolic dynamics. Homoclinic orbits are classified according to the number of turns about the primaries.

scholarly journals On the Measure-Preserving Mappings with Three-Dimensions

1993 ◽  
Vol 132 ◽  
pp. 73-89
Author(s):  
Yi-Sui Sun
Keyword(s):  
Fixed Point ◽  
Invariant Manifolds ◽  
Three Dimensional ◽  
Invariant Curve ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Kolmogorov Entropy ◽  
Numerical Exploration ◽  
Area Preserving ◽  
Dimensional Mapping

AbstractWe have systematically made the numerical exploration about the perturbation extension of area-preserving mappings to three-dimensional ones, in which the fixed points of area preserving are elliptic, parabolic or hyperbolic respectively. It has been observed that: (i) the invariant manifolds in the vicinity of the fixed point generally don’t exist (ii) when the invariant curve of original two-dimensional mapping exists the invariant tubes do also in the neighbourhood of the invariant curve (iii) for the perturbation extension of area-preserving mapping the invariant manifolds can only be generated in the subset of the invariant manifolds of original two-dimensional mapping, (iv) for the perturbation extension of area preserving mappings with hyperbolic or parabolic fixed point the ordered region near and far from the invariant curve will be destroyed by perturbation more easily than the other one, This is a result different from the case with the elliptic fixed point. In the latter the ordered region near invariant curve is solid. Some of the results have been demonstrated exactly.Finally we have discussed the Kolmogorov Entropy of the mappings and studied some applications.

Passive scalars, three-dimensional volume-preserving maps, and chaos

1988 ◽  
Vol 50 (3-4) ◽  
pp. 529-565 ◽  
Author(s):  
Mario Feingold ◽  
Leo P. Kadanoff ◽  
Oreste Piro
Keyword(s):  
Three Dimensional ◽  
Passive Scalars ◽  
Volume Preserving

scholarly journals On Embedding of the Morse-Smale Diffeomorphisms in a Topological Flow

2020 ◽  
Vol 66 (2) ◽  
pp. 160-181
Author(s):  
V. Z. Grines ◽  
E. Ya. Gurevich ◽  
O. V. Pochinka
Keyword(s):  
Invariant Manifolds ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Periodic Points ◽  
Topological Classification ◽  
Higher Dimension ◽  
Topological Information ◽  
Topology Of Manifolds ◽  
Necessary And Sufficient

This review presents the results of recent years on solving of the Palis problem on finding necessary and sufficient conditions for the embedding of Morse-Smale cascades in topological flows. To date, the problem has been solved by Palis for Morse-Smale diffeomorphisms given on manifolds of dimension two. The result for the circle is a trivial exercise. In dimensions three and higher new effects arise related to the possibility of wild embeddings of closures of invariant manifolds of saddle periodic points that leads to additional obstacles for Morse-Smale diffeomorphisms to embed in topological flows. The progress achieved in solving of Paliss problem in dimension three is associated with the recently obtained complete topological classification of Morse-Smale diffeomorphisms on three-dimensional manifolds and the introduction of new invariants describing the embedding of separatrices of saddle periodic points in a supporting manifold. The transition to a higher dimension requires the latest results from the topology of manifolds. The necessary topological information, which plays key roles in the proofs, is also presented in the survey.

Closed-Form Primitives for Generating Volume Preserving Deformations

1993 ◽  
Author(s):  
Gregory S. Chirikjian
Keyword(s):  
Closed Form ◽  
Dimensional Space ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Physical Systems ◽  
Mechanics Of Materials ◽  
Time Simulation ◽  
Infinite Variety ◽  
Volume Preserving ◽  
Three Dimensional Space

Abstract In this paper, methods for generating closed-form expressions for locally volume preserving deformations of general volumes in three dimensional space are introduced. These methods potentially have applications to computer aided geometric design, the mechanics of materials, and realistic real-time simulation and animation of physical processes. In mechanics, volume preserving deformations are intimately related to the conservation of mass. The importance of this fact manifests itself in design, and in the realistic simulation of many physical systems. Whereas volume preservation is generally written as a constraint on equations of motion in continuum mechanics, this paper develops a set of physically meaningful basic deformations which are intrinsically volume preserving. By repeated application of these primitives, an infinite variety of deformations can be written in closed form.

Fine Scale Structure of the Local Lyapunov Exponent

1997 ◽  
Vol 11 (16n17) ◽  
pp. 707-712
Author(s):  
Demin Yao
Keyword(s):  
Lyapunov Exponent ◽  
Three Dimensional ◽  
Scale Structure ◽  
Fine Scale ◽  
Spatial Function ◽  
Local Lyapunov Exponent ◽  
Chaotic Region ◽  
Scale Structures ◽  
Volume Preserving

Direct numerial calculations reveal the fine-scale structures of the local Lyapunov exponent as a spatial function in the chaotic region of a three-dimensional volume-preserving flow.

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