The Global Geometry of the Slow Manifold in the Lorenz–Krishnamurthy Model

1996 ◽  
Vol 53 (22) ◽  
pp. 3251-3264 ◽  
Author(s):  
Roberto Camassa ◽  
Siu-Kei Tin
Keyword(s):  
Slow Manifold ◽  
Global Geometry

NUMERICAL COMPUTATION OF CANARDS

2000 ◽  
Vol 10 (12) ◽  
pp. 2669-2687 ◽  
Author(s):  
JOHN GUCKENHEIMER ◽  
KATHLEEN HOFFMAN ◽  
WARREN WECKESSER
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Electrical Potential ◽  
Slow Manifold ◽  
Singularly Perturbed ◽  
Singularly Perturbed Systems ◽  
Singularly Perturbed System ◽  
Chemical Systems ◽  
Perturbed System ◽  
Physical And Chemical

Singularly perturbed systems of ordinary differential equations arise in many biological, physical and chemical systems. We present an example of a singularly perturbed system of ordinary differential equations that arises as a model of the electrical potential across the cell membrane of a neuron. We describe two periodic solutions of this example that were numerically computed using continuation of solutions of boundary value problems. One of these periodic orbits contains canards, trajectory segments that follow unstable portions of a slow manifold. We identify several mechanisms that lead to the formation of these and other canards in this example.

Global geometry optimization of (Ar)n and B(Ar)n clusters using a modified genetic algorithm

1996 ◽  
Vol 104 (7) ◽  
pp. 2684-2691 ◽  
Author(s):  
Susan K. Gregurick ◽  
Millard H. Alexander ◽  
Bernd Hartke
Keyword(s):  
Genetic Algorithm ◽  
Geometry Optimization ◽  
Global Geometry ◽  
Modified Genetic Algorithm

Geometry of the bifurcations of the normalized reduced Henon–Heiles family

1982 ◽  
Vol 382 (1783) ◽  
pp. 361-371 ◽  
Keyword(s):  
Bifurcation Diagram ◽  
Main Resonance ◽  
Fourth Degree ◽  
Global Geometry ◽  
Geometric Explanation ◽  
Hamiltonian Functions

This paper deals with the global geometry of the bifurcations of a family of Hamiltonian functions that arises from normalizing the Henon–Heiles family to fourth-degree terms and then performing a reduction. This gives a geometric explanation of the bifurcation diagram for the main resonance in the model of axisymmetric galaxies of Braun and Verhulst.

scholarly journals Some Aspects of the global Geometry of Entire Space-Like Submanifolds

2001 ◽  
Vol 40 (1-4) ◽  
pp. 233-245 ◽  
Author(s):  
J. Jost ◽  
Y. L. Xin
Keyword(s):  
Entire Space ◽  
Global Geometry

scholarly journals SYNCHRONIZATION AS A PROCESS OF SHARING AND TRANSFERRING INFORMATION

2012 ◽  
Vol 22 (11) ◽  
pp. 1250261 ◽  
Author(s):  
ERIK M. BOLLT
Keyword(s):  
Symbolic Dynamics ◽  
Slow Manifold ◽  
Transfer Entropy ◽  
The Other ◽  
Chaotic Oscillators ◽  
Smale Horseshoe ◽  
Global System ◽  
Identity Function ◽  
Transfer Operators ◽  
First Time

Synchronization of chaotic oscillators has become well characterized by errors which shrink relative to a synchronization manifold. This manifold is the identity function in the case of identical systems, or some other slow manifold in the case of generalized synchronizaton in the case of nonidentical components. On the other hand, since many decades beginning with the Smale horseshoe, chaotic oscillators can be well understood in terms of symbolic dynamics as information producing processes. We study here the synchronization of a pair of chaotic oscillators as a process for sharing information bearing bits transferred between each other, by measuring the transfer entropy tracked as the global system transitions to the synchronization state. Further, we present for the first time the notion of transfer entropy in the measure theoretic setting of transfer operators.

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