Related Experiment Video
Updated: Dec 2, 2025
11:00
Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180° Curved Artery Test Section
Published on: July 19, 2016
11.9K
Chaos in Cartan foliations.
Yaroslav V Bazaikin1, Anton S Galaev1, Nina I Zhukova2
1Faculty of Science, University of Hradec Králové, Rokitanského 62, 500 03 Hradec Králové, Czech Republic.
Chaos (Woodbury, N.Y.)
|November 3, 2020
Summary
This study explores chaotic foliations, extending chaos theory to dynamical systems. Researchers identified conditions for chaotic Cartan foliations and constructed examples, advancing the understanding of complex geometric structures.
Area of Science:
- Differential Geometry
- Dynamical Systems Theory
- Geometric Topology
Background:
- Chaotic foliations generalize Devaney's chaos concept for dynamical systems.
- The chaotic property of foliations is transversal, depending on the leaf space structure.
- Cartan foliations possess a transversal structure modeled on Cartan manifolds.
Purpose of the Study:
- To investigate chaotic Cartan foliations by analyzing their holonomy pseudogroups.
- To reduce the problem of chaotic Cartan foliations to their global holonomy groups.
- To identify non-chaotic Cartan foliations and construct examples of chaotic ones.
Main Methods:
- Modeling transversal structures of Cartan foliations on Cartan manifolds.
- Reducing the study of chaotic foliations to their holonomy pseudogroups.
- Analyzing global holonomy groups as discrete subgroups of Lie automorphism groups.
Main Results:
- The problem of chaotic Cartan foliations is reduced to analyzing their holonomy pseudogroups.
- For many Cartan foliations, this is further simplified to studying their global holonomy group.
- Several types of Cartan foliations that cannot be chaotic were identified.
- Examples of chaotic Cartan foliations were successfully constructed.
Conclusions:
- The research provides a framework for understanding chaos in Cartan foliations.
- The study successfully constructs examples of chaotic Cartan foliations.
- Key insights into the conditions preventing chaos in certain foliations were established.
More Related Videos
Related Concept Videos
01:27Gauss's Law: Planar Symmetry
9.0K
A planar symmetry of charge density is obtained when charges are uniformly spread over a large flat surface. In planar symmetry, all points in a plane parallel to the plane of charge are identical with respect to the charges. Suppose the plane of the charge distribution is the xy-plane, and the electric field at a space point P with coordinates (x, y, z) is to be determined. Since the charge density is the same at all (x, y) - coordinates in the z = 0 plane, by symmetry, the electric field at P...
9.0K
01:18Mohr's Circle for Plane Strain
994
Mohr's circle is a crucial graphical method used to analyze plane strain by plotting strain on a set of cartesian coordinates, where the abscissa is normal strain ∈ and the ordinate is shear strain γ. Similarly to Mohr’s circle for plane stress, two points X and Y are plotted. Their coordinates are (∈x, -γXY) and (∈Y, γXY), respectively.
Mohr's circle visually represents the strain states under various conditions, which is essential for...
Mohr's circle visually represents the strain states under various conditions, which is essential for...
994
01:25Coplanar Forces
5.1K
Consider an object upon which multiple forces are acting. If the lines of action of each force lie within the same plane, the system can be considered coplanar. The Cartesian vector form can be used to resolve each force into its respective components. For a coplanar system, the system will be in equilibrium if each component of the resultant force equals zero and the resultant force on the system is zero. If the sum of the forces is not equal to zero, then the object will not be in equilibrium...
5.1K
01:26Cartesian Form for Vector Formulation
992
The Cartesian form for vector formulation is a process to calculate the moment of force using the position and force vectors. The moment of force is defined as the cross-product of these vectors, making it a vector quantity. The Cartesian form of the position and force vectors involves unit vectors, which can be used to express the cross-product in determinant form.
992
01:26Divergence and Curl of Magnetic Field
3.7K
The magnetic field due to a volume current distribution given by the Biot–Savart Law can be expressed as follows:
3.7K
01:23Mohr's Circle for Plane Stress
858
Mohr's circle is a graphical method for identifying the state of stress at a point in a material, making it easier to analyze stress transformations under plane stress conditions. This two-dimensional technique visualizes both normal and shearing stresses on an element.
Consider a set of Cartesian coordinates. The horizontal and vertical axes correspond to normal stress (σ) and shearing stress (τ), respectively. Two points, points A and B, are defined by the normal and shear...
Consider a set of Cartesian coordinates. The horizontal and vertical axes correspond to normal stress (σ) and shearing stress (τ), respectively. Two points, points A and B, are defined by the normal and shear...
858