Implicit Visualization of 2D Vector Field Topology for Periodic Orbit Detection

In this paper, a 4D Lorenz-type multistable hyperchaotic system with a curve of equilibria is investigated by using differential geometry method, i.e. with KCC-theory. Due to the deviation curvature tensor and its eigenvalues, the curve of equilibria of this hyperchaotic system is proved analytically to be Jacobi unstable under a certain parameter condition, and a periodic orbit of this system is proved numerically to be also Jacobi unstable. Furthermore, the dynamics of contravariant vector field near the curve of equilibria and the periodic orbit are studied, respectively, and their results comply absolutely with the above analysis of Jacobi stability.

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Bifurcation from a Periodic Orbit for a Strongly Resonant Reversible Autonomous Vector Field

SIAM Journal on Mathematical Analysis ◽

10.1137/0524089 ◽

1993 ◽

Vol 24 (6) ◽

pp. 1577-1596 ◽

Cited By ~ 1

Author(s):

Marie-Christine Pérouème

Keyword(s):

Vector Field ◽

Periodic Orbit

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Fast Combinatorial Vector Field Topology

IEEE Transactions on Visualization and Computer Graphics ◽

10.1109/tvcg.2010.235 ◽

2011 ◽

Vol 17 (10) ◽

pp. 1433-1443 ◽

Cited By ~ 24

Author(s):

Jan Reininghaus ◽

Christian Lowen ◽

Ingrid Hotz

Keyword(s):

Vector Field ◽

Vector Field Topology ◽

Combinatorial Vector Field

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Representation and display of vector field topology in fluid flow data sets

Computer ◽

10.1109/2.35197 ◽

1989 ◽

Vol 22 (8) ◽

pp. 27-36 ◽

Cited By ~ 259

Author(s):

J. Helman ◽

L. Hesselink

Keyword(s):

Fluid Flow ◽

Vector Field ◽

Data Sets ◽

Flow Data ◽

Vector Field Topology

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Analyticity and metric transitivity on the torus

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385798108192 ◽

1998 ◽

Vol 18 (3) ◽

pp. 717-723

Author(s):

SOL SCHWARTZMAN

Keyword(s):

Vector Field ◽

Periodic Orbit ◽

Differential Equations ◽

Entire Plane ◽

Transitive Flow ◽

Divergence Free Vector ◽

Divergence Free ◽

Open Question ◽

Set Of Points

Suppose we are given an analytic divergence free vector field $(X,Y)$ on the standard torus. We can find constants $a$ and $b$ and a function $F(x,y)$ of period one in both $x$ and $y$ such that $(X,Y)=(a-F_y,b+F_x)$. For a given $F$, let $P$ be the map sending $(x,y)$ into $(F_y(x,y),-F_x(x,y))$. Let $A$ be the image of the torus under this map and let $B$ be the image under this map of the set of points $(x,y)$ at which $F_{xx}F_{yy}-(F_{xy})^2$ vanishes. For any point $(a,b)$ in the complement of the interior of $A$, the flow on the torus arising from the differential equations $dx/dt=a-F_y(x,y)$, $dy/dt=b+F_x(x,y)$ is metrically transitive if and only if $a/b$ is irrational. For any point in $A$ but not in $B$ the flow is not metrically transitive. Moreover, if $a/b$ is irrational but the flow on the torus is not metrically transitive and we use our differential equations to define a flow in the entire plane (rather than on the torus), this flow has a nonstationary periodic orbit. It is an open question whether a point $(a,b)$ in the interior of $A$ can give rise to a metrically transitive flow.

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Accurate analysis of angiograms based on 3D vector field topology

Computer Aided Geometric Design ◽

10.1016/j.cagd.2012.03.021 ◽

2013 ◽

Vol 30 (6) ◽

pp. 542-548

Author(s):

Thomas Wischgoll

Keyword(s):

Vector Field ◽

Accurate Analysis ◽

Vector Field Topology

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On holomorphic flows on Stein surfaces: Transversality, dicriticalness and stability

International Journal of Mathematics ◽

10.1142/s0129167x14500931 ◽

2014 ◽

Vol 25 (10) ◽

pp. 1450093

Author(s):

T. Ito ◽

B. Scárdua ◽

Y. Yamagishi

Keyword(s):

Vector Field ◽

Periodic Orbit ◽

Finite Number ◽

Sufficient Conditions ◽

First Integral ◽

Holomorphic Vector Field ◽

Necessary And Sufficient ◽

Holonomy Map

We study the classification of the pairs (N, X) where N is a Stein surface and X is a ℂ-complete holomorphic vector field with isolated singularities on N. We describe the role of transverse sections in the classification of X and give necessary and sufficient conditions on X in order to have N biholomorphic to ℂ2. As a sample of our results, we prove that N is biholomorphic to ℂ2 if H2(N, ℤ) = 0, X has a finite number of singularities and exhibits a singularity with three separatrices or, equivalently, a singularity with first jet of the form [Formula: see text] where λ1/λ2 ∈ ℚ+. We also study flows with many periodic orbits (i.e. orbits diffeomorphic to ℂ*), in a sense we will make clear, proving they admit a meromorphic first integral or they exhibit some special periodic orbit, whose holonomy map is a non-resonant nonlinearizable diffeomorphism map.

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