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

1993 ◽  
Vol 24 (6) ◽  
pp. 1577-1596 ◽  
Author(s):  
Marie-Christine Pérouème
Keyword(s):  
Vector Field ◽  
Periodic Orbit

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

2021 ◽  
pp. 159-180
Author(s):  
Alexander Straub ◽  
Grzegorz K. Karch ◽  
Filip Sadlo ◽  
Thomas Ertl
Keyword(s):  
Vector Field ◽  
Periodic Orbit ◽  
Vector Field Topology

The Jacobi Stability of a Lorenz-Type Multistable Hyperchaotic System with a Curve of Equilibria

2019 ◽  
Vol 29 (05) ◽  
pp. 1950062 ◽  
Author(s):  
Yuming Chen ◽  
Zongbin Yin
Keyword(s):  
Differential Geometry ◽  
Vector Field ◽  
Periodic Orbit ◽  
Curvature Tensor ◽  
Hyperchaotic System ◽  
Geometry Method ◽  
Parameter Condition

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.

Analyticity and metric transitivity on the torus

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.

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

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.

scholarly journals NONORIENTABLE MANIFOLDS IN THREE-DIMENSIONAL VECTOR FIELDS

2003 ◽  
Vol 13 (03) ◽  
pp. 553-570 ◽  
Author(s):  
HINKE M. OSINGA
Keyword(s):  
Vector Field ◽  
Periodic Orbit ◽  
Vector Fields ◽  
Invariant Manifolds ◽  
Three Dimensional ◽  
Period Doubling ◽  
Building Blocks ◽  
Two Dimensional ◽  
Dimensional Vector ◽  
Three Dimensional Vector Field

It is well known that a nonorientable manifold in a three-dimensional vector field is topologically equivalent to a Möbius strip. The most frequently used example is the unstable manifold of a periodic orbit that just lost its stability in a period-doubling bifurcation. However, there are not many explicit studies in the literature in the context of dynamical systems, and so far only qualitative sketches could be given as illustrations. We give an overview of the possible bifurcations in three-dimensional vector fields that create nonorientable manifolds. We mainly focus on nonorientable manifolds of periodic orbits, because they are the key building blocks. This is illustrated with invariant manifolds of three-dimensional vector fields that arise from applications. These manifolds were computed with a new algorithm for computing two-dimensional manifolds.

Geometrical conditions for the stability of orbits in planar systems

1996 ◽  
Vol 120 (3) ◽  
pp. 499-519 ◽  
Author(s):  
R. A. Garcia ◽  
A. Gasull ◽  
A. Guillamon
Keyword(s):  
Vector Field ◽  
Periodic Orbit ◽  
Critical Points ◽  
Real Plane ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
The Real ◽  
Planar Systems ◽  
The Stability ◽  
Derivatives Of

AbstractGiven a vector field X on the real plane, we study the influence of the curvature of the orbits of ẋ = X┴(x) in the stability of those of the system x˙ = X(x). We pay special attention to the case in which this curvature is negative in the whole plane. Under this assumption, we classify the possible critical points and give a criterion for a point to be globally asymptotically stable. In the general case, we also provide expressions for the first three derivatives of the Poincaré map associated to a periodic orbit in terms of geometrical quantities.

scholarly journals Vector Field Editing and Periodic Orbit Extraction Using Morse Decomposition

2007 ◽  
Vol 13 (4) ◽  
pp. 769-785 ◽  
Author(s):  
Guoning Chen ◽  
K. Mischaikow ◽  
R.S. Laramee ◽  
P. Pilarczyk ◽  
E. Zhang
Keyword(s):  
Vector Field ◽  
Periodic Orbit ◽  
Morse Decomposition

Periodic Orbits for Generalized Gradient Flows

1995 ◽  
Vol 38 (1) ◽  
pp. 117-119
Author(s):  
Sol Schwartzman
Keyword(s):  
Vector Field ◽  
Periodic Orbit ◽  
List Type ◽  
Dimensional Manifold ◽  
Gradient Flows ◽  
Gradient Vector ◽  
Generalized Gradient ◽  
One Dimensional ◽  
Whitney Class ◽  
Gradient Vector Field

AbstractLet Mn be an n-dimensional compact oriented connected Riemannean manifold. It is proved that either of the following conditions is sufficient to insure that the flow defined by a generalized gradient vector field in Mn has either a stationary point or a periodic orbit:a)Mn is the product of a circle with an (n — 1 ) dimensional manifold of non-zero Euler characteristic.b)The (n — 1) dimensional Stiefel-Whitney class of Mn is different from zero and in addition Mn possesses no one-dimensional 2-torsion.

The orbit of Pluto over a long interval of time

1966 ◽  
Vol 25 ◽  
pp. 227-229 ◽  
Author(s):  
D. Brouwer
Keyword(s):  
Periodic Orbit ◽  
Numerical Integration ◽  
Restricted Problem ◽  
Three Dimensional ◽  
The Third ◽  
Circular Restricted Problem ◽  
Resonance Relation

The paper presents a summary of the results obtained by C. J. Cohen and E. C. Hubbard, who established by numerical integration that a resonance relation exists between the orbits of Neptune and Pluto. The problem may be explored further by approximating the motion of Pluto by that of a particle with negligible mass in the three-dimensional (circular) restricted problem. The mass of Pluto and the eccentricity of Neptune's orbit are ignored in this approximation. Significant features of the problem appear to be the presence of two critical arguments and the possibility that the orbit may be related to a periodic orbit of the third kind.

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