THE EXISTENCE OF INVARIANT TORI FOR THE THREE-DIMENSIONAL MOTION OF A BILLIARD BALL THAT ARE CONCENTRATED IN A NEIGHBOURHOOD OF A “CLOSED GEODESIC ON THE BOUNDARY OF THE DOMAIN”

1978 ◽  
Vol 33 (4) ◽  
pp. 267-268
Author(s):  
N V Svanidze
Keyword(s):  
Closed Geodesic ◽  
Invariant Tori ◽  
Three Dimensional ◽  
Billiard Ball

Exact Torus Knot Periodic Orbits and Homoclinic Orbits in a Class of Three-Dimensional Flows Generated by a Planar Cubic System

2017 ◽  
Vol 27 (13) ◽  
pp. 1750205 ◽  
Author(s):  
Tonghua Zhang ◽  
Jibin Li
Keyword(s):  
Vector Field ◽  
Periodic Orbits ◽  
Invariant Tori ◽  
Three Dimensional ◽  
Homoclinic Orbits ◽  
Numerical Examples ◽  
Torus Knot ◽  
Cubic System ◽  
Theoretical Results

This paper considers a class of three-dimensional systems constructed by a rotating planar symmetric cubic vector field. To study its periodic orbits including homoclinic orbits, which may be knotted in space, we classify the types of periodic orbits and then calculate their exact parametric representations. Our study shows that this class of systems has infinitely many distinct types of knotted periodic orbits, which lie on three families of invariant tori. Numerical examples of [Formula: see text]-torus knot periodic orbits have also been provided to illustrate our theoretical results.

THE CUSP–HOPF BIFURCATION

2007 ◽  
Vol 17 (08) ◽  
pp. 2547-2570 ◽  
Author(s):  
J. HARLIM ◽  
W. F. LANGFORD
Keyword(s):  
Hopf Bifurcation ◽  
Equilibrium Points ◽  
Invariant Tori ◽  
Three Dimensional ◽  
Phase Portraits ◽  
Planar System ◽  
Bursting Oscillations ◽  
Codimension Two ◽  
Truncated Normal ◽  
Comprehensive Study

The coalescence of a Hopf bifurcation with a codimension-two cusp bifurcation of equilibrium points yields a codimension-three bifurcation with rich dynamic behavior. This paper presents a comprehensive study of this cusp-Hopf bifurcation on the three-dimensional center manifold. It is based on truncated normal form equations, which have a phase-shift symmetry yielding a further reduction to a planar system. Bifurcation varieties and phase portraits are presented. The phenomena include all four cases that occur in the codimension-two fold–Hopf bifurcation, in addition to bistability involving equilibria, limit cycles or invariant tori, and a fold–heteroclinic bifurcation that leads to bursting oscillations. Uniqueness of the torus family is established locally. Numerical simulations confirm the prediction from the bifurcation analysis of bursting oscillations that are similar in appearance to those that occur in the electrical behavior of neurons and other physical systems.

BREAKDOWN OF LIBRATIONAL INVARIANT SURFACES

1999 ◽  
Vol 09 (05) ◽  
pp. 975-982
Author(s):  
MIQUEL ANGEL ANDREU ◽  
ALESSANDRA CELLETTI ◽  
CORRADO FALCOLINI
Keyword(s):  
Phase Space ◽  
Numerical Investigation ◽  
Invariant Tori ◽  
Three Dimensional ◽  
Spin Orbit Coupling ◽  
Spin Orbit ◽  
The Moon ◽  
Dimensional Phase Space ◽  
Invariant Surfaces ◽  
The Stability

A numerical investigation of the stability of invariant librational tori is presented. The method has been developed for a model describing the spin-orbit coupling in Celestial Mechanics. Periodic orbits approaching the librational torus are computed by means of Newton's method. According to Greene's criterion, their stability is strictly related to the survival of invariant tori. We consider librational tori around the main spin-orbit resonances (1:1, 3:2). Their existence provides the stability of the resonances, due to the confinement properties in the three-dimensional phase space associated to our model. The results are consistent with the actual observations of the eccentricity and of the oblateness parameter. A different behavior of the Moon and Mercury around the main resonances is evidenced, providing interesting suggestions about the different probabilities of capture in a resonance.

The Coexistence of Invariant Tori and Topological Horseshoe in a Generalized Nosé–Hoover Oscillator

2017 ◽  
Vol 27 (07) ◽  
pp. 1750111 ◽  
Author(s):  
Lei Wang ◽  
Xiao-Song Yang
Keyword(s):  
Cross Section ◽  
Fractal Structure ◽  
Polynomial System ◽  
Invariant Tori ◽  
Three Dimensional ◽  
Topological Horseshoe ◽  
Periodic Trajectories ◽  
Dynamical Complexity ◽  
Chaotic Region ◽  
Stable Periodic

This paper is devoted to the study of dynamical complexity of a generalized Nosé–Hoover oscillator which is a three-dimensional quadratic polynomial system. Precisely, a lot of moderately conservative regions are found, each of which is filled with different sequences of nested tori with various knot types and is embedded in the “chaotic region”. This shows that the generalized Nosé–Hoover oscillator may possess so-called “fat fractal” structure in phase space. In addition, horseshoe chaos can be demonstrated by applying the topological horseshoe theory to a Poincaré map defined in a proper cross-section, which further shows the coexistence of infinitely stable periodic trajectories and infinite saddle periodic trajectories.

SUFFICIENT CONDITIONS FOR THE EXISTENCE OF INVARIANT TORI IN MULTIDIMENSIONAL DYNAMICAL SYSTEMS

1991 ◽  
Vol 01 (03) ◽  
pp. 681-689
Author(s):  
V. S. AFRAIMOVICH ◽  
A. L. ZHELEZNYAK ◽  
I. L. ZHELEZNYAK
Keyword(s):  
Mathematical Model ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Invariant Tori ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Strange Attractors

A method for analyzing the existence of a multidimensional torus for certain systems of ordinary differential equations is proposed. Using this method, the existence of a multidimensional torus in one of such systems is analyzed. This system is a mathematical model of the dynamics of interacting structures within the drift hydrodynamical systems. The behavior of trajectories on the multidimensional torus is numerically investigated. The existence of two- and three-dimensional tori as well as strange attractors are considered.

Crystallography of three-dimensional fluid flows with chirality in hexagonal cases

2019 ◽  
Vol 75 (6) ◽  
pp. 798-813 ◽  
Author(s):  
Takahiro Nishiyama
Keyword(s):  
Plasma Physics ◽  
Materials Science ◽  
Invariant Tori ◽  
Three Dimensional ◽  
Fluid Flows ◽  
Physics First ◽  
Free Fields ◽  
Zero Points

Magnetic groups are applied to three-dimensional fluid flows with chirality, which are called Beltrami flows (or force-free fields in plasma physics). First, six Beltrami flows are derived so that their symmetries and antisymmetries are described by six different hexagonal magnetic groups. The general Wyckoff positions are used to derive the flows. Special Wyckoff positions are shown to be useful for finding the zero points of the flows. Tube-like surfaces called invariant tori are observed to interlace and form various crystal-like structures when streamlines winding around the surfaces are numerically plotted. Next, two simpler hexagonal Beltrami flows are derived, and their zero points and invariant tori are studied. Some families of the invariant tori have arrangements similar to those observed in materials science.

scholarly journals THREE-DIMENSIONAL BILLIARDS WITH TIME MACHINE

1996 ◽  
Vol 05 (02) ◽  
pp. 179-192 ◽  
Author(s):  
MICHAEL B. MENSKY ◽  
IGOR D. NOVIKOV
Keyword(s):  
World Line ◽  
Identical Particle ◽  
Three Dimensional ◽  
The Body ◽  
Closed Trajectory ◽  
Time Machine ◽  
Billiard Ball ◽  
The Past ◽  
Small Ideal ◽  
Classical Point

Self-collision of a nonrelativistic classical point-like body, or particle, in the spacetime containing closed time-like curves (time-machine spacetime) is considered. A point-like body (particle) is an idealization of a small ideal elastic billiard ball. The known model of a time machine is used containing a wormhole leading to the past. If the body enters one of the mouths of the wormhole, it emerges from another mouth in an earlier time so that both the particle and its “incarnation” coexist during some time and may collide. Such self-collisions are considered in the case when the size of the body is much less than the radius of the mouth, and the latter is much less than the distance between the mouths. Three-dimensional configurations of trajectories with a self-collision are presented. Their dynamics is investigated in detail. Configurations corresponding to multiple wormhole traversals are discussed. It is shown that, for each world line describing self-collision of a particle, dynamically equivalent configurations exist in which the particle collides not with itself but with an identical particle having a closed trajectory (Jinnee of Time Machine).

scholarly journals Symmetric Time-Reversible Flows with a Strange Attractor

2015 ◽  
Vol 25 (05) ◽  
pp. 1550078 ◽  
Author(s):  
J. C. Sprott
Keyword(s):  
State Space ◽  
Strange Attractor ◽  
Chaotic Attractor ◽  
Invariant Tori ◽  
Three Dimensional ◽  
The State ◽  
Unusual Property ◽  
Chaotic Flow ◽  
Symmetry Of The Solution ◽  
Reversed Time

A symmetric chaotic flow is time-reversible if the equations governing the flow are unchanged under the transformation t → -t except for a change in sign of one or more of the state space variables. The most obvious solution is symmetric and the same in both forward and reversed time and thus cannot be dissipative. However, it is possible for the symmetry of the solution to be broken, resulting in an attractor in forward time and a symmetric repellor in reversed time. This paper describes the simplest three-dimensional examples of such systems with polynomial nonlinearities and a strange (chaotic) attractor. Some of these systems have the unusual property of allowing the strange attractor to coexist with a set of nested symmetric invariant tori.

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