On the Dynamics Near a Homoclinic Network to a Bifocus: Switching and Horseshoes

2015 ◽  
Vol 25 (11) ◽  
pp. 1530030 ◽  
Author(s):  
Santiago Ibáñez ◽  
Alexandre Rodrigues
Keyword(s):  
Invariant Manifolds ◽  
Switching Behavior ◽  
Nondegeneracy Condition ◽  
Two Dimensional

We study a homoclinic network associated to a nonresonant hyperbolic bifocus. It is proved that on combining rotation with a nondegeneracy condition concerning the intersection of the two-dimensional invariant manifolds of the equilibrium, switching behavior is created: close to the network, there are trajectories that visit the neighborhood of the bifocus following connections in any prescribed order. We discuss the existence of suspended horseshoes which accumulate on the network and the relation between these horseshoes and the switching behavior.

Ferroelectric resistive switching behavior in two-dimensional materials/BiFeO3 hetero-junctions

Nanoscale ◽  
2018 ◽  
Vol 10 (48) ◽  
pp. 23080-23086 ◽  
Author(s):  
Yang Li ◽  
Xue-Yin Sun ◽  
Cheng-Yan Xu ◽  
Jian Cao ◽  
Zhao-Yuan Sun ◽  
...  
Keyword(s):  
Resistive Switching ◽  
Switching Behavior ◽  
Ferroelectric Polarization ◽  
Two Dimensional ◽  
Resistive Switching Behavior ◽  
Two Dimensional Materials ◽  
Interface Modulation

We presented thickness-dependent ferroelectric resistive switching in 2D/BFO heterojunctions, which stems from ferroelectric polarization induced hetero-interface modulation.

scholarly journals On the Measure-Preserving Mappings with Three-Dimensions

1993 ◽  
Vol 132 ◽  
pp. 73-89
Author(s):  
Yi-Sui Sun
Keyword(s):  
Fixed Point ◽  
Invariant Manifolds ◽  
Three Dimensional ◽  
Invariant Curve ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Kolmogorov Entropy ◽  
Numerical Exploration ◽  
Area Preserving ◽  
Dimensional Mapping

AbstractWe have systematically made the numerical exploration about the perturbation extension of area-preserving mappings to three-dimensional ones, in which the fixed points of area preserving are elliptic, parabolic or hyperbolic respectively. It has been observed that: (i) the invariant manifolds in the vicinity of the fixed point generally don’t exist (ii) when the invariant curve of original two-dimensional mapping exists the invariant tubes do also in the neighbourhood of the invariant curve (iii) for the perturbation extension of area-preserving mapping the invariant manifolds can only be generated in the subset of the invariant manifolds of original two-dimensional mapping, (iv) for the perturbation extension of area preserving mappings with hyperbolic or parabolic fixed point the ordered region near and far from the invariant curve will be destroyed by perturbation more easily than the other one, This is a result different from the case with the elliptic fixed point. In the latter the ordered region near invariant curve is solid. Some of the results have been demonstrated exactly.Finally we have discussed the Kolmogorov Entropy of the mappings and studied some applications.

scholarly journals Unstable equilibria and invariant manifolds in quasi-two-dimensional Kolmogorov-like flow

2018 ◽  
Vol 98 (2) ◽  
Author(s):  
Balachandra Suri ◽  
Jeffrey Tithof ◽  
Roman O. Grigoriev ◽  
Michael F. Schatz
Keyword(s):  
Invariant Manifolds ◽  
Two Dimensional

scholarly journals COMPUTING TWO-DIMENSIONAL GLOBAL INVARIANT MANIFOLDS IN SLOW–FAST SYSTEMS

2007 ◽  
Vol 17 (03) ◽  
pp. 805-822 ◽  
Author(s):  
J. P. ENGLAND ◽  
B. KRAUSKOPF ◽  
H. M. OSINGA
Keyword(s):  
Invariant Manifolds ◽  
Multiple Time Scales ◽  
Test Case ◽  
Multiple Time ◽  
Two Dimensional ◽  
Stable And Unstable Manifolds ◽  
Boundary Problems ◽  
Case Examples ◽  
Unstable Manifolds ◽  
Global Invariant

We present the GLOBALIZEBVP algorithm for the computation of two-dimensional stable and unstable manifolds of a vector field. Specifically, we use the collocation routines of AUTO to solve boundary problems that are used during the computation to find the next approximate geodesic level set on the manifold. The resulting implementation is numerically very stable and well suited for systems with multiple time scales. This is illustrated with the test-case examples of the Lorenz and Chua systems, and with a slow–fast model of a somatotroph cell.

Bifurcational Mechanism of Multistability Formation and Frequency Entrainment in a van der Pol Oscillator with an Additional Oscillatory Circuit

2016 ◽  
Vol 26 (07) ◽  
pp. 1650124 ◽  
Author(s):  
Sergey Astakhov ◽  
Oleg Astakhov ◽  
Vladimir Astakhov ◽  
Jürgen Kurths
Keyword(s):  
Hopf Bifurcation ◽  
Invariant Manifolds ◽  
Van Der Pol Oscillator ◽  
Two Dimensional ◽  
Dimensional Torus ◽  
Oscillatory Circuit ◽  
Van Der Pol ◽  
Frequency Entrainment ◽  
Basin Boundaries

In this paper, the bifurcational mechanism of frequency entrainment in a van der Pol oscillator coupled with an additional oscillatory circuit is studied. It is shown that bistability observed in the system is based on two bifurcations: a supercritical Andronov–Hopf bifurcation and a sub-critical Neimark–Sacker bifurcation. The attracting basin boundaries are determined by stable and unstable invariant manifolds of a saddle two-dimensional torus.

ON GEOMETRICAL PROPERTIES OF TWO-DIMENSIONAL DIFFEOMORPHISMS WITH HOMOCLINIC TANGENCIES

1995 ◽  
Vol 05 (03) ◽  
pp. 819-829 ◽  
Author(s):  
S.V. GONCHENKO ◽  
L.P. SHIL’NIKOV
Keyword(s):  
Fixed Point ◽  
Invariant Manifolds ◽  
Two Dimensional ◽  
Geometrical Properties ◽  
Saddle Value ◽  
Almost All ◽  
Homoclinic Tangencies

Two-dimensional diffeomorphisms with a quadratic tangency of invariant manifolds of a saddle fixed point are considered in the cases where the saddle value σ is either less than 1 or equal to it. A description of the structure of hyperbolic subsets is given. In the case σ=1, it is shown that almost all such diffeomorphisms admit the complete description in distinction with the case σ<1.

scholarly journals A SURVEY OF METHODS FOR COMPUTING (UN)STABLE MANIFOLDS OF VECTOR FIELDS

2005 ◽  
Vol 15 (03) ◽  
pp. 763-791 ◽  
Author(s):  
B. KRAUSKOPF ◽  
H. M. OSINGA ◽  
E. J. DOEDEL ◽  
M. E. HENDERSON ◽  
J. GUCKENHEIMER ◽  
...  
Keyword(s):  
Vector Field ◽  
Unstable Manifold ◽  
Stable Manifold ◽  
Vector Fields ◽  
Invariant Manifolds ◽  
Lorenz System ◽  
Dimensional Manifold ◽  
Two Dimensional ◽  
The Lorenz System ◽  
Global Invariant

The computation of global invariant manifolds has seen renewed interest in recent years. We survey different approaches for computing a global stable or unstable manifold of a vector field, where we concentrate on the case of a two-dimensional manifold. All methods are illustrated with the same example — the two-dimensional stable manifold of the origin in the Lorenz system.

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.

Interlayer coupling and optoelectronic properties of ultrathin two-dimensional heterostructures based on graphene, MoS2 and WS2

2015 ◽  
Vol 3 (21) ◽  
pp. 5467-5473 ◽  
Author(s):  
Nengjie Huo ◽  
Zhongming Wei ◽  
Xiuqing Meng ◽  
Joongoo Kang ◽  
Fengmin Wu ◽  
...  
Keyword(s):  
Interlayer Coupling ◽  
Optoelectronic Properties ◽  
Switching Behavior ◽  
Two Dimensional ◽  
Phonon Modes

Graphene–WS2 heterostructures exhibit strong interlayer coupling with stiffening phonon modes, as well as ambipolar, gate-tunable rectification and enhanced photo-switching behavior.

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