scholarly journals Anti-Controlling Codimension-Two Bifurcation of Discrete Dynamical System in 1∶2 Resonance

2022 ◽  
Vol 43 (0) ◽  
pp. 1-15
Author(s):  
YANG Yujiao ◽  
◽  
◽  
XU Huidong ◽  
ZHANG Jianwen ◽  
...  
Keyword(s):  
Dynamical System ◽  
Discrete Dynamical System ◽  
Codimension Two ◽  
Codimension Two Bifurcation

Approximations of the Heteroclinic Orbits Near a Double-Zero Bifurcation with Symmetry of Order Two. Application to a Liénard Equation

2019 ◽  
Vol 29 (06) ◽  
pp. 1950074
Author(s):  
Carmen Rocşoreanu ◽  
Mihaela Sterpu
Keyword(s):  
Dynamical System ◽  
Blow Up ◽  
Heteroclinic Orbits ◽  
Liénard Equation ◽  
Heteroclinic Bifurcation ◽  
Double Zero ◽  
Codimension Two ◽  
Truncated Normal ◽  
Lienard Equation ◽  
Codimension Two Bifurcation

A dynamical system possessing an equilibrium point with two zero eigenvalues is considered. We assume that a degenerate Bogdanov–Takens bifurcation with symmetry of order two is present and, in the parameter space, a curve of heteroclinic bifurcation values emerges from the codimension two bifurcation point. Using a blow-up transformation and a perturbation method, we obtain second order approximations both for the heteroclinic orbits and for the curve of heteroclinic bifurcation values. Applications of our results for the truncated normal form and for a Liénard equation are presented. Some numerical simulations illustrating the accuracy of our results are performed.

NEW PHENOMENA IN THE NEIGHBORHOOD OF THE CODIMENSION-TWO BIFURCATION IN THE TWIN-WELL DUFFING OSCILLATOR

2000 ◽  
Vol 10 (06) ◽  
pp. 1367-1381 ◽  
Author(s):  
W. SZEMPLIŃSKA-STUPNICKA ◽  
A. ZUBRZYCKI ◽  
E. TYRKIEL
Keyword(s):  
Bifurcation Point ◽  
Chaotic Attractor ◽  
Duffing Oscillator ◽  
Codimension Two ◽  
Periodic Attractors ◽  
The Cross ◽  
Complex Phenomena ◽  
New Phenomena ◽  
Secondary Bifurcations ◽  
Codimension Two Bifurcation

In this paper, we study effects of the secondary bifurcations in the neighborhood of the primary codimension-two bifurcation point. The twin-well potential Duffing oscillator is considered and the investigations are focused on the new scenario of destruction of the cross-well chaotic attractor. The phenomenon belongs to the category of the subduction scenario and relies on the replacement of the cross-well chaotic attractor by a pair of unsymmetric 2T-periodic attractors. The exploration of a sequence of accompanying bifurcations throws more light on the complex phenomena that may occur in the neighborhood of the primary codimension-two bifurcation point. It shows that in the close vicinity of the point there appears a transition zone in the system parameter plane, the zone which separates the two so-far investigated scenarios of annihilation of the cross-well chaotic attractor.

Identification of gene interactions in fungal–plant symbiosis through discrete dynamical system modelling

2009 ◽  
Vol 3 (5) ◽  
pp. 414-428 ◽  
Author(s):  
J.G.C. Angeles ◽  
Z. Ouyang ◽  
A.M. Aguirre ◽  
P.J. Lammers ◽  
M. Song
Keyword(s):  
Dynamical System ◽  
Discrete Dynamical System ◽  
Gene Interactions ◽  
System Modelling

scholarly journals $(q,t)$-Characters of Kirillov-Reshetikhin Modules of Type $A_r$ as Quantum Cluster Variables

10.37236/7188 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Bolor Turmunkh
Keyword(s):  
Dynamical System ◽  
Discrete Dynamical System ◽  
Cluster Algebra ◽  
Quiver Varieties ◽  
Quantum Cluster ◽  
T System

Nakajima (2003) introduced a $t$-deformation of $q$-characters, $(q,t)$-characters for short, and their twisted multiplication through the geometry of quiver varieties. The Nakajima $(q,t)$-characters of Kirillov-Reshetikhin modules satisfy a $t$-deformed $T$-system. The $T$-system is a discrete dynamical system that can be interpreted as a mutation relation in a cluster algebra in two different ways, depending on the choice of direction of evolution. In this paper, we show that the Nakajima $t$-deformed $T$-system of type $A_r$ forms a quantum mutation relation in a quantization of exactly one of the cluster algebra structures attached to the $T$-system.

Geometry and Kinematics of Cylindrical Waterbomb Tessellation

2021 ◽  
Author(s):  
Rinki Imada ◽  
Tomohiro Tachi
Keyword(s):  
Dynamical System ◽  
Recurrence Relation ◽  
Discrete Dynamical System ◽  
Kinematic Model ◽  
Theoretical Reason ◽  
Finite Solution ◽  
Geometry And Kinematics ◽  
Folded Surfaces

Abstract Folded surfaces of origami tessellations have attracted much attention because they sometimes exhibit non-trivial behaviors. It is known that cylindrical folded surfaces of waterbomb tessellation called waterbomb tube can transform into wave-like surfaces, which is a unique phenomenon not observed on other tessellations. However, the theoretical reason why wave-like surfaces arise has been unclear. In this paper, we provide a kinematic model of waterbomb tube by parameterizing the geometry of a module of waterbomb tessellation and derive a recurrence relation between the modules. Through the visualization of the configurations of waterbomb tubes under the proposed kinematic model, we classify solutions into three classes: cylinder solution, wave-like solution, and finite solution. Furthermore, we give proof of the existence of a wave-like solution around one of the cylinder solutions by applying the knowledge of the discrete dynamical system to the recurrence relation.

The Double Seesaw Oscillator in a Windfield: Theory and Experiments

1997 ◽  
Author(s):  
A. H. P. van der Burgh ◽  
T. I. Haaker ◽  
B. W. van Oudheusden
Keyword(s):  
Equilibrium State ◽  
Degrees Of Freedom ◽  
Normal Modes ◽  
Typical Result ◽  
Resonant Case ◽  
Accurate Model ◽  
Codimension Two ◽  
Model Equations ◽  
Theory And Experiments ◽  
Codimension Two Bifurcation

Abstract In this paper the dynamics of an oscillator with two degrees-of-freedom of a double seesaw type in a windtunnel is studied. Model equations for this aeroelastic oscillator are derived and an analysis of these equations is given for the non-resonant case. A typical result is a (local codimension two) bifurcation which describes the transfer from the unstable equilibrium state to one of the two normal modes of the oscillator. Some experimental results are presented from which on may conclude that more accurate model equations should be developed.

scholarly journals The Limit Point of the Pentagram Map

2018 ◽  
Vol 2020 (9) ◽  
pp. 2818-2831 ◽  
Author(s):  
Max Glick
Keyword(s):  
Dynamical System ◽  
Limit Point ◽  
Convex Polygon ◽  
Discrete Dynamical System ◽  
Explicit Description ◽  
Cartesian Coordinates ◽  
The Subject ◽  
Pentagram Map

Abstract The pentagram map is a discrete dynamical system defined on the space of polygons in the plane. In the 1st paper on the subject, Schwartz proved that the pentagram map produces from each convex polygon a sequence of successively smaller polygons that converges exponentially to a point. We investigate the limit point itself, giving an explicit description of its Cartesian coordinates as roots of certain degree three polynomials.

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