NONSMOOTH AND SMOOTH BIFURCATIONS IN A DISCONTINUOUS PIECEWISE MAP

2012 ◽  
Vol 22 (05) ◽  
pp. 1250112 ◽  
Author(s):  
ZHIYING QIN ◽  
YUEJING ZHAO ◽  
JICHEN YANG
Keyword(s):  
Fixed Point ◽  
Bifurcation Diagram ◽  
Bifurcation Point ◽  
Peculiar Feature ◽  
Parameter Plane ◽  
Flip Bifurcation ◽  
Border Collision Bifurcation ◽  
Bifurcation Structures ◽  
Piecewise Map ◽  
Two Parameter

In this paper, a piecewise map with singularity of the power (-1/2) is introduced. For this piecewise map, there is an infinite discontinuous gap on the origin. The conditions of nonsmooth border-collision bifurcation and smooth fold or flip bifurcation are analytically derived. For period-1 fixed point, two-parameter-plane can be divided into seven ranges according to different bifurcation structures. For period-n orbits, codimension-2 bifurcation point may lead to different period-increment sequence, and a peculiar feature is found that there are two different period-increment sequences in the same bifurcation diagram.

ON A "CROSSROAD AREA–SPRING AREA" TRANSITION OCCURRING IN A DUFFING–RAYLEIGH EQUATION WITH A PERIODICAL EXCITATION

1993 ◽  
Vol 03 (04) ◽  
pp. 1029-1037 ◽  
Author(s):  
CHRISTIAN MIRA ◽  
MOHAMMED QRIOUET
Keyword(s):  
Excitation Frequency ◽  
Period Doubling ◽  
Qualitative Change ◽  
Parameter Plane ◽  
Rayleigh Equation ◽  
Flip Bifurcation ◽  
External Excitation ◽  
Spring Area ◽  
Bifurcation Structures ◽  
Parameter Coefficient

The bifurcation structures considered in this paper are given by a Duffing–Rayleigh equation in the presence of a periodic external excitation. The first one is related to a cascade of fold lips generated by period doubling at subharmonic oscillations, which is obtained in a parameter plane defined by the excitation frequency and its amplitude. When a third parameter (coefficient of the linear approximation of the damping) varies, a qualitative change of the parameter plane occurs. It is related to a new mechanism of "crossroad area–spring area" transition, the areas corresponding to typical arrangements of fold and flip bifurcation curves around a fold cusp.

COMPLEX BIFURCATION STRUCTURES IN THE HINDMARSH–ROSE NEURON MODEL

2007 ◽  
Vol 17 (09) ◽  
pp. 3071-3083 ◽  
Author(s):  
J. M. GONZÀLEZ-MIRANDA
Keyword(s):  
Phase Diagram ◽  
Bifurcation Diagram ◽  
Parameter Space ◽  
Neuron Model ◽  
Two Dimensional ◽  
Dimensional Parameter ◽  
Rapid Responses ◽  
Bifurcation Structures

The results of a study of the bifurcation diagram of the Hindmarsh–Rose neuron model in a two-dimensional parameter space are reported. This diagram shows the existence and extent of complex bifurcation structures that might be useful to understand the mechanisms used by the neurons to encode information and give rapid responses to stimulus. Moreover, the information contained in this phase diagram provides a background to develop our understanding of the dynamics of interacting neurons.

"CROSSROAD AREA–SPRING AREA" TRANSITION (II) FOLIATED PARAMETRIC REPRESENTATION

1991 ◽  
Vol 01 (02) ◽  
pp. 339-348 ◽  
Author(s):  
C. MIRA ◽  
J. P. CARCASSÈS ◽  
M. BOSCH ◽  
C. SIMÓ ◽  
J. C. TATJER
Keyword(s):  
Complete Information ◽  
Parametric Representation ◽  
Periodic Points ◽  
Parameter Plane ◽  
Flip Bifurcation ◽  
Cusp Point ◽  
Dimensional Representation ◽  
Bifurcation Structure ◽  
Spring Area

The areas considered are related to two different configurations of fold and flip bifurcation curves of maps, centred at a cusp point of a fold curve. This paper is a continuation of an earlier one devoted to parameter plane representation. Now the transition is studied in a thee-dimensional representation by introducing a norm associated with fixed or periodic points. This gives rise to complete information on the map bifurcation structure.

Examinations of Cross-Linked Polyvinylpyridine in Open Reactor

2005 ◽  
Vol 494 ◽  
pp. 369-374 ◽  
Author(s):  
M. Milošević ◽  
N. Pejić ◽  
Ž. Čupić ◽  
S. Anić ◽  
Lj. Kolar-Anić
Keyword(s):  
Specific Surface ◽  
Bifurcation Diagram ◽  
Surface Area ◽  
Specific Surface Area ◽  
Bifurcation Point ◽  
Particle Density ◽  
Physicochemical Characteristics ◽  
Stirred Tank ◽  
Stirred Tank Reactor ◽  
Tank Reactor

Macroporous cross-linked copolymer of 4-vinylpyridine and 25% (4:1) divinylbenzene is analyzed under open conditions, that is in a continuous well-stirred tank reactor (CSTR). With this aim the appropriate bifurcation diagram is found and the behavior of the system with and without polymer in the vicinity of the bifurcation point is used for the polymer examinations. Two different granulations of polymer are considered. Moreover, some physicochemical characteristics of the polymer, such as specific surface area, skeletal and particle density, are determined.

Constructing Keyed Strong S-Box Using an Enhanced Quadratic Map

2021 ◽  
Vol 31 (10) ◽  
pp. 2150146
Author(s):  
Yuanyuan Si ◽  
Hongjun Liu ◽  
Yuehui Chen
Keyword(s):  
Phase Diagram ◽  
Fixed Point ◽  
Lyapunov Exponent ◽  
Uniform Distribution ◽  
Bifurcation Diagram ◽  
Experimental Results ◽  
Kolmogorov Entropy ◽  
Symmetric Cryptography ◽  
Quadratic Map ◽  
Nonlinear Component

As the only nonlinear component for symmetric cryptography, S-Box plays an important role. An S-Box may be vulnerable because of the existence of fixed point, reverse fixed point or short iteration cycles. To construct a keyed strong S-Box, first, a 2D enhanced quadratic map (EQM) was constructed, and its dynamic behaviors were analyzed through phase diagram, Lyapunov exponent, Kolmogorov entropy, bifurcation diagram and randomness testing. The results demonstrated that the state points of EQM have uniform distribution, ergodicity and better randomness. Then a keyed strong S-Box construction algorithm was designed based on EQM, and the fixed point, reverse fixed point, and short cycles were eliminated. Experimental results verified the algorithm’s feasibility and effectiveness.

scholarly journals A Study on the double crisis vertex in a twoparameter plane*

2002 ◽  
Vol 51 (12) ◽  
pp. 2694
Author(s):  
Hong Ling ◽  
Xu Jian-Xue
Keyword(s):  
Parameter Plane ◽  
Two Parameter

scholarly journals Discrete-time phytoplankton–zooplankton model with bifurcations and chaos

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
A. Q. Khan ◽  
M. B. Javaid
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Discrete Time ◽  
Control Method ◽  
Chaotic Behavior ◽  
Discrete Analogue ◽  
Linear Stability Theory ◽  
Flip Bifurcation ◽  
Time Model ◽  
Local Dynamics

AbstractThe local dynamics with different topological classifications, bifurcation analysis, and chaos control for the phytoplankton–zooplankton model, which is a discrete analogue of the continuous-time model by a forward Euler scheme, are investigated. It is proved that the discrete-time phytoplankton–zooplankton model has trivial and semitrivial fixed points for all involved parameters, but it has an interior fixed point under the definite parametric condition. Then, by linear stability theory, local dynamics with different topological classifications are investigated around trivial, semitrivial, and interior fixed points. Further, for the discrete-time phytoplankton–zooplankton model, the existence of periodic points is also investigated. The existence of possible bifurcations around trivial, semitrivial, and interior fixed points is also investigated, and it is proved that there exists a transcritical bifurcation around a trivial fixed point. It is also proved that around trivial and semitrivial fixed points of the phytoplankton–zooplankton model there exists no flip bifurcation, but around an interior fixed point there exist both Neimark–Sacker and flip bifurcations. From the viewpoint of biology, the occurrence of Neimark–Sacker implies that there exist periodic or quasi-periodic oscillations between phytoplankton and zooplankton populations. Next, the feedback control method is utilized to stabilize chaos existing in the phytoplankton–zooplankton model. Finally, simulations are presented to validate not only obtained results but also the complex dynamics with orbits of period-8, 9, 10, 11, 14, 15 and chaotic behavior of the discrete-time phytoplankton–zooplankton model.

Bifurcation and evolution of a forced and damped Duffing system in two-parameter plane

2018 ◽  
Vol 93 (2) ◽  
pp. 749-766 ◽  
Author(s):  
Jian-Fei Shi ◽  
Yan-Long Zhang ◽  
Xiang-Feng Gou
Keyword(s):  
Parameter Plane ◽  
Duffing System ◽  
Two Parameter

scholarly journals Discrete-Time Predator-Prey Model with Bifurcations and Chaos

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
K. S. Al-Basyouni ◽  
A. Q. Khan
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Discrete Time ◽  
Control Method ◽  
Linear Stability Theory ◽  
Flip Bifurcation ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Positive Fixed Point

In this paper, local dynamics, bifurcations and chaos control in a discrete-time predator-prey model have been explored in ℝ + 2 . It is proved that the model has a trivial fixed point for all parametric values and the unique positive fixed point under definite parametric conditions. By the existing linear stability theory, we studied the topological classifications at fixed points. It is explored that at trivial fixed point model does not undergo the flip bifurcation, but flip bifurcation occurs at the unique positive fixed point, and no other bifurcations occur at this point. Numerical simulations are performed not only to demonstrate obtained theoretical results but also to tell the complex behaviors in orbits of period-4, period-6, period-8, period-12, period-17, and period-18. We have computed the Maximum Lyapunov exponents as well as fractal dimension numerically to demonstrate the appearance of chaotic behaviors in the considered model. Further feedback control method is employed to stabilize chaos existing in the model. Finally, existence of periodic points at fixed points for the model is also explored.

Sign in / Sign up

Export Citation Format

Share Document