Codimension-3 Bifurcation in the p53 Regulatory Network Model

2021 ◽  
Vol 31 (07) ◽  
pp. 2150104
Author(s):  
Cuicui Jiang ◽  
Yongxin Zhang ◽  
Wendi Wang
Keyword(s):  
Limit Cycle ◽  
Complex Dynamics ◽  
Stable Limit Cycle ◽  
Feedback Loops ◽  
Stable Limit ◽  
Biological Applications ◽  
Stable States ◽  
Heteroclinic Loop ◽  
Stable Equilibria ◽  
New Phenomena

In this paper, a p53-Mdm2 mathematical model is analyzed to understand the biological implications of feedback loops in a p53 system. Results show that the model can undergo four types of codimension-3 Bogdanov–Takens bifurcations, including cusp, saddle, focus and elliptic. Specifically, we find new phenomena including the coexistence of four positive equilibria, two limit cycles, the coexistence of three stable states (two stable equilibria and one stable limit cycle, or three stable equilibria), a heteroclinic loop enclosing a smaller stable limit cycle and a larger stable limit cycle. These findings extend the understanding of the complex dynamics of the p53 system, and can provide some potential biological applications.

Bifurcation of Codimension 3 in a Predator–Prey System of Leslie Type with Simplified Holling Type IV Functional Response

2016 ◽  
Vol 26 (02) ◽  
pp. 1650034 ◽  
Author(s):  
Jicai Huang ◽  
Xiaojing Xia ◽  
Xinan Zhang ◽  
Shigui Ruan
Keyword(s):  
Functional Response ◽  
Limit Cycle ◽  
Stable Limit Cycle ◽  
Positive Equilibrium ◽  
Stable Limit ◽  
Homoclinic Loop ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Type Iv ◽  
Stable Equilibria

It was shown in [Li & Xiao, 2007] that in a predator–prey model of Leslie type with simplified Holling type IV functional response some complex bifurcations can occur simultaneously for some values of parameters, such as codimension 1 subcritical Hopf bifurcation and codimension 2 Bogdanov–Takens bifurcation. In this paper, we show that for the same model there exists a unique degenerate positive equilibrium which is a degenerate Bogdanov–Takens singularity (focus case) of codimension 3 for other values of parameters. We prove that the model exhibits degenerate focus type Bogdanov–Takens bifurcation of codimension 3 around the unique degenerate positive equilibrium. Numerical simulations, including the coexistence of three hyperbolic positive equilibria, two limit cycles, bistability states (one stable equilibrium and one stable limit cycle, or two stable equilibria), tristability states (two stable equilibria and one stable limit cycle), a stable limit cycle enclosing a homoclinic loop, a homoclinic loop enclosing an unstable limit cycle, or a stable limit cycle enclosing three unstable hyperbolic positive equilibria for various parameter values, confirm the theoretical results.

scholarly journals STUDYING THE STABILITY BY USING LOCAL LINEARIZATION METHOD

2020 ◽  
Vol 2 (1) ◽  
pp. 27-31
Author(s):  
Abdulghafoor Jasim Salim ◽  
Kais Ismail Ebrahem ◽  
Suhirman
Keyword(s):  
Singular Point ◽  
Limit Cycle ◽  
Autoregressive Model ◽  
Stable Limit Cycle ◽  
Linearization Method ◽  
Stable Limit ◽  
Local Linearization ◽  
Non Linear ◽  
The Stability ◽  
Local Linearization Method

Abstract: In this paper we study the stability of one of a non linear autoregressive model with trigonometric term  by using local linearization method proposed by Tuhro Ozaki .We find the singular point ,the stability of the singular point and the limit cycle. We conclude  that the proposed model under certain conditions have a non-zero singular point which is  a asymptotically salable ( when  0 ) and have an  orbitaly stable limit cycle . Also we give some examples in order to explain the method. Key Words : Non-linear Autoregressive model; Limit cycle; singular point; Stability.

scholarly journals Height control for a two-mass hopping robot based on stable limit cycle

2016 ◽  
Vol 13 (6) ◽  
pp. 172988141665774
Author(s):  
Taihui Zhang ◽  
Honglei An ◽  
Qing Wei ◽  
Wenqi Hou ◽  
Hongxu Ma
Keyword(s):  
Limit Cycle ◽  
Control Algorithm ◽  
Stable Limit Cycle ◽  
Stable Limit ◽  
Inverted Pendulum Model ◽  
Hopping Robot ◽  
Control Objective ◽  
Upper Level ◽  
Mass Spring ◽  
And Control

Differing from the commonly used spring loaded inverted pendulum model, this paper makes use of a two-mass spring model considering impact between the foot and ground which is closer to the real hopping robot. The height of upper mass which includes the upper leg and body is the main control objective. Then we develop a new kind of control algorithm acting on two levels: The upper level aims to achieve the desired velocity of the upper mass based on a stable limit cycle, where three different controllers are used to regulate the limit cycle; the target of the lower level is to drive the system to converge to the desired state and control the contact force between the foot and ground within an appropriate range based on the inner force control at the same time. Simulation results presented in this paper confirm the efficiency of this control algorithm.

On the Poincaré–Hill cycle map of rotational random walk: locating the stochastic limit cycle in a reversible Schnakenberg model

2009 ◽  
Vol 466 (2115) ◽  
pp. 771-788 ◽  
Author(s):  
Melissa Vellela ◽  
Hong Qian
Keyword(s):  
Limit Cycle ◽  
Stable Limit Cycle ◽  
Quantitative Measure ◽  
Power Spectral Analysis ◽  
Chemical System ◽  
Deterministic System ◽  
Stable Limit ◽  
Novel Device ◽  
Schnakenberg Model ◽  
Cycle Map

Recent studies on stochastic oscillations mostly focus on the power spectral analysis. However, the power spectrum yields information only on the frequency of oscillation and cannot differentiate between a stable limit cycle and a stable focus. The cycle flux, introduced by Hill (Hill 1989 Free energy transduction and biochemical cycle kinetics ), is a quantitative measure of the net movement over a closed path, but it is impractical to compute for all possible cycles in systems with a large state space. Through simple examples, we introduce concepts used to quantify stochastic oscillation, such as the cycle flux, the Hill–Qian stochastic circulation and rotation number. We introduce a novel device, the Poincaré–Hill cycle map (PHCM), which combines the concept of Hill’s cycle flux with the Poincaré map from nonlinear dynamics. Applying the PHCM to a reversible extension of an oscillatory chemical system, the Schnakenberg model, reveals stable oscillations outside the Hopf bifurcation region in which the deterministic system contains a limit cycle. Bistable behaviour is found on the small volume scale with high probabilities around both the fixed point and the limit cycle. Convergence to the deterministic system is found in the thermodynamic limit.

The Effects of Friction in Axial Splines on Rotor System Stability

1993 ◽  
Vol 115 (2) ◽  
pp. 272-278
Author(s):  
A. F. Artiles
Keyword(s):  
Friction Coefficient ◽  
Limit Cycle ◽  
Stable Limit Cycle ◽  
Threshold Value ◽  
System Stability ◽  
Stable Limit ◽  
Spin Speed ◽  
Side Load ◽  
Unstable Limit Cycle ◽  
Rotor Bearing

An in-depth parametric evaluation of the effects of Coulomb friction in an axial spline joint on the stability of the rotor-bearing system was conducted through time transient integration of the equations of motion. The effects of spin speed, friction coefficient, spline torque, external damping, imbalance, and side load as well as asymmetric bearing stiffnesses were investigated. A subsynchronous instability is present at the bending critical speed when the spin speed is above this critical. The limit cycle orbit is circular, proportional to the product of the friction coefficient and spline torque (μT), inversely proportional to the external damping, and independent of spin speed. When imbalance is applied to the rotor, beating between the subsynchronous natural frequency and the synchronous (spin speed) frequency occurs. The subsynchronous component of the orbit is proportional to μT, while the synchronous component is proportional to the imbalance. When a static side load is applied, the unstable node at the center of the orbitally stable limit cycle grows into an ellitpical orbitally unstable limit cycle, separating stable from unstable regions of the phase plane. Below a threshold value of side load, the transient motion approaches one of two asymptotic solutions depending on the initial conditions: the larger stable limit cycle or a point at the center of the smaller unstable limit cycle. Beyond the threshold value of side load, the rotor-bearing is stable and all motions decay to a point. Asymmetry in the bearing stiffnesses reduces the size of the subsynchronous whirl orbit.

Energy harvesting for actuators and sensors using free-floating flaps

2016 ◽  
Vol 28 (2) ◽  
pp. 163-177 ◽  
Author(s):  
Lars O Bernhammer ◽  
Roeland De Breuker ◽  
Moti Karpel
Keyword(s):  
Limit Cycle ◽  
Vibrational Energy ◽  
Stable Limit Cycle ◽  
Electric Circuit ◽  
Electronic System ◽  
Rotational Axis ◽  
Stable Limit ◽  
Limit Cycle Oscillations ◽  
Trailing Edge ◽  
The Stability

A novel configuration of an energy harvester for local actuation and sensing devices using limit cycle oscillations has been modeled, designed and tested. A wing section has been designed with two trailing-edge free-floating flaps. A free-floating flap is a flap that can freely rotate around a hinge axis and is driven by trailing edge tabs. In the rotational axis of each flap a generator is mounted that converts the vibrational energy into electricity. It has been demonstrated numerically how a simple electronic system can be used to keep such a system at stable limit cycle oscillations by varying the resistance in the electric circuit. Additionally, it was shown that the stability of the system is coupled to the charge level of the battery, with increasing charge level leading to a less stable system. The system has been manufactured and tested in the Open Jet Wind Tunnel Facility of the Technical University Delft. The numerical results could be validated successfully and voltage generation could be demonstrated at cost of a decrease in lift of 2%.

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