PROPERTIES OF A LORENZ MODEL FOR CONVECTION IN A ROTATING FLUID LAYER

1992 ◽  
Vol 06 (16n17) ◽  
pp. 1055-1061
Author(s):  
GOVINDAN RAJESH ◽  
SUPREETI DAS ◽  
JAYANTA K. BHATTACHARJEE
Keyword(s):  
Limit Cycle ◽  
Fluid Layer ◽  
Stable Limit Cycle ◽  
Period Doubling ◽  
Finite Amplitude ◽  
Stable Limit ◽  
Rotating Fluid ◽  
Onset Of Convection ◽  
Rayleigh Numbers ◽  
Period Doubling Bifurcations

A Lorenz-like model due to Veronis for onset of convection in a rotating fluid layer is analysed for Rayleigh numbers higher than the point at which stationary convection occurs. The most outstanding feature is that for Taylor numbers above a critical value, the Hopf bifurcation does lead to a finite amplitude stable limit cycle via a hysteretic transition. This limit cycle undergoes a sequence of period doubling bifurcations to form a Feigenbaum attractor which then makes a transition to the Lorenz-like attractor.

scholarly journals A Study of Period Doubling in a One-Zone Pulsating Stellar Model

1989 ◽  
Vol 111 ◽  
pp. 265-265
Author(s):  
A.B. Foken
Keyword(s):  
Degrees Of Freedom ◽  
Stable Limit Cycle ◽  
Period Doubling ◽  
Stellar Model ◽  
Zone Model ◽  
Stable Limit ◽  
Hydrodynamic Models ◽  
Excitation Rate ◽  
Return Map ◽  
Period Doubling Bifurcations

AbstractA simple one-zone model for nonlinear stellar pulsations is outlined and applied to the study of period doubling observed in some W Virginis and RV Tauri stars. The model reveals a number of period doubling bifurcations as the parameters are varied, similar to those found by Buchler & Kovacs in their series of hydrodynamic models. In the vicinity of a stable limit cycle, despite its large number of degrees of freedom, the model develops an essentially one-dimensional Poincaré’s return map, determining the modulation of the amplitude. The analysis of these maps confirmed that period doubling has its origin in a strong nonlinear increase of total energy losses per period as radial amplitude, δr/r, increases. An additional study of nearly periodic hydrodynamic models with P > 15 days, calculated including radiative transfer effects, shows that the rate of energy dissipation per period by shocks in the atmospheres increases rapidly with δr/r, whereas the excitation rate, δo, remains rather stable. This permitted us to construct an analytic return map for maxima of the total kinetic energy which clearly demonstrates the mechanism of successive period doubling as δo, the sole bifurcation parameter, grows monotonically.

scholarly journals Hopf Bifurcation, Cascade of Period-Doubling, Chaos, and the Possibility of Cure in a 3D Cancer Model

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Marluci Cristina Galindo ◽  
Cristiane Nespoli ◽  
Marcelo Messias
Keyword(s):  
Immune System ◽  
Hopf Bifurcation ◽  
Limit Cycle ◽  
Chaotic Attractor ◽  
Stable Limit Cycle ◽  
Period Doubling ◽  
Tumour Cells ◽  
Dimensional System ◽  
Stable Limit ◽  
Cancer Model

We study a cancer model given by a three-dimensional system of ordinary differential equations, depending on eight parameters, which describe the interaction among healthy cells, tumour cells, and effector cells of immune system. The model was previously studied in the literature and was shown to have a chaotic attractor. In this paper we study how such a chaotic attractor is formed. More precisely, by varying one of the parameters, we prove that a supercritical Hopf bifurcation occurs, leading to the creation of a stable limit cycle. Then studying the continuation of this limit cycle we numerically found a cascade of period-doubling bifurcations which leads to the formation of the mentioned chaotic attractor. Moreover, analyzing the model dynamics from a biological point of view, we notice the possibility of both the tumour cells and the immune system cells to vanish and only the healthy cells survive, suggesting the possibility of cure, since the interactions with the immune system can eliminate tumour cells.

Finite-amplitude convection in the presence of finitely conducting boundaries

2001 ◽  
Vol 432 ◽  
pp. 351-367 ◽  
Author(s):  
M. WESTERBURG ◽  
F. H. BUSSE
Keyword(s):  
Thermal Conductivity ◽  
Prandtl Number ◽  
Fluid Layer ◽  
Finite Amplitude ◽  
Onset Of Convection ◽  
Theoretical Predictions ◽  
Square Cells ◽  
Convection Rolls ◽  
Rayleigh Numbers ◽  
Horizontal Fluid Layer

Finite-amplitude convection in the form of rolls and their stability with respect to infinitesimal disturbances is investigated in the case of boundaries of the horizontal fluid layer which exhibit a thermal conductivity comparable to that of the fluid. It is found that even when rolls represent the preferred mode at the onset of convection a transition to square cells may occur at slightly supercritical Rayleigh numbers. The phenomenon of inertial convection in low Prandtl number fluids appears to become more pronounced as the conductivity of the boundaries is reduced. Modulated convection rolls have also been found as solutions of the problem. But they appear to be unstable in general. Experimental observations have been made and are found in general agreement with the theoretical predictions.

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.

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 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.

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