Bifurcation Analysis of a Two-DoF System Subject to Digital Position Control

2012 ◽  
Author(s):  
Giuseppe Habib ◽  
Giuseppe Rega ◽  
Gabor Stépán
Keyword(s):  
Numerical Simulations ◽  
Bifurcation Analysis ◽  
Position Control ◽  
Control Force ◽  
Piecewise Constant ◽  
The Stability

This paper analyzes the stability of a two-DoF system, subject to PD digital position control. In the model the control force is considered piecewise constant. Introducing the nonlinearity related to the saturation of the control force, the bifurcations occurring in the system are analyzed. The system is generally loosing stability through Neimark-Sacker bifurcations, with a relatively simple dynamics. However, the interaction of two different Neimark-Sacker bifurcations steers the system to much more complicated behaviors. About this kind of bifurcation, namely double Neimark-Sacker bifurcation, there are very few studies in the literature. Our analysis is carried out using the method proposed by Kuznetsov. The performed investigation shows the appearance of quasiperiodic motions and the existence of regions with coexisting periodic stable attractors, in the space of the control gains. Numerical simulations validate the results obtained analytically.

Nonlinear Bifurcation Analysis of a Single-DoF Model of a Robotic Arm Subject to Digital Position Control

2012 ◽  
Vol 8 (1) ◽  
Author(s):  
Giuseppe Habib ◽  
Giuseppe Rega ◽  
Gabor Stepan
Keyword(s):  
Center Manifold ◽  
Position Control ◽  
High Accuracy ◽  
Industrial Applications ◽  
Critical Parameters ◽  
Robotic Arm ◽  
Control Force ◽  
Local Center ◽  
The Stability ◽  
Stability Limits

Precision and stability in position control of robots are critical parameters in many industrial applications where high accuracy is needed. It is well known that digital effect is destabilizing and can cause instabilities. In this paper, we analyze a single DoF model of a robotic arm and we present the stability limits in the parameter space of the control gains. Furthermore we introduce a nonlinearity relative to the saturation of the control force in the model, reduce the dynamics of the nonlinear map to its local center manifold, study the bifurcation along the stability border and identify conditions under which a supercritical or subcritical bifurcation occurs. The obtained results explain some of the typical instabilities occurring in industrial applications. We verify the obtained results through numerical simulations.

scholarly journals Chaos and Hopf Bifurcation Analysis of the Delayed Local Lengyel-Epstein System

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Qingsong Liu ◽  
Yiping Lin ◽  
Jingnan Cao ◽  
Jinde Cao
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Numerical Simulations ◽  
Bifurcation Analysis ◽  
Center Manifold ◽  
Local Reaction ◽  
Reaction Diffusion ◽  
Critical Value ◽  
Hopf Bifurcations ◽  
The Stability

The local reaction-diffusion Lengyel-Epstein system with delay is investigated. By choosingτas bifurcating parameter, we show that Hopf bifurcations occur when time delay crosses a critical value. Moreover, we derive the equation describing the flow on the center manifold; then we give the formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. Finally, numerical simulations are performed to support the analytical results and the chaotic behaviors are observed.

Nonlinear Bifurcation Analysis of a Robotic Arm Subject to Digital Position Control

2011 ◽  
Author(s):  
Giuseppe Habib ◽  
Giuseppe Rega ◽  
Gabor Stepan
Keyword(s):  
Center Manifold ◽  
Position Control ◽  
High Accuracy ◽  
Industrial Applications ◽  
Critical Parameters ◽  
Robotic Arm ◽  
Control Force ◽  
Local Center ◽  
The Stability ◽  
Stability Limits

Precision and stability in position control of robots are critical parameters in many industrial applications where high accuracy is needed. It is well known that digital effect is destabilizing and can cause instabilities. In this paper, we analyze a single DoF system and we present the stability limits in the parameter space of the control gains. Furthermore we introduce a nonlinearity relative to the saturation of the control force in the model, reduce the dynamics of the nonlinear map to its local center manifold, study the bifurcation along the stability border and identify conditions under which a supercritical or subcritical bifurcation occurs. The obtained results explain some of the typical instabilities occurring in industrial applications. We verify the obtained results through numerical simulations.

Bifurcation Analysis in a Lotka-Volterra Model with Delay

2012 ◽  
Vol 594-597 ◽  
pp. 2693-2696
Author(s):  
Chang Jin Xu
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Numerical Simulations ◽  
Bifurcation Analysis ◽  
Volterra Model ◽  
The Stability ◽  
Theoretical Results

In this paper, a Lotka-Volterra model with time delay is considered. The stability of the equilibrium of the model is investigated and the existence of Hopf bifurcation is proved. Numerical simulations are performed to justify the theoretical results. Finally, main conclusions are included.

scholarly journals Bifurcation Analysis of a Generic Reaction–Diffusion Turing Model

2014 ◽  
Vol 24 (04) ◽  
pp. 1450042 ◽  
Author(s):  
Ping Liu ◽  
Junping Shi ◽  
Rui Wang ◽  
Yuwen Wang
Keyword(s):  
Steady State ◽  
Numerical Simulations ◽  
Bifurcation Analysis ◽  
Taylor Expansion ◽  
Reaction Diffusion ◽  
Spatiotemporal Dynamics ◽  
Hopf Bifurcations ◽  
Bifurcation Direction ◽  
The Rich ◽  
The Stability

A generic Turing type reaction–diffusion system derived from the Taylor expansion near a constant equilibrium is analyzed. The existence of Hopf bifurcations and steady state bifurcations is obtained. The bifurcation direction and the stability of the bifurcating periodic obits are calculated. Numerical simulations are included to show the rich spatiotemporal dynamics.

Stability and Bifurcation Analysis in a Diffusive Brusselator-Type System

2016 ◽  
Vol 26 (07) ◽  
pp. 1650119 ◽  
Author(s):  
Maoxin Liao ◽  
Qi-Ru Wang
Keyword(s):  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Periodic Solutions ◽  
Bifurcation Analysis ◽  
Equilibrium Solution ◽  
Dynamic Properties ◽  
Type System ◽  
Ode System ◽  
Stability And Bifurcation Analysis ◽  
The Stability

In this paper, the dynamic properties for a Brusselator-type system with diffusion are investigated. By employing the theory of Hopf bifurcation for ordinary and partial differential equations, we mainly obtain some conditions of the stability and Hopf bifurcation for the ODE system, diffusion-driven instability of the equilibrium solution, and the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions for the PDE system. Finally, some numerical simulations are presented to verify our results.

Bifurcation analysis of modeling biodegradation of Microcystins

2019 ◽  
Vol 12 (03) ◽  
pp. 1950028
Author(s):  
Keying Song ◽  
Wanbiao Ma ◽  
Zhichao Jiang
Keyword(s):  
Time Delay ◽  
Normal Form ◽  
Numerical Simulations ◽  
Bifurcation Analysis ◽  
Center Manifold ◽  
Positive Equilibrium ◽  
Hopf Bifurcations ◽  
Normal Form Theory ◽  
Form Theory ◽  
The Stability

In this paper, a model with time delay describing biodegradation of Microcystins (MCs) is investigated. Firstly, the stability of the positive equilibrium and the existence of Hopf bifurcations are obtained. Furthermore, an explicit algorithm for determining the direction and the stability of the bifurcating periodic solutions is derived by using the normal form theory and center manifold argument. Finally, some numerical simulations are carried out to illustrate the applications of the results.

scholarly journals Fast-Scale Instability and Stabilization by Adaptive Slope Compensation of a PV-Fed Differential Boost Inverter

2021 ◽  
Vol 11 (5) ◽  
pp. 2106
Author(s):  
Abdelali El Aroudi ◽  
Mohamed Debbat ◽  
Mohammed Al-Numay ◽  
Abdelmajid Abouloiafa
Keyword(s):  
Numerical Simulations ◽  
Full Range ◽  
Weather Conditions ◽  
Current Loop ◽  
Point Tracking ◽  
Bifurcation Phenomena ◽  
Slope Compensation ◽  
Power Point ◽  
The Stability ◽  
Detailed Circuit

Numerical simulations reveal that a single-stage differential boost AC module supplied from a PV module under an Maximum Power Point Tracking (MPPT) control at the input DC port and with current synchronization at the AC grid port might exhibit bifurcation phenomena under some weather conditions leading to subharmonic oscillation at the fast-switching scale. This paper will use discrete-time approach to characterize such behavior and to identify the onset of fast-scale instability. Slope compensation is used in the inner current loop to improve the stability of the system. The compensation slope values needed to guarantee stability for the full range of operating duty cycle and leading to an optimal deadbeat response are determined. The validity of the followed procedures is finally validated by a numerical simulations performed on a detailed circuit-level switched model of the AC module.

scholarly journals Spatiotemporal Patterns Formed by a Discrete Nutrient-Phytoplankton Model with Time Delay

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-18
Author(s):  
Feifan Zhang ◽  
Wenjiao Zhou ◽  
Lei Yao ◽  
Xuanwen Wu ◽  
Huayong Zhang
Keyword(s):  
Steady State ◽  
Time Delay ◽  
Numerical Simulations ◽  
Functional Response ◽  
Bifurcation Analysis ◽  
Discrete Model ◽  
Continuous Model ◽  
Spatiotemporal Patterns ◽  
Homogeneous Steady State ◽  
Simulation Results

In this research, a continuous nutrient-phytoplankton model with time delay and Michaelis–Menten functional response is discretized to a spatiotemporal discrete model. Around the homogeneous steady state of the discrete model, Neimark–Sacker bifurcation and Turing bifurcation analysis are investigated. Based on the bifurcation analysis, numerical simulations are carried out on the formation of spatiotemporal patterns. Simulation results show that the diffusion of phytoplankton and nutrients can induce the formation of Turing-like patterns, while time delay can also induce the formation of cloud-like pattern by Neimark–Sacker bifurcation. Compared with the results generated by the continuous model, more types of patterns are obtained and are compared with real observed patterns.

scholarly journals Earth-size planet formation in the habitable zone of circumbinary stars

2020 ◽  
Vol 494 (1) ◽  
pp. 1045-1057 ◽  
Author(s):  
G O Barbosa ◽  
O C Winter ◽  
A Amarante ◽  
A Izidoro ◽  
R C Domingos ◽  
...  
Keyword(s):  
Numerical Simulations ◽  
Planet Formation ◽  
Terrestrial Planet ◽  
Close Binary ◽  
Habitable Zone ◽  
Density Profiles ◽  
Habitable Planets ◽  
Habitable Zones ◽  
Test Particles ◽  
The Stability

ABSTRACT This work investigates the possibility of close binary (CB) star systems having Earth-size planets within their habitable zones (HZs). First, we selected all known CB systems with confirmed planets (totaling 22 systems) to calculate the boundaries of their respective HZs. However, only eight systems had all the data necessary for the computation of HZ. Then, we numerically explored the stability within HZs for each one of the eight systems using test particles. From the results, we selected five systems that have stable regions inside HZs, namely Kepler-34,35,38,413, and 453. For these five cases of systems with stable regions in HZ, we perform a series of numerical simulations for planet formation considering discs composed of planetary embryos and planetesimals, with two distinct density profiles, in addition to the stars and host planets of each system. We found that in the case of the Kepler-34 and 453 systems, no Earth-size planet is formed within HZs. Although planets with Earth-like masses were formed in Kepler-453, they were outside HZ. In contrast, for the Kepler-35 and 38 systems, the results showed that potentially habitable planets are formed in all simulations. In the case of the Kepler-413system, in just one simulation, a terrestrial planet was formed within HZ.

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