Hopf Bifurcation in PD Controlled Pendulum or Manipulator

2002 ◽  
Vol 124 (2) ◽  
pp. 327-332 ◽  
Author(s):  
Tom Bucklaew ◽  
Ching-Shi Liu
Keyword(s):  
Steady State ◽  
Hopf Bifurcation ◽  
Dynamic Behavior ◽  
Limit Cycle ◽  
Initial Conditions ◽  
Pendulum System ◽  
Periodic Attractors ◽  
Control Failure ◽  
Damped Systems ◽  
Strongly Damped

In this brief the dynamic behavior of a parametrically forced manipulator, or pendulum, system with PD control is examined. For an excitation of sufficient amplitude or frequency a Hopf bifurcation to a steady-state limit cycle is shown to result, appearing as a precursor to instability. The parameter space is mapped in order to illustrate regions where control failure will likely occur, even in the strongly damped case. For weakly damped systems, the Hopf bifurcation can additionally exhibit a dependence on initial conditions. The resulting case of competing point and periodic attractors is discussed.

STOCHASTIC SIMULATION OF CHEMICAL CHUA SYSTEM

2004 ◽  
Vol 14 (03) ◽  
pp. 1053-1057 ◽  
Author(s):  
QIAN SHU LI ◽  
RUI ZHU
Keyword(s):  
Master Equation ◽  
Dynamic Behavior ◽  
Stochastic Simulation ◽  
Limit Cycle ◽  
Initial Conditions ◽  
Deterministic Chaos ◽  
The Other ◽  
Initial Condition ◽  
Rapid Separation ◽  
Random Nature

A master equation for a scheme of chemical Chua system has been simulated stochastically. Two pairs of reaction simulations, one pair corresponding to deterministic chaos and the other pair to limit cycle, are carried out. The initial conditions differ with only one molecule for the former, while one hundred molecules for the latter. The rapid separation of time traces is shown only in the former case. This marked difference of dynamic behavior between the two cases might reveal the intrinsic random nature of deterministic chemical chaos, i.e. initial-condition sensitivity based on random collision events.

scholarly journals Hopf bifurcation in a delayed model for tumor-immune system competition with negative immune response

2006 ◽  
Vol 2006 ◽  
pp. 1-9 ◽  
Author(s):  
Radouane Yafia
Keyword(s):  
Immune Response ◽  
Asymptotic Behavior ◽  
Steady State ◽  
Immune System ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Limit Cycle ◽  
Numerical Study ◽  
Bifurcation Problem ◽  
Delayed Model

The dynamics of the model for tumor-immune system competition with negative immune response and with one delay investigated. We show that the asymptotic behavior depends crucially on the time delay parameter. We are particularly interested in the study of the Hopf bifurcation problem to predict the occurrence of a limit cycle bifurcating from the nontrivial steady state, by using the delay as a parameter of bifurcation. The obtained results provide the oscillations given by the numerical study in M. Gałach (2003), which are observed in reality by Kirschner and Panetta (1998).

scholarly journals Dynamic Behavior for an SIRS Model with Nonlinear Incidence Rate and Treatment

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Junhong Li ◽  
Ning Cui
Keyword(s):  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Incidence Rate ◽  
Dynamic Behavior ◽  
Limit Cycle ◽  
Nonlinear Incidence Rate ◽  
Existence Of Equilibrium ◽  
Nonlinear Incidence ◽  
Sirs Model ◽  
Asymptotical Stable

This paper considers an SIRS model with nonlinear incidence rate and treatment. It is assumed that susceptible and infectious individuals have constant immigration rates. We investigate the existence of equilibrium and prove the global asymptotical stable results of the endemic equilibrium. We then obtained that the model undergoes a Hopf bifurcation and existences a limit cycle. Some numerical simulations are given to illustrate the analytical results.

Steady-State Dynamic Response of a Limited Slip System

1968 ◽  
Vol 35 (2) ◽  
pp. 322-326 ◽  
Author(s):  
W. D. Iwan
Keyword(s):  
Steady State ◽  
Dynamic Response ◽  
Slip System ◽  
External Load ◽  
Initial Conditions ◽  
Viscous Damping ◽  
Steady State Response ◽  
Response Curves ◽  
State Response ◽  
System Nonlinearity

The steady-state response of a system constrained by a limited slip joint and excited by a trigonometrically varying external load is discussed. It is shown that the system may possess such features as disconnected response curves and jumps in response depending on the strength of the system nonlinearity, the level of excitation, the amount of viscous damping, and the initial conditions of the system.

Converging evidence in support of common dynamical principles for speech and movement coordination

1984 ◽  
Vol 246 (6) ◽  
pp. R928-R935 ◽  
Author(s):  
J. A. Kelso ◽  
B. Tuller
Keyword(s):  
Dynamical System ◽  
Speech Production ◽  
Limit Cycle ◽  
Movement Coordination ◽  
Rate Dependent ◽  
Mode Of Operation ◽  
Periodic Attractors ◽  
Point Attractor ◽  
Production Research ◽  
Dynamical Mode

We suggest that a principled analysis of language and action should begin with an understanding of the rate-dependent, dynamical processes that underlie their implementation. Here we present a summary of our ongoing speech production research, which reveals some striking similarities with other work on limb movements. Four design themes emerge for articulatory systems: 1) they are functionally rather than anatomically specific in the way they work; 2) they exhibit equifinality and in doing so fall under the generic category of a dynamical system called point attractor; 3) across transformations they preserve a relationally invariant topology; and 4) this, combined with their stable cyclic nature, suggests that they can function as nonlinear, limit cycle oscillators (periodic attractors). This brief inventory of regularities, though not mean to be inclusive, hints strongly that speech and other movements share a common, dynamical mode of operation.

A novel methodology to explore the periodic gait of a biped walker under uncertainty using a machine learning algorithm

Robotica ◽  
2021 ◽  
pp. 1-16
Author(s):  
Namjung Kim ◽  
Bongwon Jeong ◽  
Kiwon Park
Keyword(s):  
Limit Cycle ◽  
Systematic Approach ◽  
Learning Algorithm ◽  
Initial Conditions ◽  
Angular Displacement ◽  
External Disturbance ◽  
Rehabilitation Robotics ◽  
Stable Region ◽  
Gait Rehabilitation ◽  
External Disturbances

Abstract In this paper, we present a systematic approach to improve the understanding of stability and robustness of stability against the external disturbances of a passive biped walker. First, a multi-objective, multi-modal particle swarm optimization (MOMM-PSO) algorithm was employed to suggest the appropriate initial conditions for a given biped walker model to be stable. The MOMM-PSO with ring topology and special crowding distance (SCD) used in this study can find multiple local minima under multiple objective functions by limiting each agent’s search area properly without determining a large number of parameters. Second, the robustness of stability under external disturbances was studied, considering an impact in the angular displacement sampled from the probabilistic distribution. The proposed systematic approach based on MOMM-PSO can find multiple initial conditions that lead the biped walker in the periodic gait, which could not be found by heuristic approaches in previous literature. In addition, the results from the proposed study showed that the robustness of stability might change depending on the location on a limit cycle where immediate angular displacement perturbation occurs. The observations of this study imply that the symmetry of the stable region about the limit cycle will break depending on the accelerating direction of inertia. We believe that the systematic approach developed in this study significantly increased the efficiency of finding the appropriate initial conditions of a given biped walker and the understanding of robustness of stability under the unexpected external disturbance. Furthermore, a novel methodology proposed for biped walkers in the present study may expand our understanding of human locomotion, which in turn may suggest clinical strategies for gait rehabilitation and help develop gait rehabilitation robotics.

Analysis of the non-periodic oscillations of a self-excited friction-damped system with closely-spaced modes

2021 ◽  
Author(s):  
Lukas Woiwode ◽  
Alexander F. Vakakis ◽  
Malte Krack
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycle ◽  
Dry Friction ◽  
The Self ◽  
Perturbation Approach ◽  
Friction Damping ◽  
Periodic Oscillations ◽  
Analytical Results ◽  
Fold Bifurcation ◽  
Phase Motion

Abstract It is widely known that dry friction damping can bound the self-excited vibrations induced by negative damping. The vibrations typically take the form of (periodic) limit cycle oscillations. However, when the intensity of the self-excitation reaches a condition of maximum friction damping, the limit cycle loses stability via a fold bifurcation. The behavior may become even more complicated in the presence of any internal resonance conditions. In this work, we consider a two-degree-of-freedom system with an elastic dry friction element (Jenkins element) having closely spaced natural frequencies. The symmetric in-phase motion is subjected to self-excitation by negative (viscous) damping, while the symmetric out-of-phase motion is positively damped. In a previous work, we showed that the limit cycle loses stability via a secondary Hopf bifurcation, giving rise to quasi-periodic oscillations. A further increase of the self-excitation intensity may lead to chaos and finally divergence, long before reaching the fold bifurcation point of the limit cycle. In this work, we use the method of Complexification-Averaging to obtain the slow flow in the neighborhood of the limit cycle. This way, we show that chaos is reached via a cascade of period doubling bifurcations on invariant tori. Using perturbation calculus, we establish analytical conditions for the emergence of the secondary Hopf bifurcation and approximate analytically its location. In particular, we show that non-periodic oscillations are the typical case for prominent nonlinearity, mild coupling (controlling the proximity of the modes) and sufficiently light damping. The range of validity of the analytical results presented herein is thoroughly assessed numerically. To the authors' knowledge, this is the first work that shows how the challenging Jenkins element can be treated formally within a consistent perturbation approach in order to derive closed-form analytical results for limit cycles and their bifurcations.

Robust Stabilizing Controller Design for Inverted Pendulum System

2014 ◽  
Vol 71 (1) ◽  
Author(s):  
Hazem I. Ali
Keyword(s):  
Steady State ◽  
Time Domain ◽  
State Feedback ◽  
Inverted Pendulum ◽  
Controller Design ◽  
H Infinity ◽  
Pendulum System ◽  
Stabilizing Controller ◽  
Inverted Pendulum System ◽  
System Uncertainties

In this paper the design of robust stabilizing state feedback controller for inverted pendulum system is presented. The Ant Colony Optimization (ACO) method is used to tune the state feedback gains subject to different proposed cost functions comprise of H-infinity constraints and time domain specifications. The steady state and dynamic characteristics of the proposed controller are investigated by simulations and experiments. The results show the effectiveness of the proposed controller which offers a satisfactory robustness and a desirable time response specifications. Finally, the robustness of the controller is tested in the presence of system uncertainties and disturbance.

Exact Dynamics in a Model of the Bimolecular Reaction A+B→ 0

1992 ◽  
Vol 290 ◽  
Author(s):  
Eric Clément ◽  
Patrick Leroux-Hugon ◽  
Leonard M. Sander
Keyword(s):  
Exact Solution ◽  
Steady State ◽  
Initial Conditions ◽  
Finite Size ◽  
Bimolecular Reaction ◽  
Reaction Model

AbstractWe have previously given an exact solution [1] for the steady state of a model of the bimolecular reaction model A+B→ 0 due to Fichthorn et al. [2]. The dimensionality of the substrate plays a central role, and below d=2 segregation on macroscopic scales becomes important: above d=2 saturation sets in for finite size systems. Here we extend our treatment to give an exact account of the dynamics and show how various initial conditions develop into the segregated and saturated regimes. In certain conditions we find logarithmic relaxation which is related to the dimensionality.

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