Papers by Keyword: Limit Cycle

Abstract: This work studies the effect of time delayed feedback on stationary solutions in a van derPol type system. We consider the case where the feedback gain is harmonically modulated with a resonantfrequency. Perturbation analysis is conducted to obtain the modulation equations near primaryresonance, the stability analysis for stationary solutions is performed and bifurcation diagram is determined.It is shown that the modulated feedback gain position can influence significantly the steadystates behavior of the delayed van der Pol oscillator. In particular, for appropriate values of the modulateddelay parameters, the existence region of the limit cycle (LC) can be increased or quenched.Moreover, new regions of quasiperiodic vibration may emerge for certain values of the modulatedgain. Numerical simulation was conducted to validate the analytical predictions.

Abstract: A dimensionless nonlinear state-space model was established considering the structural particularities of digital hydraulic cylinder, and dynamic bifurcation characteristics of the system were analyzed and validated based on MATCONT. The results show that, when the piston diameter, valve orifice area gradient, ball-screw pitch and the maximum desired speed are not designed appropriately, digital hydraulic cylinder is prone to Hopf bifurcation. Limit cycles that the stable and the unstable neutralize each other at subcritical Hopf bifurcation points, causing the system tracking outputs divergence from continuous oscillation to increasing oscillation. Losing synchronism of stepper motor is essentially a form of the instability caused by system supercritical Hopf bifurcation.

Abstract: In order to investigate self-excited vibration mechanism of wheel-rail lateral contact system, a two DOF elasticity position wheelset lateral vibration model is established which considers the dry friction; the mechanism of the wheelset lateral self-excited vibration is investigated from the energy point of view. It shows that: the bifurcation diagram of this wheel-rail lateral contact system has a supercritical Hopf bifurcation. The energy of self-excited vibration derives from a part of traction energy; the creep rate in the wheel-rail system act as a feedback mechanism in the wheelset lateral self-excited vibration system. The stability of the wheelset self-excited vibration system depends mainly on the total energy removed from and imported into the system.

Abstract: To make the theoretical analysis of the microbial continuous culture more close to the experimental results, we consider a chemostat model with Watt type functional response and variable yield. The existence of limit cycles and Hopf bifurcation is investigated, which is useful in the further study of the oscillatory behaviors of the microbial growth in the vessel. The conditions for the global asymptotical stability of the model are obtained by Dulac criterion.

Abstract: A challenging task in wireless sensor network (WSN) is to deliver authentic data between source nodes and sink nodes. The collision or dead lock occurs when two or more close nodes are attempted to send data at the same time to the others node. To avoid such dead lock situation in the network we propose a nonlinear mathematical model. The effect of nonlinearity often renders a periodic solution unstable for certain parametric choices even a very small change in initial conditions can lead to different result in chaotic systems which appears to exhibit chaos for a range of parametric values when long time behavior studied. The local stability conditions for the system have been discussed and analyzed. Numerically simulations have been carried out to study the complex behavior of the system for reasonable ranges of parameters in WSN.

Abstract: It proves that steering wheel shimmy is a vibration of stable limit cycle occurring after Hopf bifurcation, which is elaborated by nonlinear dynamics theory, and the control objectives of shimmy are proposed according to its bifurcation properties. Numerical analysis of bifurcation characteristics has been conducted with a nonlinear shimmy model whose parameters come from a domestic automobile with independent suspension. The results indicate that when the speed reaches 49.98Km/h, supercritical Hopf bifurcation occurs to the system and stable limit cycle appears, i.e. wheels oscillate around the kingpin at the same amplitude; when the speed comes to 76.30 Km/h, Hopf bifurcation occurs again and limit cycle disappears. The bifurcation speed and amplitude of limit cycle match the shimmy speed and amplitude measured from road experiments very well, which confirms the conclusions that shimmy is a vibration of stable limit cycle occurring after Hopf bifurcation at critical speed.

Abstract: In this paper, a continuous bioprocess model with single species for an inhibitory growth-limiting nutrient is considered, where the dilution rate is time varying and controlled by a critical cell concentration in the bioreactor. It is shown that the model has rich dynamics. A coexistence equilibrium and the washout equilibrium can be asymptotically stable simultaneously so that coexistence may depend on initial conditions.

Abstract: In this paper, the ratio-dependent chemostat model with Holling-(n+1) type functional response is considered. The model develops the Monod model and the ratio-dependent model. By use of the Poincar -Bendixson theory we prove the existence of limit cycle. Detailed qualitative analysis about the global asymptotic stability of its equilibria is carried out by using the Lyapunov-LaSalle invariant principle and the method of Dulac criterion.

Abstract: A perturbation procedure, in which the elliptic perturbation method and the hyperbolic perturbation method are applied, is presented for predicting heteroclinic connection of limit cycle or self-excited ocsillator. The limit cycle can be analytically　constructed first by the elliptic perturbation method after Hopf bifurcation, in which the amplitude of limit cycle can be　controlled by the modulus of elliptic functions. The heteroclinic trajectories, which are formed by the heteroclinic connection of limit cycle, can also be constructed by similar perturbation　procedure but adopting the hyperbolic functions instead of　elliptic functions. And the criterion of heteroclinic connection is given in the perturbation procedure. A typical self-excited oscillator is studied in detail to assess the present method.

Abstract: A method to find the optimized parameter values of passive dynamic walking biped is presented. The effects of biped physical parameters on the stability property of passive gaits are studied by simulation experiments. The chosen parameters include the mass distribution, length of leg and slope angle. The stability property of passive walking limit cycles is used as criterion of optimization calculation, including the orbital stability described by eigen-values of linearized Poincaré map and the global property described by size of attraction region. The simulation results show how the stability of limit cycle varies when physical parameters of the passive biped change. The work is useful to explore the inherent property of passive dynamic walking and can be used as an important instruction in the mechanical design of biped robots based on principle of passive dynamic walking.