Subcritical bifurcation and bistability in thermoacoustic systems

2013 ◽  
Vol 715 ◽  
pp. 210-238 ◽  
Author(s):  
Priya Subramanian ◽  
R. I. Sujith ◽  
P. Wahi
Keyword(s):  
Hopf Bifurcation ◽  
Linear Stability ◽  
Multiple Scales ◽  
Landau Equation ◽  
Linear Instability ◽  
Numerical Continuation ◽  
Rijke Tube ◽  
Subcritical Hopf Bifurcation ◽  
Fold Bifurcation ◽  
Parameter Values

AbstractThis paper analyses subcritical transition to instability, also known as triggering in thermoacoustic systems, with an example of a Rijke tube model with an explicit time delay. Linear stability analysis of the thermoacoustic system is performed to identify parameter values at the onset of linear instability via a Hopf bifurcation. We then use the method of multiple scales to recast the model of a general thermoacoustic system near the Hopf point into the Stuart–Landau equation. From the Stuart–Landau equation, the relation between the nonlinearity in the model and the criticality of the ensuing bifurcation is derived. The specific example of a model for a horizontal Rijke tube is shown to lose stability through a subcritical Hopf bifurcation as a consequence of the nonlinearity in the model for the unsteady heat release rate. Analytical estimates are obtained for the triggering amplitudes close to the critical values of the bifurcation parameter corresponding to loss of linear stability. The unstable limit cycles born from the subcritical Hopf bifurcation undergo a fold bifurcation to become stable and create a region of bistability or hysteresis. Estimates are obtained for the region of bistability by locating the fold points from a fully nonlinear analysis using the method of harmonic balance. These analytical estimates help to identify parameter regions where triggering is possible. Results obtained from analytical methods compare reasonably well with results obtained from both experiments and numerical continuation.

Study of Dynamic Characteristics of Friction: A Comparative Analysis of the Velocity Dependent and the LuGre Friction Model

2005 ◽  
Author(s):  
Firoz Ali Jafri ◽  
David F. Thompson
Keyword(s):  
Hopf Bifurcation ◽  
Friction Model ◽  
Continuation Methods ◽  
Bifurcation Parameter ◽  
Lugre Model ◽  
Lugre Friction Model ◽  
Subcritical Hopf Bifurcation ◽  
Fold Bifurcation ◽  
Dependent Model ◽  
Lugre Friction

In this paper, we conduct numerical analysis to study the effects of friction on the dynamic response of a single degree of freedom mechanical system. Two different friction models, the velocity dependent friction model and the LuGre friction model, have been used to model the friction interface. Bifurcation analysis has been conducted using equilibrium and limit cycle continuation methods. With system viscous damping as the bifurcation parameter, a reverse subcritical Hopf bifurcation is observed in the case of velocity dependent model. In the case of the LuGre model for the same bifurcation parameter, a reverse supercritical Hopf bifurcation is observed at lower velocities but at higher velocities it changes to a reverse subcritical Hopf bifurcation. A fold bifurcation of the limit cycles is also seen at higher velocities for the LuGre model.

scholarly journals Bifurcation Type Change of AC Electrostatically Actuated MEMS Resonators due to DC Bias

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Kyle N. Taylor
Keyword(s):  
Multiple Scales ◽  
Dc Voltage ◽  
Electrostatically Actuated ◽  
Mems Resonators ◽  
Subcritical Hopf Bifurcation ◽  
Fold Bifurcation ◽  
Type Change ◽  
Ground Plate ◽  
Dc Bias ◽  
And Shifts

This paper investigates the nonlinear response of microelectromechanical system (MEMS) cantilever resonator electrostatically actuated by applying a soft alternating current (AC) voltage and an even softer direct current (DC) voltage between the resonators and a parallel fixed ground plate. The AC frequency is near natural frequency. This drives the resonator into nonlinear parametric resonance. The method of multiple scales (MMS) is used to solve the dimensionless differential equation of motion of the resonator and find the steady-state solutions. The reduced order model (ROM) method is used to validate the results obtained using MMS. The effect of the soft DC voltage (bias) component on the frequency response is reported. It is shown that the DC bias changes the subcritical Hopf bifurcation into a cyclic fold bifurcation and shifts the bifurcation point (where the system loses stability) to lower frequencies and larger amplitudes.

Stability Analysis of A Thin Micropolar Fluid Flowing on A Rotating Circular Disk

2011 ◽  
Vol 27 (1) ◽  
pp. 95-105 ◽  
Author(s):  
C. K. Chen ◽  
M. C. Lin ◽  
C. I. Chen
Keyword(s):  
Stability Analysis ◽  
Linear Stability ◽  
Micropolar Fluid ◽  
Film Flow ◽  
Multiple Scales ◽  
Landau Equation ◽  
Kinematic Model ◽  
Circular Disk ◽  
Necessary Condition ◽  
Thin Film Flow

ABSTRACTThe stability analysis of a thin micropolar fluid flowing on a rotating circular disk is investigated numerically. The target is restricted to some neighborhood of critical value in the linear stability analysis. First, a generalized nonlinear kinematic model is derived by the long wave perturbation method. The method of normal mode is applied to the linear stability. After the weakly nonlinear dynamics of a film flow is studied by using the method of multiple scales, the Ginzburg-Landau equation is determined to discuss the necessary condition in terms of the various states of subcritical stability, subcritical instability, supercritical stability, and supercritical explosion for the existence of such flow pattern. The modeling results indicate that the rotation number and the radius of circular disk play the significant roles in destabilizing the flow. Furthermore, the micropolar parameter K serves as the stabilizing factor in the thin film flow.

Hopf Bifurcation and Numerical Simulation in a Calcium Oscillation Model

2012 ◽  
Vol 226-228 ◽  
pp. 510-515 ◽  
Author(s):  
Hong Kun Zuo ◽  
Quan Bao Ji ◽  
Yi Zhou
Keyword(s):  
Hopf Bifurcation ◽  
Intracellular Signaling ◽  
Calcium Release ◽  
Dimensional Space ◽  
Calcium Oscillations ◽  
Calcium Oscillation ◽  
Subcritical Hopf Bifurcation ◽  
Calcium Induced Calcium Release ◽  
Parameter Values ◽  
Centre Manifold Theorem

Calcium oscillations play a very important role in providing the intracellular signaling, and many mathematical models have been proposed to describe calcium oscillations. The Shen-Larter model presented here is based on calcium-induced calcium release (CICR) and the inositol trisphosphate cross-coupling (ICC). Nonlinear dynamics of this model is investigated by using the centre manifold theorem and bifurcation theory, including the variation in classification and stability of equilibria with different parameter values. The results show that the appearance and disappearance of calcium oscillations are due to subcritical Hopf bifurcation of equilibria. The numerical simulations are performed in order to illustrate the correctness of our theoretical analysis, including the bifurcation diagram of fixed points, the phase diagram of the system in two dimensional space and time series.

Oscillatory Dynamics of P53 Network With Time Delays

2018 ◽  
Vol 21 (6) ◽  
pp. 411-419 ◽  
Author(s):  
Conghua Wang ◽  
Fang Yan ◽  
Yuan Zhang ◽  
Haihong Liu ◽  
Linghai Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Cell Fate ◽  
Time Delays ◽  
Sustained Oscillation ◽  
Multiple Time ◽  
Cell Fate Decisions ◽  
Oscillatory Dynamics ◽  
Main Components ◽  
Parameter Values ◽  
P53 Network

Aims and Objective: A large number of experimental evidences report that the oscillatory dynamics of p53 would regulate the cell fate decisions. Moreover, multiple time delays are ubiquitous in gene expression which have been demonstrated to lead to important consequences on dynamics of genetic networks. Although delay-driven sustained oscillation in p53-based networks is commonplace, the precise roles of such delays during the processes are not completely known. Method: Herein, an integrated model with five basic components and two time delays for the network is developed. Using such time delays as the bifurcation parameter, the existence of Hopf bifurcation is given by analyzing the relevant characteristic equations. Moreover, the effects of such time delays are studied and the expression levels of the main components of the system are compared when taking different parameters and time delays. Result and Conclusion: The above theoretical results indicated that the transcriptional and translational delays can induce oscillation by undergoing a super-critical Hopf bifurcation. More interestingly, the length of these delays can control the amplitude and period of the oscillation. Furthermore, a certain range of model parameter values is essential for oscillation. Finally, we illustrated the main results in detail through numerical simulations.

TRAVELLING WAVES AND OSCILLATIONS IN SAL’NIKOV’S COMBUSTION REACTION IN A COMPRESSIBLE GAS

2015 ◽  
Vol 56 (3) ◽  
pp. 233-247 ◽  
Author(s):  
RHYS A. PAUL ◽  
LAWRENCE K. FORBES
Keyword(s):  
Asymptotic Solution ◽  
Multiple Scales ◽  
Travelling Waves ◽  
Method Of Lines ◽  
Travelling Wave ◽  
Reaction Scheme ◽  
Compressible Viscous Gas ◽  
The Method Of Lines ◽  
Temperature Waves ◽  
Parameter Values

We consider a two-step Sal’nikov reaction scheme occurring within a compressible viscous gas. The first step of the reaction may be either endothermic or exothermic, while the second step is strictly exothermic. Energy may also be lost from the system due to Newtonian cooling. An asymptotic solution for temperature perturbations of small amplitude is presented using the methods of strained coordinates and multiple scales, and a travelling wave solution with a sech-squared profile is derived. The method of lines is then used to approximate the full system with a set of ordinary differential equations, which are integrated numerically to track accurately the evolution of the reaction front. This numerical method is used to verify the asymptotic solution and investigate behaviours under different conditions. Using this method, temperature waves progressing as pulsatile fronts are detected at appropriate parameter values.

Subcritical Hopf bifurcation in NeNd hollow cathode discharge

1994 ◽  
Vol 196 (3-4) ◽  
pp. 191-194 ◽  
Author(s):  
P.R. Sasi Kumar ◽  
V.P.N. Nampoori ◽  
C.P.G. Vallabhan
Keyword(s):  
Hopf Bifurcation ◽  
Hollow Cathode ◽  
Hollow Cathode Discharge ◽  
Subcritical Hopf Bifurcation

Feedback control of a nonlinear aeroelastic system with non-semi-simple eigenvalues at the critical point of Hopf bifurcation

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Licai Wang ◽  
Yudong Chen ◽  
Chunyan Pei ◽  
Lina Liu ◽  
Suhuan Chen
Keyword(s):  
Hopf Bifurcation ◽  
Feedback Control ◽  
Critical Point ◽  
Control Parameter ◽  
Eigenvalue Problems ◽  
Multiple Scales ◽  
Approximate Solutions ◽  
State Matrix ◽  
Aeroelastic System ◽  
Center Manifold Reduction

Abstract The feedback control of Hopf bifurcation of nonlinear aeroelastic systems with asymmetric aerodynamic lift force and nonlinear elastic forces of the airfoil is discussed. For the Hopf bifurcation analysis, the eigenvalue problems of the state matrix and its adjoint matrix are defined. The Puiseux expansion is used to discuss the variations of the non-semi-simple eigenvalues, as the control parameter passes through the critical value to avoid the difficulty for computing the derivatives of the non-semi-simple eigenvalues with respect to the control parameter. The method of multiple scales and center-manifold reduction are used to deal with the feedback control design of a nonlinear system with non-semi-simple eigenvalues at the critical point of the Hopf bifurcation. The first order approximate solutions are developed, which include gain vector and input. The presented methods are based on the Jordan form which is the simplest one. Finally, an example of an airfoil model is given to show the feasibility and for verification of the present method.

Nonlinear Vibration Analysis of a Flexible Rotor Supported by a Flexure Pivot Tilting Pad Journal Bearing in Comparison to a Fixed Profile Journal Bearing with Experimental Verification

2021 ◽  
Author(s):  
Nuntaphong Koondilogpiboon ◽  
Tsuyoshi Inoue
Keyword(s):  
Hopf Bifurcation ◽  
Journal Bearing ◽  
Flexible Rotor ◽  
Maximum Test ◽  
Operating Region ◽  
Stable Operating ◽  
Subcritical Hopf Bifurcation ◽  
Tilting Pad ◽  
Calculation Results ◽  
Tilting Pad Journal Bearing

Abstract In this paper, an efficient numerical method consisting of the real mode component mode synthesis (CMS) model reduction, shooting method with parallel computing, and Floquet analysis was developed for nonlinear rotordynamics analysis of a flexible rotor supported by a 4-lobe flexure pivot tilting pad journal bearing (FPTPJB) in load-on-pad (LOP) and load-between-pad (LBP) orientations in comparison to a fixed profile journal bearing (JB) of the same pad geometry. The method used the rotor's finite elements and bearing forces obtained from directly solving the Reynolds equation to determine the limit cycles and Hopf bifurcation types. For the investigated rotor and bearing parameters, the numerical results indicated that the onset speed of instability (OSI) of FPTPJB is considerably higher than that of JB of the same orientation. Also, FPTPJB in LOP orientation yielded higher OSI than the LBP one, whereas the OSI of JB in LOP orientation was substantially higher than the LBP counterpart. Nonlinear calculation results indicated that all bearing types and orientations gave subcritical Hopf bifurcation. The FPTPJB in LOP orientation produced the largest stable operating region, whereas the JB in LBP configuration yield the smallest one. The experiment showed subcritical Hopf bifurcation occurred at speed close to the calculated OSI in all cases except FPTPJB in LOP orientation that the OSI is higher than the maximum test rig speed. The whirling orbit had the same frequency as the first critical speed and precessed in the direction of shaft rotation.

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