A fifth-order multiple-scale solution for Hopf bifurcations

2004 ◽  
Vol 82 (31-32) ◽  
pp. 2723-2731 ◽  
Author(s):  
D. Dessi ◽  
F. Mastroddi ◽  
L. Morino
Keyword(s):  
Multiple Scale ◽  
Hopf Bifurcations ◽  
Fifth Order

scholarly journals Weakly nonlinear analysis of thermoacoustic bifurcations in the Rijke tube

2016 ◽  
Vol 805 ◽  
pp. 523-550 ◽  
Author(s):  
Alessandro Orchini ◽  
Georgios Rigas ◽  
Matthew P. Juniper
Keyword(s):  
Dynamical Evolution ◽  
Hopf Bifurcations ◽  
Solvability Conditions ◽  
Rijke Tube ◽  
Nonlinear Dynamical ◽  
Weakly Nonlinear ◽  
The Cost ◽  
Fifth Order ◽  
Thermoacoustic Oscillations ◽  
Good Agreement

In this study we present a theoretical weakly nonlinear framework for the prediction of thermoacoustic oscillations close to Hopf bifurcations. We demonstrate the method for a thermoacoustic network that describes the dynamics of an electrically heated Rijke tube. We solve the weakly nonlinear equations order by order, discuss their contribution on the overall dynamics and show how solvability conditions at odd orders give rise to Stuart–Landau equations. These equations, combined together, describe the nonlinear dynamical evolution of the oscillations’ amplitude and their frequency. Because we retain the contribution of several acoustic modes in the thermoacoustic system, the use of adjoint methods is required to derive the Landau coefficients. The analysis is performed up to fifth order and compared with time domain simulations, showing good agreement. The theoretical framework presented here can be used to reduce the cost of investigating oscillations and subcritical phenomena close to Hopf bifurcations in numerical simulations and experiments and can be readily extended to consider, e.g. the weakly nonlinear interaction of two unstable thermoacoustic modes.

scholarly journals On the Internal Resonance of a Spinning Disk Under Space-Fixed Pulsating Edge Loads

2001 ◽  
Vol 68 (6) ◽  
pp. 854-859 ◽  
Author(s):  
Jen-San Chen
Keyword(s):  
Natural Frequency ◽  
Trivial Solution ◽  
Internal Resonance ◽  
Excitation Frequency ◽  
Single Mode ◽  
Multiple Scale ◽  
Hopf Bifurcations ◽  
Solution Path ◽  
Nontrivial Solutions ◽  
Spinning Disk

Internal resonance between a pair of forward and backward modes of a spinning disk under space-fixed pulsating edge loads is investigated by means of multiple scale method. It is found that internal resonance can occur only at certain rotation speeds at which the natural frequency of the forward mode is close to three times the natural frequency of the backward mode and the excitation frequency is close to twice the frequency of the backward mode. For a light damping case the trivial solution can lose stability via both pitchfork as well as Hopf bifurcations when frequency detuning of the edge load is varied. On the other hand, nontrivial solutions experience both saddle-node and Hopf bifurcations. When the damping is increased, the Hopf bifurcations along the trivial solution path disappear. Furthermore, there exists a certain value of damping beyond which no nontrivial solution is possible. Single-mode resonance is also briefly discussed for comparison.

MULTIPLE-SCALE AND NUMERICAL ANALYSES FOR THE NONLINEAR OSCILLATIONS OF A GAS BUBBLE SURROUNDED BY A MAXWELL'S FLUID

2019 ◽  
Vol 46 (3) ◽  
pp. 261-275
Author(s):  
César Yepes ◽  
Jorge Naude ◽  
Federico Mendez ◽  
Margarita Navarrete ◽  
Fátima Moumtadi
Keyword(s):  
Nonlinear Oscillations ◽  
Multiple Scale ◽  
Numerical Analyses ◽  
Gas Bubble
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