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Hopf Bifurcation (hopf + bifurcation)

Distribution by Scientific Domains

Engineering	50%

Selected Abstracts

Bifurcation and stability analysis of laminar flow in curved ducts

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 4 2010
Werner Machane
Abstract The development of viscous flow in a curved duct under variation of the axial pressure gradient q is studied. We confine ourselves to two-dimensional solutions of the Dean problem. Bifurcation diagrams are calculated for rectangular and elliptic cross sections of the duct. We detect a new branch of asymmetric solutions for the case of a rectangular cross section. Furthermore we compute paths of quadratic turning points and symmetry breaking bifurcation points under variation of the aspect ratio , (,=0.8,1.5). The computed diagrams extend the results presented by other authors. We succeed in finding two origins of the Hopf bifurcation. Making use of the Cayley transformation, we determine the stability of stationary laminar solutions in the case of a quadratic cross section. All the calculations were performed on a parallel computer with 32×32 processors. Copyright © 2009 John Wiley & Sons, Ltd. [source]

Linear stability analysis of flow in a periodically grooved channel

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 6 2003
T. Adachi1
Abstract We have conducted the linear stability analysis of flow in a channel with periodically grooved parts by using the spectral element method. The channel is composed of parallel plates with rectangular grooves on one side in a streamwise direction. The flow field is assumed to be two-dimensional and fully developed. At a relatively small Reynolds number, the flow is in a steady-state, whereas a self-sustained oscillatory flow occurs at a critical Reynolds number as a result of Hopf bifurcation due to an oscillatory instability mode. In order to evaluate the critical Reynolds number, the linear stability theory is applied to the complex laminar flow in the periodically grooved channel by constituting the generalized eigenvalue problem of matrix form using a penalty-function method. The critical Reynolds number can be determined by the sign of a linear growth rate of the eigenvalues. It is found that the bifurcation occurs due to the oscillatory instability mode which has a period two times as long as the channel period. Copyright © 2003 John Wiley & Sons, Ltd. [source]

Numerical investigation of the first instabilities in the differentially heated 8:1 cavity

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 8 2002
F. Auteri
Abstract We present a new Galerkin,Legendre spectral projection solver for the simulation of natural convection in a differentially heated cavity. The projection method is applied to the study of the first non-stationary instabilities of the flow in a 8:1 cavity. Statistics of the periodic solution are reported for a Rayleigh number of 3.4×105. Moreover, we investigate the location and properties of the first Hopf bifurcation and of the three successive bifurcations. The results confirm the previous finding in the range of Rayleigh numbers investigated that the flow instabilities originate in the boundary layer on the vertical walls. A peculiar phenomenon of symmetry breaking and symmetry restoring is observed portraying the first steps of the transition to chaos for this flow. Copyright © 2002 John Wiley & Sons, Ltd. [source]

Feedback stabilization of bifurcations in multivariable nonlinear systems,Part II: Hopf bifurcations

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, Issue 4 2007
Yong Wang
Abstract In this paper we derive necessary and sufficient conditions of stabilizability for multi-input nonlinear systems possessing a Hopf bifurcation with the critical mode being linearly uncontrollable, under the non-degeneracy assumption that stability can be determined by the third order term in the normal form of the dynamics on the centre manifold. Stabilizability is defined as the existence of a sufficiently smooth state feedback such that the Hopf bifurcation of the closed-loop system is supercritical, which is equivalent to local asymptotic stability of the system at the bifurcation point. We prove that under the non-degeneracy conditions, stabilizability is equivalent to the existence of solutions to a third order algebraic inequality of the feedback gains. Explicit conditions for the existence of solutions to the algebraic inequality are derived, and the stabilizing feedback laws are constructed. Part of the sufficient conditions are equivalent to the rank conditions of an augmented matrix which is a generalization of the Popov,Belevitch,Hautus (PBH) rank test of controllability for linear time invariant (LTI) systems. We also apply our theory to feedback control of rotating stall in axial compression systems using bleed valve as actuators. Copyright © 2006 John Wiley & Sons, Ltd. [source]

The 3-D bifurcation and limit cycles in a food-chain model

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 1 2009
Lemin Zhu
Abstract In this paper, by using a corollary to the center manifold theorem, we show that the 3-D food-chain model studied by many authors undergoes a 3-D Hopf bifurcation, and then we obtain the existence of limit cycles for the 3-D differential system. The methods used here can be extended to many other 3-D differential equation models. Copyright © 2008 John Wiley & Sons, Ltd. [source]

Global stability and the Hopf bifurcation for some class of delay differential equation

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 10 2008
Marek Bodnar
Abstract In this paper, we present an analysis for the class of delay differential equations with one discrete delay and the right-hand side depending only on the past. We extend the results from paper by U. Fory, (Appl. Math. Lett. 2004; 17(5):581,584), where the right-hand side is a unimodal function. In the performed analysis, we state more general conditions for global stability of the positive steady state and propose some conditions for the stable Hopf bifurcation occurring when this steady state looses stability. We illustrate the analysis by biological examples coming from the population dynamics. Copyright © 2007 John Wiley & Sons, Ltd. [source]

A predator,prey model with disease in the prey species only

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 8 2007
David Greenhalgh
Abstract A predator,prey model with transmissible disease in the prey species is proposed and analysed. The essential mathematical features are analysed with the help of equilibrium, local and global stability analyses and bifurcation theory. We find four possible equilibria. One is where the populations are extinct. Another is where the disease and predator populations are extinct and we find conditions for global stability of this. A third is where both types of prey exist but no predators. The fourth has all three types of individuals present and we find conditions for limit cycles to arise by Hopf bifurcation. Experimental data simulation and brief discussion conclude the paper. Copyright © 2006 John Wiley & Sons, Ltd. [source]

First Liapunov coefficient for coupled identical oscillators.

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 17 2006
Application to coupled demand, supply model
Abstract A general formula for the computation of the first Liapunov coefficient corresponding to the Hopf bifurcation in a four-dimensional system of two coupled identical oscillators is performed for two cases. Only bi-dimensional vectors are involved. Then a model of two coupled demand,supply systems, depending on four parameters is considered. A study of the Hopf bifurcation is done around one of the symmetrical equilibrium, as the parameters vary. The loci in the parameter space of the parameters values corresponding to subcritical, supercritical or degenerated Hopf bifurcation are found. The computation of the Liapunov coefficients is done using the derived formula. Numerical plots emphasizing the existence of different types of limit cycles are developed. Copyright © 2006 John Wiley & Sons, Ltd. [source]

Classical predator,prey system with infection of prey population,a mathematical model

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 14 2003
J. Chattopadhyay
Abstract The present paper deals with the problem of a classical predator,prey system with infection of prey population. A classical predator,prey system is split into three groups, namely susceptible prey, infected prey and predator. The relative removal rate of the susceptible prey due to infection is worked out. We observe the dynamical behaviour of this system around each of the equilibria and point out the exchange of stability. It is shown that local asymptotic stability of the system around the positive interior equilibrium ensures its global asymptotic stability. We prove that there is always a Hopf bifurcation for increasing transmission rate. To substantiate the analytical findings, numerical experiments have been carried out for hypothetical set of parameter values. Our analysis shows that there is a threshold level of infection below which all the three species will persist and above which the disease will be epidemic. Copyright © 2003 John Wiley & Sons, Ltd. [source]

Symmetry and bifurcation in vestibular system

PROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2007
Marty Golubitsky
The vestibular system in almost all vertebrates, humans included, controls balance by employing a set of six semicircular canals, three in each inner ear, to detect angular accelerations of the head. Signals from the canals are transmitted to neck motoneurons and activate eight corresponding muscle groups. These signals may be either excitatory or inhibitory, depending on the direction of acceleration. McCollum and Boyle have observed that in the cat the network of neurons concerned possesses octahedral symmetry, a structure deduced from the known innervation patterns (connections) from canals to muscles. We re-derive the octahedral symmetry from mathematical features of the probable network architecture, and model the movement of the head in response to the activation patterns of the muscles concerned. We assume that connections among neck muscles can be modeled by a ,coupled cell network', a system of coupled ODEs whose variables correspond to the eight muscles, and that network also has octahedral symmetry. The network and its symmetries imply that these ODEs must be equivariant under a suitable action of the octahedral group. Using results of Ashwin and Podvigina, we show that with the appropriate group actions, there are six possible spatiotemporal patterns of time-periodic states that can arise by Hopf bifurcation from an equilibrium corresponding to natural head motions. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

Internal Noise Coherent Resonance for Mesoscopic Chemical Oscillations: A Fundamental Study

CHEMPHYSCHEM, Issue 7 2006
Zhonghuai Hou
Abstract The effect of internal noise for a mesoscopic chemical oscillator is studied analytically in a parameter region outside, but close to, the supercritical Hopf bifurcation. By normal form calculation and a stochastic averaging procedure, we obtain stochastic differential equations for the oscillation amplitude r and phase , that is solvable. Noise-induced oscillation and internal noise coherent resonance, which has been observed in many numerical experiments, are reproduced well by the theory. [source]