BIFURCATION ANALYSIS OF DIFFERENTIAL-DIFFERENCE-ALGEBRAIC EQUATIONS

2004 ◽  
Vol 14 (08) ◽  
pp. 2853-2865
Author(s):  
DONGYUN DU ◽  
YUN TANG
Keyword(s):  
Bifurcation Analysis ◽  
Pitchfork Bifurcation ◽  
Algebraic Equations ◽  
Bifurcation Theorem ◽  
Local Bifurcations ◽  
Singularity Induced Bifurcation

The dynamics of differential-difference-algebraic equations is studied. The paper extends the study of the local bifurcations to transcritical and pitchfork bifurcation under certain nondegenerate conditions using Lyapunov–Schmidt reduction. Furthermore, an improved version of singularity induced bifurcation theorem is given in this paper.

Stability and Bifurcation Analysis of a Forced Cylinder Wake

2005 ◽  
Author(s):  
N. W. Mureithi ◽  
M. Rodriguez
Keyword(s):  
Bifurcation Analysis ◽  
Interaction Mechanism ◽  
Period Doubling ◽  
Pitchfork Bifurcation ◽  
Mode Interaction ◽  
Cylinder Wake ◽  
Modal Interaction ◽  
Amplitude Equations ◽  
Vortex Merging ◽  
Interaction Amplitude

We present a study on the dynamics of a cylinder wake subjected to forced excitation. Williams et al. (1992) discovered that the spatial symmetry of the excitation plays a crucial role in determining the resulting wake dynamics. Reflection-symmetric forcing was found to affect the Karman wake much more strongly compared to Z2(κ, π) asymmetric forcing. For low forcing amplitudes, the existence of a nonlinear mode interaction mechanism was postulated to explain the observed “beating” phenomenon observed in the wake. Previous work by the authors (Mureithi et al. 2002, 2003) presented general forms of the modal interaction amplitude equations governing the dynamics of the periodically forced wake. In the present work, numerical CFD computations of the forced cylinder wake are presented. It is shown that the experimentally observed wake bifurcations can be reproduced by numerical simulations with reasonable accuracy. The CFD computations show that the forced wake first looses reflection symmetry followed by a bifurcation associated with vortex merging as the forcing amplitude is increased. A bifurcation analysis of a simplified amplitude equation shows that these two transitions are due to a pitchfork bifurcation and a period-doubling bifurcation of mixed mode solutions.

scholarly journals SELF-OSCILLATIONS AND SLIDING IN RELAY FEEDBACK SYSTEMS: SYMMETRY AND BIFURCATIONS

2001 ◽  
Vol 11 (04) ◽  
pp. 1121-1140 ◽  
Author(s):  
MARIO DI BERNARDO ◽  
KARL HENRIK JOHANSSON ◽  
FRANCESCO VASCA
Keyword(s):  
Dynamical Systems ◽  
Periodic Solutions ◽  
Bifurcation Analysis ◽  
Feedback System ◽  
Feedback Systems ◽  
Relay Feedback ◽  
Linear Dynamical Systems ◽  
Local Bifurcations

This paper is concerned with the bifurcation analysis of linear dynamical systems with relay feedback. The emphasis is on the bifurcations of the system periodic solutions and their symmetry. It is shown that, despite what has been conjectured in the literature, a symmetric and unforced relay feedback system can exhibit asymmetric periodic solutions. Moreover, the occurrence of periodic solutions characterized by one or more sections lying within the system discontinuity set is outlined. The mechanisms underlying their formation are carefully studied and shown to be due to an interesting, novel class of local bifurcations.

scholarly journals Hopf Bifurcation Analysis of a Predator-Prey Biological Economic System with Nonselective Harvesting

2015 ◽  
Vol 2015 ◽  
pp. 1-10
Author(s):  
Biwen Li ◽  
Zhenwei Li ◽  
Boshan Chen ◽  
Gan Wang
Keyword(s):  
Hopf Bifurcation ◽  
Economic System ◽  
Bifurcation Analysis ◽  
Economic Perspective ◽  
Modified Model ◽  
Bifurcation Theorem ◽  
Predator Prey ◽  
Economic Profit ◽  
Prey System ◽  
Nonselective Harvesting

A modified predator-prey biological economic system with nonselective harvesting is investigated. An important mathematical feature of the system is that the economic profit on the predator-prey system is investigated from an economic perspective. By using the local parameterization method and Hopf bifurcation theorem, we analyze the Hopf bifurcation of the proposed system. In addition, the modified model enriches the database for the predator-prey biological economic system. Finally, numerical simulations illustrate the effectiveness of our results.

Multiple Bifurcations of Synchronized Oscillators With Delays

2005 ◽  
Author(s):  
Yuan Y. Yuan
Keyword(s):  
Local Stability ◽  
Time Delays ◽  
Center Manifold ◽  
Zero Solution ◽  
Pitchfork Bifurcation ◽  
Bifurcation Theorem ◽  
Center Manifold Theorem ◽  
System A ◽  
Normal Form Method ◽  
Theoretical Results

The synchronized oscillator with two discrete time delays is considered. The local stability of the zero solution of this system is investigated by studying the distributions of the eigenvalues of the system. A complete bifurcation analysis is given by employing the center manifold theorem, normal form method and bifurcation theorem. It is shown that the trivial fixed point may lose stability via a transcritical/pitchfork bifurcation, Hopf bifurcation or Bogdanov-Takens bifurcation. Some numerical simulation examples are given for justify the theoretical results.

BIFURCATION ANALYSIS OF A DELAYED DYNAMIC SYSTEM VIA METHOD OF MULTIPLE SCALES AND SHOOTING TECHNIQUE

2005 ◽  
Vol 15 (02) ◽  
pp. 425-450 ◽  
Author(s):  
HUAILEI WANG ◽  
HAIYAN HU
Keyword(s):  
Time Delay ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Duffing Oscillator ◽  
Time History ◽  
Pitchfork Bifurcation ◽  
Periodic Motions ◽  
Bifurcation Diagrams ◽  
Shooting Technique ◽  
Local Bifurcations

This paper presents a detailed study on the bifurcation of a controlled Duffing oscillator with a time delay involved in the feedback loop. The first objective is to determine the bifurcating periodic motions and to obtain the global diagrams of local bifurcations of periodic motions with respect to time delay. In order to determine the bifurcation point, an analysis on the stability switches of the trivial equilibrium is first performed for all possible parametric combinations. Then, by means of the method of multiple scales, an analysis on the local bifurcation of periodic motions is given. The static bifurcation diagrams on the amplitude-delay plane exhibit two kinds of local bifurcations of periodic motions, namely the saddle-node bifurcation and the pitchfork bifurcation, indicating a sudden emergence of two periodic motions with different stability and a Hopf bifurcation, respectively, in the sense of dynamic bifurcation. The second objective is to develop a shooting technique to locate both stable and unstable periodic motions of autonomous delay differential equations such that the periodic motions and their stability predicted using the method of multiple scales could be verified. The efficacy of the shooting scheme is well illustrated by some examples via phase trajectory and time history. It is shown that periodic motions located by the shooting method agree very well with those predicted on the bifurcation diagrams. Finally, the paper presents some interesting features of this simple, but dynamics-rich system.

scholarly journals Two-Parameters Bifurcation in Quasilinear Dierential-Algebraic Equations

2016 ◽  
Vol 12 (1) ◽  
pp. 5786-5796
Author(s):  
Kamal H Yasir ◽  
Abbas M Al_husenawe
Keyword(s):  
Normal Form ◽  
Basic Principle ◽  
Bifurcation Theory ◽  
Taylor Expansion ◽  
Pitchfork Bifurcation ◽  
Algebraic Equations ◽  
Saddle Node ◽  
Two Parameters

In this paper, bifurcation of solution of guasilinear dierential-algebraic equations (DAEs) is studied. Whereas basic principle that quasilinear DAE is eventually reducible to an ordinary dierential equation (ODEs) and that this reduction so we can apply the classical bifurcation theory of the (ODEs). The taylor expansion applied to the reduced DAEs to prove that is equivalent to an ODE which is a normal form under some non-degeneracy conditions theorems given in this work deal with the saddle node,transcritical and pitchfork bifurcation with two-parameters. Some illustrated examples are given to explain the idea of the paper.

scholarly journals Hopf Bifurcation Analysis for a Two Species Periodic Chemostat Model with Discrete Delays

2020 ◽  
pp. 93-105
Author(s):  
Jane Ireri ◽  
Ganesh Pokhariyal ◽  
Stephene Moindi
Keyword(s):  
Periodic Solution ◽  
Fourier Series ◽  
Hopf Bifurcation ◽  
Bifurcation Analysis ◽  
Nutrient Input ◽  
Chemostat Model ◽  
Bifurcation Theorem ◽  
Interior Equilibrium ◽  
Limiting Nutrient ◽  
Discrete Delays

In this paper we analyze a Chemostat model of two species competing for a single limiting nutrient input varied periodically using a Fourier series with discrete delays. To understand global aspects of the dynamics we use an extension of the Hopf bifurcation theorem, a method that rigorously establishes existence of a periodic solution. We show that the interior equilibrium point changes its stability and due to the delay parameter it undergoes a Hopf bifurcation.Numerical results shows that coexistence is possible when delays are introduced and Fourier series produces the required seasonal variations. We also show that for small delays periodic variations of nutrients has more influence on species density variations than the delay.

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