scholarly journals Feedback control of unstable periodic orbits in equivariant Hopf bifurcation problems

2013 ◽  
Vol 371 (1999) ◽  
pp. 20120467 ◽  
Author(s):  
C. M. Postlethwaite ◽  
G. Brown ◽  
M. Silber
Keyword(s):  
Hopf Bifurcation ◽  
Unstable Periodic Orbits ◽  
Feedback Controls ◽  
Bifurcation Theorem ◽  
Non Invasive ◽  
Subcritical Hopf Bifurcation ◽  
Bifurcation Problems ◽  
Spatio Temporal ◽  
The Stability ◽  
Equivariant Hopf Bifurcation

Symmetry-breaking Hopf bifurcation problems arise naturally in studies of pattern formation. These equivariant Hopf bifurcations may generically result in multiple solution branches bifurcating simultaneously from a fully symmetric equilibrium state. The equivariant Hopf bifurcation theorem classifies these solution branches in terms of their symmetries, which may involve a combination of spatial transformations and temporal shifts. In this paper, we exploit these spatio-temporal symmetries to design non-invasive feedback controls to select and stabilize a targeted solution branch, in the event that it bifurcates unstably. The approach is an extension of the Pyragas delayed feedback method, as it was developed for the generic subcritical Hopf bifurcation problem. Restrictions on the types of groups where the proposed method works are given. After addition of the appropriately optimized feedback term, we are able to compute the stability of the targeted solution using standard bifurcation theory, and give an account of the parameter regimes in which stabilization is possible. We conclude by demonstrating our results with a numerical example involving symmetrically coupled identical nonlinear oscillators.

EFFECTS OF SHORT-CUT IN A DELAYED RING NETWORK

2012 ◽  
Vol 22 (03) ◽  
pp. 1250054 ◽  
Author(s):  
JUANJUAN MAN ◽  
SHANGJIANG GUO ◽  
YIGANG HE
Keyword(s):  
Hopf Bifurcation ◽  
Ring Network ◽  
Distribution Of Zeros ◽  
Bifurcation Theorem ◽  
Trivial Equilibrium ◽  
Secondary Bifurcation ◽  
Steady State Bifurcation ◽  
Spatio Temporal ◽  
Equivariant Hopf Bifurcation ◽  
Absolute Synchronization

This paper presents a detailed analysis on the dynamics of a ring network with short-cut. We first investigate the absolute synchronization on the basis of Lyapunov stability approach, and then discuss the linear stability of the trivial solution by analyzing the distribution of zeros of the characteristic equation. Based on the equivariant branching lemma, we not only obtain the existence of primary steady state bifurcation but also analyze the patterns and stability of the bifurcated nontrivial equilibria. Moreover, by means of the equivariant Hopf bifurcation theorem, we not only investigate the effect of connection strength on the spatio-temporal patterns of periodic solutions emanating from the trivial equilibrium, but also derive the formula to determine the direction and stability of Hopf bifurcation. In particular, we further consider the secondary bifurcation of the nontrivial equilibria. These studies show that short-cut may be used as a simple but efficient switch to control the dynamics of a system.

scholarly journals Delayed feedback control of three diffusively coupled Stuart–Landau oscillators: a case study in equivariant Hopf bifurcation

2013 ◽  
Vol 371 (1999) ◽  
pp. 20120472 ◽  
Author(s):  
Isabelle Schneider
Keyword(s):  
Hopf Bifurcation ◽  
Feedback Control ◽  
Delayed Feedback ◽  
Delayed Feedback Control ◽  
Non Invasive ◽  
Selective Stabilization ◽  
Spatio Temporal ◽  
The Individual ◽  
Equivariant Hopf Bifurcation

The modest aim of this case study is the non-invasive and pattern-selective stabilization of discrete rotating waves (‘ponies on a merry-go-round’) in a triangle of diffusively coupled Stuart–Landau oscillators. We work in a setting of symmetry-breaking equivariant Hopf bifurcation. Stabilization is achieved by delayed feedback control of Pyragas type, adapted to the selected spatio-temporal symmetry pattern. Pyragas controllability depends on the parameters for the diffusion coupling, the complex control amplitude and phase, the uncontrolled super-/sub-criticality of the individual oscillators and their soft/hard spring characteristics. We mathematically derive explicit conditions for Pyragas control to succeed.

scholarly journals Nonlinear Dynamics of a PI Hydroturbine Governing System with Double Delays

2017 ◽  
Vol 2017 ◽  
pp. 1-14
Author(s):  
Hongwei Luo ◽  
Jiangang Zhang ◽  
Wenju Du ◽  
Jiarong Lu ◽  
Xinlei An
Keyword(s):  
Hopf Bifurcation ◽  
Equilibrium Points ◽  
Bifurcation Theorem ◽  
Center Manifold Theorem ◽  
Governing System ◽  
Normal Form Method ◽  
The Stability ◽  
System Frequency ◽  
Realistic Significance ◽  
Theoretical Results

A PI hydroturbine governing system with saturation and double delays is generated in small perturbation. The nonlinear dynamic behavior of the system is investigated. More precisely, at first, we analyze the stability and Hopf bifurcation of the PI hydroturbine governing system with double delays under the four different cases. Corresponding stability theorem and Hopf bifurcation theorem of the system are obtained at equilibrium points. And then the stability of periodic solution and the direction of the Hopf bifurcation are illustrated by using the normal form method and center manifold theorem. We find out that the stability and direction of the Hopf bifurcation are determined by three parameters. The results have great realistic significance to guarantee the power system frequency stability and improve the stability of the hydropower system. At last, some numerical examples are given to verify the correctness of the theoretical results.

Turing Instability and Hopf Bifurcation in Cellular Neural Networks

2021 ◽  
Vol 31 (08) ◽  
pp. 2150143
Author(s):  
Zunxian Li ◽  
Chengyi Xia
Keyword(s):  
Neural Networks ◽  
Hopf Bifurcation ◽  
Turing Instability ◽  
Cellular Neural Networks ◽  
Bifurcation Theorem ◽  
Decoupling Method ◽  
Dynamical Behaviors ◽  
The Stability ◽  
Approximate Expressions ◽  
Zero Equilibrium

In this paper, we explore the dynamical behaviors of the 1D two-grid coupled cellular neural networks. Assuming the boundary conditions of zero-flux type, the stability of the zero equilibrium is discussed by analyzing the relevant eigenvalue problem with the aid of the decoupling method, and the conditions for the occurrence of Turing instability and Hopf bifurcation at the zero equilibrium are derived. Furthermore, the approximate expressions of the bifurcating periodic solutions are also obtained by using the Hopf bifurcation theorem. Finally, numerical simulations are provided to demonstrate the theoretical results.

Graphical Hopf Bifurcation of a Filippov HTLV-1 Model With Delay in Cytotoxic T Cells Response

2018 ◽  
Vol 140 (9) ◽  
Author(s):  
Elham Shamsara ◽  
Zahra Afsharnezhad ◽  
Elham Javidmanesh
Keyword(s):  
T Cells ◽  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Bifurcation Theory ◽  
Cytotoxic T Cells ◽  
Dynamical Behavior ◽  
Bifurcation Theorem ◽  
Nyquist Criterion ◽  
The Stability ◽  
Frequency Domain Approach

In this paper, we present a discontinuous cytotoxic T cells (CTLs) response for HTLV-1. Moreover, a delay parameter for the activation of CTLs is considered. In fact, a system of differential equation with discontinuous right-hand side with delay is defined for HTLV-1. For analyzing the dynamical behavior of the system, graphical Hopf bifurcation is used. In general, Hopf bifurcation theory will help to obtain the periodic solutions of a system as parameter varies. Therefore, by applying the frequency domain approach and analyzing the associated characteristic equation, the existence of Hopf bifurcation by using delay immune response as a bifurcation parameter is determined. The stability of Hopf bifurcation periodic solutions is obtained by the Nyquist criterion and the graphical Hopf bifurcation theorem. At the end, numerical simulations demonstrated our results for the system of HTLV-1.

STABILITY AND HOPF BIFURCATION ON A TWO-NEURON SYSTEM WITH TIME DELAY IN THE FREQUENCY DOMAIN

2007 ◽  
Vol 17 (04) ◽  
pp. 1355-1366 ◽  
Author(s):  
WENWU YU ◽  
JINDE CAO
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Frequency Domain ◽  
Harmonic Balance ◽  
Neuron System ◽  
Bifurcation Theorem ◽  
Nyquist Criterion ◽  
The Stability ◽  
Frequency Domain Approach ◽  
System With Time Delay

In this paper, a general two-neuron model with time delay is considered, where the time delay is regarded as a parameter. It is found that Hopf bifurcation occurs when this delay passes through a sequence of critical value. By analyzing the characteristic equation and using the frequency domain approach, the existence of Hopf bifurcation is determined. The stability of bifurcating periodic solutions are determined by the harmonic balance approach, Nyquist criterion and the graphic Hopf bifurcation theorem. Numerical results are given to justify the theoretical analysis.

An algebraic criterion for symmetric Hopf bifurcation

1993 ◽  
Vol 440 (1910) ◽  
pp. 727-732 ◽  
Keyword(s):  
Fixed Point ◽  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Spherical Harmonics ◽  
Irreducible Representations ◽  
Two Dimensional ◽  
Bifurcation Theorem ◽  
Algebraic Criterion ◽  
Equivariant Hopf Bifurcation

The equivariant Hopf bifurcation theorem states that bifurcating branches of periodic solutions with certain symmetries exist when the fixed-point subspace of that subgroup of symmetries is two dimensional. We show that there is a group-theoretic restriction on the subgroup of symmetries in order for that subgroup to have a two-dimensional fixed-point subspace in any representation. We illustrate this technique for all irreducible representations of SO(3) on the space V l of spherical harmonics for l even.

scholarly journals Global Hopf Bifurcation on Two-Delays Leslie-Gower Predator-Prey System with a Prey Refuge

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Qingsong Liu ◽  
Yiping Lin ◽  
Jingnan Cao
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Functional Differential Equations ◽  
Global Hopf Bifurcation ◽  
Bifurcation Theorem ◽  
Predator Prey ◽  
Prey System ◽  
Two Delays ◽  
Functional Differential ◽  
The Stability

A modified Leslie-Gower predator-prey system with two delays is investigated. By choosingτ1andτ2as bifurcation parameters, we show that the Hopf bifurcations occur when time delay crosses some critical values. Moreover, we derive the equation describing the flow on the center manifold; then we give the formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the theoretical results and chaotic behaviors are observed. Finally, using a global Hopf bifurcation theorem for functional differential equations, we show the global existence of the periodic solutions.

scholarly journals Control of Limit Cycle Oscillations of a Two-Dimensional Aeroelastic System

2010 ◽  
Vol 2010 ◽  
pp. 1-13 ◽  
Author(s):  
M. Ghommem ◽  
A. H. Nayfeh ◽  
M. R. Hajj
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycle ◽  
Limit Cycle Oscillations ◽  
Feedback Controls ◽  
Aeroelastic System ◽  
Freestream Velocity ◽  
Nonlinear Springs ◽  
Subcritical Hopf Bifurcation ◽  
Static Feedback ◽  
Nonlinear Static

Linear and nonlinear static feedback controls are implemented on a nonlinear aeroelastic system that consists of a rigid airfoil supported by nonlinear springs in the pitch and plunge directions and subjected to nonlinear aerodynamic loads. The normal form is used to investigate the Hopf bifurcation that occurs as the freestream velocity is increased and to analytically predict the amplitude and frequency of the ensuing limit cycle oscillations (LCO). It is shown that linear control can be used to delay the flutter onset and reduce the LCO amplitude. Yet, its required gains remain a function of the speed. On the other hand, nonlinear control can be effciently implemented to convert any subcritical Hopf bifurcation into a supercritical one and to significantly reduce the LCO amplitude.

HOPF BIFURCATION ANALYSIS OF TWO SUNFLOWER EQUATIONS

2012 ◽  
Vol 05 (01) ◽  
pp. 1250001 ◽  
Author(s):  
JINGNAN WANG ◽  
WEIHUA JIANG
Keyword(s):  
Hopf Bifurcation ◽  
Global Hopf Bifurcation ◽  
Bifurcation Theorem ◽  
Local Hopf Bifurcation ◽  
Sunflower Equation ◽  
Nonlinear Relation ◽  
Bifurcation Direction ◽  
The Stability ◽  
Bifurcating Periodic Solution ◽  
Modified Equation

In this paper, two sunflower equations are considered. Using delay τ as a parameter and applying the global Hopf bifurcation theorem, we investigate the existence of global Hopf bifurcation for the sunflower equation. Furthermore, we analyze the local Hopf bifurcation of the modified equation with nonlinear relation about stem's increase, including the occurrence, the bifurcation direction, the stability and the approximation expression of the bifurcating periodic solution using the theory of normal form and center manifold. Finally, the obtained results of these two equations are compared, which finds that the result about the period of their bifurcating periodic solutions is obviously different, while the bifurcation direction and stability are identical.

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