Calculation of Coefficients of Simplest Normal Forms of Hopf and Generalized Hopf Bifurcations

The coefficients of the simplest normal forms of both high-dimensional generalized Hopf and high-dimensional Hopf bifurcation systems were discussed using the adjoint operator method. A particular nonlinear scaling and an inner product were introduced in the space of homogeneous polynomials. Theorems were established for the explicit expression of the simplest normal forms in terms of the coefficients of both the conventional normal forms of Hopf and generalized Hopf bifurcation systems. A symbolic manipulation was designed to perform the calculation of the coefficients of the simplest normal forms using Mathematica. The original ordinary differential equation was required in the input and the simplest normal form could be obtained as the output. Finally, the simplest normal forms of 6-dimensional generalized Hopf singularity of type 2 and 5-dimensional Hopf bifurcation system were discussed by executing the program. The output showed that the 5th- and 9th-order terms remained in 6-dimensional generalized Hopf singularity of type 2 and the 3rd- and 5th-order terms remained in 5-dimensional Hopf bifurcation system.

WAMS-based monitoring and control of Hopf bifurcations in multi-machine power systems

A method is proposed to monitor and control Hopf bifurcations in multi-machine power systems using the information from wide area measurement systems (WAMSs). The power method (PM) is adopted to compute the pair of conjugate eigenvalues with the algebraically largest real part and the corresponding eigenvectors of the Jacobian matrix of a power system. The distance between the current equilibrium point and the Hopf bifurcation set can be monitored dynamically by computing the pair of con- jugate eigenvalues. When the current equilibrium point is close to the Hopf bifurcation set, the approximate normal vector to the Hopf bifurcation set is computed and used as a direction to regulate control parameters to avoid a Hopf bifurcation in the power system described by differential algebraic equations (DAEs). The validity of the proposed method is demonstrated by regulating the reactive power loads in a 14-bus power system.

HOPF BIFURCATIONS OF NONAUTONOMOUS SYSTEMS AT RESONANCE

Hop/bifurcations of periodic nonautonomous systems at resonance are studied and the results similar to those at nonresonance are drawn.

Hopf Bifurcations of a Chemostat System with Bi-parameters

We study a chemostat system with two parameters, So-initial density and D-flow-speed of the solution. At first, a generalization of the traditional Hopf bifurcation theorem is given. Then, an existence theorem for the Hopf bifurcation of the chemostat system is presented.

Hopf Bifurcations, Drops in the Lid–Driven Square Cavity Flow

The lid-driven square cavity flow is investigated by numerical experiments.It is found that from Re=5,000 to Re=7,307.75 the solution is stationary,but at Re=7,308 the solution is time periodic.So the critical Reynolds number for the first Hopf bifurcation localizes between Re=7,307.75 and Re=7,308.Time periodical behavior begins smoothly,imperceptibly at the bottom left corner at a tiny tertiary vortex;all other vortices stay still,and then it spreads to the three relevant corners of the square cavity so that all small vortices at all levels move periodically.The primary vortex stays still.At Re=13,393.5 the solution is time periodic;the long-term integration carried out past t_(∞)=126,562.5 and the fluctuations of the kinetic energy look periodic except slight defects.However at Re=13,393.75 the solution is not time periodic anymore:losing unambiguously,abruptly time periodicity,it becomes chaotic.So the critical Reynolds number for the second Hopf bifurcation localizes between Re=13,393.5 and Re=13,393.75.At high Reynolds numbers Re=20,000 until Re=30,000 the solution becomes chaotic.The long-term integration is carried out past the long time t_(∞)=150,000,expecting the time asymptotic regime of the flow has been reached.The distinctive feature of the flow is then the appearance of drops:tiny portions of fluid produced by splitting of a secondary vortex,becoming loose and then fading away or being absorbed by another secondary vortex promptly.At Re=30,000 another phenomenon arises—the abrupt appearance at the bottom left corner of a tiny secondary vortex,not produced by splitting of a secondary vortex.

LOCAL AND GLOBAL HOPF BIFURCATIONS FOR A PREDATOR-PREY SYSTEM WITH TWO DELAYS

In this paper, the Leslie predator-prey system with two delays is studied. The stability of the positive equilibrium is discussed by analyzing the associated characteristic transcendental equation. The direction and stability of the bifurcating periodic solutions are determined by applying the center manifold theorem and normal form theory. The conditions to guarantee the global existence of periodic solutions are given.

LOCAL AND GLOBAL HOPF BIFURCATIONS IN A DELAYED HUMAN RESPIRATORY SYSTEM

This paper considers a delayed human respiratory model. Firstly, the stability of the equilibrium of the model is investigated and the occurrence of a sequence of Hopf bifurcations of the model is proved. Secondly, the explicit algorithms which determine the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived by applying the normal form method and the center manifold theory. Finally, the existence of the global periodic solutions is showed under some assumptions on the model.

Mechanism analysis of regulating Turing instability and Hopf bifurcation of malware propagation in mobile wireless sensor networks

A dynamical model is constructed to depict the spatial-temporal evolution of malware in mobile wireless sensor networks(MWSNs). Based on such a model, we design a hybrid control scheme combining parameter perturbation and state feedback to effectively manipulate the spatiotemporal dynamics of malware propagation. The hybrid control can not only suppress the Turing instability caused by diffusion factor but can also adjust the occurrence of Hopf bifurcation induced by time delay. Numerical simulation results show that the hybrid control strategy can efficiently manipulate the transmission dynamics to achieve our expected desired properties, thus reducing the harm of malware propagation to MWSNs.

Generalized Hopf Bifurcation in a Delay Model of Neutrophil Cells Model

The DDE-Biftool software is applied to solve the dynamical stability and bifurcation problem of the neutrophil cells model. Based on Hopf point finding with the stability property of the equilibrium solution loss, the continuation of the bifurcating periodical solution starting from Hopf point is exploited. The generalized Hopf point is tracked by seeking for the critical value of free parameter of the switching phenomena of the open loop, which describes the lineup of bifurcating periodical solutions from Hopf point. The normal form near the generalized Hopf point is computed by Lyapunov-Schimdt reduction scheme combined with the center manifold analytical technique. The near dynamics is classified by geometrically different topological phase portraits.

Hopf bifurcation analysis and circuit implementation for a novel four-wing hyper-chaotic system

In the paper, a novel four-wing hyper-chaotic system is proposed and analyzed. A rare dynamic phenomenon is found that this new system with one equilibrium generates a four-wing-hyper-chaotic attractor as parameter varies. The system has rich and complex dynamical behaviors, and it is investigated in terms of Lyapunov exponents, bifurcation diagrams, Poincare maps, frequency spectrum, and numerical simulations. In addition, the theoretical analysis shows that the system undergoes a Hopf bifurcation as one parameter varies, which is illustrated by the numerical simulation. Finally, an analog circuit is designed to implement this hyper-chaotic system.

HOPF BIFURCATION OF AN OSCILLATOR WITH QUADRATIC AND CUBIC NONLINEARITIES AND WITH DELAYED VELOCITY FEEDBACK

This paper studies the local dynamics of an SDOF system with quadratic and cubic stiffness terms,and with linear delayed velocity feedback.The analysis indicates that for a sufficiently large velocity feedback gain,the equilibrium of the system may undergo a number of stability switches with an increase of time delay,and then becomes unstable forever.At each critical value of time delay for which the system changes its stability,a generic Hopf bifurcation occurs and a periodic motion emerges in a one-sided neighbourhood of the critical time delay.The method of Fredholm alternative is applied to determine the bifurcating periodic motions and their stability.It stresses on the effect of the system parameters on the stable regions and the amplitudes of the bifurcating periodic solutions.

HOPF BIFURCATION OF A NONLINEAR RESTRAINED CURVED PIPE CONVEYING FLUID BY DIFFERENTIAL QUADRATURE METHOD

This paper proposes a new method for investigating the Hopf bifurcation of a curved pipe conveying fluid with nonlinear spring support.The nonlinear equation of motion is derived by forces equilibrium on microelement of the system under consideration.The spatial coordinate of the system is discretized by the differential quadrature method and then the dynamic equation is solved by the Newton-Raphson method.The numerical solutions show that the inner fluid velocity of the Hopf bifurcation point of the curved pipe varies with different values of the parameter, nonlinear spring stiffness.Based on this,the cycle and divergent motions are both found to exist at specific fluid flow velocities with a given value of the nonlinear spring stiffness.The results are useful for further study of the nonlinear dynamic mechanism of the curved fluid conveying pipe.

Double Hopf bifurcation of composite laminated piezoelectric plate subjected to external and internal excitations

The double Hopf bifurcation of a composite laminated piezoelectric plate with combined external and internal excitations is studied. Using a multiple scale method, the average equations are obtained in two coordinates. The bifurcation response equations of the composite laminated piezoelectric plate with the primary parameter resonance, i.e., 1：3 internal resonance, are achieved. Then, the bifurcation feature of bifurcation equations is considered using the singularity theory. A bifurcation diagram is obtained on the parameter plane. Different steady state solutions of the average equations are analyzed. By numerical simulation, periodic vibration and quasi-periodic vibration responses of the Composite laminated piezoelectric plate are obtained.

Stability and Hopf bifurcation of a nonlinear electromechanical coupling system with time delay feedback

The stability and the Hopf bifurcation of a nonlinear electromechanical coupling system with time delay feedback are studied. By considering the energy in the air-gap field of the AC motor, the dynamical equation of the electromechanical coupling transmission system is deduced and a time delay feedback is introduced to control the dynamic behaviors of the system. The characteristic roots and the stable regions of time delay are determined by the direct method, and the relationship between the feedback gain and the length summation of stable regions is analyzed. Choosing the time delay as a bifurcation parameter, we find that the Hopf bifurcation occurs when the time delay passes through a critical value.A formula for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is given by using the normal form method and the center manifold theorem. Numerical simulations are also performed, which confirm the analytical results.

Hopf Bifurcation Control of a Hyperchaotic Circuit System

This paper is concerned with the Hopf bifurcation control of a new hyperchaotic circuit system. The stability of the hyperchaotie circuit system depends on a selected control parameter is studied, and the critical value of the system parameter at which Hopf bifurcation occurs is investigated. Theoretical analysis give the stability of the Hopf bifurcation. In particular, washout filter aided feedback controllers are designed for delaying the bifurcation point and ensuring the stability of the bifurcated limit cycles. An important feature of the control laws is that they do not result in any change in the set of equilibria. Computer simulation results are presented to confirm the analytical predictions.

A modified averaging scheme with application to the secondary Hopf bifurcation of a delayed van der Pol oscillator

In this paper, a modified averaging scheme is presented for a class of time-delayed vibration systems with slow variables. The new scheme is a combination of the averaging techniques proposed by Hale and by Lehman and Weibel, respectively. The averaged equation obtained from the modified scheme is simple enough but it retains the required information for the local nonlinear dynamics around an equilibrium. As an application of the present method, the delay value for which a secondary Hopf bifurcation occurs is successfully located for a delayed van der Pol oscillator.

Diffusion-driven instability and Hopf bifurcation in Brusselator system

The Hopfbifurcation for the Brusselator ordinary-differential-equation （ODE） model and the corresponding partial-differential-equation （PDE） model are investigated by using the Hopf bifurcation theorem. The stability of the Hopf bifurcation periodic solution is discussed by applying the normal form theory and the center manifold theorem. When parameters satisfy some conditions, the spatial homogenous equilibrium solution and the spatial homogenous periodic solution become unstable. Our results show that if parameters are properly chosen, Hopf bifurcation does not occur for the ODE system, but occurs for the PDE system.

STOCHASTIC HOPF BIFURCATION IN QUASIINTEGRABLE-HAMILTONIAN SYSTEMS

A new procedure is developed to study the stochastic Hopf bifurcation in quasi- integrable-Hamiltonian systems under the Gaussian white noise excitation.Firstly,the singular bound- aries of the first-class and their asymptotic stable conditions in probability are given for the averaged Ito differential equations about all the sub-system's energy levels with respect to the stochastic aver- aging method.Secondly,the stochastic Hopf bifurcation for the coupled sub-systems are discussed by defining a suitable bounded torus region in the space of the energy levels and employing the theory of the torus region when the singular boundaries turn into the unstable ones.Lastly,a quasi-integrable- Hamiltonian system with two degrees of freedom is studied in detail to illustrate the above procedure. Moreover,simulations by the Monte-Carlo method are performed for the illustrative example to verify the proposed procedure.It is shown that the attenuation motions and the stochastic Hopf bifurcation of two oscillators and the stochastic Hopf bifurcation of a single oscillator may occur in the system for some system's parameters.Therefore,one can see that the numerical results are consistent with the theoretical predictions.

Analytical Hopf Bifurcation and Stability Analysis of T System

Complex dynamics are studied in the T system, a three-dimensional autonomous nonlinear system. In particular, we perform an extended Hopf bifurcation analysis of the system. The periodic orbit immediately following the Hopf bifurcation is constructed analytically for the T system using the method of multiple scales, and the stability of such orbits is analyzed. Such analytical results complement the numerical results present in the literature. The analytical results in the post-bifurcation regime are verified and extended via numerical simulations, as well as by the use of standard power spectra, autocorrelation functions, and fractal dimensions diagnostics. We find that the T system exhibits interesting behaviors in many parameter regimes.

Effects of viscoelasticity on the stability and bifurcations of nonlinear energy sinks

Due to the increasing use of passive absorbers to control unwanted vibrations,many studies have been done on energy absorbers ideally,but the lack of studies of real environmental conditions on these absorbers is felt.The present work investigates the effect of viscoelasticity on the stability and bifurcations of a system attached to a nonlinear energy sink(NES).In this paper,the Burgers model is assumed for the viscoelasticity in an NES,and a linear oscillator system is considered for investigating the instabilities and bifurcations.The equations of motion of the coupled system are solved by using the harmonic balance and pseudo-arc-length continuation methods.The results show that the viscoelasticity affects the frequency intervals of the Hopf and saddle-node branches,and by increasing the stiffness parameters of the viscoelasticity,the conditions of these branches occur in larger ranges of the external force amplitudes,and also reduce the frequency range of the branches.In addition,increasing the viscoelastic damping parameter has the potential to completely eliminate the instability of the system and gradually reduce the amplitude of the jump phenomenon.