Bifurcations, Analytical and Non-Analytical Traveling Wave Solutions of (2 + 1)-Dimensional Nonlinear Dispersive Boussinesq Equation

For the (2 + 1)-dimensional nonlinear dispersive Boussinesq equation, by using the bifurcation theory of planar dynamical systems to study its corresponding traveling wave system, the bifurcations and phase portraits of the regular system are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of analytical and non-analytical solutions of the singular system are given by using singular traveling wave theory. For certain special cases, some explicit and exact parametric representations of traveling wave solutions are derived such as analytical periodic waves and non-analytical periodic cusp waves. Further, two-dimensional wave plots of analytical periodic solutions and non-analytical periodic cusp wave solutions are drawn to visualize the dynamics of the equation.

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.

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.

Bifurcations for counterintuitive post-inhibitory rebound spike related to absence epilepsy and Parkinson disease

Seizures are caused by increased neuronal firing activity resulting from reduced inhibitory effect and enhancement of inhibitory modulation to suppress this activity is used as a therapeutic tool.However,recent experiments have shown a counterintuitive phenomenon that inhibitory modulation does not suppress but elicit post-inhibitory rebound(PIR)spike along with seizure to challenge the therapeutic tool.The nonlinear mechanism to avoid the PIR spike can present theoretical guidance to seizure treatment.This paper focuses on identifying credible bifurcations that underlie PIR spike by modulating multiple parameters in multiple theoretical models.The study identifies a codimension-2 bifurcation called saddle-node homoclinic orbit(SNHOB),which is an intersection between saddle node bifurcation on invariant cycle(SNIC)and other two bifurcations.PIR spike cannot be evoked for the SNIC far from the SNHOBbut induced for the SNIC close to the SNHOB,which extends the bifurcation condition for PIR spike from the well-known Hopf to SNIC.Especially,in a thalamic neuron model,increases of conductance of T-type Ca^(2+)(TC a)channel induce SNIC bifurcation approaching to the SNHOB to elicit PIR spikes,closely matching experimental results of the absence seizure or Parkinson diseases.Such results imply that,when inhibition is employed to relieve absence seizure and Parkinson diseases related to PIR spike,modulating SNIC to get far from the SNHOBto avoid PIR spike is the principle.The study also addresses the complex roles of TCacurrent and comprehensive relationships between PIR spike and nonlinear conceptions such as bifurcation types and shapes of threshold curve.

HETEROCLINIC ORBIT AND SUBHARMONIC BIFURCATIONS AND CHAOS OF NONLINEAR OSCILLATOR

Dynamical behavior of nonlinear oscillator under combined parametric and forcing excitation, which includes yon der Pol damping, is very complex. In this paper, Melnikov's method is used to study the heteroclinic orbit bifurcations, subharmonic bifurcations and chaos in this system. Smale horseshoes and chaotic motions can occur from odd subharmonic bifurcation of infinite order in this system-far various resonant cases finally the numerical computing method is used to study chaotic motions of this system. The results achieved reveal some new phenomena.

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.

Singular Hopf Bifurcations in DAE Models of Power Systems

We investigate an important relationship that exists between the Hopf bifurcation in the singularly perturbed nonlinear power systems and the singularity induced bifurcations (SIBs) in the corresponding different- tial-algebraic equations (DAEs). In a generic case, the SIB phenomenon in a system of DAEs signals Hopf bifurcation in the singularly perturbed systems of ODEs. The analysis is based on the linear matrix pencil theory and polynomials with parameter dependent coefficients. A few numerical examples are included.

Circuit Implementations,Bifurcations and Chaos of a Novel Fractional-Order Dynamical System

Linear transfer function approximations of the fractional integrators 1Is~ with m ^- 0.80-0.99 with steps of 0.01 are calculated systemically from the fractional order calculus and frequency-domain approximation method. To illustrate the effectiveness for fractional functions, the magnitude Bode diagrams of the actual and approximate transfer functions 1Ism with a slope of -20m dB//decade are depicted. By using the transfer function approxima- tions of the fractional integrators, a new fractional-order nonlinear system is investigated through the bifurcation diagram and Lyapunov exponent. The corresponding circuit of the fractional-order system is designed and the experimental results match perfectly with the numerical simulations.

On Bifurcations of an Ordinary Differential Equation

Biforations of an ordinary differential equation with two-point boundary value condition are investigated. Using the singularity theory based on the Liapunov-Schmidt reduction, we have obtained some characterization results.

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.

HOPE-LANDAU BIFURCATIONS OF HIGHER DIMENSIONAL TORI

The existence of degenerate bifurcations from Tm to Pm+1 is provedunder the condition of quasi-periodic critical points.

On the Bifurcations of Dean Flow between Porous Horizontal Cylinders with a Radial Flow and a Radial Temperature Gradient

The stability of a heat-conducting flow due to the pumping of a fluid around the annulus of horizontal porous cylinders is studied. The basic flow is under the action of radial flow and a radial temperature gradient. The objects of investigations are different regimes and bifurcations which may arise in this flow.

Local Stability Analysis and Bifurcations of a Discrete-Time Host-Parasitoid Model

In this paper, we examine a discrete-time Host-Parasitoid model which is a non-dimensionalized Nicholson and Bailey model. Phase portraits are drawn for different ranges of parameters and display the complicated dynamics of this system. We conduct the bifurcation analysis with respect to intrinsic growth rate r and searching efficiency a. Many forms of complex dynamics such as chaos, periodic windows are observed. Transition route to chaos dynamics is established via period-doubling bifurcations. Conditions of occurrence of the period-doubling, Neimark-Sacker and saddle-node bifurcations are analyzed for b≠a where a,b are searching efficiency. We study stable and unstable manifolds for different equilibrium points and coexistence of different attractors for this non-dimensionalize system. Without the parasitoid, the host population follows the dynamics of the Ricker model.

BIFURCATIONS OF A CANTILEVERED PIPE CONVEYING STEADY FLUID WITH A TERMINAL NOZZLE

This paper studies interactions of pipe and fluid and deals with bifurcations of a cantilevered pipe conveying a steady fluid, clamped at one end and having a nozzle subjected to nonlinear constraints at the free end. Either the nozzle parameter or the flow velocity is taken as a variable parameter. The discrete equations of the system are obtained by the Ritz-Galerkin method. The static stability is studied by the Routh criteria. The method of averaging is employed to investigate the stability of the periodic motions. A Runge-Kutta scheme is used to examine the analytical results and the chaotic motions. Three critical values are given. The first one makes the system lose the static stability by pitchfork bifurcation. The second one makes the system lose the dynamical stability by Hopf bifurcation. The third one makes the periodic motions of the system lose the stability by doubling-period bifurcation.

Bifurcations in penetrative Rayleigh-Be′nard convection in a cylindrical container

The bifurcations of penetrative Rayleigh-B′enard convection in cylindrical containers are studied by the linear stability analysis(LSA) combined with the direct numerical simulation(DNS) method. The working ?uid is cold water near 4?C, where the Prandtl number P r is 11.57, and the aspect ratio(radius/height) of the cylinder ranges from 0.66 to 2. It is found that the critical Rayleigh number increases with the increase in the density inversion parameter θ_m. The relationship between the normalized critical Rayleigh number(Rac(θ_m)/Rac(0)) and θ_m is formulated, which is in good agreement with the stability results within a large range of θ_m. The aspect ratio has a minor effect on Rac(θ_m)/Rac(0). The bifurcation processes based on the axisymmetric solutions are also investigated. The results show that the onset of axisymmetric convection occurs through a trans-critical bifurcation due to the top-bottom symmetry breaking of the present system.Moreover, two kinds of qualitatively different steady axisymmetric solutions are identi?ed.

Bifurcations and the penetrating rate analysis of a model for percussive drilling

In this paper, we investigate a low dimensional model of percussive drilling with vibro-impact to mimic the nonlinear dynamics of the bounded progression. Non- holonomity which arises in the stick-slip caused by the impact during drilling fails to be correctly identified via the classical techniques. A reduced model without non-holono- mity is derived by the introduction of a new state variable, of which averaging technique is employed successfully to detect the periodic motions. Local bifurcations are presented directly by using C-L method. Numerical simulations and the penetrating rate analysis along different choices of parame- ters have been carried out to probe the nonlinear behaviour and the optimal penetrating rate of the drilling system.

Controlling bifurcations and chaos in discrete small-world networks

We propose an impulsive hybrid control method to control the period-doubling bifurcations and stabilize unstable periodic orbits embedded in a chaotic attractor of a small-world network. Simulation results show that the bifurcations can be delayed or completely eliminated. A periodic orbit of the system can be controlled to any desired periodic orbit by using this method.

CO-DIMENSION 2 BIFURCATIONS AND CHAOS IN CANTILEVERED PIPE CONVEYING TIME VARYING FLUID WITH THREE-TO-ONE INTERNAL RESONANCES

The nonlinear behavior of a cantilevered fluid conveying pipe subjected to principal parametric and internal resonances is investigated in this paper.The flow velocity is divided into constant and sinusoidal parts.The velocity value of the constant part is so adjusted such that the system exhibits 3:1 internal resonances for the first two modes.The method of multiple scales is employed to obtain the response of the system and a set of four first-order nonlinear ordinary- differential equations for governing the amplitude of the response.The eigenvalues of the Jacobian matrix are used to assess the stability of the equilibrium solutions with varying parameters.The co- dimension 2 derived from the double-zero eigenvalues is analyzed in detail.The results show that the response amplitude may undergo saddle-node,pitchfork,Hopf,homoclinic loop and period- doubling bifurcations depending on the frequency and amplitude of the sinusoidal flow.When the frequency of the sinusoidal flow equals exactly half of the first-mode frequency,the system has a route to chaos by period-doubling bifurcation and then returns to a periodic motion as the amplitude of the sinusoidal flow increases.

Effects of supported angle on stability and dynamical bifurcations of cantilevered pipe conveying fluid

The effects of the supported angle on the stability and dynamical bifurcations of an inclined cantilevered pipe conveying fluid are investigated. First, a theoretical model of the pipe is developed through the force balance and stress-strain relationship. Second, the response surfaces, stability, and critical lines of the typical hanging system （H-S） and standing system （S-S） are discussed based on the modal analysis. Last, the bifurcation diagrams of the pipe are presented for different supported angles. It is shown that pipes will undergo a series of bifurcation processes and show rich dynamic phenomena such as buckling, Hopf bifurcation, period-doubling bifurcation, chaotic motion, and divergence motion.

Bifurcations and chaos control in discrete small-world networks

An impulsive delayed feedback control strategy to control period-doubling bifurcations and chaos is proposed. The control method is then applied to a discrete small-world network model. Qualitative analyses and simulations show that under a generic condition, the bifurcations and the chaos can be delayed or eliminated completely. In addition, the periodic orbits embedded in the chaotic attractor can be stabilized.