Stability and bifurcation in an electromechanical system with time delays

The effect of time delays occurring in a proportional-integral-derivative feedback controller on the linear stability of a simple electromechanical system is investigated by analyzing the characteristic transcendental equation. It is found that the trivial fixed point of the system can lose its stability through Hopf bifurcations when the time delay crosses certain critical values. Codimension two bifurcations, which result from non-resonant and resonant Hopf–Hopf bifurcation interactions, are also found to exist in the system.     

Control of the nonlinear oscillator bifurcation under a superharmonic resonance

A weakly nonlinear oscillator is modeled by a differential equation. A superharmonic resonance system can have a saddle-node bifurcation, with a jumping transition from one state to another. To control the jumping phenomena and the unstable region of the nonlinear oscillator, a combination of feedback controllers is designed. Bifurcation control equations are derived by using the method of multiple scales. Furthermore, by performing numerical simulations and by comparing the responses of the uncontrolled system and the controlled system, we clarify that a good controller can be obtained by changing the feedback control gain. Also, it is found that the linear feedback gain can delay the occurrence of saddle-node bifurcations, while the nonlinear feedback gain can eliminate saddle-node bifurcations. Feasible ways of further research of saddle-node bifurcations are provided. Finally, we show that an appropriate nonlinear feedback control gain can suppress the amplitude of the steady-state response.     

Hopf bifurcation in a delayed food-limited model with feedback control

In this paper, we consider a delayed food-limited model with feedback control. By regarding the delay as the bifurcation parameter and analyzing the corresponding characteristic equations, the linear stability of the system is discussed, and Hopf bifurcations are demonstrated. By the normal form and the center manifold theory, the explicit formulae are derived to determine the stability, direction and other properties of bifurcating periodic solutions. Finally, some examples are presented to verify our main results.     

Fractional-order complex T system: bifurcations,chaos control,and synchronization

In this paper, the fractional-order complex T system is proposed. The dynamics of the system including symmetry, the stability of equilibrium points, bifurcations with variation of system parameters, and derivative orders are investigated. Period-doubling and tangent bifurcations with appropriate derivative orders and system parameters are observed. Besides, the control problem of the system is examined by using the feedback control technique. Furthermore, based on the stability theory of fractional-order systems, the scheme of function projective synchronization for the fractional-order complex T system is presented. The function projective synchronization for the system is realized by designing an appropriate synchronization controller. Numerical simulations are used to demonstrate the effectiveness and feasibility of the proposed scheme.     

Lyapunov and LMI analysis and feedback control of border collision bifurcations

Feedback control of piecewise smooth discrete-time systems that undergo border collision bifurcations is considered. These bifurcations occur when a fixed point or a periodic orbit of a piecewise smooth system crosses or collides with the border between two regions of smooth operation as a system parameter is quasistatically varied. The class of systems studied is piecewise smooth maps that depend on a parameter, where the system dimension n can take any value. The goal of the control effort in this work is to replace the bifurcation so that in the closed-loop system, the steady state remains locally attracting and locally unique (“nonbifurcation with persistent stability”). To achieve this, Lyapunov and linear matrix inequality (LMI) techniques are used to derive a sufficient condition for nonbifurcation with persistent stability. The derived condition is stated in terms of LMIs. This condition is then used as a basis for the design of feedback controls to eliminate border collision bifurcations in piecewise smooth maps and to produce the desirable behavior noted earlier. Numerical examples that demonstrate the effectiveness of the proposed control techniques are given.     

Local Bifurcation Control of a Forced Single-Degree-of-Freedom Nonlinear System: Saddle-Node Bifurcation

It is well known that saddle-node bifurcations can occur in the steady-state response of a forced single-degree-of-freedom (SDOF) nonlinear system in the cases of primary and superharmonic resonances. This discontinuous or catastrophic bifurcation can lead to jump and hysteresis phenomena, where at a certain interval of the control parameter, two stable attractors exist with an unstable one in between. A feedback control law is designed to control the saddle-node bifurcations taking place in the resonance response, thus removing or delaying the occurrence of jump and hysteresis phenomena. The structure of candidate feedback control law is determined by analyzing the eigenvalues of the modulation equations. It is shown that three types of feedback – linear, nonlinear, and a combination of linear and nonlinear – are adequate for the bifurcation control. Finally, numerical simulations are performed to verify the effectiveness of the proposed feedback control.     

Bifurcation analysis in a delayed Lokta�CVolterra predator�Cprey model with two delays

In this paper, a class of delayed Lokta?CVolterra predator?Cprey model with two delays is considered. By analyzing the associated characteristic transcendental equation, its linear stability is investigated and Hopf bifurcation is demonstrated. Some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using normal form theory and center manifold theory. Some numerical simulations for supporting the theoretical results are also provided. Finally, main conclusions are given.     

Bifurcation Control of a Parametrically Excited Duffing System

A linear time-delayed feedback control is used to delay the occurrenceof pitchfork bifurcations and to eliminate saddle-node bifurcations,which may arise in the nonlinear response of a parametrically excitedDuffing system under the principal parametric resonance. The feedbackgains and the time delay are chosen by analyzing the modulationequations of the amplitude and the phase. It is shown that by using anappropriate feedback control, the stable region of the trivial solutionscan be broadened, a discontinuous bifurcation can be transformed into acontinuous one, and the jump phenomenon in the resonance response can beremoved.     

Multiple scales for two-parameter bifurcations in a neutral equation

Van der Pol??s equation with extended delay feedback is investigated as a neutral differential-difference equation. Normal forms near codimension two bifurcations, including Hopf?Cpitchfork and Hopf?CHopf bifurcation, are determined by the method of multiple scales. Through analyzing the associated amplitude equations, we obtain the detailed bifurcation sets and find some interesting phenomena such as quasi-periodic oscillations and strange attractor, which are confirmed by several numerical simulations.     

One-to-three resonant Hopf bifurcations of a maglev system

This paper studies the dynamics of a maglev system around 1:3 resonant Hopf–Hopf bifurcations. When two pairs of purely imaginary roots exist for the corresponding characteristic equation, the maglev system has an interaction of Hopf–Hopf bifurcations at the intersection of two bifurcation curves in the feedback control parameter and time delay space. The method of multiple time scales is employed to drive the bifurcation equations for the maglev system by expressing complex amplitudes in a combined polar-Cartesian representation. The dynamics behavior in the vicinity of 1:3 resonant Hopf–Hopf bifurcations is studied in terms of the controller’s parameters (time delay and two feedback control gains). Finally, numerical simulations are presented to support the analytical results and demonstrate some interesting phenomena for the maglev system.     

The impact of delayed feedback on the pulsating oscillations of class-B lasers

Class-B laser systems can be described by a set of slow-fast differential equations with a small parameter, and they exhibit pulsating oscillations in common. In this paper, the impact of the delayed feedback on the pulsating solutions is investigated. At first, a careful analysis of the local stability and bifurcation shows the existence of a series of Hopf bifurcations and double Hopf bifurcations when the strength of the delayed feedback or the delay increases, by means of stability switches. Then, an application of the geometric singular perturbation theory reveals the evolutional mechanism of the pulsating solutions from transients to stationary states, and the effect of the delayed feedback upon the pulsating solutions is examined.     

Nonlinear Effects on Coexistence Phenomenon in Parametric Excitation

We investigate the effect of nonlinearites on a parametrically excited ordinary differential equation whose linearization exhibits the phenomenon of coexistence. The differential equation studied governs the stability of a mode of vibration in an unforced conservative two degree of freedom system used to model thefree vibrations of a thin elastica. Using perturbation methods, we show thatat parameter values corresponding to coexistence, nonlinear terms can cause the origin to become nonlinearly unstable,even though linear stability analysis predicts the origin to be stable.We also investigate the bifurcations associated with this instability.     

Simply supported elastic beams under parametric excitation

In this paper, the nonlinear characteristics of the parametric resonance of simply supported elastic beams are investigated. Considering a geometrically exact formulation for unsharable and inextensible elastic beams subject to support motions, the integral-partial-differential equation of motion is obtained. The third-order perturbation of the equation of motion is then determined in a form amenable to an asymptotic treatment. The method of multiple scales is used to obtain the equations that describe the modulation of the amplitude and phase of parametric-resonance motions. The stability and bifurcations of the system are investigated considering, in particular, the frequency-response function. Furthermore, experimental results are shown to confirm the theoretically predicted stability and bifurcations.     

Non-linear dynamics of a slender beam carrying a lumped mass with principal parametric and internal resonances

The non-linear behaviour of a slender beam carrying a lumped mass subjected to principal parametric base excitation is investigated. The dimension of the beam–mass system and the position of the attached mass are so adjusted that the system exhibits 3 : 1 internal resonance. Multi-mode discretization of the governing equation which retains the cubic non-linearities of geometrical and inertial type is carried out using Galerkin’s method. The method of multiple scales is used to reduce the second-order temporal differential equation to a set of first-order differential equations which is then solved numerically to obtain the steady-state response and the stability of the system. The linear first-order perturbation results show new zones of instability due to the presence of internal resonance. For low amplitude of excitation and damping Hopf bifurcations are observed in the trivial steady-state response. The multi-branched non-trivial response curves show turning point, pitch-fork and Hopf bifurcations. Cascade of period and torus doubling, crises as well as the Shilnikov mechanism for chaos are observed. This is the first natural physical system exhibiting a countable infinity of horseshoes in a neighbourhood of the homoclinic orbit.     

Robust Stability Test for Dynamic Systems with Short Delays by Using Padé Approximation

The paper presents a simple approach to testifying the asymptotic stability and interval stability (robust stability against the change of system parameters in given intervals) for linear dynamic systems involving short time delays. The stability analysis starts with the study of the characteristic roots of a transcendental equation having exponential functions. By means of the Padé approximation to the exponential functions, the transcendental characteristic equation is approximated as an algebraic equation. Then, the test of asymptotic stability and interval stability of the system is completed in a very simple way. The stability analysis of a vibration system with short time delays in the feedback paths of displacement and velocity, taken as an example, is given in detail. The analysis and numerical examples indicate that the approach gives excellent accuracy for linear dynamic systems with short time delays.     

Stability and bifurcations analysis of rotating shafts with base excitations

Base excitation in a rotating machinery such as turbo generators, aircraft engines, etc could occur when these systems are subjected to the base movements. This paper investigates the nonlinear behavior of a symmetrical rotating shaft under base excitation when the system is in the vicinity of the main resonance. Dynamic imbalances and harmonic base excitations are the sources of excitation in this system. The equations of motion are derived using the extended Hamilton principle and are mapped into the complex plane. The complex partial differential equation of motion is transformed to the ordinary one utilizing mode shape of the linear system. The method of multiple scales is used to solve the equation of motion. The steady state solutions and their stability are determined, and the effects of damping coefficient, base excitations, and eccentricities of shaft on the stability and bifurcations of the system are investigated. The numerical integration is performed to validate the perturbation results. It is shown that the achieved results from two over-mentioned methods are in accordance with each other.     

Stability and bifurcation analysis in the delay-coupled nonlinear oscillators

This paper investigates the dynamical behavior of two oscillators with nonlinearity terms, which are coupled with finite delay parameters. Each oscillator is a general class of second-order nonlinear delay-differential equations. The system of delay differential equations is analyzed by reducing the delay equations to a system of ordinary differential equations on a finite-dimensional center manifold, the corresponding to an infinite-dimensional phase space. In addition, the characteristic equation for the linear stability of the trivial equilibrium is completely analyzed and the stability region is illustrated in the parameters space. Our analysis reveals necessary coefficients of the reduced vector field on the center manifold for studying the bifurcations of the trivial equilibrium such as transcritical, pitchfork, and Hopf bifurcation. Finally, we consider the delay-coupled van der Pol equations.     

Linear and non-linear interactions between static and dynamic bifurcations of damped planar beams

The critical and post-critical behavior of a non-conservative non-linear structure, undergoing statical and dynamical bifurcations, is analyzed. The system consists of a purely flexible planar beam, equipped with a lumped visco-elastic device, loaded by a follower force. A unique integro-differential equation of motion in the transversal displacement, with relevant boundary conditions, is derived. Then, the linear stability diagram of the trivial rectilinear configuration is built-up in the parameter space. Particular emphasis is given to the role of the damping on the critical scenario. The occurrence of different mechanisms of instability is highlighted, namely, of divergence, Hopf, double zero, resonant and non-resonant double Hopf, and divergence-Hopf bifurcation. Attention is then focused on the two latter (codimension-two) bifurcations. A multiple scale analysis is carried-out directly on the continuous model, and the relevant non-linear bifurcation equations in the amplitudes of the two interactive modes are derived. The fixed-points of these equations are numerically evaluated as functions of two bifurcation parameters and some equilibrium paths illustrated. Finally, the bifurcation diagrams, illustrating the system behavior around the critical points of the parameter space, are obtained.     

Stability and Synchronization of a Ring of Identical Cells with Delayed Coupling

We consider a ring of identical elements with time delayed, nearest neighbour coupling. The individual elements are modelled by a scalar delay differential equation which includes linear decay and nonlinear delayed feedback. The linear stability of the trivial solution is completely analyzed and illustrated in the parameter space of the coupling strength and the coupling delay. Conditions for global stability of the trivial solution are also given. The bifurcation and stability of nontrivial synchronous solutions from the trivial solution is analyzed using a centre manifold construction.Dedicated to Professor Shui-Nee Chow on the occasion of his 60th birthday.     

Hopf Bifurcation and Stability of Periodic Solutions for van der Pol Equation with Distributed Delay

The van der Pol equation with a distributed time delay is analyzed. Itslinear stability is investigated by employing the Routh–Hurwitzcriteria. Moreover, the local asymptotic stability conditions are alsoderived. By using the mean time delay as a bifurcation parameter, themodel is found to undergo a sequence of Hopf bifurcations. The directionand the stability criteria of the bifurcating periodic solutions areobtained by the normal form theory and the center manifold theorem. Somenumerical simulation examples for justifying the theoretical analysisare also given.