Search

Under some transient disturbance, the grid-forming voltage source converter may lose its synchronization with the grid, inducing low-frequency oscillation in the instantaneous power, current and angle. The physical origin of such oscillations is found to be a homoclinic bifurcation in this paper. Before the system runs into a homoclinic bifurcation, a stable equilibrium point and a stable periodic orbit co-exist. When a large disturbance is applied, the system exhibits a periodic orbit, which manifests itself as low-frequency oscillation. Moreover, no periodic orbit exists after the homoclinic bifurcation. On the basis of bifurcation analysis, boundaries of stable operation are derived in the parameter space which serve as practical design guidelines to avoid oscillation.

In this paper, a dry friction oscillator with periodic excitation is investigated. Melnikov method is used to study chaos of this model. The parameter regions of chaos are obtained. Numerical simulations are also given, which verify the analytical results.

The normal forms of homoclinic bifurcation for a rotor-active magnetic bearings (AMB) system with the time-varying stiffness are computed through considering the zeros of Melnikov equation by using the method of singularity theory. The results indicate that the bifurcation of limit point, simple bifurcation, isola center, hysteresis and quartic fold occurs.

Special techniques have been developed and bifurcation processes in nonlinear circuit with laser-arc discharge and inertial feedback was investigated. It was found that Hopf bifurcation can be both supercritical and subcritical. The boundary between the two types of bifurcation has been found. The interaction of Hopf bifurcation with other ones (the bifurcation of the twin limit cycle and bifurcation of saddle separatrix) was described.

A new current-tunable chaotic jerk equation is revealed for an existing minimal chaotic circuit. Such a circuit has employed six minimum numbers of electronic components in the category of its kind. The chaotic jerk attractors on an (X, Ẋ) plane and an (X, ẊẌ) plane are illustrated including the current-tunable bifurcations and the largest Lyapunov exponent. A new homoclinic orbit of the circuit is also demonstrated on an (X, ẊẌ) plane.

In this paper, we analyze effects of frequency of periodic perturbation on a nonlinear system with subcritical hopf bifurcation, and find that the relationship between system dynamics and perturbation frequency is complexity. Forcing frequency affects transitions, such as bifurcation and hysteresis, in diagram of periodic solution amplitude (with the same frequency as forcing frequency) versus bifurcation parameter. Forcing frequency also has influence on system stability boundary. Under certain condition, system has worst stability performance when forced by periodic perturbation with hopf frequency. The results gotten in general nonlinear system are applied into rotating stall of axial compressor.

The symmetry breaking bifurcation of a sine map is discussed when the control parameter in the sine map is chosen as a bifurcation parameter. Based on the sine map, the bifurcation points can be derived by the iterative map. Then, the stability of the system is enhanced by employing a cubic and a linear chaotic controller to exactly control the locations of the bifurcation points. Moreover, the universal constants of the chaotic system have been obtained by numerical simulation. The validity of the theoretical analysis is proved by the diagrams of bifurcation and Lyapunov exponent.

Bilinear systems are widely used in different engineering systems and incomplete research of these systems can lead to disaster. The main task of this paper is to study the dynamics of valve system by conducting complete bifurcation analysis, using the method of complete bifurcation groups (MCBG).

It is well-known that for vibro-impact systems an existence of a periodic solution with a low-velocity impact (so-called grazing) may yield a complex behavior of solutions. Slightly changing the parameters of a system with grazing, we may obtain a periodic solution which passes near the delimiter without touching it. In this paper we show that sometimes existence of these solutions can provide chaos.

The methods and approaches of the global bifurcation analysis of nonlinear dynamical systems are under consideration. The main idea of new approaches is a concept of complete bifurcation groups, periodic skeletons, protuberances and rare attractors (RA). The method of complete bifurcation groups (MCBG) allows obtaining new qualitative results in well-known dynamical models. MCBG is based on the method of stable and unstable periodic regimes continuation on a parameter. New results obtained by the method are discussed in this paper for the pendulum vibration absorbers with one and several degrees-of-freedom. The concept of RA and protuberances illustrated by typical bifurcation groups allows finding new important applications for absorbers of vibration. Obtained results are important for explanations of some catastrophic or good-luck events in engineering.

In this paper, we propose a numerical method to calculate the grazing bifurcation points in an impact oscillator with periodic boundaries. First, we illustrate the n-dimensional autonomous impact oscillator with the moving boundaries. The boundaries consist of the scalar function involving the periodic function. When the trajectory hits one boundary, the solution jumps to the other immediately. Next, we construct the composite Poincare map and show its derivatives. Using the derivatives, we calculate the location of the periodic point, bifurcation parameter and time that elapses before the trajectory hitting the boundary. Finally, we apply the method to a Rayleigh-type oscillator with the sinusoidal boundaries to confirm its validity.

The dynamics of a discrete-time predator-prey system is researched in the closed first quadrant R + 2 in this paper. It is shown that the system undergoes flip bifurcation and Neimark-Sacker bifurcation in the interior of the closed first quadrant R + 2 by using the center manifold theorem and bifurcation theory.

The phase response properties of the Hindmarsh-Rose model in different blocks are investigated in this paper. We find: (1) the location of the largest magnitude of the BPRCs can only be in the penultimate spike or the final spike which depends on the position of the system in the block. (2) the location of the largest magnitude of the BPRCs changes sharply when the system transits from one block to the neighbor one. Then we use bifurcation diagrams of fast subsystem with the slow variable as the bifurcation parameter to analyze the structure of the BPRCs, and draw the conclusions that the slow variable is critical in determining the properties of BPRCs, especially the location of the largest magnitude of the BPRCs, which depends on the difference between the values of slow variable for the spike maxima and for the homoclinic bifurcation.

In this paper, we study a discrete-time prey-predator system under the feedback control. By using the center manifold theorem and bifurcation theory, the conditions of the positive equilibrium undergoing flip bifurcation and Hopf bifurcation are given regarding the constant carrying capacity k as the parameter, which is more practical.

In this paper, we study the bifurcation of an epidemic model with sub-optimal immunity and saturated treatment/recovery rate. Different from classical models, sub-optimal models are more realistic to explain the microparasite infections disease such as Pertussis and Influenza A. By carrying out the bifurcation analysis of the model, we show that for certain values of the model parameters, Hopf bifurcation, Bogdonov-Takens bifurcation and its associated homoclinic bifurcation occur. By studying the bifurcation curves, we can predict the persistence or extinction of diseases.

Impact oscillators appear in various fields such as nervous system, ecological system, and mechanical system. These systems have a characteristic property that the dynamics discontinuously behaves due to jumps at hitting borders in the state space. In general, it is difficult to obtain analytical solutions in this class. Thus a numerical method is indispensable for the bifurcation analysis in the impact oscillators; however, unfortunately, it has not been established. Therefore, we proposed a numerical method for the bifurcation analyses in the impact oscillator with a fixed border and applied the proposed method to the Rayleigh-type oscillator.

Two coupled logistic maps whose parameters are forced into periodic varying are investigated. From the investigation of bifurcation in this system, nonexistence of odd periodic orbit except fixed point and existence of many coexisting attractors, which consist of periodic orbits or chaotic orbits, are observed. Basins where boundary depends on the invariant manifold of saddle points are numerically analyzed by considering second order iteration and using superposition with Newton method, although the system has discontinuity.

In this paper, the bifurcation and chaos motion in a stable Duffing system is examined by means of both of the phase plot and the Lyapunov exponents. The relationship between the Lyapunov exponents and both of the periods doubling bifurcation and symmetry breaking bifurcation are shown in detail.

In this paper n-dimensional discrete-time systems with cusp bifurcation is considered from the viewpoint of control theory. This paper presents a formula for codimension-2 bifurcation-cusp bifurcation analysis using a frequency domain approach. The analyzed bifurcation is one of the building blocks to understand other more complex singularities and to propose certain methods for controlling the bifurcation behavior in nonlinear maps in the future.

This By using some analysis techniques and the bifurcation theory, this paper studies Hopf bifurcation problems of a kind of Rosenzweig-Macarthur system with time delay. A condition is obtained for the existence of a Hopf bifurcation of periodic Solution of the following Rosenzweig-Macarthur system with time delay: equations where all parameters R , c , d , e , α , β are positive constants.

We consider a delayed predator-prey system with Holling II functional response. Firstly, the paper considers the stability and local Hopf bifurcation for a delayed prey-predator model using the basic theorem on zeros of generaltranscendental function, which was established by Cook etc‥ Secondly, special attention is paid to the global existence of periodic solutions bifurcating from Hopf bifurcations. By using a global Hopf bifurcation result due to Wu, we show that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of delay. Finally, several numerical simulations supporting the theoretical analysis are given.

In this paper, the global bifurcation for a rotor-active magnetic bearings (AMB) system with the time-varying stiffness is investigated through discussing the zeros of the Melnikov function. Some critical conditions of chaos occurring are given. According to the results, the case that the order of Melnikov function is more than or equal to 2 is more general than other case (no zero or simple zero). In other word, homoclinic tangency may occur in several resonance relations.

The paper analyzes a new hyper-chaotic Lu attractor which has rich and complex dynamical behaviors, by utilizing Lyapunov exponent spectrum, bifurcation diagram and phase portraits. And the local bifurcation is investigated by the centre manifold theorem. With the variation of parameters, the system will undergo pitchfork bifurcation and Hopf bifurcation at zero equilibrium, respectively. Finally, numerical simulations are showed to verify the theoretical analyses.

The dynamics of a discrete-time predator-prey system is investigated in the closed first quadrant . First, the existence and local stability of the fixed points in the closed first quadrant are discussed in detail. Then, by using center manifold theorem and bifurcation theory, it is proved rigorously that when the bifurcation parameters vary in the small neighborhood of the corresponding bifurcation set, flip bifurcation and Hopf bifurcation can emerge near the unique positive fixed point of the system. To suppress the undesirable chaos induced by the period-doubling bifurcation and stabilize the unstable period-1 point embedded in the chaotic attractor, periodic feedback control scheme is proposed. For this system, it is easy to verify that 1 is not the eigenvalue of the Jacobian matrix near the period-1 point, hence, the gain matrix can be obtained analytically. Numerical simulations are presented to illustrate our results with the theoretical analysis and show the effect of the control method.

New symmetric property, composed of equivariant condition, was generated near each bifurcation point in numerical simulation of Brusslator reaction diffusion model. It was found that when the equivariant condition is added to Brusselator model, numerical results can exhibit symmetry phenomena. Different bifurcation subgroups were verified to be the cause of the generation of this symmetry breaking property. This manifests the existence of equivariant nontrivial bifurcation solution in Brusselator model.Furthermore, it is shown that when bifurcation parameter is increased, different nontrivial bifurcation solution branches bifurcating from steady solution also appears correspondingly.

We investigate numerically the nonsmooth bifurcation induced by dry-friction for a braking system and find that there exists grazing-sliding bifurcation with multi-sliding parts which can lead to chaos suddenly.

DC-DC Converters have been reported as exhibiting a wide range of bifurcations and chaos under certain conditions [2]. This paper analyses the bifurcations in current controlled SEPIC and boost topologies operating in the continuous conduction mode by means of a discrete time model. Stroboscopic mapping is used to investigate possible bifurcation phenomena that periodic-1 orbits can undergo bifurcation in the SEPIC and boost converters when the reference current is taken as bifurcation parameter [8]. The characteristic multipliers locate the onset of the flip bifurcation [7]. Such a jump from stable operation to chaos has been verified by the computation of Lyapunov exponent. The periodic-1 orbit loses its stability via flip bifurcation and the resulting attractor is a periodic-2 orbit. This later bifurcates to chaos via border collision bifurcation. A computer simulation using MATLAB SIMULINK confirms the predicted bifurcations. It has been inferred that margin of system stability decreases with decrease in input voltage as well as increase in reference current and the load.

It is a well-known result that in a versal deformation of the Takens-Bogdanov bifurcation is possible to find dynamical systems that undergo saddle-node, homoclinic and Hopf bifurcations. In this document a nonlinear control system in the plane is considered, whose nominal vector field undergoes the Takens-Bogdanov bifurcation, and then the idea is to design a scalar control law such that the closed-loop system undergoes the called controllable Hopf bifurcation.

In this paper, the rigorous analysis is presented for the bifurcation phenomena of a hybrid system containing time delay. First, we introduce a simple system and its behavior of the orbit, and derive a return map explicitly. The fundamental bifurcation phenomenon is analyzed by using one and two parameter bifurcation diagrams. Even in a simple system containing delay time, we discover the two stable orbits coexist, m-piece chaotic attractor, and so on. Finally, we consider the fundamental differences between ideal and time delay system.

This paper is devoted to investigating the quasiperiodic route to chaos in the four-dimensional piecewise-smooth autonomous model of a control system with hysteresis. We show how the quasiperiodic trajectory may lie on different kinds of two dimensional tori, including single-, double-, and quintuple-turn tori. We also present a detailed analysis of a scenario for the destruction of two-dimensional torus through a homoclinic bifurcation.

In this paper, a tabu learning single neuron model is investigated. By applying the frequency domain approach and analyzing the associated characteristic equation, the existence of bifurcation parameter for this model is determined. Furthermore, we found that if the memory decay rate is used as a bifurcation parameter, Hopf bifurcation occurs in the neuron. This means that a family of periodic solutions bifurcates out from the equilibrium when the bifurcation parameter exceeds a critical value. The direction and stability of the bifurcating periodic solutions are determine by the Nyquist criterion and the graphical Hopf bifurcation theorem. Some numerical simulations for justifying the theoretical analysis are also given.

Considering a pulse-width modulated DC/DC converter as an example, this paper describes a border-collision bifurcation that can lead to the appearance of quasiperiodicity in piecewise-smooth dynamical systems. We demonstrate how a two-dimensional torus can arise from a periodic orbit through a bifurcation in which two complex-conjugated Poincare characteristic multipliers abruptly jump from the inside to the outside of the unit circle. The torus may be ergodic or resonant. In both cases, however, the diameter of the torus develops approximately linearly with the distance to the bifurcation point as opposed to the characteristic parabolic form of the well-known Neimark-Sacker bifurcation.

Global bifurcations and global phenomena are difficult to analyze by numerical means in a large power system, but they are important for recognizing complex system behavior. Using standard results from bifurcation theory and center manifold theory, the authors show the presence of homoclinic orbits in the large system, near the common boundary between the saddle node and Hopf bifurcation segments. >

In this paper, we investigate various bifurcation and related dynamics of certain periodic attractors in a three-coupled oscillator system with hard type nonlinearity. The periodic attractors exist for comparatively large /spl epsiv/(=parameter showing the degree of nonlinearity), and they disappear via saddle-node (S-N) bifurcation when /spl epsiv/ becomes small. Sometimes, there exist a heteroclinic and a homoclinic cycle near the bifurcation parameter value. In such cases, a quasi-periodic attractor appears generally after the S-N bifurcation. In particular, it presents intermittent phenomenon just after the S-N bifurcation. We clarify the existence of the heteroclinic and homoclinic cycles by drawing an unstable manifold of saddles on the Poincare section, and demonstrate the intermittent phenomenon by simulation.

In this paper we investigate the practical significance of homoclinic points in the PLL equation. We obtained parameter regions of homoclinic points in the PLL equation in our previous papers. It is well-known that if a system has homoclinic points, flows of the system present chaotic behavior at least in its transient state. We confirm this fact concretely by drawing initial condition planes for various parameter values with and without homoclinic points. As a result, it is found that for parameters without homoclinic points, the basin boundary of each attractor is smooth, however for those with homoclinic points it is more or less fractal. Further, for initial conditions chosen in fractal regions flows present chaotic transient behavior before settling in an attractor.

The goal of this work is to investigate the nonlinear oscillations of the forced, damped rotating pendulum. When the damping coefficient and the amplitude of the excitation force are zero, the system is autonomous with an explicitly known homoclinic and heteroclinic orbits. The homoclinic and the heteroclinic orbits are calculated. Melnikov functions due to the homoclinic and the heteroclinic orbits are calculated to detect the transverse homoclinic and heteroclinic orbits. Regular and chaotic motions are shown to be possible in the damped case. Numerical methods are used to obtain time history, phase portrait, Laypunov exponents, Poincare' maps and their fractal dimensions.

This paper presents the bifurcation control of longitudinal flight dynamic, via washout filter based design. Based on the mathematical model proposed by Garrard and Jordial (1977). a rich variety of bifurcation phenomena, such as saddle-node bifurcation, Hopf bifurcation, and cycle fold bifurcations are observed in the numerical simulations of longitudinal flight dynamics of F-8 aircraft. The occurrence of saddle-node bifurcation and Hopf bifurcation might result in jump behavior and pitch oscillations of flight dynamics. Bifurcation control laws are proposed to change the property of bifurcation phenomena and hence improve system stability.

In this paper, the present authors investigate a scenario for the mechanism of taming chaos via bifurcation theory, and assert that this phenomenon is caused by the slight shift of the saddle node bifurcation curves.

The authors present a simulation of the equations of a two-machine swing, each component of which has nonzero damping. The simulation confirms the nonempty transversal intersection of the stable and unstable manifolds of a saddle point and its replica. This intersection means that a homoclinic orbit exists in the dynamics. This in turn implies the existence of chaotic dynamics in the two-machine system. >

Bifurcation phenomena in cellular neural networks are investigated. In a two-cell autonomous system, Hopf-like bifurcation has been found, at which the flow around the origin, an equilibrium point of the system, changes from asymptotically stable to periodic. As the parameter grows further, by reaching another bifurcation value, the generated limit cycle disappears and the network becomes convergent again. >

We found a novel connecting orbit in the averaged Duffing-Rayleigh equation. The orbit starts from an unstable manifold of a saddle type equilibrium point and reaches to a stable manifold of a node type equilibrium. Although the connecting orbit is structurally stable by means of the conventional definition of structural stability, a one-dimensional manifold into which the connecting orbit flows is unstable. We can consider the orbit is one of global bifurcations governing the differentiability of the closed orbit.

In this paper we calculate the homoclinic bifurcation set for large damping phase-locked loop with a five segment piecewise linear phase detector via Melnikov's method. We can obtain results with small numerical errors, since we can calculate homoclinic orbits analytically by using the piecewise-linear method. Finally, we consider the extinction mechanism of the running chaotic attractor.

We study the dynamics of a weakly nonlinear single-degree-of-freedom system subjected to parametric and external excitations. An asymptotic method is used to analyze the bifurcations of quasi-periodic solutions near strong resonances. By applying the Melnikov technique to the reduced system, it is shown that there exists transversal homoclinic orbits resulting in chaotic dynamics. By introducing a nonlinear resonant parametric perturbation in the system we analyze how chaos can be suppressed.

Oscillatory regimes and bifurcation transitions in system with phase control are investigated. The bifurcation diagrams and phase portraits of the attractors are presented with various scenarios of evolution of the oscillatory motions versus the system parameters. New features of the system's chaotic behaviour are observed.

We investigate Shilnikov homoclinic bifurcation from a new type of phase-locked loop (PLL) which uses second-order loop filter with no modulation carrier input. This system can be represented as a 3rd-order autonomous system with piecewise linear characteristics. We have found many Shilnikov orbits, and draw a bifurcation diagram in gain K/sub 0/ versus detuning /spl delta/ parameter plane.

We investigate bifurcation and chaos observed in coupled BVP neurons with external impulsive forces. Although the single neuron without the external force has only one equilibrium point, combining these n (n/spl ges/3) neurons unidirectionally in a ring, n-phase periodic solutions are generated. Applying the impulsive forces, successive period-doubling bifurcations of the n-phase solution occur and chaotic states, namely n-phase chaos, appear. When n=2 and 3, we find the parameter regions in which the switching phenomena of burst firing are observed. Moreover, the mechanisms of the switching phenomena are clarified by numerical bifurcation analysis.

The bifurcation structure of fractional harmonic entrainments in the forced Rayleigh oscillator is studied. In each fractional harmonic entrainment, complex bifurcation occurs when the saddle node bifurcation curve, which forms fractional harmonic entrainments, approaches the Neimark-Sacker bifurcation curve which follows the fundamental harmonic entrainment. Two types of fractional harmonic entrainments are observed. We call the entrainments Type I and Type II. In Type I, the attractor which is asymmetric with respect to the origin is borne by a saddle node bifurcation. Subsequently, period doubling bifurcation occurs and chaos is generated. In Type II, the attractor which is symmetric with respect to the origin is borne by a saddle node bifurcation. Symmetry breaking transition occurs in the tube of the saddle node bifurcation curve. Subsequently, period doubling bifurcation occurs and chaos is generated.