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In this paper, a commensal model with additive Allee effect on the first species and nonlinear growth rate on the second species is proposed. The classification, type and stability of all possible equilibria of the system are investigated. We also prove the presence of saddle-node bifurcation and transcritical bifurcation by Sotomayor’s theorem. At the end of this study, an example and its simulations are given to illustrate our main results.

In this paper, an attempt has been made to understand the role of predator’s interference and additional food on the dynamics of a diffusive population model. We have studied a predator–prey interaction system with mutually interfering predator by considering additional food and Crowley–Martin functional response (CMFR) for both the reaction–diffusion model and associated spatially homogeneous system. The local stability analysis ensures that as the quantity of alternative food decreases, predator-free equilibrium stabilizes. Moreover, we have also obtained a condition providing a threshold value of additional food for the global asymptotic stability of coexisting steady state. The nonspatial model system changes stability via transcritical bifurcation and switches its stability through Hopf-bifurcation with respect to certain ranges of parameter determining the quantity of additional food. Conditions obtained for local asymptotic stability of interior equilibrium solution of temporal system determines the local asymptotic stability of associated diffusive model. The global stability of positive equilibrium solution of diffusive model system has been established by constructing a suitable Lyapunov function and using Green’s first identity. Using Harnack inequality and maximum modulus principle, we have established the nonexistence of nonconstant positive equilibrium solution of the diffusive model system. A chain of patterns on increasing the strength of additional food as spots[Formula: see text][Formula: see text][Formula: see text]stripes[Formula: see text][Formula: see text][Formula: see text]spots has been obtained. Various kind of spatial-patterns have also been demonstrated via numerical simulations and the roles of predator interference and additional food are established.

The increase of nutrients in lakes typically stimulates the growth of algae in this environment. Therefore, it is important to understand the connection between nutrient concentration and algal biomass to manage the water pollution caused by excessive plant nutrients. It is worth observing that phosphorus and nitrogen are often considered as the principal limiting nutrients for aquatic algal production due to their short supply compared to cellular growth requirements. In freshwaters, phosphorus is the least abundant among the nutrients needed in large quantity by photosynthetic organisms, hence this is the primary nutrient that limits their growth. The purpose of this work is to compare the effects of nitrogen and phosphorus on the growth of algae in lakes. By using a sensitivity analysis technique, we found that the sources of phosphorus provide a greater risk for bloom of algae than that of nitrogen. Therefore, to reduce the occurrence of algal bloom more attention should be paid for the control of phosphorus input into the lake but the inflow of nitrogen cannot be ignored. The existence of a transcritical bifurcation is discussed and its direction is investigated by applying the projection method technique. Further, to make the system more realistic, time delay involved in the conversion of detritus into nutrients is considered. We show that for increasing values of time delay, the system undergoes an Andronov–Hopf-bifurcation. Some simulations are presented to verify the analytical findings. The results of our study can be helpful for the policy makers to mitigate algal blooms from lakes.

We propose a novel impulsive chemostat model with the substrate concentration as the basis for the implementation of control strategies, and then investigate the model’s global dynamics. The exact domains of the impulsive and phase sets are discussed in the light of phase portraits of the model, and then we define the Poincaré map and study its complex properties. Furthermore, the existence and stability of the microorganism eradication periodic solution are addressed, and the analysis of a transcritical bifurcation reveals that an order-1 periodic solution is generated. We also provide the conditions for the global stability of an order-1 periodic solution and show the existence of order-[Formula: see text] [Formula: see text] periodic solutions. Moreover, the PRCC results and bifurcation analyses not only substantiate our results, but also indicate that the proposed system exists with complex dynamics. Finally, biological implications related to the theoretical results are discussed.

An application of the new method of complete bifurcation groups (MCBG) in parametrically excited pendulum systems with several equilibrium positions and with the periodically vibrating point of suspension in both directions is introduced. Construction of complete bifurcation groups is based on the method of stable and unstable periodic regimes continuation on a parameter. Global bifurcation analysis of the parametrically excited pendulum systems with several equilibrium positions allows finding new bifurcation groups with rare attractors and chaotic regimes.

A general method for formulating first-order time-delayed chaotic systems with simple linear time-delayed term is proposed. The formulated systems are realized with electronic circuit experiments. In order to determine the unknown coefficients in a general delayed differential equations for having chaotic solutions, we follow the route of period-doubling bifurcation to chaos. Firstly, the conditions for a time-delayed system having a stable periodic solution, generating from a destablized steady state, is analyzed with Hopf bifurcation theory. Then the delay time parameter is changed according to the bifurcation direction to search the chaotic state, which is identified by the Lyapunov exponents spectra. The theoretical analysis is well confirmed by numerical simulations and circuit experiments.

In this paper, a control method developed earlier is employed to consider controlling bifurcations in a class of Lorenz-like systems. Particular attention is focused on Hopf bifurcation control via linear and nonlinear stability analyses. The Lorenz system, Chen system and Lü system are studied in detail. Simple feedback controls are designed for controlling the stability of equilibrium solutions, limit cycles and chaotic motions. All formulas are derived in general forms including the system parameters. Computer simulation results are presented to confirm the analytical predictions.

In this paper, a tri-neuron network with bidirectionally delay and self-feedback is considered. We derive some sufficient conditions dependent or independent of delays for the local stability and instability of this model. Regarding the self-connection delay as the parameter, the Hopf bifurcation analysis is carried out. The direction and stability of the Hopf bifurcation are worked out by applying the normal form theory and the center manifold theory. An example is given and numerical simulations are presented to illustrate the obtained results.

In this work, we study the dynamics of a three-dimensional, continuous, piecewise smooth map. Much of the nontrivial dynamics of this map occur when its fixed point or periodic orbit hits the switching manifold resulting in the so-called border collision bifurcation. We study the local and global bifurcation phenomena resulting from such borderline collisions. The conditions for the occurrence of nonsmooth period-doubling, saddle-node, and Neimark–Sacker bifurcations are derived. We show that dangerous border collision bifurcation can also occur in this map. Global bifurcations arise in connection with the occurrence of nonsmooth Neimark–Sacker bifurcation by which a spiral attractor turns into a saddle focus. The global dynamics are systematically explored through the computation of resonance tongues and numerical continuation of mode-locked invariant circles. We demonstrate the transition to chaos through the breakdown of mode-locked torus by degenerate period-doubling bifurcation, homoclinic tangency, etc. We show that in this map a mode-locked torus can be transformed into a quasiperiodic torus if there is no global bifurcation.

An effective and simple method, called perturbation scheme (PS), is proposed to study fold-Hopf bifurcation for delayed neural systems qualitatively and quantitatively when time delay and connection weight are considered as two bifurcation parameters. As an illustration, the proposed method is employed to investigate a delayed bidirectional associative memory (BAM) neural network. Dynamics arising from fold-Hopf bifurcation are classified qualitatively and expressed approximately in a closed form for periodic solution. We also give analytical results on the existence, stability and bifurcation of nontrivial equilibria of the delayed system. Our investigations reveal that secondary Hopf bifurcation and pitchfork bifurcation of limit cycle may emanate from the pitchfork-Hopf point. In addition, the secondary Hopf bifurcation can lead to multistability between equilibrium points and periodic solution in some region of parameter space. The validity of analytical predictions is shown by their consistency with the results from the center manifold reduction (CMR) with normal form and numerical simulation. As an analytical tool, the advantage of the PS also lies in its simplicity and ease of implementation.

We study the stability and Hopf bifurcation of a neutral functional differential equation (NFDE) which is transformed from an amplitude equation with multiple-delay feedback control. By analyzing the distribution of the eigenvalues, the stability and existence of Hopf bifurcation are obtained. Furthermore, the direction and stability of the Hopf bifurcation are determined by using the center manifold and normal form theories for NFDEs. Finally, we carry out some numerical simulations to illustrate the results.

The dynamics of a linearly coupled Mackey–Glass system with delay are investigated. Based on the distribution of eigenvalues, we prove that a sequence of Hopf bifurcation occurs at the positive equilibrium as the delay increases and obtain the bifurcation set in the parameter plane. The explicit algorithm for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are derived, using the theories of normal form and center manifold. The global existence of periodic solutions is established using a global Hopf bifurcation result due to Wu [1998] and a Bendixson's criterion for higher dimensional ordinary differential equations due to [Li & Muldowney, 1993].

The local bifurcation boundary and steady-state security boundary of parameter-dependent electric power system models are computed and studied. A computational procedure for numerically constructing the local bifurcation boundary and the steady-state security boundary is proposed. Then the proposed computational procedure is applied to large power systems to compute the local bifurcation boundary and steady-state security boundary. Numerical studies reveal the characteristics and the convexity properties of these boundaries. The impact of the physical limitation of the generator reactive capability on the local bifurcation boundary and the steady-state security boundary are also investigated.

The complex bifurcation structure in the parameter space of the general piecewise-linear scalar map with a single discontinuity — nowadays known as nested period adding structure — was completely studied analytically by N. N. Leonov already 50 years ago. He used an elegant and very efficient recursive technique, which allows the analytical calculation of the border-collision bifurcation curves, causing the nested period adding structure to occur. In this work, we have demonstrated that the application of Leonov's technique is not resticted to that particular bifurcation structure. On the contrary, the presented map replacement approach, which is an extension of Leonov's technique, allows the analytical calculation of border-collision bifurcation curves for periodic orbits with high periods and complex symbolic sequences using appropriate composite maps and the bifurcation curves for periodic orbits with much lower periods.

The goal of this paper is to analyze the character of the first Hopf bifurcation (subcritical versus supercritical) that appears in a one-dimensional reaction–diffusion equation with nonlinear boundary conditions of logistic type with delay. We showed in the previous work [Arrieta et al., 2010] that if the delay is small, the unique non-negative equilibrium solution is asymptotically stable. We also showed that, as the delay increases and crosses certain critical value, this equilibrium becomes unstable and undergoes a Hopf bifurcation. This bifurcation is the first one of a cascade occurring as the delay goes to infinity. The structure of this cascade will depend on the parameters appearing in the equation. In this paper, we show that the first bifurcation that occurs is supercritical, that is, when the parameter is bigger than the delay bifurcation value, stable periodic orbits branch off from the constant equilibrium.

In this paper, we have explored the bifurcation behavior and chaos of a two-coupled pendulum system with a coupling energy of the form: κ(θ1 - θ2)2. Using fixed point analysis, we have determined the bifurcation map, which provides a pictorial view of the number and stability properties of the fixed points with respect to the coupling parameter. The bifurcation map shows that the two-coupled pendulum system can exhibit three forms of bifurcation: pitchfork; saddle-node; and a new bifurcation in which a fixed point of the mixed type changes to a center, while a second fixed point of mixed type is born. In order to analyze the chaotic dynamics of the two-coupled pendulum system, we have introduced an equivalent model to the system. This model enables us to investigate the system dynamics in terms of the motion of a particle interacting with a potential landscape. Through analyzing the geometry of the landscape, we are able to determine the dynamical transition points from regular to locally chaotic, and then to globally chaotic behavior. The validity of our analytical prediction of E0 = 4ε for the onset of chaos, and ET = π2ε/2 for the global transition to chaos, has been duly verified through numerical simulations.

Some observations are made on a class of one-predator, two-prey models via numerical analysis. The simulations are performed with the aid of an adaptive grid method for constructing bifurcation diagrams and cell-to-cell mapping for global analysis. A two-dimensional bifurcation diagram is constructed to show that regions of coexistence of all three species, which imply the balance of competitive and predatory forces, are surrounded by regions of extinction of one or two species. Two or three coexisting attractors which may have a chaotic member are found in some regions of the bifurcation diagram. Their separatrices are computed to show the domains of attraction. The bifurcation diagram also contains codimension-two bifurcation points including Bogdanov–Takens, Gavrilov–Guckenheimer, and Bautin bifurcations. The dynamics in the vicinity of these codimension-two bifurcation points are discussed. Some global bifurcations including homoclinic and heteroclinic bifurcations are investigated. They can account for the disappearance of chaotic attractors and limit cycles. Bifurcations of limit cycles such as transcritical and saddle-node bifurcations are also studied in this work. Finally, some relevant calculations of Lyapunov exponents and power spectra are included to support the chaotic properties.

We show that in an artificial dynamic neural network that depends on a real parameter μ, steady states do not exist for μ ≤ -2, and positive and negative steady states exist for μ > -2. We hope that such a bifurcation phenomenon in our network model may explain some of the real observations in nature.

We consider parameter-dependent, continuous-time dynamical systems under discretizations. It is shown that fold-Hopf singularities are O(hp)-shifted and turned into fold-Neimark–Sacker points by one-step methods of order p. Then we analyze the effect of discretizations methods on the local bifurcation diagram near Bogdanov–Takens and fold-Hopf singularities. In particular, we prove that the discretized codimension one curves intersect at the singularities in a generic manner. The results are illustrated by a numerical example.

The aim of this work is twofold — on the one hand, to perform a theoretical analysis of the global behavior organized by a T-point–Hopf in ℤ2-symmetric systems; on the other hand, to apply the obtained results for a numerical study of Chua's equation, where for the first time this bifurcation is considered.In a parameterized three-dimensional system of autonomous differential equations, a T-point is a point of the parameter space where a special kind of codimension-two heteroclinic cycle occurs. A more degenerate scenario appears when one of the equilibria involved in such a cycle undergoes a Hopf bifurcation. This degeneration, which corresponds to a codimension-three bifurcation, is called T-point–Hopf and has been recently studied for a generic system. However, the presence of ℤ2-symmetry may lead to the existence of a double T-point–Hopf heteroclinic cycle, which is responsible for the appearance of interesting global behavior that we will study in this paper.The theoretical models proposed for two different situations are based on the construction of a Poincaré map. The existence of certain kinds of homoclinic and heteroclinic connections between equilibria and/or periodic orbits is proved and their organization close to the T-point–Hopf bifurcation is described. The numerical phenomena found in Chua's equation strongly agree with the results deduced from the models.

A reaction–diffusion Gierer–Meinhardt model of morphogenesis subject to Dirichlet fixed boundary condition in the one-dimensional spatial domain is considered. We perform a detailed Hopf bifurcation analysis and steady state bifurcation analysis to the system. Our results suggest the existence of spatially nonhomogenous periodic orbits and nonconstant positive steady state solutions, which imply the possibility of complex spatiotemporal patterns of the system. Numerical simulations are carried out to support our theoretical analysis.

Given an m-parameterized family of n-dimensional vector fields, such that: (i) for some value of the parameters, the family has an equilibrium point, (ii) its linearization has a double zero eigenvalue and no other eigenvalue on the imaginary axis, sufficient conditions on the vector field are given such that the dynamics on the two-dimensional center manifold is locally topologically equivalent to the versal deformation of the planar Takens–Bogdanov bifurcation.

In this paper, bifurcation of the generalized Swift–Hohenberg equation is considered. We first study the bifurcation of the generalized Swift–Hohenberg equation in one spatial dimension with three kinds of boundary conditions. With the help of Liapunov–Schmidt reduction, the original equation is transformed to the reduced system, and then the bifurcation analysis is carried out. Secondly, bifurcation of the generalized Swift–Hohenberg equation in two spatial dimensions with periodic boundary conditions is also considered, using the perturbation method, asymptotic expressions of the nontrivial solutions bifurcated from the trivial solution are obtained. Moreover, the stability of the bifurcated solutions is discussed.

This paper addresses bifurcation properties of equilibria in lumped electrical circuits. The goal is to tackle these properties in circuit-theoretic terms, characterizing the bifurcation conditions in terms of the underlying network digraph and the electrical features of the circuit devices. The attention is mainly focused on so-called singular bifurcations, resulting from the semistate (differential-algebraic) nature of circuit models, but the scope of our approach seems to extend to other types of bifurcations. The bifurcation analysis combines different tools coming from graph theory (such as proper trees in circuit digraphs, Maxwell's determinantal expansions or the colored branch theorem) with several results from linear algebra (matrix pencils, the Cauchy–Binet formula, Schur complements). Several examples illustrate the results.

In the present paper, we investigate the impact of fear in a tri-trophic food chain model. We propose a three-species food chain model, where the growth rate of middle predator is reduced due to the cost of fear of top predator, and the growth rate of prey is suppressed due to the cost of fear of middle predator. Mathematical properties such as equilibrium analysis, stability analysis, bifurcation analysis and persistence have been investigated. We also describe the global stability analysis of the equilibrium points. Our numerical simulations reveal that cost of fear in basal prey may exhibit bistability by producing unstable limit cycles, however, fear in middle predator can replace unstable limit cycles by a stable limit cycle or a stable interior equilibrium. We observe that fear can stabilize the system from chaos to stable focus through the period-halving phenomenon. We conclude that chaotic dynamics can be controlled by the fear factors. We apply basic tools of nonlinear dynamics such as Poincaré section and maximum Lyapunov exponent to identify the chaotic behavior of the system.

In this paper, bifurcations of heterodimensional cycle with one nonhyperbolic equilibrium and one saddle-focus in three-dimensional vector fields are investigated. We study the interaction of a transcritical bifurcation with a codimension-0/codimension-2 heteroclinic cycle. Based on the construction of a Poincaré return map, we obtain the expressions of parametric curves of homoclinic and heteroclinic connections around the heterodimensional cycle as well as periodic orbits. Furthermore, the configurations of the parametric curves corresponding to different bifurcations are illustrated.

This paper is concerned with the asymptotic behavior of the solutions u(x,t) of the Swift–Hohenberg equation with quintic nonlinearity on a one-dimensional domain (0, L). With α and the length L of the domain regarded as bifurcation parameters, branches of nontrivial solutions bifurcating from the trivial solution at certain points are shown. Local behavior of these branches are also studied. Global bounds for the solutions u(x,t) are established and then the global attractor is investigated. Finally, with the help of a center manifold analysis, two types of structures in the bifurcation diagrams are presented when the bifurcation points are closer, and their stabilities are analyzed.

Parametrized contact-equivalence is a successful theory for the understanding and classification of the qualitative local behavior of bifurcation diagrams and their perturbations. Path formulation is an alternative point of view making explicit the singular behavior due to the core of the bifurcation germ (when the parameters vanish) from the effects of the way parameters enter. We show how to use path formulation to classify and structure efficiently multiparameter bifurcation problems in corank 2 problems. In particular, the nondegenerate umbilics singularities are the generic cores in four situations: the general or gradient problems, with or without ℤ2 symmetry where ℤ2 acts on the second component of ℝ2 via κ(x,y) = (x,-y). The universal unfolding of the umbilic singularities have an interesting "Russian doll" type of structure of miniversal unfoldings in all those categories. With the path formulation approach we can handle one, or more, parameter situations using the same framework. We can even consider some special parameter structure (for instance, some internal hierarchy of parameters). We classify the generic bifurcations with 1, 2 or 3 parameters that occur in those cases. Some results are known with one bifurcation parameter, but the others are new. We discuss some applications to the bifurcation of a loaded cylindrical panel. This problem has many natural parameters that provide concrete examples of our generic diagrams around the first interaction of the buckling modes.

The objective of this paper is to study the dynamical properties of a Holling-type II predator–prey system with constant rate harvesting. It is shown that the model has at most three equilibria in the first quadrant and can exhibit numerous kinds of bifurcation phenomena, including the saddle-node bifurcation, the degenerate Bogdanov–Takens bifurcation of codimension 3, the supercritical and subcritical Hopf bifurcation, the generalized Hopf bifurcation. These results reveal far richer dynamics than that of the model with no harvesting.

In this paper, we use the method of displacement functions to study the existence, stability and bifurcation of periodic solutions of scalar periodic impulsive differential equations. We obtain some new and interesting results on saddle-node bifurcation and double-period bifurcation of periodic solution.

A logistic equation with piecewise constant arguments is investigated. Firstly, the linear stability of the model is studied. It is found that there exists a Hopf bifurcation when the parameter passes a critical value. Then the explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solution is derived by using the normal form method and center manifold theorem. Finally, computer simulations are performed to illustrate the analytical results found.

Starting from previous analytical results assuring the existence of a saddle-node bifurcation curve of periodic orbits for continuous piecewise linear systems, numerical continuation is done to get some primary bifurcation curves for the piecewise linear Chua's oscillator in certain dimensionless parameter plane. The primary period doubling, homoclinic and saddle-node of periodic orbits' bifurcation curves are computed. A Belyakov point is detected in organizing the connection of these curves. In the parametric region between period-doubling, focus-center-limit cycle and homoclinic bifurcation curves, chaotic attractors coexist with stable nontrivial equilibria. The primary saddle-node bifurcation curve plays a leading role in this coexistence phenomenon.

As we know, Hopf bifurcation is an important part of bifurcation theory of dynamical systems. Almost all known works are concerned with the bifurcation and number of limit cycles near a nondegenerate focus or center. In the present paper, we study a general near-Hamiltonian system on the plane whose unperturbed system has a nilpotent center. We obtain an expansion for the first order Melnikov function near the center together with a computing method for the first coefficients. Using these coefficients, we obtain a new bifurcation theorem concerning the limit cycle bifurcation near the nilpotent center. An interesting application example & a cubic system having five limit cycles & is also presented.

A generic stationary instability that arises in quasi-reversible systems is studied. It is characterized by the confluence of three eigenvalues at the origin of complex plane with only one eigenfunction. We characterize the dynamics through the normal form that exhibits in particular, Shilnikov chaos, for which we give an analytical prediction. We construct a simple mechanical system, Shilnikov particle, which exhibits this quasi-reversal instability and displays its chaotic behavior.

The bifurcations of generic heteroclinic loop with one nonhyperbolic equilibrium p1and one hyperbolic saddle p2are investigated, where p1is assumed to undergo transcritical bifurcation. Firstly, we discuss bifurcations of heteroclinic loop when transcritical bifurcation does not happen, the persistence of heteroclinic loop, the existence of homoclinic loop connecting p1(resp. p2) and the coexistence of one homoclinic loop and one periodic orbit are established. Secondly, we analyze bifurcations of heteroclinic loop accompanied by transcritical bifurcation, namely, nonhyperbolic equilibrium p1splits into two hyperbolic saddles [Formula: see text] and [Formula: see text], a heteroclinic loop connecting [Formula: see text] and p2, homoclinic loop with [Formula: see text] (resp. p2) and heteroclinic orbit joining [Formula: see text] and [Formula: see text] (resp. [Formula: see text] and p2; p2and [Formula: see text]) are found. The results achieved here can be extended to higher dimensional systems.

In this work we study some properties associated with the border-collision bifurcations in a two-dimensional piecewise-linear map in canonical form, related to the case where a fixed point of one of the linear maps has complex eigenvalues and undergoes a center bifurcation when its eigenvalues pass through the unit circle. This problem is faced in several applied piecewise-smooth models, such as switching electrical circuits, impacting mechanical systems, business cycle models in economics, etc. We prove the existence of an invariant region in the phase space for parameter values related to the center bifurcation and explain the origin of a closed invariant attracting curve after the bifurcation. This problem is related also to particular border-collision bifurcations leading to such curves which may coexist with other attractors. We show how periodicity regions in the parameter space differ from Arnold tongues occurring in smooth models in case of the Neimark–Sacker bifurcation, how so-called dangerous border-collision bifurcations may occur, as well as multistability. We give also an example of a subcritical center bifurcation which may be considered as a piecewise-linear analogue of the subcritical Neimark–Sacker bifurcation.

The paper presents a detailed analysis on the dynamics of a two-neuron network with time-delayed connections between the neurons and time-delayed feedback from each neuron to itself. On the basis of characteristic roots method and Hopf bifurcation theorems for functional differential equations, we investigate the existence of local Hopf bifurcation. In addition, the direction of Hopf bifurcation and stability of the periodic solutions bifurcating from the trivial equilibrium are determined based on the normal form theory and center manifold theorem. Moreover, employing the global Hopf bifurcation theory due to [Wu, 1998], we study the global existence of periodic solutions. It is shown that the local Hopf bifurcation indicates the global Hopf bifurcation after the second group critical value of the delay.

This paper provides further results on bifurcation analysis of indirect field oriented control of induction motors. Previous results presented on this subject [Bazanella & Reginatto, 2000, 2001; Gordillo et al., 2002] are summarized and extended by means of a codimension-two bifurcation analysis. It is shown that codimension-two bifurcation phenomena, such as a Bogdanov–Takens and zero–Hopf bifurcations, occur in IFOC as a result of parameter mismatch and certain setting of the proportional-integral speed controller. Conditions for the existence of such bifurcations are derived analytically, as long as possible, and bifurcation diagrams are presented with the help of simulation and numerical bifurcation analysis.

Time-delayed feedback has been introduced as a powerful tool to control unstable periodic orbits or control unstable steady states. In the present paper, regarding the delay as a parameter, we investigate the effect of delay on the dynamics of Lü system with delayed feedback. After the effect of delay on the steady states is analyzed, Hopf bifurcation is studied, where the direction, stability and other properties of the bifurcating periodic solutions are determined by using the normal form theory and the center manifold theorem. Finally, we provide several numerical simulations, which indicate that when the delay passes through certain critical values, chaotic oscillation is converted into a stable steady state or a stable periodic orbit.

There is a kind of correspondence between stochastic resonance and bifurcation of the moment equations of a noisy nonlinear system with the same noise intensity as the resonance independent variable and the bifurcation parameter, respectively. In this paper, this correspondence is examined and revealed in the noisy one-dimensional bistable system and the noisy two-dimensional Duffing oscillator. The bifurcation of the moment equations of each noisy system is the bifurcation with double-branch of fixed-point shift. Besides classical stochastic resonance, a kind of complex stochastic resonance corresponds to the bifurcation of moment equations. This complex stochastic resonance is induced by the stochastic transitions of system motion among the three fixed point attractors on both sides of the bifurcation point of the original system, which is predicted semi-analytically. Finally, due to this correspondence being examined, the mechanism of stochastic resonance can be provided through analyzing the change of the energy transfer induced by the bifurcation of the moment equations.

The coalescence of a Hopf bifurcation with a codimension-two cusp bifurcation of equilibrium points yields a codimension-three bifurcation with rich dynamic behavior. This paper presents a comprehensive study of this cusp-Hopf bifurcation on the three-dimensional center manifold. It is based on truncated normal form equations, which have a phase-shift symmetry yielding a further reduction to a planar system. Bifurcation varieties and phase portraits are presented. The phenomena include all four cases that occur in the codimension-two fold–Hopf bifurcation, in addition to bistability involving equilibria, limit cycles or invariant tori, and a fold–heteroclinic bifurcation that leads to bursting oscillations. Uniqueness of the torus family is established locally. Numerical simulations confirm the prediction from the bifurcation analysis of bursting oscillations that are similar in appearance to those that occur in the electrical behavior of neurons and other physical systems.

In this paper, a novel method named pseudo-oscillator analysis is developed for the local dynamics near a Hopf bifurcation of scalar nonlinear dynamical systems with time delays. For this purpose, a pseudo-oscillator that is slightly perturbed from an undamped oscillator is firstly constructed, its fundamental frequency is the same as the frequency at the bifurcation point, and the disturbance is associated with the original system. Next, the pseudo-power function, defined as the power function of the pseudo-oscillator, is estimated along a harmonic function. Then we conclude that the local dynamics near the Hopf bifurcation can be justified from the variation of the averaged pseudo-power function. The new method features a clear physical intuition and easy computation, and it yields very accurate prediction for the periodic solution resulted from the Hopf bifurcation, as shown in three illustrative examples.

This paper focuses on resonance phenomena that occur in a vicinity of a linear degeneracy corresponding to a triple-zero eigenvalue of an equilibrium point in an autonomous tridimensional system. Namely, by means of blow-up techniques that relate the triple-zero bifurcation to the Kuramoto–Sivashinsky system, we characterize the resonances that appear near the triple-zero bifurcation. Using numerical tools, the results are applied to the Rössler equation, showing a number of interesting bifurcation behaviors associated to these resonance phenomena. In particular, the merging of the periodic orbits appeared in resonances, the existence of two types of Takens–Bogdanov bifurcations of periodic orbits and the presence of Feigenbaum cascades of these bifurcations, joined by invariant tori curves, are pointed out.

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.

In this paper, a possible degeneration of the focus-center-limit cycle bifurcation for piecewise smooth continuous systems is analyzed. The case of continuous piecewise linear systems with two zones is considered, and the coexistence of two limit cycles for certain values of parameters is justified. Finally, the Chua's circuit is shown to exhibit the analyzed bifurcation. The obtained bifurcation set in the parameter plane is similar to the degenerate Hopf bifurcation for differentiable systems.

The control of the saddle-node, transcritical and pitchfork bifurcations are analyzed in nonlinear control systems with one zero eigenvalue. It is shown that, provided some conditions on the vector fields are satisfied, it is possible to design a control law such that the bifurcation properties can be modified in some desirable way. To simplify the analysis to dimension one, the center manifold theory is used.

Gluing bifurcation in a modified Chua's oscillator is reported. Keeping other parameters fixed when a control parameter is varied in the modified oscillator model, two symmetric homoclinic orbits to saddle focus at origin, which are mirror images of each other, are glued together for a particular value of the control parameter. In experiments, two asymmetric limit cycles are homoclinic to the saddle focus origin for different values of the control parameter. However, imperfect gluing bifurcation has been observed, in experiments, when one stable and unstable limit cycles merge to the saddle focus origin via saddle-node bifurcation.

A synchronized neural network model with delays is considered. The bifurcations arising from the zero root of the corresponding characteristic equation have been studied by employing the center manifold theorem, normal form method and bifurcation theory. It is shown that the system may exhibit transcritical/pitchfork bifurcation, or Bogdanov–Takens bifurcation. Some numerical simulation examples are given to justify the theoretical results.

In this paper a bifurcation analysis of a piecewise-affine discrete-time dynamical system is carried out. Such a system derives from a well-known map which has good features from its circuit implementation point of view and good statistical properties in the generation of pseudo-random sequences. The considered map is a generalization of it and the bifurcation parameters take into account some common circuit implementation nonidealities or mismatches. It will be shown that several different dynamic situations may arise, which will be completely characterized as a function of three parameters. In particular, it will be shown that chaotic intervals may coexist, may be cyclical, and may undergo several global bifurcations. All the global bifurcation curves and surfaces will be obtained either analytically or numerically by studying the critical points of the map (i.e. extremum points and discontinuity points) and their iterates. In view of a robust design of the map, this bifurcation analysis should come before a statistical analysis, to find a set of parameters ensuring both robust chaotic dynamics and robust statistical properties.

In this paper, a novel method of energy analysis is developed for dynamical systems with time delays that are slightly perturbed from undamped SDOF/MDOF vibration systems. Being served frequently as the mathematical models in many applications, such systems undergo Hopf bifurcation including the classic "Hopf bifurcation" for SDOF systems and "multiple Hopf bifurcation" for MDOF systems, under certain conditions. An interesting observation of this paper is that the local dynamics near a Hopf bifurcation, including the stability of the trivial equilibrium and the bifurcating periodic solutions, of such systems, can be justified simply by the change of the total energy function. The key idea is that for the systems of concern, the total power (the total derivative of the energy function) can be estimated along an approximated solution with harmonic entries, the main part of the solution near the Hopf bifurcation. It shows that the present method works effectively for stability prediction of the trivial equilibrium and the bifurcating periodic solutions, and that it provides a high accurate estimation of the amplitudes of the bifurcating periodic solutions. Compared with the current methods such as the center manifold reduction which involves a great deal of symbolic computation, the energy analysis features a clear physical intuition and easy computation. Two illustrative examples are given to demonstrate the effectiveness of the present method.