Showing papers on "Bifurcation diagram published in 2007"
TL;DR: In this paper, the dynamics of a discrete-time predator-prey system is investigated in the closed first quadrant R + 2, and it is shown that the system undergoes flip bifurcation and Hopf bifurbation in the interior of R+2 by using center manifold theorem and bifurlcation theory.
Abstract: The dynamics of a discrete-time predator–prey system is investigated in the closed first quadrant R + 2 . It is shown that the system undergoes flip bifurcation and Hopf bifurcation in the interior of R + 2 by using center manifold theorem and bifurcation theory. Numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, such as the period-5, 6, 9, 10, 14, 18, 20, 25 orbits, cascade of period-doubling bifurcation in period-2, 4, 8, quasi-periodic orbits and the chaotic sets. These results reveal far richer dynamics of the discrete model compared with the continuous model. The Lyapunov exponents are numerically computed to confirm further the complexity of the dynamical behaviors.
300 citations
TL;DR: In this article, a hyperchaotic Lorenz system is constructed via state feedback control using the Lyapunov exponents, Poincare section and bifurcation diagram.
207 citations
TL;DR: A bidirectional associative memory NN with four neurons and multiple delays is considered and analysis of its linear stability and Hopf bifurcation is performed by applying the normal form theory and the center manifold theorem.
Abstract: Various local periodic solutions may represent different classes of storage patterns or memory patterns, and arise from the different equilibrium points of neural networks (NNs) by applying Hopf bifurcation technique. In this paper, a bidirectional associative memory NN with four neurons and multiple delays is considered. By applying the normal form theory and the center manifold theorem, analysis of its linear stability and Hopf bifurcation is performed. An algorithm is worked out for determining the direction and stability of the bifurcated periodic solutions. Numerical simulation results supporting the theoretical analysis are also given
178 citations
TL;DR: New insights are provided into the classical problem of a one-dimensional superconducting wire exposed to an applied electric current using the time-dependent Ginzburg-Landau model and the evident collision of real eigenvalues of the associated PT-symmetric linearization leads to the emergence of complex elements of the spectrum.
Abstract: We provide here new insights into the classical problem of a one-dimensional superconducting wire exposed to an applied electric current using the time-dependent Ginzburg-Landau model. The most striking feature of this system is the well-known appearance of oscillatory solutions exhibiting phase slip centers (PSC's) where the order parameter vanishes. Retaining temperature and applied current as parameters, we present a simple yet definitive explanation of the mechanism within this nonlinear model that leads to the PSC phenomenon and we establish where in parameter space these oscillatory solutions can be found. One of the most interesting features of the analysis is the evident collision of real eigenvalues of the associated PT-symmetric linearization, leading as it does to the emergence of complex elements of the spectrum.
152 citations
TL;DR: In this article, the authors investigated the global dynamics of an SIRS model with a nonlinear incidence rate and established a threshold for a disease to be extinct or endemic, analyzed the existence and asymptotic stability of equilibria, and verified the existence of bistable states, i.e., a stable disease free equilibrium and a stable endemic equilibrium or a stable limit cycle.
Abstract: The global dynamics of an SIRS model with a nonlinear incidence rate is investigated. We establish a threshold for a disease to be extinct or endemic, analyze the existence and asymptotic stability of equilibria, and verify the existence of bistable states, i.e., a stable disease free equilibrium and a stable endemic equilibrium or a stable limit cycle. In particular, we find that the model admits stability switches as a parameter changes. We also investigate the backward bifurcation, the Hopf bifurcation and Bogdanov–Takens bifurcation and obtain the Hopf bifurcation criteria and Bogdanov–Takens bifurcation curves, which are important for making strategies for controlling a disease.
132 citations
TL;DR: In this paper, the influence of random fluctuations in environmental parameters (e.g., nutrient input and rainfall) on the behavior of two simple bistable, ecological models with a single dynamical variable was studied.
125 citations
TL;DR: In this paper, a new hyperchaotic system by introducing an additional state feedback into a three-dimensional quadratic chaotic system is presented. But the system only has one equilibrium, but it can evolve into periodic, quasi-periodic, chaotic and hyperchaotics.
115 citations
TL;DR: In this paper, a new hyper-chaotic system was obtained by adding a nonlinear quadratic controller to the second equation of the three-dimensional autonomous modified Lorenz chaotic system.
112 citations
TL;DR: In this paper, the authors investigated the relaxation of a dewetting contact line in the so-called Landau-Levich geometry, in which a vertical solid plate is withdrawn from a bath of partially wetting liquid.
Abstract: The relaxation of a dewetting contact line is investigated theoretically in the so-called ‘Landau–Levich’ geometry in which a vertical solid plate is withdrawn from a bath of partially wetting liquid. The study is performed in the framework of lubrication theory, in which the hydrodynamics is resolved at all length scales (from molecular to macroscopic). We investigate the bifurcation diagram for unperturbed contact lines, which turns out to be more complex than expected from simplified ‘quasi-static’ theories based upon an apparent contact angle. Linear stability analysis reveals that below the critical capillary number of entrainment, Cac, the contact line is linearly stable at all wavenumbers. Away from the critical point, the dispersion relation has an asymptotic behaviour σ∝|q| and compares well to a quasi-static approach. Approaching Cac, however, a different mechanism takes over and the dispersion evolves from ∼|q| to the more common ∼q2. These findings imply that contact lines cannot be described using a universal relation between speed and apparent contact angle, but viscous effects have to be treated explicitly.
108 citations
TL;DR: This diagram shows the existence and extent of complex bifurcation structures that might be useful to understand the mechanisms used by the neurons to encode information and give rapid responses to stimulus and provides a background to develop the understanding of the dynamics of interacting neurons.
Abstract: The results of a study of the bifurcation diagram of the Hindmarsh–Rose neuron model in a two-dimensional parameter space are reported. This diagram shows the existence and extent of complex bifurcation structures that might be useful to understand the mechanisms used by the neurons to encode information and give rapid responses to stimulus. Moreover, the information contained in this phase diagram provides a background to develop our understanding of the dynamics of interacting neurons.
98 citations
TL;DR: In this paper, the effect of fractional order of damping on the dynamic behaviors of the motion of the fractionally damped Duffing equation is examined. And the authors show that the size of the attractor trends to enlarge when fractional ordering α increases.
Abstract: Vibration phenomena of the fractionally damped systems have attracted increasing attentions in recent years. In this paper, dynamics of the fractionally damped Duffing equation is examined. The fractionally damped Duffing equation is transformed into a set of fractional integral equations solved by a predictor–corrector method. The effect of fractional order of damping on the dynamic behaviors of the motion is the main subject of the study. In this work, bifurcation of the parameter-dependent system is drawn numerically. The time evolutions of the nonlinear dynamical system responses are also described in phase portraits and the Poincare map technique. In addition, the occurrence and the nature of chaotic attractors are verified by evaluating the largest Lyapunov exponents. Results obtained from this study illustrates that the fractional order of damping has a significant effect on the dynamic behaviors of the motion. The size of the attractor trends to enlarge when fractional order α increases. Regular motions (including period-3 motion) and chaotic motions are examined. Moreover, a period doubling route to chaos is also found. Many period-3 windows are also observed in bifurcation diagram.
Book•
01 Oct 2007
TL;DR: Groups Group Actions and Representations Smooth G-Manifolds Equivariant Bifurcation Theory: Steady State Bifurlcation Equivariants Bifurbcation Theory as mentioned in this paper.
Abstract: Groups Group Actions and Representations Smooth G-Manifolds Equivariant Bifurcation Theory: Steady State Bifurcation Equivariant Bifurcation Theory: Dynamics Equivariant Transversality Applications of G-Transversality to Bifurcation Theory I Equivariant Dynamics Dynamical Systems on G-Manifolds Applications of G-Transversality to Bifurcation Theory II.
TL;DR: The computations show that unstable limit cycles with an odd number of positive Floquet exponents can be stabilized by time-delayed feedback control, contrary to incorrect claims in the literature.
Abstract: We investigate the normal form of a subcritical Hopf bifurcation subjected to time-delayed feedback control. Bifurcation diagrams which cover time-dependent states as well are obtained by analytical means. The computations show that unstable limit cycles with an odd number of positive Floquet exponents can be stabilized by time-delayed feedback control, contrary to incorrect claims in the literature. The model system constitutes one of the few examples where a nonlinear time delay system can be treated entirely by analytical means.
TL;DR: This paper shows that, in a voltage-mode-controlled dc-dc converter, if the switching is governed by pulse-width modulation of the first kind (PWM-1), an explicit form of the stroboscopic map can be obtained and analyses the bifurcation behavior using the explicit map.
Abstract: Nonlinear phenomena in power electronic circuits are generally studied through discrete-time maps. However, there exist very few circuit configurations (like, for example, the current-mode-controlled dc-dc converters or current programmed H-bridge inverter) for which the map can be obtained in closed form. In this paper, we show that, in a voltage-mode-controlled dc-dc converter, if the switching is governed by pulse-width modulation of the first kind (PWM-1), an explicit form of the stroboscopic map can be obtained. The resulting discrete-time state space is piecewise smooth, divided into five regions, each with a different functional form. We then analyze the bifurcation behavior using the explicit map and demonstrate the different types of border collision bifurcations that may occur in this system as a fixed point moves from one region to another. This includes the very interesting case of a direct transition from periodicity to quasi-periodicity through the route of border collision bifurcation. Mode-locking periodic windows are also obtained at certain ranges of the parameters. The two-parameter bifurcation diagram is presented, showing the domains of existence of different oscillatory modes in the system parameter plane
TL;DR: In this paper, the 3D dynamics of a cantilevered pipe conveying fluid is explored when an additional "point" mass is attached at the free end, and the theoretical results are compared with the results of a set of experiments done previously and good qualitative and quantitative agreement is observed.
TL;DR: New and improved algorithms for the bifurcation analysis of fixed points and periodic orbits (cycles) of maps and their implementation in matcont are discussed and two examples illustrating the developed techniques are provided: a generalized Henon map and a juvenile/adult competition model from mathematical biology.
Abstract: We discuss new and improved algorithms for the bifurcation analysis of fixed points and periodic orbits (cycles) of maps and their implementation in matcont, a MATLAB toolbox for continuation and bifurcation analysis of dynamical systems. This includes the numerical continuation of fixed points of iterates of the map with one control parameter, detecting and locating their bifurcation points (i.e., limit point, period-doubling, and Neimark-Sacker) and their continuation in two control parameters, as well as detection and location of all codimension 2 bifurcation points on the corresponding curves. For all bifurcations of codim 1 and 2, the critical normal form coefficients are computed, both numerically with finite directional differences and using symbolic derivatives of the original map. Using a parameter-dependent center manifold reduction, explicit asymptotics are derived for bifurcation curves of double and quadruple period cycles rooted at codim 2 points of cycles with arbitrary period. These asymptotics are implemented into the software and allow one to switch at codim 2 points to the continuation of the double and quadruple period bifurcations. We provide two examples illustrating the developed techniques: a generalized Henon map and a juvenile/adult competition model from mathematical biology.
TL;DR: The aim of this paper is the study of the long-term behavior of population communities described by piecewise smooth models (known as Filippov systems) by proposing a relatively simple method, called the puzzle method, to construct the complete bifurcation diagram step-by-step.
TL;DR: In this article, the critical and post-critical behavior of a non-conservative nonlinear structure undergoing statical and dynamical bifurcations is analyzed, with particular emphasis on the role of damping on the critical scenario.
Abstract: 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.
TL;DR: The stability of positive equilibrium and the existence of Hopf bifurcation are demonstrated with a delayed Lotka–Volterra prey–predator system with diffusion effects and Neumann boundary conditions.
TL;DR: In this article, the perturbations of transcritical and pitchfork bifurcations at a simple eigenvalue are examined, and two-parameter unfoldings of singularities are rigorously established.
TL;DR: The perturbation‐incremental scheme is employed to investigate the delay‐induced weak resonant double Hopf bifurcation and dynamics in the van der Pol–Duffing and the Stuart–Landau systems with delayed feedback.
Abstract: An efficient method, called the perturbation‐incremental scheme (PIS), is proposed to study, both qualitatively and quantitatively, the delay‐induced weak or high‐order resonant double Hopf bifurcation and the dynamics arising from the bifurcation of nonlinear systems with delayed feedback. The scheme is described in two steps, namely, the perturbation and the incremental steps, when the time delay and the feedback gain are taken as the bifurcation parameters. As for applications, the method is employed to investigate the delay‐induced weak resonant double Hopf bifurcation and dynamics in the van der Pol–Duffing and the Stuart–Landau systems with delayed feedback. For bifurcation parameters close to a double Hopf point, all solutions arising from the resonant bifurcation are classified qualitatively and expressed approximately in a closed form by the perturbation step of the PIS. Although the analytical expression may not be accurate enough for bifurcation parameters away from the double Hopf point, it is...
TL;DR: In this article, the authors studied the codimension two and three Hopf bifurcations and the pertinent Lyapunov stability coefficients in the Watt governor system.
Abstract: This paper pursues the study carried out by the authors in Stability and Hopf bifurcation in the Watt governor system [14], focusing on the codimension one Hopf bifurcations in the centrifugal Watt governor differential system, as presented in Pontryagin's book Ordinary Differential Equations, [13]. Here are studied the codimension two and three Hopf bifurcations and the pertinent Lyapunov stability coefficients and bifurcation diagrams, illustrating the number, types and positions of bifurcating small amplitude periodic orbits, are determined. As a consequence it is found a region in the space of parameters where an attracting periodic orbit coexists with an attracting equilibrium.
TL;DR: In this paper, a perturbation-incremental (PI) method is presented for the computation, continuation and bifurcation analysis of limit cycle oscillations (LCO) of a two-degree-of-freedom aeroelastic system containing a freeplay structural nonlinearity.
TL;DR: In this paper, it was shown that if the eigenvalues of the Jacobian limit to λ L ± i ω L on one side of the discontinuity and − λ R ± I ω R on the other side, with λ l, λ r > 0, and the quantity Λ = λ ǫ l / ω l − ρ r / ρ R is nonzero, then a periodic orbit is created or destroyed as the equilibrium crosses the discontinuities.
TL;DR: In this article, the dynamics of a structurally non-linear two-dimensional airfoil in turbulent flow are investigated numerically using a Monte Carlo approach, and the results are examined in terms of the probability structure of the response and the largest Lyapunov exponent.
Abstract: The dynamics of a structurally non-linear two-dimensional airfoil in turbulent flow is investigated numerically using a Monte Carlo approach. Both the longitudinal and vertical components of turbulence, corresponding to parametric (multiplicative) and external (additive) excitation, respectively, are modelled. The properties of the airfoil are chosen such that the underlying non-excited, deterministic system exhibits binary flutter; the loss of stability of the equilibrium point due to flutter then leads to a limit cycle oscillation (LCO) via a supercritical Hopf bifurcation. For the random system, the results are examined in terms of the probability structure of the response and the largest Lyapunov exponent. The airfoil response is interpreted from the point of view of the concepts of D- and P-bifurcations, as defined in random bifurcation theory. It is found that the bifurcation is characterized by a change in shape of the response probability structure, while no discontinuity in the variation of the largest Lyapunov exponent with airspeed is observed. In this sense, the trivial bifurcation obtained for the deterministic airfoil, where the D- and P-bifurcations coincide, appears only as a P-bifurcation for the random case. At low levels of turbulence intensity, the Gaussian-like bell-shaped bi-dimensional PDF bifurcates into a crater shape; this is interpreted as a random fixed point bifurcating into a random LCO. At higher levels of turbulence intensity, the post-bifurcation PDF loses its underlying deterministic LCO structure. The crater is transformed into a two-peaked shape, with a saddle at the origin. From a more universal point of view, the robustness of the random bifurcation scenario is critiqued in light of the relative importance of the two components of turbulent excitation.
TL;DR: In this article, the stability and direction of Hopf bifurcation are determined by applying the normal form theory and the center manifold argument, and numerical simulations are performed to illustrate the analytical results.
Abstract: A class of three level food chain system is studied. With the theory of delay equations and Hopf bifurcation, the conditions of the positive equilibrium undergoing Hopf bifurcation is given regarding τ as the parameter. The stability and direction of Hopf bifurcation are determined by applying the normal form theory and the center manifold argument, and numerical simulations are performed to illustrate the analytical results.
TL;DR: In this paper, a continuation method was used to calculate flow bifurcation with/without heat transfer in a two-sided lid-driven cavity with an aspect ratio of 1.96.
TL;DR: In this article, a four-dimensional hyperchaotic Lorenz system was obtained by adding a nonlinear controller to the Lorenz chaotic system, which is studied by bifurcation diagram, Lyapunov exponent spectrum and phase diagram.
Abstract: This paper presents a four-dimensional hyperchaotic Lorenz system, obtained by adding a nonlinear controller to Lorenz chaotic system. The hyperchaotic Lorenz system is studied by bifurcation diagram, Lyapunov exponent spectrum and phase diagram. Numerical simulations show that the new system's behavior can be convergent, divergent, periodic, chaotic and hyperchaotic as the parameter varies.
TL;DR: This work investigates the behavior of a neural network model consisting of three neurons with delayed self and nearest-neighbor connections that gives analytical results on the existence, stability and bifurcation of nontrivial equilibria of the system and shows the existence of codimension two bifURcation points involving both standard and D3-equivariant, Hopf and pitchfork bIfurcation points.
Abstract: We investigate the behavior of a neural network model consisting of three neurons with delayed self and nearest-neighbor connections. We give analytical results on the existence, stability and bifurcation of nontrivial equilibria of the system. We show the existence of codimension two bifurcation points involving both standard and D3-equivariant, Hopf and pitchfork bifurcation points. We use numerical simulation and numerical bifurcation analysis to investigate the dynamics near the pitchfork–Hopf interaction points. Our numerical investigations reveal that multiple secondary Hopf bifurcations and pitchfork bifurcations of limit cycles may emanate from the pitchfork–Hopf points. Further, these secondary bifurcations give rise to ten different types of periodic solutions. In addition, the secondary bifurcations can lead to multistability between equilibrium points and periodic solutions in some regions of parameter space. We conclude by generalizing our results into conjectures about the secondary bifurcations that emanate from codimension two pitchfork–Hopf bifurcation points in systems with Dn symmetry.
TL;DR: In this article, the stability and bifurcation of the distributed delays Cohen-Grossberg neural networks with two neurons were discussed. And the authors proved that Hopf-bifurcation occurs.
Abstract: In this paper, we discuss the stability and bifurcation of the distributed delays Cohen–Grossberg neural networks with two neurons. By choosing the average delay as a bifurcation parameter, we prove that Hopf bifurcation occurs. The stability of bifurcating periodic solutions and the direction of Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. Finally, numerical simulation results are given to support the theoretical predictions.