Showing papers on "Infinite-period bifurcation published in 2017"
TL;DR: This paper investigates an issue of bifurcation control for a novel incommensurate fractional-order predator-prey system with time delay and it is shown that the control effort is markedly influenced by feedback gain.
116 citations
TL;DR: In this paper, the existence and uniqueness of crossing limit cycles for pseudo-Hopf bifurcation was proved under generic conditions, and a crossing limit cycle for this family was presented.
Abstract: The creation or destruction of a crossing limit cycle when a sliding segment changes its stability, is known as pseudo-Hopf bifurcation. In this paper, under generic conditions, we find an unfolding for such bifurcation, and we prove the existence and uniqueness of a crossing limit cycle for this family.
46 citations
TL;DR: This study unveils another, relevant and so far underestimated transition point: the saddle-node-loop bifurcation, which can be reached by several parameters, including capacitance, leak conductance, and temperature, and turns out to induce even more drastic changes in synchronization than the BT transition.
Abstract: Prominent changes in neuronal dynamics have previously been attributed to a specific switch in onset bifurcation, the Bogdanov-Takens (BT) point. This study unveils another, relevant and so far underestimated transition point: the saddle-node-loop bifurcation, which can be reached by several parameters, including capacitance, leak conductance, and temperature. This bifurcation turns out to induce even more drastic changes in synchronization than the BT transition. This result arises from a direct effect of the saddle-node-loop bifurcation on the limit cycle and hence spike dynamics. In contrast, the BT bifurcation exerts its immediate influence upon the subthreshold dynamics and hence only indirectly relates to spiking. We specifically demonstrate that the saddle-node-loop bifurcation (i) ubiquitously occurs in planar neuron models with a saddle node on invariant cycle onset bifurcation, and (ii) results in a symmetry breaking of the system's phase-response curve. The latter entails an increase in synchronization range in pulse-coupled oscillators, such as neurons. The derived bifurcation structure is of interest in any system for which a relaxation limit is admissible, such as Josephson junctions and chemical oscillators.
31 citations
TL;DR: S spatiotemporal bifurcation analysis in a ratio-dependent predator prey model is performed and explicit conditions for the existence of non-constant steady states that emerge from the interactions between predator and prey are derived.
30 citations
TL;DR: In this article, a detailed study of excursive (or Ledinegg) instability and density wave oscillations (DWOs) is carried out for two-phase flow in a natural circulation loop.
26 citations
TL;DR: In this paper, the existence and stability of the 1/n impact periodic motions near grazing points in single-degree-of-freedom impact oscillators were analyzed with the aid of the discontinuity mapping technique to determine the degenerate grazing bifurcation points.
Abstract: Co-dimension-one grazing bifurcations of 1 / n impact periodic motions are often accompanied by saddle-node bifurcations or period-doubling bifurcations. The presence of certain degenerate grazing bifurcation points plays an important role in transitions between these two different co-dimension-one grazing bifurcation scenarios. When the saddle-node bifurcation line and period-doubling bifurcation line meet at a grazing bifurcation point, a degenerate grazing bifurcation occurs. By considering the existence and stability of the 1 / n impact periodic motions near grazing points in single-degree-of-freedom impact oscillators, an analytical method is developed with the aid of the discontinuity mapping technique to determine the certain degenerate grazing bifurcation points of 1 / n motions. It is found that the linear term in the local discontinuity mapping does not affect the distribution of certain degenerate grazing points. The unfolding in the neighborhood of degenerate grazing bifurcation points is verified by numerical simulations.
23 citations
TL;DR: In this paper, a nonlinear analysis of a two-dimensional airfoil oscillating in pitch and plunge degrees of freedom is performed to investigate the effects of a discontinuous free-play nonlinearity in pitch on the response of the air-foil system.
Abstract: This paper is devoted to study a two-dimensional airfoil oscillating in pitch and plunge degrees of freedom. A nonlinear analysis is performed to investigate the effects of a discontinuous freeplay nonlinearity in pitch on the response of the airfoil system. In fact we show that in the presence of freeplay, the air velocity has a direct effect on the pitch vibrations of the airfoil system. Namely, it can generate the flutter leading to the limit cycle oscillation for the airfoil. With the aid of a fixed point of the Poincar
$$\acute{\text {e}}$$
map of the system and numerical findings, we determine the flutter and the limit cycle oscillation of that. The frequency, period of the limit cycle oscillation of pitch motion and the flutter speed are calculated. Tangent points are also computed, and it is shown that these points cannot be two-fold singularities for the system. Furthermore, by using the theoretical techniques of discontinuous systems, we will obtain parametric regions for the existence of grazing bifurcation (global bifurcation). The existence of grazing bifurcation helps us to display that for some values of the air velocity, different transitions or sudden jumps can occur in the system’s response. Numerical results demonstrate that these transitions are accompanied by the appearance and disappearance of a tangential contact between the trajectory and the switching boundaries. Also they can cause a change in the response of the pitch motion from simply periodic to double periodic (periodic-2). Moreover, stability regions for the airfoil system with freeplay will be found. The property of these stability regions is that inside them there exist no flutter and limit cycle oscillation. Some numerical examples are given which are in good agreement with our theoretical results.
22 citations
TL;DR: To overcome the difficulty that originates from the classical bifurcation criteria, the explicit critical criteria without using eigenvalues calculation of high-dimensional map are applied to determine bIfurcation points of Co-dimension-one period doubling biforcation and Neimark–Sacker bifircation.
18 citations
TL;DR: It is proved that all solutions of this model are bounded, and there exist some values of the parameters such that the model has a global attractor.
Abstract: In this paper, we study the global dynamics and bifurcations of a two-dimensional discrete time host–parasitoid model with strong Allee effect. The existence of fixed points and their stability are analysed in all allowed parametric region. The bifurcation analysis shows that the model can undergo fold bifurcation and Neimark–Sacker bifurcation. As the parameters vary in a small neighbourhood of the Neimark–Sacker bifurcation condition, the unique positive fixed point changes its stability and an invariant closed circle bifurcates from the positive fixed point. From the viewpoint of biology, the invariant closed curve corresponds to the periodic or quasi-periodic oscillations between host and parasitoid populations. Furthermore, it is proved that all solutions of this model are bounded, and there exist some values of the parameters such that the model has a global attractor. These theoretical results reveal the complex dynamics of the present model.
17 citations
TL;DR: An susceptible-infective-removed epidemic model incorporating media coverage with time delay with results that show that the time delay in media coverage can not affect the stability of the disease-free equilibrium when the basic reproduction number is less than unity.
Abstract: An susceptible-infective-removed epidemic model incorporating media coverage with time delay is proposed. The stability of the disease-free equilibrium and endemic equilibrium is studied. And then, the conditions which guarantee the existence of local Hopf bifurcation are given. Furthermore, we show that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of delay. The obtained results show that the time delay in media coverage can not affect the stability of the disease-free equilibrium when the basic reproduction number is less than unity. However, the time delay affects the stability of the endemic equilibrium and produces limit cycle oscillations while the basic reproduction number is greater than unity. Finally, some examples for numerical simulations are included to support the theoretical prediction.
17 citations
TL;DR: In this article, the authors investigate the local and global bifurcation behaviors of an archetypal self-excited smooth and discontinuous oscillator driven by moving belt friction.
Abstract: In this paper, we investigate the local and global bifurcation behaviors of an archetypal self-excited smooth and discontinuous oscillator driven by moving belt friction. The belt friction is described in the sense of Stribeck characteristic to formulate the mathematical model of the proposed system. For such a friction characteristic, the complicated bifurcation behaviors of the system are discussed. The bifurcation of the multiple sliding segments for this self-excited system is exhibited by analytically exploring the collision of tangent points. The Hopf bifurcation of this self-excited system with viscous damping is analyzed by making the examination of the eigenvalues at the steady state and discussing the stability of the limit cycles. The bifurcation diagrams and the corresponding phase portraits are depicted to demonstrate the complicated dynamical behaviors of double tangency bifurcation, the bifurcation of sliding homoclinic orbit to a saddle, subcritical Hopf bifurcation and grazing bifurcation for this system.
TL;DR: In this paper, the spatiotemporal properties of waves in the Lorenz-96 model and their dependence on the dimension parameter n and the forcing parameter F were studied and compared.
Abstract: . In this paper we study the spatiotemporal properties of waves in the Lorenz-96 model and their dependence on the dimension parameter n and the forcing parameter F. For F > 0 the first bifurcation is either a supercritical Hopf or a double-Hopf bifurcation and the periodic attractor born at these bifurcations represents a traveling wave. Its spatial wave number increases linearly with n, but its period tends to a finite limit as n → ∞. For F
TL;DR: This study of interplay of two time delays of different origins will shed light on the control of bifurcation-delay and improve the knowledge of time-delayed systems.
Abstract: The slow passage effect in a dynamical system generally induces a delay in bifurcation that imposes an uncertainty in the prediction of the dynamical behaviors around the bifurcation point. In this paper, we investigate the influence of linear time-delayed self-feedback on the slow passage through the delayed Hopf and pitchfork bifurcations in a parametrically driven nonlinear oscillator. We perform linear stability analysis to derive the Hopf bifurcation point and its stability as a function of self-feedback time delay. Interestingly, the bifurcation-delay associated with Hopf bifurcation behaves differently in two different edges. In the leading edge of the modulating signal, it decreases with increasing self-feedback delay, whereas in the trailing edge, it behaves in an opposite manner. We also show that the linear time-delayed self-feedback can reduce bifurcation-delay in pitchfork bifurcation. These results are illustrated numerically and corroborated experimentally. We also propose a mechanistic explanation of the observed behaviors. In addition, we show that our observations are robust in the presence of noise. We believe that this study of interplay of two time delays of different origins will shed light on the control of bifurcation-delay and improve our knowledge of time-delayed systems.
TL;DR: The bisection procedure and an improved stagger-and-step method are employed to present evidence of visual chaotic saddles on the fractal basin boundary and in the internal basin, respectively and show that the period saddles play an important role in the evolution of chaotic saddle.
TL;DR: In this paper, the linear stability and bifurcation analysis of two-phase flow in the single heated channel for the natural circulation loop is explored and a novel case of stability boundary (listed as Type A in the present work) emerging for higher subcooling numbers on Npch - Nsub parameter plane, which has not been noted in literature earlier and lies near to the Ledinegg stability boundary, which is a static instability.
TL;DR: It is shown that feedback time delay can induce nonhomogeneous periodic oscillatory patterns and Hopf bifurcation near the positive steady-state solution is proved to occur at a sequence of critical values.
TL;DR: In this paper, the structure of the set of possible solutions of a degenerate boundary value problem was studied, and numerical examples were given showing each of these possibilities can occur, for solutions with one interior zero.
Abstract: In [ 12 ], the structure of the set of possible solutions of a degenerate boundary value problem was studied. For solutions with one interior zero, there were two possibilities for the solution set. In this paper, numerical examples are given showing each of these possibilities can occur.
TL;DR: In this article, the effect of subcritical and supercritical Hopf bifurcations on the stability boundary of boiling water reactors (BWRs) has been discussed.
TL;DR: In this article, the existence and stability of the equilibrium points of the discrete predator-prey model with the prey subject to the Allee effect are investigated in detail, and the basins of attraction of the single population model are given.
Abstract: In this paper, complex dynamics of the discrete predator–prey model with the prey subject to the Allee effect are investigated in detail. Firstly, when the prey intrinsic growth rate is not large, the basins of attraction of the equilibrium points of the single population model are given. Secondly, rigorous results on the existence and stability of the equilibrium points of the model are derived, especially, by analyzing the higher order terms, we obtain that the non-hyperbolic extinction equilibrium point is locally asymptotically stable. The existences and bifurcation directions for the flip bifurcation, the Neimark–Sacker bifurcation and codimension-two bifurcations with 1:2 resonance are derived by using the center manifold theorem and the bifurcation theory. We derive that the model only exhibits a supercritical flip bifurcation and it is possible for the model to exhibit a supercritical or subcritical Neimark–Sacker bifurcation at the larger positive equilibrium point. Chaos in the sense of ...
TL;DR: It is concluded that time delay plays a vital role in the generation of MMOs because the occurrence and evolution of such complex MMOs depend on the magnitude of the delay.
TL;DR: In this paper, the authors considered bifurcation of limit cycles for planar cubic-order systems with an isolated nilpotent critical point and proved the existence of at least 9 small-amplitude limit cycles in the neighborhood of the critical point.
TL;DR: The first Lyapunov coefficient is introduced to judge the limit cycles of 3D IS-LM macroeconomics, i.e. from a practical view of the business cycle, to demonstrate the stability of limit cycles when Hopf bifurcation occurs.
TL;DR: In this paper, the effects of a perturbation which is applied to a linear control law and, due to the perturbations, the control changes from purely positional to position-velocity control are analyzed.
TL;DR: In this article, the authors proposed a method to construct Poincare-Bendixson regions by using transversal curves that enables them to prove the existence of a limit cycle that has been numerically detected.
Abstract: This paper deals with the problem of location and existence of limit cycles for real planar polynomial differential systems. We provide a method to construct Poincare-Bendixson regions by using transversal curves, that enables us to prove the existence of a limit cycle that has been numerically detected. We apply our results to several known systems, like the Brusselator one or some Lienard systems, to prove the existence of the limit cycles and to locate them very precisely in the phase space. Our method, combined with some other classical tools can be applied to obtain sharp bounds for the bifurcation values of a saddle-node bifurcation of limit cycles, as we do for the Rychkov system.
TL;DR: The center manifold theorem and the normal form technique were used to study the stability and instability of limit cycles emerging from the Hopf bifurcation to prove that with IDA-PBC the authors can control not only the unstable equilibrium but also some trajectories such as limit cycles.
Abstract: This paper deals with the problem of obtaining stable and robust oscillations of underactuated mechanical systems. It is concerned with the Hopf bifurcation analysis of a Controlled Inertia Wheel Inverted Pendulum (C-IWIP). Firstly, the stabilization was achieved with a control law based on the Interconnection, Damping, Assignment Passive Based Control method (IDA-PBC). Interestingly, the considered closed-loop system exhibits both supercritical and subcritical Hopf bifurcation for certain gains of the control law. Secondly, we used the center manifold theorem and the normal form technique to study the stability and instability of limit cycles emerging from the Hopf bifurcation. Finally, numerical simulations were conducted to validate the analytical results in order to prove that with IDA-PBC we can control not only the unstable equilibrium but also some trajectories such as limit cycles.
TL;DR: The main result of this paper is the central limit theorem for bifurcation ratio of general branch order, a generalized form of the centrallimit theorem for the lowest bifURcation ratio, which was previously proved.
Abstract: The Horton-Strahler ordering method, originating in hydrology, formulates the hierarchical structure of branching patterns using a quantity called the bifurcation ratio. The main result of this paper is the central limit theorem for bifurcation ratio of general branch order. This is a generalized form of the central limit theorem for the lowest bifurcation ratio, which was previously proved. Some useful relations are also derived in the proofs of the main theorems.
TL;DR: In this article, a memristor-based oscillator with cubic nonlinearity is studied, and it is shown that oscillation excitation has distinctive features of the supercritical Andronov-Hopf bifurcation and can be achieved by changing of a parameter value as well as by variation of initial conditions.
Abstract: The model of a memristor-based oscillator with cubic nonlinearity is studied. The considered system has infinitely many equilibrium points, which build a line of equilibria in the phase space. Numerical modeling of the dynamics is combined with bifurcational analysis. It is shown that oscillation excitation has distinctive features of the supercritical Andronov--Hopf bifurcation and can be achieved by changing of a parameter value as well as by variation of initial conditions. Therefore the considered bifurcation is called Andronov-Hopf bifurcation with and without parameter.
TL;DR: In this paper, the stability and local bifurcation behaviors for the nonlinear aeroelastic model of an airfoil with external store are investigated using both analytical and numerical methods.
Abstract: In this paper, stability and local bifurcation behaviors for the nonlinear aeroelastic model of an airfoil with external store are investigated using both analytical and numerical methods. Three kinds of degenerated equilibrium points of bifurcation response equations are considered. They are characterized as (1) one pair of purely imaginary eigenvalues and two pairs of conjugate complex roots with negative real parts; (2) two pairs of purely imaginary eigenvalues in nonresonant case and one pair of conjugate complex roots with negative real parts; (3) three pairs of purely imaginary eigenvalues in nonresonant case. With the aid of Maple software and normal form theory, the stability regions of the initial equilibrium point and the explicit expressions of the critical bifurcation curves are obtained, which can lead to static bifurcation and Hopf bifurcation. Under certain conditions, 2-D tori motion may occur. The complex dynamical motions are considered in this paper. Finally, the numerical solutions achieved by the fourth-order Runge–Kutta method agree with the analytic results.
TL;DR: In this paper, the stability and bifurcation behaviors of a predator-prey model with piecewise constant arguments and time delay are investigated, and the model will produce two different branches by limiting branch parameters of different intervals.
TL;DR: It was found that in the range of sufficiently small R ~c, bifurcation cannot be initiated, and a simple criterion to compute the threshold R~c for the onset of b ifurcation is proposed.
Abstract: Channel head bifurcation is a key factor for generating complexity of channel networks. Here, we investigate incipient channel head bifurcation using linear stability analysis. Channel heads are simplified as circular hollows, toward which surface sheet flow accelerates in the radial direction. Sinusoidal perturbations in the angular direction with different angular wavenumbers k are imposed on the bed, and their growth rates Ω~ are computed. Because the channel head radius R~c is extending over time, the base state (circular hollow in the absence of perturbations) also evolves continuously. With the use of the momentary stability concept, the evolving base state is defined as momentarily unstable to the imposed perturbation if the disturbance is growing faster than the evolution of the base state. It was found that in the range of sufficiently small R~c, bifurcation cannot be initiated. As R~c increases, bifurcation starts to be possible with k≈ 3–5. A higher k implies bifurcation with a narrower channel junction angle (θ = 2π/k). The average junction angle of the Colorado High Plains for the smallest drainage area is about 85∘ with a standard deviation of 35∘ [Solyom and Tucker, 2007]. Our predicted angles (75∘-120∘ ) agree qualitatively with the observed angles. Finally, we propose a simple criterion to compute the threshold R~c for the onset of bifurcation.