Showing papers on "Infinite-period bifurcation published in 2022"
TL;DR: In this paper , a modified van der Pol-Duffing system with periodic parametric excitation is considered, and the authors reveal the mechanism of certain special phenomena in bursting oscillations such as the sudden increase of the spiking amplitude.
Abstract: The main purpose of the paper is to reveal the mechanism of certain special phenomena in bursting oscillations such as the sudden increase of the spiking amplitude. When multiple equilibrium points coexist in a dynamical system, several types of stable attractors via different bifurcations from these points may be observed with the variation of parameters, which may interact with each other to form other types of bifurcations. Here we take the modified van der Pol–Duffing system as an example, in which periodic parametric excitation is introduced. When the exciting frequency is far less than the natural frequency, bursting oscillations may appear. By regarding the exciting term as a slow-varying parameter, the number of the equilibrium branches in the fast generalized autonomous subsystem varies from one to five with the variation of the slow-varying parameter. The equilibrium branches may undergo different types of bifurcations, such as Hopf and pitchfork bifurcations. The limit cycles, including the cycles via Hopf bifurcations and the cycles near the homo-clinic orbit, may interact with each other to form the fold limit cycle bifurcations. With the increase of the exciting amplitude, different stable attractors and bifurcations of the generalized autonomous fast subsystem involve the full system, leading to different types of bursting oscillations. Fold limit cycle bifurcations may cause the sudden change of the spiking amplitude, since at the bifurcation points, the trajectory may oscillate according to different stable limit cycles with obviously different amplitudes. At the pitchfork bifurcation point, two possible jumping ways may result in two coexisted asymmetric bursting attractors, which may expand in the phase space to interact with each other to form an enlarged symmetric bursting attractor with doubled period. The inertia of the movement along the stable equilibrium may cause the trajectory to pass across the related bifurcations, leading to the delay effect of the bifurcations. Not only the large exciting amplitude, but also the large value of the exciting frequency may increase inertia of the movement, since in both the two cases, the change rate of the slow-varying parameter may increase. Therefore, a relative small exciting frequency may be taken in order to show the possible influence of all the equilibrium branches and their bifurcations on the dynamics of the full system.
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TL;DR: In this article, the local cyclicity of polynomial vector fields in R 3 was studied and a quadratic system with 11 limit cycles, a cubic system with 31 limit cycles and a quartic system with 54 limit cycles were given.
2 citations
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TL;DR: In this paper , the authors consider a model describing the length of two queues that incorporates customer choice behavior based on delayed queue length information and examine the non-symmetric case, wherein the values of the time-delay parameter in each queue are different.
Abstract: We consider a model describing the length of two queues that incorporates customer choice behavior based on delayed queue length information. The symmetric case, where the values of the time-delay parameter in each queue are the same, was recently studied. It was shown that under some conditions, the stable equilibrium solution becomes unstable as the common time delay passes a threshold value. This one-time stability switch occurs only at a symmetry-breaking Hopf bifurcation where a family of stable asynchronous limit-cycle solutions arise. In this paper, we examine the non-symmetric case, wherein the values of the time-delay parameter in each queue are different. We show that, in contrast to the symmetric case, the non-symmetric case allows bubbling, multiple stability switches and coexistence of distinct families of stable limit cycles. An investigation of the dynamical behavior of the non-symmetric system in a neighborhood of a double-Hopf bifurcation using numerical continuation explains the occurrence of the bistable limit cycles. Quasi-periodic oscillations were also observed due to the presence of torus bifurcations near the double-Hopf bifurcation. These identifications of the underlying mechanisms that cause unwanted oscillations in the system give a better understanding of the effects of providing delayed information and consequently help in better management of queues.
2 citations
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TL;DR: In this paper , an impulsive state feedback control for a three-degree-of-freedom vibro-impact system with clearances was designed to create a stable Neimark-Sacker bifurcation.
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TL;DR: In this article , a review of Hopf-like bifurcations for two-dimensional ODE systems with state-dependent switching rules is presented, including boundary equilibrium and limit cycle creation via hysteresis or time delay.
2 citations
TL;DR: In this article , a system of coupled phase oscillators near a saddle-node on invariant circle bifurcation and driven by random intrinsic frequencies is studied, and it is shown that the system undergoes a phase transition changing the qualitative properties of collective dynamics.
Abstract: We study a system of coupled phase oscillators near a saddle-node on invariant circle bifurcation and driven by random intrinsic frequencies. Under the variation of control parameters, the system undergoes a phase transition changing the qualitative properties of collective dynamics. Using Ott-Antonsen reduction and geometric techniques for ordinary differential equations, we identify heteroclinic bifurcation in a family of vector fields on a cylinder, which explains the change in collective dynamics. Specifically, we show that heteroclinic bifurcation separates two topologically distinct families of limit cycles: contractible limit cycles before bifurcation from noncontractibile ones after bifurcation. Both families are stable for the model at hand.
2 citations
TL;DR: In this paper , the bifurcation analysis in a discrete-time Leslie-Gower predator-prey model with constant yield predator harvesting was investigated, and the stability analysis for the fixed points of the discretized model was shown briefly.
Abstract: This work investigates the bifurcation analysis in a discrete-time Leslie–Gower predator–prey model with constant yield predator harvesting. The stability analysis for the fixed points of the discretized model is shown briefly. In this study, the model undergoes codimension-1 bifurcation such as fold bifurcation (limit point), flip bifurcation (period-doubling) and Neimark–Sacker bifurcation at a positive fixed point. Further, the model exhibits codimension-2 bifurcations, including Bogdanov–Takens bifurcation and generalized flip bifurcation at the fixed point. For each bifurcation, by using the critical normal form coefficient method, various critical states are calculated. To validate our analytical findings, the bifurcation curves of fixed points are drawn by using MATCONTM. The system exhibits interesting rich dynamics including limit cycles and chaos. Moreover, it has been shown that the predator harvesting may control the chaos in the system.
TL;DR: In this paper , a complete analysis of the limit cycle bifurcation from infinity in 3D relay systems, which belong to the class of three-dimensional symmetric discontinuous piecewise linear systems with two zones, is presented.
TL;DR: In this article , the local cyclicity of polynomial vector fields in R3 was studied and a quadratic system with 11 limit cycles, a cubic system with 31 limit cycles and a quartic system with 54 limit cycles were given.
01 Jan 2022
TL;DR: In this article , the authors show all the global bifurcation diagrams of stationary solutions to a phase field model in the 1-dimensional case, and show that bifurlcation diagrams are surprisingly rich in variety depending on the latent heat and the initial total enthalpy.
Abstract: We are concerned with bifurcation diagrams of stationary solutions to a phase field model proposed by Fix and followed by Caginalp. We show all the global bifurcation diagrams of stationary solutions to the model in the 1-dimension case. We see that bifurcation diagrams are surprisingly rich in variety depending on the latent heat and the initial total enthalpy. For instance, bifurcation diagrams include the secondary bifurcation point where symmetric breaking occurs, and curves which connect a limit of boundary layer solutions to the other limit of internal layer solutions.
TL;DR: In this article , the authors considered an autonomous three-dimensional quadratic ODE system with nine parameters and derived conditions under which this system has infinitely many limit cycles, and proved the nonlocal existence of an infinite set of limit cycles emerging by means of Andronov-Hopf bifurcation.
Abstract: We consider an autonomous three-dimensional quadratic ODE system with nine parameters, which is a generalization of the Langford system. We derive conditions under which this system has infinitely many limit cycles. First, we study the equilibrium points of such systems and their eigenvalues. Next, we prove the non-local existence of an infinite set of limit cycles emerging by means of Andronov – Hopf bifurcation.
TL;DR: In this article , the stability conditions of fixed points in the Bazykin-Berezovskaya predator-prey model were investigated using analytical and numerical bifurcation analysis.
Abstract: This paper investigates multiple bifurcations analyses and strong resonances of the Bazykin–Berezovskaya predator–prey model in depth using analytical and numerical bifurcation analysis. The stability conditions of fixed points, codim-1 and codim-2 bifurcations to include multiple and generic bifurcations are studied. This model exhibits transcritical, flip, Neimark–Sacker, and [Formula: see text], [Formula: see text], [Formula: see text] strong resonances. The normal form coefficients and their scenarios for each bifurcation are examined by using the normal form theorem and bifurcation theory. For each bifurcation, various types of critical states are calculated, such as potential transformations between the one-parameter bifurcation point and different bifurcation points obtained from the two-parameter bifurcation point. To validate our analytical findings, the bifurcation curves of fixed points are determined by using MatcontM.
TL;DR: In this paper , the number of limit cycles for a class of cubic systems with general quadratic polynomial perturbations was studied and it was shown that five limit cycles can be bifurcated from a period annulus.
Abstract: In this paper, we study the number of limit cycles for a class of cubic systems with general quadratic polynomial perturbations. By using the Melnikov function theory we obtain that five limit cycles can be bifurcated from a period annulus. We also study the Hopf bifurcation at the center surrounded by the annulus.
TL;DR: In this paper , the complex dynamical disease of the hematopoietic stem cells model based on Mackey's mathematical description is analyzed and the bifurcating periodical oscillation solutions of the system are continued by applying numerical simulation method.
Abstract: Underlying the state feedback control, the complex dynamical disease of the hematopoietic stem cells model based on Mackey’s mathematical description is analyzed. The bifurcating periodical oscillation solutions of the system are continued by applying numerical simulation method. The limit point cycle bifurcation and period doubling bifurcation are observed frequently in the continuation process. The attraction basins of the positive equilibrium solution shrink as the differentiate rate is ascending and the observed Mobiüs strain is simulated with boundary as the period-2 solution. The period doubling bifurcation leads to period-2, period-4, and period-8 solutions which are simulated. Starting from period doubling bifurcation point, the continuation of the bifurcating solution routes to homoclinic solution is finished. The simulation results improve the comprehension related to the spontaneous dynamical character manifested in the hematopoietic stem cells model.
TL;DR: In this article , the authors investigate global dynamics of the discontinuous limit case of an archetypal oscillator with a constant excitation that is a model of an arch bridge with viscous damping subjected to a sinusoidally varying central load.
25 Nov 2022
TL;DR: In this article , a nonlinear controller is set up to the system, and the period doubling bifurcation control is carried out at four period points of the system and the control parameters value are inversely calculated.
Abstract: In this paper, taking one-dimensional quadratic discrete chaotic system as the research object, the nonlinear dynamic characteristics of the system such as period doubling bifurcation and chaos are studied. Using the state feedback control method, a nonlinear controller is set up to the system, and the period doubling bifurcation control is carried out at four period point of the system. By specifying the bifurcation parameter in advance, the control parameters value are inversely calculated. The four period bifurcation point appears at the prespecified location, and the purpose of period doubling bifurcation control is realized. The correctness of the theoretical analysis is verified by numerical simulation.
TL;DR: The degenerate Neimark-Sacker bifurcation curve is not a traditional parabolic shape, but an unbounded extended line as mentioned in this paper , which means that this system undergoes a degenerate NEI.
Abstract: ABSTRACT In this paper, we investigate the fold bifurcation, flip bifurcation and degenerate Neimark–Sacker bifurcation of a second-order rational difference equation. As we know, many scholars used the first-order Poincaré-Lyapunov constant σ to determine the type of Neimark–Sacker bifurcation. However, by computing, we find that which means this system undergoes a degenerate Neimark–Sacker bifurcation. Therefore, using the centre manifold theorem, the Normal Form theory and the bifurcation theory, we calculate the fifth-order term of this system and give the conditions of the degenerate Neimark–Sacker bifurcation. Simultaneously, the Neimark–Sacker bifurcation curve is not a traditional parabolic shape, but an unbounded extended line. Finally, the numerical simulations are provided to illustrate theoretical results.
TL;DR: In this paper , the authors considered a particular position of the antiparallel vectors to yield a two-parametric unfolding of the Teixeira Singularity, which is known as pseudo-Hopf bifurcation.
Abstract: Consider a piecewise linear dynamical system in [Formula: see text], divided by a switching plane, with the Teixeira Singularity (TS). Under the antiparallelism hypothesis, it is known that the system undergoes the so-called pseudo-Hopf bifurcation (pH). In this paper, we consider a particular position of the antiparallel vectors to yield a two-parametric unfolding of the TS. The bifurcation analysis gives us, besides the curve of pH bifurcation points, a curve of saddle-node bifurcation points for crossing limit cycles. We call this phenomenon the pseudo-Bautin bifurcation, since it exhibits the exact same behavior as the Bautin bifurcation for smooth dynamical systems.
TL;DR: In this article , a typical switched model alternating between a Duffing oscillator and van der Pol oscillator is established to explore the typical dynamical behaviors as well as the mechanism of the switched system.
Abstract: By introducing a switching scheme related to the state and time, a typical switched model alternating between a Duffing oscillator and van der Pol oscillator is established to explore the typical dynamical behaviors as well as the mechanism of the switched system. Shooting methods to locate the limit cycle and specify bifurcation sets are described by defining an appropriate Poincaré map. Different types of multiple-Focus/Cycle and single-Focus/Cycle period oscillations in the system can be observed. Symmetry-breaking, period-doubling, and grazing bifurcation curves are obtained in the plane of bifurcation parameters, dividing the parameters plane into several regions corresponding to different kinds of oscillations. Meanwhile, based on the numerical simulation and bifurcation analysis, the mechanisms of several typical dynamical behaviors observed in different regions are presented.
TL;DR: In this paper , the mathematical model of four differential equations for organisms that describe the effect of anti-predation behavior, age stage and toxicity have been analyzed, and the results of the bifurcation behavior analysis have been fully presented using numerical simulation.
Abstract: In this study, the mathematical model of four differential equations for organisms that describe the effect of anti-predation behavior, age stage and toxicity have been analyzed. Local bifurcation and Hopf bifurcation have been studied by changing a parameter of a model to study the dynamic behavior determined by bifurcation curves and the occurrence states of bifurcation saddle node, transcritical and pitch fork bifurcation. The potential equilibrium point at which Hopf bifurcation occurs has been determined and the results of the bifurcation behavior analysis have been fully presented using numerical simulation.
TL;DR: In this paper , the authors considered the problem of construction of real autonomous quadratic systems of three differential equations with the nonlocal existence of an infinite number of limit cycles, emerging from the focus due to the Andronov-Hopf bifurcation.
Abstract: In this paper, we consider the problem of construction of real autonomous quadratic systems of three differential equations with the nonlocal existence of an infinite number of limit cycles. This means that an infinite number of limit cycles, emerging from the focus due to the Andronov – Hopf bifurcation, can exist in the phase space not only in the vicinity of the focus and not only for parameter values close to the bifurcation value. To solve this problem we use the method of determination of limit cycles as the curves of intersection of an invariant plane with a family of invariant elliptic paraboloids. Then the study of the limit cycles of the constructed system of the third order is reduced to the study of the corresponding system of the second order on each of the invariant elliptic paraboloids. The proof of the nonlocal existence of the limit cycle and the investigation of its stability for such a second-order system is carried out by constructing a topographic system of Poincaré functions or by transforming to polar coordinates.
TL;DR: In this paper , the qualitative properties and bifurcations of a discrete-time logistic type model for the competitive interaction of two species were investigated, where polynomial symbolic algebraic theory was applied to deal with complex high-order semi-algebraic systems.
Abstract: In this paper, we investigate the qualitative properties and bifurcations of a discrete-time logistic type model for the competitive interaction of two species. Applying polynomial symbolic algebraic theory to deal with complex high-order semi-algebraic systems, and using the bifurcation theory, we give not only the topological structure of the orbits near each fixed point but also the parameter conditions such that the model produces transcritical bifurcation, supercritical (or subcritical) flip bifurcation and supercritical (or subcritical) Neimark-Sacker bifurcation, respectively. Besides, the corresponding mapping is proven to be chaotic in the sense of Marotto. At last, we simulate the stable orbits of period 2 produced from the supercritical flip bifurcation, the stable invariant circle resulting from the Neimark-Sacker bifurcation and the chaos in the sense of Marotto to verify our results.
TL;DR: In this article , the authors investigated the dynamical behaviors of a prey-predator model with multiple strong Allee effect and examined the fixed points of the model for existence and topological classification.
Abstract: In this study, the dynamical behaviors of a prey–predator model with multiple strong Allee effect are investigated. The fixed points of the model are examined for existence and topological classification. By selecting as the bifurcation parameter $\beta$, it is demonstrated that the model can experience a Neimark-Sacker bifurcation at the unique positive fixed point. Bifurcation theory is used to present the Neimark-Sacker bifurcation conditions of existence and the direction of the bifurcation. Additionally, some numerical simulations are provided to back up the analytical result. Following that, the model's bifurcation diagram and the triangle-shaped stability zone are provided.
TL;DR: In this article , the authors present a general mechanism of generating limit cycles in planar piecewise polynomial differential systems with two zones by means of a transcritical bifurcation at infinity and from a global centre.
Abstract: We present a general mechanism of generation of limit cycles in planar piecewise polynomial differential systems with two zones by means of a transcritical bifurcation at infinity and from a global centre. This study justifies the existence of limit cycles that arise through the intersection of the separation boundary with the one that characterizes the global centre.
TL;DR: In this paper, the mathematical model of four differential equations for organisms that describe the effect of anti-predation behavior, age stage and toxicity have been analyzed, and the results of the bifurcation behavior analysis have been fully presented using numerical simulation.
Abstract: In this study, the mathematical model of four differential equations for organisms that describe the effect of anti-predation behavior, age stage and toxicity have been analyzed. Local bifurcation and Hopf bifurcation have been studied by changing a parameter of a model to study the dynamic behavior determined by bifurcation curves and the occurrence states of bifurcation saddle node, transcritical and pitch fork bifurcation. The potential equilibrium point at which Hopf bifurcation occurs has been determined and the results of the bifurcation behavior analysis have been fully presented using numerical simulation.