Bifurcation Of Limit Cycles Research Articles

Bifurcation of limit cycles from a Liénard system with asymmetric figure eight-loop case

<p style='text-indent:20px;'>In this paper, we consider the number of limit cycles of the Li<inline-formula><tex-math id="M2">\begin{document}$ \acute{e} $\end{document}</tex-math></inline-formula>nard system of the form <inline-formula><tex-math id="M3">\begin{document}$ \dot{x} = y, \ \dot{y} = -x(x^2+bx-1)+\varepsilon f_{m}(x)y $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M4">\begin{document}$ b>0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ f_{m}(x) = \sum_{i = 0}^m a_{i}x^{i} $\end{document}</tex-math></inline-formula> is a polynomial of <inline-formula><tex-math id="M6">\begin{document}$ x $\end{document}</tex-math></inline-formula> with degree not greater than <inline-formula><tex-math id="M7">\begin{document}$ m $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M8">\begin{document}$ 0<\varepsilon \ll 1 $\end{document}</tex-math></inline-formula>. By studying the number of isolated zeros of the corresponding Abelian integral <inline-formula><tex-math id="M9">\begin{document}$ I(h) = \oint_{L_{h}}f_{m}(x)ydx, $\end{document}</tex-math></inline-formula> we obtain the upper bound of the number of limit cycles that bifurcated from periodic orbits of the unperturbed system for <inline-formula><tex-math id="M10">\begin{document}$ \varepsilon = 0 $\end{document}</tex-math></inline-formula>.</p>

Hopf bifurcation problems near double positive equilibrium points for a class of quartic Kolmogorov model

<abstract><p>The Kolmogorov model is a class of significant ecological models and is initially introduced to describe the interaction between two species occupying the same ecological habitat. Limit cycle bifurcation problem is close to Hilbertis 16th problem. In this paper, we focus on investigating bifurcation of limit cycle for a class of quartic Kolmogorov model with two positive equilibrium points. Using the singular values method, we obtain the Lyapunov constants for each positive equilibrium point and investigate their limit cycle bifurcations behavior. Furthermore, based on the analysis of their Lyapunov constants' structure and Hopf bifurcation, we give the condition that each one positive equilibrium point of studied model can bifurcate 5 limit cycles, which include 3 stable limit cycles.</p></abstract>

Limit cycle bifurcations of near-Hamiltonian systems with multiple switching curves and applications

<p style='text-indent:20px;'>In the present paper, we are devoted to the study of limit cycle bifurcations in piecewise smooth near-Hamiltonian systems with multiple switching curves, obtaining a formula of the first order Melnikov function in general case. As an application, we give lower bounds of the number of limit cycles for a piecewise smooth near-Hamiltonian system with a closed switching curve passing through the origin under piecewise polynomial perturbations.</p>

Third-order bifurcation of limit cycles for a perturbed quartic isochronous centre

Third-order bifurcation of limit cycles for a perturbed quartic isochronous centre

Third-Order Bifurcation of Limit Cycles for a Perturbed Quartic Isochronous Center

Third-Order Bifurcation of Limit Cycles for a Perturbed Quartic Isochronous Center

A stage structured predator-prey model with periodic orbits

<p style='text-indent:20px;'>This paper studies the existence and multiplicity of periodic orbits of a stage-structured predator-prey model with Beddington-DeAngelis functional response. We derive an existence criterion of periodic orbit in terms of inequalities of system's nine parameters and prove that the system admits at least two limit cycles or three limit cycles via subcritical Hopf bifurcation or generalized Hopf bifurcation theory and hypersurface theory. We also prove that each one-parameter system possesses at least one limit cycle when it is larger or smaller than its Hopf bifurcating value, or between its Hopf bifurcating values. Combining theoretic analysis and numerical simulations, we investigate global bifurcations of limit cycles for a varying parameter system, which provides a plenty of bifurcating properties and again suggests that the system has at least three limit cycles. Similar results are obtained for the system without mutual interference.</p>

Limit cycles in piecewise smooth perturbations of a class of cubic differential systems

In this paper, we study the bifurcation of limit cycles from a class of cubic integrable non-Hamiltonian systems under arbitrarily small piecewise smooth perturbations of degree n . By using the averaging theory and complex method, the lower and upper bounds for the maximum number of limit cycles bifurcating from the period annulus of the unperturbed systems are given at first order in ε . It is also shown that in this case, the maximum number of limit cycles produced by piecewise smooth perturbations is almost twice the upper bound of the maximum number of limit cycles produced by smooth perturbations for the considered systems.

Dynamic mechanism of epileptic seizures induced by excitatory pyramidal neuronal population

<abstract><p>The pyramidal neuronal population (PY) in the cerebral cortex is closely related to epilepsy, while the excitability of PY is directly affected by the excitatory interneurons (EIN), the inhibitory interneurons (IN), and the thalamic relay nucleus (TC). Based on this, we use the thalamocortical neural field model to explore the dynamic mechanism of system transition by taking the synaptic connection strengths of the above three nuclei on PY as the main factor affecting seizures. The results show that the excitatory effects of EIN on PY induce transitions from 1-spike and wave discharges (SWDs) to 2-spike and wave discharges (2-SWDs), the inhibitory effects of IN on PY induce transitions from saturated state to tonic oscillation state, and the excitatory effects of TC on PY induce transitions from clonic oscillation state to saturated state. According to the single-parameter bifurcation analysis, it is found that Hopf and fold limit cycle bifurcations are the key factors leading to the state transition. In addition, the state analysis of the three pathways is carried out in pairs. The results show that the system produces more types of epileptic seizures with the combined action of EIN and TC on PY. According to the two-parameter bifurcation curve, we obtain the stable parameter areas of tonic-clonic oscillations, SWDs, 2-SWDs and saturated discharges, and clearly find the reasonable transition path between tonic-clonic seizures and absence seizures. This may provide some theoretical guidance for the transmission and evolution of seizures.</p></abstract>

Periodic Solutions of the Forest Pest System Via Hopf Bifurcation and Averaging Theory

This work aims to analyse the dynamic behaviours of the forest pest system. We confirm the forest pest system in plane for limit cycles bifurcating existence from a Hopf bifurcation under certain conditions by using the first Lyapunov coefficient and the second-order of averaging theory. It is shown that all stationary points in this system have Hopf bifurcation points and provide an estimation of the bifurcating limit cycles.

All possible bursting oscillations in neighborhood of a Hopf bifurcation point induced by low-frequency excitation

Hopf bifurcation may be one of most common bifurcations, which results in the transitions between quiescence and spiking states in a bursting attractor, since it causes the alternation between an equilibrium point and a limit cycle. Though a lot of bursting attractors related to Hopf bifurcation have been obtained, while how many types of bursting attractors may appear in the neighborhood of a Hopf bifurcation point still remains an open problem. Here we try to answer the question based on the normal form up to the seventh order of a vector field with Hopf bifurcation at the origin. By introducing a low-frequency external excitation to the normal form, we derive the equilibrium branches and their bifurcations of the fast subsystem with the variation of the slow-varying excitation, based on which four typical cases corresponding to qualitatively different evolution of the behaviors are presented. Accordingly, four types of quasi-periodic bursting oscillations, i.e. Hopf/Hopf bursting attractors with (1) and (2) without saddle on the limit cycle bifurcation in the spiking oscillations, respectively, (3) fold/Hopf/fold/Hopf bursting attractors, and (4) bursting attractor with saddle on limit cycle bifurcation at the transition, are obtained with the increase of the exciting amplitude, the mechanism of which is obtained by overlapping equilibrium branches together with bifurcations in generalized autonomous system and transformed phase portrait. It is found that when two coexisted stable attractors are separated by an unstable attractor on the phase plane in generalized autonomous system, the influence of one of the stable attractors on the dynamics of full system may disappear. Furthermore, when the window for a stable attractor between two bifurcation points along slow-varying parameter axis is relatively short, the effect of the attractor as well as the bifurcation can only appear when the external excitation changes slowly enough. The amplitude of the repetitive spiking oscillations may vary according to two different limit cycles separated by saddle on limit cycle bifurcation, at which a sudden change of the oscillating amplitude can be observed.

General study on limit cycle bifurcation near a double homoclinic loop

In this paper, we study the bifurcation problem of limit cycles near a general double homoclinic loop. We establish a general theory to obtain a lower bound of the maximal number of limit cycles near the double homoclinic loop. As an application, we prove that a near-Hamiltonian system of the form x˙=y(y2−1)+ε∑i=0naix2i+1,y˙=−x has at least [52n] limit cycles for 1≤n≤56. This number is maximal that we can find so far for the system.

Limit cycles in piecewise polynomial Hamiltonian systems allowing nonlinear switching boundaries

This paper aims to study the limit cycles of planar piecewise polynomial Hamiltonian systems of degree n with the switching boundary y=xm, where m and n are positive integers. We answer a version of the 16th Hilbert problem for such systems, providing an upper bound for the maximum number of limit cycles in the function of m and n. We also are devoted to giving a lower bound via perturbing piecewise linear Hamiltonian systems having at the origin a global center. For this, a complete classification of the center conditions is established. In pursuit of a better lower bound, we require that this global center is nonlinear induced by the piecewise linearity, instead of the normal linear differential center considered in most of the existing articles. This renders that the traveling time of the unperturbed periodic orbits in each smooth zone is not calculable explicitly. Thus it is difficult to use the known Melnikov functions and averaged functions to study the bifurcating limit cycles. To overcome this difficulty, we develop an arbitrary order Melnikov-like function, which does not depend on the traveling time, for general d-dimensional piecewise smooth integrable systems allowing nonlinear switching boundaries. Finally, employing the new Melnikov-like function to the considered perturbation problem, we obtain a lower bound and an upper bound for the maximum number of limit cycles bifurcating from the unperturbed periodic orbits up to any order.

Hopf bifurcation at infinity in 3D Relay systems

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. A criticality parameter is found, whose sign determines the character of the bifurcation. When such non-degeneracy parameter vanishes, a higher co-dimension bifurcation takes place, giving rise to the emergence of a curve of saddle–node bifurcations of periodic orbits, which allows to determine parameter regions where two limit cycles coexist.The existence of a large amplitude limit cycle that bifurcates from infinity is justified through a suitable adaptation of the closing equations method, and analytical expressions for its amplitude and period are provided. Derivatives of the corresponding transition maps are rigorously studied in order to characterize the stability of the bifurcating limit cycle.The theoretical results are applied to a specific family of 3D relay systems, where several high co-dimension bifurcation points are detected, organizing the bifurcation set of the family.

Cyclicity of period annulus for a class of quadratic reversible systems with a nonrational first integral

In this paper, we study the quadratic perturbations of a one-parameter family of reversible quadratic systems whose first integral contains the logarithmic function. By the criterion function for determining the lowest upper bound of the number of zeros of Abelian integrals, we obtain that the cyclicity of either period annulus is two. To the best of our knowledge, this is the first result for the cyclicity of period annulus of the one-parameter family of reversible quadratic systems whose first integral contains the logarithmic function. Moreover, the simultaneous bifurcation and distribution of limit cycles from two-period annuli are considered.

Global studies on a continuous planar piecewise linear differential system with three zones

This paper is concerned with the global dynamics of a continuous planar piecewise linear differential system with three zones, where the dynamic of the one of the exterior linear zones is saddle and the remaining one is anti-saddle. We give all global phase portraits in the Poincaré disc and the complete bifurcation diagram including boundary equilibrium bifurcation curves, degenerate boundary equilibrium bifurcation curves, homoclinic bifurcation curves and double limit cycle bifurcation curves. Its application in a second-order memristor oscillator is shown. Finally, some numerical phase portraits are demonstrated to illustrate our theoretical results.

Bifurcation of Limit Cycles of a Piecewise Smooth Hamiltonian System

Bifurcation of Limit Cycles of a Piecewise Smooth Hamiltonian System

Bifurcation and Path-Following Analysis of Periodic Orbits of a Fermi Oscillator Model

In this research, we offer a bifurcation analysis to describe impacting, stick, and grazing between a particle and the piston to better understand the nonlinear dynamics of a Fermi oscillator. The principles of hybrid dynamical systems will be utilized to explain the moving process in such a Fermi oscillator. The motion complexity and stick mechanism of such an oscillator are demonstrated using periodic and chaotic motions. We introduce the hybrid dynamical system and numerical continuation detection tool COCO. Physical and mathematical models are used to construct the obtained bounded mathematical model. The frequency, amplitude in oscillating base displacement, and the gap between the stationary boundary and the piston’s equilibrium position are the major parameters. We employ path-following analysis to illustrate the bifurcation that leads to solution destabilization. From the viewpoint of eigenvalue analysis, the essence of period-doubling and Fold bifurcation is revealed. Numerical continuation methods are used to perform a complete one-parameter bifurcation analysis of the Fermi oscillator. The presence of codimension-1 bifurcations of limit cycles, like period-doubling, fold, and grazing bifurcations, is shown in this work. Then, the one-parameter continuation analysis is tracked in two-parameter bifurcation diagrams that include a second important factor of interest. Bifurcations cause all solutions to lose stability, according to our findings. The acquired results provide a better understanding of the Fermi oscillator mechanism and demonstrate that we may control the system dynamics by modifying the parameters.

An Investigation of the Bifurcation Behavior of an F-18 Aircraft Model

Bifurcations of equilibria and limit cycles of an F-18 aircraft model have been investigated in this paper based on one or two continuation parameters. First, it is shown that the aircraft can be in a stable straight-and-level flight state by coordinating the elevator deflection and the engine thrust and fixing the other parameters. Second, the aircraft may be disturbed by perturbations out of the longitudinal plane and get into the lateral-directional motion mode by a branch point bifurcation. Third, when the straight flight state loses its stability, body axis angular rates will present periodic or quasi-periodic motion pattern by the Hopf or Neimark–Sacker bifurcation. Finally, bifurcation structure of limit cycles has been exhibited, including the generalized Hopf bifurcation, the Neimark–Sacker bifurcation, the Hopf-Hopf bifurcation and the fold-Neimark–Sacker bifurcation. Meanwhile corresponding bifurcation curves in two-parameter plane have been depicted with the help of numerical continuation techniques.

Optimizing energy dissipation in gas foil bearings to eliminate bifurcations of limit cycles in unbalanced rotor systems

High-speed rotor systems mounted on gas foil bearings present bifurcations which change the quality of stability, and may compromise the operability of rotating systems, or increase noise level when amplitude of specific harmonics drastically increases. The paper identifies the dissipating work in the gas film to be the source of self-excited motions driving the rotor whirling close to bearing’s surface. The energy flow among the components of a rotor gas foil bearing system with unbalance is evaluated for various design sets of bump foil properties, rotor stiffness and unbalance magnitude. The paper presents a methodology to retain the dissipating work at positive values during the periodic limit cycle motions caused by unbalance. An optimization technique is embedded in the pseudo-arc length continuation of limit cycles, those evaluated (when exist) utilizing an orthogonal collocation method. The optimization scheme considers the bump foil stiffness and damping as the variables for which bifurcations do not appear in a certain speed range. It is found that secondary Hopf (Neimark–Sacker) bifurcations, which trigger large limit cycle motions, do not exist in the unbalanced rotors when bump foil properties follow the optimization pattern. Period-doubling (flip) bifurcations are possible to occur, without driving the rotor in high response amplitude. Different design sets of rotor stiffness and unbalance magnitude are investigated for the efficiency of the method to eliminate bifurcations. The quality of the optimization pattern allows optimization in real time, and gas foil bearing properties shift values during operation, eliminating bifurcations and allowing operation at higher speed margins. Compliant bump foil is found to enhance the stability of the system.

Maximum number of limit cycles bifurcating from the period annulus of cubic polynomial systems

This paper is devoted to the limit cycle bifurcation problem for some cubic polynomial systems, whose unperturbed systems have a period annulus and two invariant lines. Using the first order Melnikov function and Chebyshev criterion, we obtain the maximum number of limit cycles bifurcating from the period annulus. It improves a known result given by Sui and Zhao [Internat. J. Bifur. Chaos Appl. Sci. Engrg. 28 (2018), 1850063].