Bifurcation Of Limit Cycles Research Articles

Limit Cycles and Integrability of a Class of Quintic System

In this paper, a class of quintic systems is investigated. The first 13 focal values are computed with the aid of MATHEMATICA. Then the necessary conditions of integrability and linearizability are obtained and the sufficiency of every condition is proved. Meanwhile, bifurcation of limit cycles is discussed, 13 limit cycles can be bifurcated from the origin. As far as the number of limit cycles enclosing an isolated singular point is concerned, this is so far the best result for elementary singular points.

Dynamic behavior of self-acting gas-lubricated long journal bearing

Planar dynamics of a rotor supported by self-acting gas-lubricated journal bearing is investigated theoretically. Whirling of the shaft is one of the common problems in a gas-lubricated journal bearing system. As whirling increases, it becomes large enough to cause a destructive contact between the bearing surfaces. It is essential to predict the parameter range for the safe operation of the gas-lubricated journal bearing systems. An analytical model of the long, self-acting gas-lubricated journal bearing system is numerically integrated for analysis of steady-state solutions and limit cycle oscillations. Dynamic behavior of the system is analyzed with the variation of dimensionless bearing compressibility parameter Λ and dimensionless bearing stability parameter Λ2Ω. For a set of stability parameters Λ2Ω ranging from 0.01 to 1000, the critical value of Λ is determined which corresponds to angular speed ω of the rotor. With the decrease of parameter Λ2Ω, critical value Λ decreases. Limit cycles bifurcating from the equilibrium point at critical values are unstable for a low value of Λ2Ω=0.01 and stable for other higher values. In both the cases of stable and unstable limit cycles, limit cycles disappear beyond a compressibility parameter Λ and no further fold bifurcation of the limit cycle is detected. Results are presented as equilibrium point bifurcation and bifurcation of the limit cycle. The dynamics of the system and safe operating range are presented with diagrams. The results obtained are validated through trajectories of the journal center by numerical integration of the system of equations for different initial conditions presented in the figures.

Finite-amplitude stability in regenerative chatter: The effect of process damping nonlinearity and intermittent cutting in turning

The damping forces generated at the tool and workpiece interface, known as process damping, greatly influence the dynamics of regenerative chatter in machining. In addition to the increased stability against chatter at low cutting speeds, which is the well-known effect of process damping, post-chatter dynamics are also affected significantly by process damping nonlinearity, which is the less-studied effect of this phenomenon. Once the process vibrations become unstable after chatter occurrence, the process becomes intermittent in the sense that the tool loses contact with the workpiece and they are no longer in continuous engagement due to large-amplitude chatter vibrations. When vibration amplitudes are relatively large, process damping nonlinearity is shown to create stable limit cycles around which vibrations stabilize at a finite-amplitude. The relationship between the amplitude of the resulting limit cycles and the applied machining parameters is only known approximately from simplified models. Besides, the bifurcation of the resulting limit cycles by machining parameters such as cutting depth and spindle speed has not been studied. In this work, we use numerical continuation to develop the bifurcation diagrams of an SDOF model of regenerative chatter that includes process damping nonlinearity and non-smooth dynamics due to the disengagement of the tool and workpiece in post-chatter large-amplitude oscillations. The developed bifurcation diagrams accurately determine the relationship between the applied machining parameters and the amplitude of the resulting limit cycles. They also show that the stable limit cycles lose their stability by a subcritical non-smooth fold bifurcation when the amplitude is large enough to cause tool-workpiece disengagement. We also show that the static forces in the process, which are normally neglected in existing studies, become dynamic due to the intermittent tool-workpiece disengagement and alter the nature of limit cycle bifurcations. The findings of this study are validated numerically and by comparison to experimental results.

Analysis of Neuronal Oscillations of Fractional-Order Morris-Lecar Model

Fractional calculus is a new approach for modeling biological and physical phenomena with memory effects. Fractional calculus uses differential and integral operators including non-integer orders to study the non-linear behavior of physical and biological systems with some degrees of fractionality or fractality. Since the long memory properties of neuronal responses can be better explained using fractional derivative, in this study we generalize the integer-order Morris-Lecar model in the fractional-order domain to better modeling of neuron dynamics. To investigate the complex spiking patterns of fractional-order Morris-Lecar neural system the fractional calculus has been applied to build this new mathematical model. We compare the results with integer-order Morris-Lecar model. The analytical solutions of these equations cannot explicitly be obtained. Therefore, to find the dynamical behaviors of solutions, we used approximation and numerical schemes. Depending on the different parameters values for 0<η≤1, the fractional-order Morris-Lecar reproduces quiescent, spiking and bursting activities the same as its original model but for higher input current. We numerically discover the hopf bifurcation, saddle node bifurcation of limit cycle and homoclinic bifurcation for this model for different input current and derivative orders. Taking the advantages of the fractional order derivative, for a variety of orders, we define different classes of this model which helps to better extract all the complicated dynamics of this single neuron model.

Bifurcation of limit cycles for a class of quintic Hamiltonian systems with quartic perturbed terms

Bifurcation of limit cycles for a class of quintic Hamiltonian systems with quartic perturbed terms

Nonlinear analysis of forced mechanical systems with internal resonance using spectral submanifolds, Part II: Bifurcation and quasi-periodic response

In Part I of this paper, we have used spectral submanifold (SSM) theory to construct reduced-order models for harmonically excited mechanical systems with internal resonances. In that setting, extracting forced response curves formed by periodic orbits of the full system was reduced to locating the solution branches of equilibria of the corresponding reduced-order model. Here, we use bifurcations of the equilibria of the reduced-order model to predict bifurcations of the periodic response of the full system. Specifically, we identify Hopf bifurcations of equilibria and limit cycles in reduced models on SSMs to predict the existence of two-dimensional and three-dimensional quasi-periodic attractors and repellers in periodically forced mechanical systems of arbitrary dimension. We illustrate the accuracy and efficiency of these computations on finite-element models of beams and plates.

Cluster Oscillation of a Fractional-Order Duffing System with Slow Variable Parameter Excitation

The complicated dynamic behavior of a fractional-order Duffing system with slow variable parameter excitation is investigated. The stability and bifurcation behavior of the fast subsystem are analyzed by using the dynamic theory of fractional-order systems. The pitchfork bifurcation, Hopf bifurcation and limit cycle bifurcation are discussed in detail, and it was found that Hopf bifurcation only happens while the fractional order is bigger than 1. On the other hand, the influence of the amplitude of parametric excitation on cluster oscillation models is discussed. The results show that amplitude regulates cluster oscillation models with different bifurcation types. The point–point cluster oscillation only relates to pitchfork bifurcation. The point–cycle cluster oscillation includes pitchfork bifurcation and Hopf bifurcation. The point–cycle–cycle cluster oscillation involves three kinds of bifurcation, i.e., the pitchfork bifurcation, Hopf bifurcation and limit cycle bifurcation. The larger the amplitude, the more bifurcation types are involved. The research results of cluster oscillation and its generation mechanism will provide valuable theoretical basis for mechanical manufacturing and engineering practice.

On the number of limit cycles of a class of Liénard–Rayleigh oscillators

This paper is devoted to the study of limit cycle bifurcations for a class of cubic near-Hamiltonian systems, which are in the form of Liénard–Rayleigh oscillators. By using geometric techniques to the first order Melnikov functions, we provide a complete study on the maximum number of limit cycles in Poincaré bifurcation from a period annulus.

The discontinuous limit case of an archetypal oscillator with a constant excitation and van der Pol damping

We 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. The discontinuous dynamical system can exhibit three limit cycles and rich nonlinear phenomena, including boundary equilibrium bifurcation, Hopf bifurcation, grazing bifurcation, pseudo saddle–node loop bifurcation and double crossing limit cycle bifurcation. Because of destroyed symmetry for this system, it exhibits more complex and abundant dynamics than the smooth oscillator with van der Pol damping and can appear new dynamical phenomena, such as coexistence of a crossing limit cycle and a pseudo homoclinic loop, coexistence of two crossing limit cycles and the existence of pseudo saddle–node loops. The bifurcation diagram of the discontinuous system is shown in parameter space and all global phase portraits in the Poincaré disk are sketched in the phase plane.

Configuration of Ten Limit Cycles in a Class of Cubic Switching Systems

The bifurcation of limit cycles is an important part in the study of switching systems. The investigation of limit cycles includes the number and configuration, which are related to Hilbert’s 16th problem. Most researchers studied the number of limit cycles, and only few works focused on the configuration of limit cycles. In this paper, we develop a general method to determine the configuration of limit cycles based on the Lyapunov constants. To show our method by an example, we study a class of cubic switching systems, which has three equilibria: (0,0) and (±1,0), and compute the Lyapunov constants based on Poincaré return map, then find at least 10 small-amplitude limit cycles that bifurcate around (1,0) or (−1,0). Using our method, we determine the location distribution of these ten limit cycles.

On occurrence of mixed-torus bursting oscillations induced by non-smoothness

The paper devotes to the slow–fast behaviors of a higher-dimensional non-smooth system with the coupling of two scales. Some novel bursting attractors and interesting phenomena are presented, especially the so-called mixed-torus bursting oscillations, in which the trajectory moves along different tori in turn, though the bursting attractor still behaves in quasi-periodic form. Based on a 4-D hyper-chaotic model with two scrolls, a modified non-smooth slow–fast version is established. Upon the smooth and non-smooth bifurcations analysis, the coexisted attractors with the variation of the slow-varying parameter are derived. With the increase in the exciting amplitude, bursting oscillations may change from periodic to quasi-periodic attractor, the mechanism of which is obtained by employing the overlap of the transformed phase portrait and coexisted attractors as well as the bifurcations. Sliding along the boundary on the trajectory can be observed on the bursting attractor, the mechanism of which can be accounted for by employing the non-smooth theory or by the in-turn influence of two pseudo-attractors in different regions. Super-cone defined by an unstable focus may lead to the period-adding or period-decreasing phenomena in the spiking oscillations, while fold limit cycle bifurcation may result in the dramatical change of the spiking amplitude. Different limit cycles may involve the full vector field, resulting in the so-called mixed-torus bursting oscillations, in which the quiescent states seem to be pivots of the oscillations.

Limit Cycle Bifurcations Near a Heteroclinic Loop with Two Nilpotent Cusps of General Order

In this paper, we study the expansion of the first order Melnikov function near a heteroclinic loop with two nilpotent cusps of general order. More precisely, the order of the two cusps is [Formula: see text] and [Formula: see text] respectively, where [Formula: see text] [Formula: see text]. For general [Formula: see text] and [Formula: see text], we give the expansion of the first order Melnikov function and the formulas for the first few coefficients. We further give a general theorem on the number of limit cycles bifurcated from the heteroclinic loop. These results extend the existing results for [Formula: see text], [Formula: see text] and [Formula: see text] [Formula: see text]. As an application, these results are applied to study the number of limit cycles near a heteroclinic loop with two cusps of different order.

Bifurcation of limit cycles in piecewise quadratic differential systems with an invariant straight line

We solve the center-focus problem in a class of piecewise quadratic polynomial differential systems with an invariant straight line. The separation curve is also a straight line which is not invariant. We provide families having at the origin a weak-foci of maximal order. In the continuous class, the cyclicity problem is also solved, being 3 such maximal number. Moreover, for the discontinuous class but without sliding segment, we prove the existence of 7 limit cycles of small amplitude.

Optimization of stimulation waveforms for regulating spike-wave discharges in a thalamocortical model

Based on our modified thalamocortical model considering excitatory interneuron, this study incorporates deep brain stimulation (DBS) applied to pyramidal cluster and focuses on optimizing stimulus waveforms of DBS. The epileptic activities can be reproduced and modulated by inhibitory and excitatory interneurons, which are closely related to the generation of absence seizures. Under the bifurcation mechanism analysis, fold limit cycle and Hopf bifurcations well explain the termination and transition of various discharge states. To offer a potential way to improve control effects and energy savings, we investigate a range of waveforms covering rectangular, linear increase and decrease, triangular, exponential increase and decrease, Gaussian and sinusoidal pulse in detail. These single pulses all can effectively remove spike-wave discharges (SWD) with appropriate stimulus parameters, except for exponential decrease pulse. Additionally, stimulation outcomes are uncovered by adjusting stimulus timing and pulse width, and the optimal pulse width of each waveform has been suggested. By comprehensively analyzing the control rate, charge injected ability and energy requirement, the sinusoidal pulse achieves a better impact on eliminating SWD. From the perspective of rectangular and sinusoidal stimulations, the single pulse is of great advantage comparing to the traditional DBS, which may give some enlightenment for the clinical treatment, especially extend the application of DBS in neurological disorders.

Bifurcation of limit cycles from a parabolic–parabolic type critical point in a class of planar piecewise smooth quadratic systems

In this paper we study small amplitude limit cycles bifurcated from a parabolic–parabolic (PP) type critical point of planar piecewise smooth quadratic systems having exactly one switching line given by the x-axis. Besides the non-smoothness, more difficulties arise when the critical point has a parabolic contact. For such kind of critical point, to make sure that the return map is analytic, one has to use the generalized polar coordinates instead of the classic polar coordinates to compute the corresponding Lyapunov constants. Consequently, much more complicated computations are involved. By the recent results of Novaes and Silva, the index of the first nonzero Lyapunov constant for a PP type critical point is always an even number 2ℓ+2 with ℓ≥1. We call the corresponding weak focus (0,0) is of order ℓ. We obtain eight center conditions and six conditions under which (0,0) is a weak focus of order 6. Furthermore, we prove that at least seven limit cycles can bifurcate from (0,0).

On occurrence of sudden increase of spiking amplitude via fold limit cycle bifurcation in a modified Van der Pol–Duffing system with slow-varying periodic excitation

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.

Bifurcation of Limit Cycles for a Perturbed Piecewise Quadratic Differential Systems

Bifurcation of Limit Cycles for a Perturbed Piecewise Quadratic Differential Systems

Mechanism analysis for excitatory interneurons dominating poly-spike wave and optimization of electrical stimulation.

In addition to inhibitory interneurons, there exist excitatory interneurons (EINs) in the cortex, which mainly have excitatory projections to pyramidal neurons. In this study, we improve a thalamocortical model by introducing EIN, investigate the dominant role of EIN in generating spike and slow wave discharges (SWDs), and consider a non-rectangular pulse to control absence seizures. First, we display here that the improved model can reproduce typical SWDs of absence seizures. Moreover, we focus on the function of EIN by means of bifurcation analysis and find that EIN can induce transition behaviors under Hopf-type and fold limit cycle bifurcations. Specifically, the system has three stable solutions composing a tri-stable region. In this region, there are three attraction basins, which hints that external stimulation can drive the system trajectory from one basin to another, thereby eliminating abnormal oscillations. Furthermore, we compare the increasing ramp with rectangular pulse and optimize stimulation waveforms from the perspective of electrical charges input. The controlling role of the single increasing ramp to absence seizures is remarkable and the optimal stimulus parameters have been found theoretically. This work provides a computational model containing EIN and a theoretical basis for future physiological experiments and clinical research studies.

Novel bursting patterns and the bifurcation mechanism in a piecewise smooth Chua’s circuit with two scales

The aim of this paper is to investigate the influence of the coupling of two scales on the dynamics of a piecewise smooth dynamical system. A relatively simple model with two switching boundaries is taken as an example by introducing a nonlinear piecewise smooth resistor and a harmonically changed electric source into a typical Chua’s circuit. Taking suitable values of the parameters, four different types of bursting oscillations are observed corresponding to different values of the exciting amplitude. Regarding the periodic excitation as a slow-varying parameter, equilibrium branches of the fast subsystem as well as the related bifurcations, such as fold bifurcation, Hopf bifurcation, period doubling bifurcation, nonsmooth Hopf bifurcation and nonsmooth fold limit cycle bifurcation, are explored with theoretical and numerical methods. With the help of the overlap of the transformed phase portrait and the equilibrium branches, the mechanism of the bursting oscillations can be analyzed in detail. It is found that for relatively small exciting amplitude, since the trajectory is governed by a smooth subsystem, only conventional bifurcations take place, leading to the transitions between the spiking states and quiescent states. However, with an increase of the exciting amplitude so that the trajectory passes across the switching boundaries, nonsmooth bifurcations occurring at the boundaries may involve the structures of attractors, leading to complicated bursting oscillations. Further increasing the exciting amplitude, the number of the spiking states decreases although more bifurcations take place, which can be explained by the delay effect of bifurcation.

Bursting oscillations with adding-sliding structures in a Filippov-type Chua’s circuit

In this paper, a modified Chua’s circuit, possessing an external excitation current and a piecewise nonlinear resistor, is considered to investigate bursting oscillations and the dynamical mechanism in the Filippov system, focusing on the effects of sliding bifurcations on the bursting dynamics. Five typical representative bursting oscillations are observed when there is a gap between the external excitation frequency and the natural one. Codimension-1 bifurcations of the fast subsystem are discussed by regarding the whole excitation term as a bifurcation parameter. The necessary conditions of conventional bifurcations of the equilibria and the bifurcations of boundary equilibria are obtained via theoretical analysis. Both the local adding-sliding bifurcation, crossing-sliding bifurcation, non-smooth fold bifurcation, the fold bifurcation of non-smooth limit cycle in the fast subsystem and the adding-sliding bifurcation in the slow–fast coupling system are observed via numerical method. Based on slow–fast analysis method and the bifurcation analysis, the dynamical mechanism is discovered. Research found that the non-smooth fold bifurcation of the boundary equilibrium can lead to jumping behaviors of the trajectory from the switching manifold to the attractors of the subsystem. The crossing-sliding bifurcation may not cause the transition between spiking state and quiescent state. The local adding-sliding bifurcation in the fast subsystem can lead to adding-sliding bifurcation of non-smooth limit cycle in the coupling system. There may exist adding-sliding structure in bursting oscillations before or after the adding-sliding bifurcation.