Bifurcation Of Limit Cycles Research Articles

Limit cycle bifurcations by perturbing a kind of quadratic Hamiltonian system having a homoclinic loop

Limit cycle bifurcations by perturbing a kind of quadratic Hamiltonian system having a homoclinic loop

Melnikov functions and limit cycle bifurcations for a class of piecewise Hamiltonian systems

<abstract><p>This study evaluated the number of limit cycles for a class of piecewise Hamiltonian systems with two zones separated by two semi-straight lines. First, we obtained explicit expressions of higher Melnikov functions. Then we applied these expressions to find the upper bounds of the number of limit cycles bifurcated from a period annulus of a piecewise polynomial Hamiltonian system.</p></abstract>

Hopf bifurcation for a class of predator-prey system with small immigration

<p>The subject of this paper concerns with the bifurcation of limit cycles for a predator-prey model with small immigration. Since, in general, the biological systems are not isolated, taking into account immigration in the model becomes more realistic. In this context, we deal with a model with a Holling type Ⅰ function response and study, using averaging theory of second order, the Hopf bifurcation that can emerge under small perturbation of the biological parameters.</p>

Analysis of mixed rhythm and its dynamics in closed-loop respiratory control system driven by electromagnetic induction

The pre-Bötzinger complex is a crucial region for generating respiratory rhythms in mammals. Peripheral chemoreceptors have a significant influence on respiratory rhythm by monitoring changes in blood oxygen concentration and carbon dioxide concentration. This study introduces a closed-loop respiratory control model, which is driven by electromagnetic induction and based on the activation of pre-Bötzinger complex neurons. The model incorporates various factors including the motor pool, lung volume, lung oxygen, blood oxygen, and chemoreceptors. The response of the system subjected to the same hypoxic perturbation under different electromagnetic induction is studied, and the control effect of magnetic flux feedback coefficient on the recovery of mixed rhythms is investigated. Using bifurcation analysis and numerical simulations, it is found that the magnetic flux feedback coefficient has a significant influence on the ability to recover respiratory rhythm. The dynamic mechanism of the magnetic flux feedback coefficient on different hypoxic responses in closed-loop systems are revealed. Dynamic analysis indicates that under certain electromagnetic induction, the mixed bursting rhythm in the closed-loop system can autoresuscitate if the bifurcation structure before and after applying hypoxia perturbation are completely identical. However, when the bifurcation structure before and after applying hypoxia perturbation are different, the mixed bursting rhythm in the system cannot autoresuscitate. In addition, for the cases where automatic recovery is not achieved under mild electromagnetic induction, increasing the magnetic flux feedback coefficient appropriately can lead the system to autoresuscitate, which is closely related to the Hopf bifurcation and fold bifurcation of limit cycle. This study contributes to understanding the influence of the interaction between the central respiratory and peripheral chemoreceptive feedback on respiratory rhythm, as well as the control effect of external induction on the hypoxic response.

Global stability of the interior equilibrium and the stability of Hopf bifurcating limit cycle in a model for crop pest control

<p>Mathematical modeling and analysis of a crop-pest interacting system helps us to understand the dynamical properties of the system such as stability, bifurcations and chaos. In this article, a predator-prey type mathematical model for pest control using bio-pesticides has been analysed to study the global stability property of the interior equilibrium point. Moreover, the occurrence and orbital stability of Hopf bifurcating limit cycle solutions have been studied using ref30's conditions. Analytical and numerical results show that the interior equilibrium of the pest control model is globally asymptotically stable. Also, Hopf bifurcating occurs when the bifurcation parameter crosses the critical value, and the bifurcating periodic solution is found to be stable.</p>

Double homoclinic bifurcations by perturbing a class of cubic Z2-equivariant polynomial systems with nilpotent singular points

This paper focuses on the study of the limit cycle bifurcation of a general cubic Z2-equivariant Hamiltonian differential system with two nilpotent singular points. Initially, the necessary and sufficient conditions for the occurrence of two nilpotent singularities are provided in the unperturbed system, along with a complete description of all possible phase portraits on the plane. Subsequently, using a combination of the Melnikov function method and various analytical techniques, we derive the approximate expansion of the first order Melnikov function at the interval endpoint, as well as formula for its coefficients. These results can be utilized to analyze double homoclinic loop bifurcation of the perturbed system, finding the lower bound of limit cycles.

Grazing-Induced Dynamics of the Piecewise-Linear Chua’s Circuit

In this paper, we consider the piecewise-linear Chua’s circuit, which is well known for its rich variety of bifurcation, chaotic and other nonlinear phenomena. Suitable switching boundaries are introduced based on the piecewise-linear representation of Chua’s diode. In this way, we derive analytical conditions for a grazing bifurcation to occur, when one or two families of periodic orbits have a zero-velocity contact with the switching boundaries. In connection to this phenomenon, we also study the focus-center-limit cycle bifurcation and its implications regarding the system dynamics, from both analytical and numerical points of view. Furthermore, a detailed parametric study of Chua’s circuit is carried out via path-following techniques for nonsmooth dynamical systems, implemented via the continuation software COCO. This study reveals the presence of codimension-one bifurcations of limit cycles, such as those mentioned above, as well as classical (fold and period-doubling) bifurcations. The analysis confirms the presence of coexisting attractors, which are produced by a hysteresis loop induced by the interaction of a fold and a focus-center-limit cycle bifurcation.

Bifurcation of Limit Cycles from Boundary Equilibria in Impacting Hybrid Systems

Bifurcation of Limit Cycles from Boundary Equilibria in Impacting Hybrid Systems

Limit cycle bifurcations near a double 2-polycycle on the cylinder

In this paper, we consider the bifurcation problem of limit cycles near a double 2-polycycle for a cylinder near-Hamiltonian system. We first study the stability of a general homoclinic loop of type II on the cylinder, and then obtain certain results on the number and distribution of limit cycles for the cylinder near-Hamiltonian system near the double 2-polycycle by using the Melnikov function method and the method of stability-changing of a homoclinic loop of type I or II. Furthermore, we provide a way to find alien limit cycles for cylinder near-Hamiltonian system. As applications, we obtain the number of limit cycles of a class of cylinder near-Hamiltonian systems.

Bifurcation analysis in a predator–prey model with strong Allee effect on prey and density‐dependent mortality of predator

The influences of Allee effect and density‐dependent mortality on population growth are of great significance in ecology. In this paper, we first consider a Gause‐type predator–prey model with simplified Holling type IV functional response, strong Allee effect on prey, and density‐dependent mortality of predator. It is shown that the system exhibits rich and complex dynamics like bistability, local and global bifurcations such as transcritical bifurcation, saddle‐node bifurcation, Hopf bifurcation, cusp bifurcation, Bogdanov–Takens bifurcation, Bautin bifurcation, homoclinic bifurcation, saddle‐node bifurcation of limit cycle, and heteroclinic bifurcation. Next, we separately explore the influences of Allee effect and density‐dependent mortality on the dynamics of the same model. The results show that the strong Allee effect induces the occurrence of heteroclinic bifurcation and the reduction of the number of limit cycles, while the density‐dependent mortality stabilizes the system. Since the analytical expressions of the interior equilibria are difficult to derive, the results are verified numerically.

Modeling cooperative evolution in prey species using the snowdrift game with evolutionary impact on prey–predator dynamics

Cooperative evolution refers to the collaborative process by which two or more individuals or species interact and adapt together some specific behavior, leading to mutual benefits for their survival and reproduction. In this article, we employ the snowdrift game to model cooperative evolution in prey species and examine its evolutionary impact on the dynamics of prey–predator interactions. The model system consists of a replicator equation that captures the evolution of cooperative behavior in prey species, along with two growth equations of prey and predator incorporating the factors that influence cooperative evolution on prey reproduction. Our analysis reveals that in long time run, the prey population becomes polymorphic, allowing cooperative and defective individuals to coexist and persist over time. However, a very low cooperative proportion poses a risk of extinction for both prey and predator. Again if majority of prey species exhibit cooperative behavior, the predator species still faces possible extinction due to the strong group defense mechanism employed by prey. At some intermediate level of cooperative proportions, the proposed model system exhibits more rich and complex dynamics such as oscillatory patterns, bi-stable phenomena, homoclinic orbit, and saddle–node bifurcations of limit cycles. Further, we explore the transient dynamics of the proposed system through which one can anticipate the potential shifts in population size and can identify critical thresholds in promoting sustainable coexistence.

An Investigation of Dynamical Behavior of a Wing Model

AbstractBifurcations of equilibria of a wing model have been investigated in this paper. It is shown that the quintic nonlinear terms in the pitch and the plunge coordinate have affected the bifurcation structure of nontrivial equilibria in different degree. In contrast with the quintic stiffening parameter in plunge, the quintic parameter in pitch has a relatively significant effect, which will affect the number, position and stability of nontrivial equilibria. Therein two pairs of nontrivial equilibria with opposite stability coexist or disappear by two fold bifurcations. If the freestream velocity has been taken as a continuation parameter, it will affect the bifurcation structure of all the equilibria, including the trivial and the nontrivial. Wherein the equilibria vary from a trivial to two nontrivial ones by a pitchfork bifurcation. Then one of nontrivial equilibria experiences a supercritical Hopf bifurcation and the bifurcated limit cycles form an ellipsoidal structure with the limit cycles bifurcated from the trivial equilibrium.

On the number of limit cycles near a homoclinic loop with a nilpotent cusp of order m

This paper investigates the expansions of Melnikov functions near a homoclinic loop with a nilpotent cusp of order m. It presents a methodology for calculating all coefficients in these expansions, which can be employed to study the problem of limit cycle bifurcation. As an application, by utilizing the obtained results, the paper rigorously establishes that a polynomial Liénard system of degree n+1 has at least n+[n4] limit cycles near the homoclinic loop with a nilpotent cusp of order one. This work not only updates and generalizes existing results, but also provides a rigorous application of the obtained findings in the context of limit cycle bifurcation.

Bifurcating Limit Cycles with a Perturbation of Systems Composed of Piecewise Smooth Differential Equations Consisting of Four Regions

Systems composed of piecewise smooth differential (PSD) mappings have quantitatively been searched for answers to a substantial issue of limit cycle (LC) bifurcations. In this paper, LC numbers (LCNs) of a PSD system (PSDS) consisting of four regions are dealt with. A Melnikov mapping whose order is one is implicitly obtained by finding its originators when the system is perturbed under any nth degree of real polynomials. Then, the approach employing the Picard–Fuchs mapping is utilized to attain a higher boundary of bifurcation LCNs of systems composed of PSD functions with a global center. The method we used could be implemented to examine the problems related to the LC of other PSDS.

The important role of glial transmitters released by astrocytes in Alzheimer's disease: A perspective from dynamical modeling.

This paper aims to establish a coupling model of neuronal populations and astrocytes and, on this basis, explore the possible mechanism of electroencephalography (EEG) slowing in Alzheimer's disease (AD) from the viewpoint of dynamical modeling. First and foremost, excitatory and inhibitory time constants are shown to induce the early symptoms of AD. The corresponding dynamic nature is mainly due to changes in the amplitude and frequency of the oscillatory behavior. However, there are also a few cases that can be attributed to the change of the oscillation mode caused by the limit cycle bifurcation and birhythmicity. Then, an improved neural mass model influenced by astrocytes is proposed, considering the important effects of glutamate and adenosine triphosphate (ATP) released by astrocytes on the synaptic transmission process reported in experiments. The results show that a dysfunctional astrocyte disrupts the physiological state, causing three typical EEG slowing phenomena reported clinically: the decreased dominant frequency, the decreased rhythmic activity in the α band, and the increased rhythmic activity in the δ+θ band. In addition, astrocytes may control AD when the effect of ATP on synaptic connections is greater than that of glutamate. The control rate depends on the ratio of the effect of glutamate on excitatory and inhibitory synaptic connections. These modeling results can not only reproduce some experimental and clinical results, but, more importantly, may offer a prediction of some underlying phenomena, helping to inspire the disease mechanisms and therapeutic methods of targeting astrocytes.

Blowup Analysis of a Hysteresis Model Based Upon Singular Perturbations

In this paper, we provide a geometric analysis of a new hysteresis model that is based upon singular perturbations. Here hysteresis refers to a type of regularization of piecewise smooth differential equations where the past of a trajectory, in a small neighborhood of the discontinuity set, determines the vector-field at present. In fact, in the limit where the neighborhood of the discontinuity vanishes, hysteresis converges in an appropriate sense to Filippov’s sliding vector-field. Recently (2022), however, Bonet and Seara showed that hysteresis, in contrast to regularization through smoothing, leads to chaos in the regularization of grazing bifurcations, even in two dimensions. The hysteresis model we analyze in the present paper—which was developed by Bonet et al in a paper from 2017 as an attempt to unify different regularizations of piecewise smooth systems—involves two singular perturbation parameters and includes a combination of slow–fast and nonsmooth effects. The description of this model is therefore—from the perspective of singular perturbation theory—challenging, even in two dimensions. Using blowup as our main technical tool, we prove existence of an invariant cylinder carrying fast dynamics in the azimuthal direction and a slow drift in the axial direction. We find that the slow drift is given by Filippov’s sliding vector-field to leading order. Moreover, in the case of grazing, we identify two important parameter regimes that relate the model to smoothing (through a saddle-node bifurcation of limit cycles) and hysteresis (through chaotic dynamics, due to a folded saddle and a novel return mechanism).

Nilpotent center conditions in cubic switching polynomial Liénard systems by higher-order analysis

The aim of this paper is to investigate two important problems related to nilpotent center conditions and bifurcation of limit cycles in switching polynomial systems. Due to the difficulty in calculating the Lyapunov constants of switching polynomial systems at non-elementary singular points, it is extremely difficult to use the existing Poincaré-Lyapunov method to study these two problems. In this paper, we develop a higher-order Poincaré-Lyapunov method to consider the nilpotent center problem in switching polynomial systems, with particular attention focused on cubic switching Liénard systems. With proper perturbations, explicit center conditions are derived for switching Liénard systems at a nilpotent center. Moreover, with Bogdanov-Takens bifurcation theory, the existence of five limit cycles around the nilpotent center is proved for a class of switching Liénard systems, which is a new lower bound of cyclicity for such polynomial systems around a nilpotent center.

Complex dynamics of the passive biped robot with flat feet: Gait bifurcation, intermittency and crisis

During the past decades, the walking dynamics of different types of passive biped robots have garnered significant attention. This paper investigates the complex dynamics of a passive flat-footed robot, encompassing both the local bifurcation of limit cycles and global bifurcation, resulting in significant changes in the system’s walking pattern. To identify the robot’s dynamic behavior, the Lyapunov Exponents (LEs) are calculated using the small orthogonal perturbation vectors method. The obtained LEs diagram also unveils several subtle periodic windows scattered within the chaotic region, where interior crisis and intermittency occur. Moreover, two new kinds of gaits that coexist with the traditional period-1 ones are found. Of particular significance is the discovery of the double boundary crisis that caused by the critical period-3 unstable periodic orbit, and this crisis serves as the fundamental reason to the falling down on slopes of the flat-footed robots. By gaining a better understanding of the nature of passive dynamic walking, it becomes possible to design more efficient controllers that impose less stringent torque requirements on biped walking robots.

Degenerate codimension-2 cusp of limit cycles in a Holling–Tanner model with harvesting and anti-predator behavior

In this paper, a Holling–Tanner predator–prey model with constant-yield prey harvesting and anti-predator behavior is investigated. It is shown that the codimension of a nilpotent cusp is 3 and a degenerate Bogdanov–Takens bifurcation of codimension 3 acts as an organizing center for rich dynamical behaviors including saddle–node bifurcation, Hopf bifurcation, saddle–node bifurcation of limit cycles, and homoclinic bifurcation. Moreover, a Hopf bifurcation of codimension 2 is found, a codimension-2 cusp of limit cycles is firstly observed, and the corresponding normal form coefficients are also obtained, which indicate that there exists an acute angle region of three coexistent limit cycles. In particular, a reversed S-shaped saddle–node bifurcation curve of limit cycles is found. In addition, we observe that the anti-predator behavior may cause nearly extinction of the predator population while the prey population reaches its maximum which is less than the carrying capacity due to the effect of harvesting; however, the constant-yield prey harvesting may drive both species to extinction suddenly. Finally, numerical simulations including bifurcation diagrams and phase portraits are given to illustrate the theoretical results.

Exploring Limit Cycle Bifurcations in the Presence of a Generalized Heteroclinic Loop

This work revisits the number of limit cycles (LCs) in a piecewise smooth system of Hamiltonian with a heteroclinic loop generalization, subjected to perturbed functions through polynomials of degree m. By analyzing the asymptotic expansion (AE) of the Melnikov function with first-order M(h) near the generalized heteroclinic loop (HL), we utilize the expansions of the corresponding generators. This approach allows us to establish both lower and upper bounds for the quantity of limit cycles in the perturbed system. Our analysis involves a combination of expansion techniques, derivations, and divisions to derive these findings.