Limit Cycle Research Articles

Bifurcation Analysis of a Holling–Tanner Model with Generalist Predator and Constant-Yield Harvesting

In this paper, we introduce constant-yield prey harvesting into the Holling–Tanner model with generalist predator. We prove that the unique positive equilibrium is a cusp of codimension 4. As the parameter values change, the system exhibits degenerate Bogdanov–Takens bifurcation of codimension 4. Using the resultant elimination method, we show that the positive equilibrium is a weak focus of order 2, and the system undergoes degenerate Hopf bifurcation of codimension 2 and has two limit cycles. By numerical simulations, we demonstrate that the system exhibits homoclinic bifurcation and saddle–node bifurcation of limit cycles as the parameters are varied. The main results show that constant-yield prey harvesting and generalist predator can lead to complex dynamic behavior of the model.

Control of large amplitude limit cycle of a multi-dimensional nonlinear dynamic system of a composite cantilever beam

For the first time, a control strategy based on Fuzzy Sliding Mode Control is implemented in the control of a large amplitude limit cycle of a composite cantilever beam in a multi-dimensional nonlinear form. In the dynamic model establishment of the investigated structure, the higher-order shearing effect is applied, as well as the second-order discretization. Numerical simulation demonstrates that a multi-dimensional nonlinear dynamic system of the investigated structure is demanded for accurate estimation of large amplitude limit cycle responses. Therefore, a control strategy is employed to effectively suppress such responses of the beam in multi-dimensional nonlinear form.

Pulsing in the Ahu‘ail‘au pond–spillway system during the 2018 Klauea eruption: a dynamical systems perspective

During the 2018 K $\unicode{x012B}$ lauea lower East Rift Zone eruption, lava from 24 fissures inundated more than 8000 acres of land, destroying more than 700 structures over three months. Eruptive activity eventually focused at a single vent characterized by a continuously fed lava pond that was drained by a narrow spillway into a much wider, slower channelized flow. The spillway exhibited intervals of ‘pulsing’ behaviour in which the lava depth and velocity were observed to oscillate on time scales of several minutes. At the time, this was attributed to variations in vesiculation originating at depth. Here, we construct a toy fluid dynamical model of the pond–spillway system, and present an alternative hypothesis in which pulsing is generated at the surface, within this system. We posit that the appearance of pulsing is due to a supercritical Hopf bifurcation driven by an increase in the Reynolds number. Asymptotics for the limit cycle near the bifurcation point are derived with averaging methods and compare favourably with the cycle periodicity. Because oscillations in the pond were not observable directly due to the elevation of the cone rim and an obscuring volcanic plume, we model the observations using a spatially averaged Saint-Venant model of the spillway forced by the pond oscillator. The predicted spillway cycle periodicity and waveforms compare favourably with observations made during the eruption. The unusually well-documented nature of this eruption enables estimation of the viscosity of the erupting lava.

Nonsmooth dynamics of a Filippov predator–prey ecological model with antipredator behavior

This article proposes a class of nonsmooth Filippov pest–predator ecosystems with intermittent control strategies based on the pest’s antipredator behavior. aiming to investigate the influence of control strategies and switching thresholds on pest control. First, a comprehensive theoretical analysis of various equilibria within the Filippov system is undertaken, emphasizing the presence and stability of sliding mode dynamics and pseudoequilibrium. Secondly, through numerical simulations, the article discusses boundary-focus, boundary-node, and boundary-saddle bifurcation. Finally, the nonexistence of limit cycles in the Filippov system is theoretically studied. The research indicates that the solution trajectories of the model ultimately stabilize either at the real equilibria or at pseudoequilibrium on the model’s switching surface. Moreover, when the model has multiple coexisting real equilibrium and pseudoequilibrium, the pest-control strategy is correlated with the initial density of both the pest and the predator population.

Hunting stability control of high-speed bogie based on active yaw damper

Hunting stability is an inherent property of railway vehicles that determines the operational speed. This paper establishes a half-vehicle model with nonlinear wheel/rail equivalent conicity and interactive forces. Additionally, the dynamic performances of vehicle under various levels of wheel wear with passive suspension are compared and analyzed. Importantly, the investigation delves into the hunting stability of the vehicle system, employing both linear and nonlinear control approaches. The results demonstrate a notable reduction in critical speed during the end-worn period with a passive suspension. However, this reduction can be substantially countered through the application of active control, resulting in a significant speed increase. The implementation of stiffness control raises the frequency of limit cycles, whereas damping control serves to diminish it. Notably, an appropriate linear cubic stiffness control effectively mitigates the amplitudes of limit cycles during instances of instability. Moreover, the control strategy derived from the simplified model is extended to enhance the stability of the entire vehicle system. The research findings hold the potential to offer a promising strategy for the active control of high-speed vehicles, particularly during periods of wheel wear.

Four crossing limit cycles of a family of discontinuous piecewise linear systems with three zones separated by two parallel straight lines

Four crossing limit cycles of a family of discontinuous piecewise linear systems with three zones separated by two parallel straight lines

Role of Allee and Fear for Controlling Chaos in a Predator–Prey System with Circulation of Disease in Predator

This paper explores a predator–prey system featuring fear and disease within the predator population,utilizing the Rosenzweig–MacArthur model with Holling type-II functional response. The primary focus lies in investigating the impact of a fear factor, wherein the prey’s growth rate is hindered due to predator-induced fear. Additionally, the model accounts for the spread of disease among predators,leading to a division between susceptible and infected predator subpopulations. The inclusion of an Allee effect in the susceptible predator further enriches the model. The study involves a thorough examination, encompassing local and global stability analysis as well as Hopf bifurcation analysis around the interior equilibrium point. Numerical simulations underscore a noteworthy observation: an escalation in interaction force propels the system into chaotic dynamics,marked by stable focus, limit cycles and period-doubling phenomena. A noteworthy finding pertains to the influence of the Allee parameter ([Formula: see text]) on chaotic dynamics. As the Allee parameter values increase, the system tends to stable focus through a sequence of chaotic states, period-doubling and limit cycles. Subsequently, the paper introduces the role of another pivotal parameter, the fear factor, into the chaotic dynamics. Intriguingly, chaos transforms into stable focus through diverse nonlinear phenomena, including period-doubling and limit cycles. This nuanced exploration of parameters sheds light on the intricate dynamics governing the predator–prey system, offering a comprehensive understanding of the interplay between fear, disease and ecological factors. So our observation throughout this paper that how chaos behaves here after one by one injection of our new features: fear factor and Allee parameter?

IMPACT OF HUNTING COOPERATION AND FEAR EFFECT IN A GENERALIST PREDATOR–PREY MODEL

Predator foraging facilitation or hunting cooperation and the antipredator behavior of prey are essential mechanisms in evolutionary biology and ecology and may strongly influence the predator–prey dynamics. In a real-world scenario, this behavioral tendency is well documented, but less is known about how it could affect the dynamics between predator and prey. Here, we investigate the impact of the fear of predator on prey and the hunting cooperation in predator on the predator–prey dynamics, where the predator is assumed to be of generalist type. We observe that without fear, even with the high level of hunting cooperation, both populations may coexist, though the increasing level of hunting cooperation reduces the prey density at coexistence equilibrium. Moreover, increasing level of fear also destabilizes the system with and without hunting cooperation. Further, in the presence of hunting cooperation and fear effect, the model shows three different types of bistability phenomena: bistability between two coexisting equilibria, bistability between coexisting equilibria and prey-free equilibrium, and bistability between stable limit cycle and coexisting equilibria. In addition, saddle-node, Hopf, transcritical bifurcation of codimension one, Bautin (generalized Hopf), Bogdanov–Takens, and cusp bifurcation of codimension two are observed.

Influence of the magnetic flux on the dynamics of a self-sustaining System: Analytical, numerical and analogical investigations

Abstract This paper investigates the nonlinear dynamics of a ferroelectric enzyme-substrate
reaction modeled by \textcolor{red}{the} bi-rhythmic \emph{van der Pol} oscillator coupled to \textcolor{red}{the} magnetic
flux. We derive the equilibrium points and study their stability.
We analyze some bifurcation structures and the variation of the Lyapunov exponents.
The phenomena of symmetric attractors and the
 anti-monotonicity are observed. \textcolor{red}{By} increasing the magnetic flux,
 \textcolor{red}{we find that the equilibrium points are stable},
 tends to control chaotic regimes, and affects regular and quasi-regular ones.
 As the magnetic flux increases, the amplitude of the oscillations around the equilibrium
 points decreases and the \textcolor{red}{amplitude of the}
 limit cycles at the Hopf bifurcation \textcolor{red}{tends} to disappear.
 Further \textcolor{red}{increasing the magnetic flux gives} rise to chaotic dynamics.
 The electrical circuit and analogical simulations are \textcolor{red}{derived} using the PSpice software.
\textcolor{red}{The agreement between analogical and numerical results is acceptable}.

The Limit Cycles for a Class of Non-autonomous Piecewise Differential Equations

The Limit Cycles for a Class of Non-autonomous Piecewise Differential Equations

Estimates for the Number of Limit Cycles in Discontinuous Generalized Liénard Equations

Estimates for the Number of Limit Cycles in Discontinuous Generalized Liénard Equations

Non-Linear Dynamics of Simple Elastic Systems Undergoing Friction-Ruled Stick–Slip Motions

The stick–slip phenomenon is a jerking motion that can occur while two objects slide over each other with friction. There are several situations in which this phenomenon can be observed: between the slabs of the friction dampers used to mitigate vibrations in buildings, as well as between the components of the base isolation systems used for seismic protection. The systems of this kind are usually designed to work in a smooth and flawless manner, but under particular conditions undesired jerking motions may develop, yielding complex dynamic behavior even when only a few degrees of freedom are involved. A simplified approach to the problems of this kind leads to the mechanical model of a rigid block connected elastically to a rigid support and at the same time with friction to a second rigid support, both the supports having a prescribed motion. Despite the apparent simplicity of this model, it is very useful for studying important features of the non-linear dynamics of many physical systems. In this work, after a suitable formulation of the problem, the equations of motion are solved analytically in the sticking and sliding phases, and the influence of the main parameters of the system on its dynamics and limit cycles is investigated and discussed.

Limit Cycles and Local Bifurcation of Critical Periods in a Class of Switching Z2 Equivariant Quartic System

In this paper, the limit cycles and local bifurcation of critical periods for a class of switching Z2 equivariant quartic system with two symmetric singularities are investigated. First, through the computation of Lyapunov constants, the conditions of the two singularities to become the centers are determined. Then, we prove that there are at most 18 limit cycles with a distribution pattern of 9-9 around the two symmetric singular points of the system. Numerical simulation is conducted to validate the obtained results. Furthermore, by calculating the period constants, we determine the conditions for the critical point to be a weak center of finite order. Finally, the number of local critical periods that bifurcate from the equilibrium point under the center conditions is discussed. This study presents the first example of a quartic switching smooth system with 18 limit cycles and 4 local critical periods bifurcating from two symmetric singular points.

Sliding dynamics of a Filippov ecological system with nonlinear threshold control and pest resistance

By applying the integrated pest management strategy to address pest control problems, Filippov pest-natural enemy systems have been extensively studied. However, little attention has been paid to pest changing rate and pest resistance in these studies. Therefore, this paper proposes a complex and more realistic Filippov pest-natural enemy system with curved discontinuity boundary by considering Beddington–DeAngelis functional response and pest resistance, as well as incorporating pest changing rate in the threshold control index. Based on the relevant theory of the Filippov system, we analyze a sextic equation and obtain that the proposed Filippov system can have at most six sliding segments. By using qualitative techniques, we discuss the sliding segment and pseudo-equilibrium bifurcations. Numerically, we investigate the local and global sliding bifurcations. It can be detected that one new local sliding bifurcation occurs, which could be called the pseudo-Hopf bifurcation but differs from the traditional pseudo-Hopf bifurcation as it involves two sliding segments with opposite stabilities and generates a sliding limit cycle. We also discover two interesting global sliding bifurcations with novel bifurcation processes, which may be termed triple limit cycle bifurcations as they both relate to the appearance of three nested limit cycles.

Dynamically selected steady states and criticality in non-reciprocal networks

We consider a simple neural network model, evolving via non-linear coupled stochastic differential equations, where neural couplings are random Gaussian variables with non-zero mean and arbitrary degree of reciprocity. Using a path-integral approach, we analyse the dynamics, averaged over the network ensemble, in the thermodynamic limit. Our results show that for any degree of reciprocity in the couplings, two types of criticality emerge, corresponding to ferromagnetic and spin-glass order, respectively. The critical lines separating the disordered from the ordered phases is consistent with spectral properties of the coupling matrix, as derived from random matrix theory. As the non-reciprocity (or asymmetry) in the couplings increases, both ordered phases diminish in size, ultimately resulting in the disappearance of the spin-glass phase when the couplings become anti-symmetric. We investigate non-fixed point steady-state solutions for uncorrelated interactions. For such solutions the time-lagged correlation function evolves according to a gradient-descent dynamics on a potential, which depends on the stationary variance. Our analysis shows that in the spin-glass region, the variance dynamically selected by the system leads the correlation function to evolve on the separatrix curve, limiting different realizable steady states, whereas in the ferromagnetic region, a fixed point solution is selected as the only realizable steady state. In the spin-glass region, stationary solutions are unstable against perturbations that break time-translation invariance, indicating chaotic behaviour in large single network instances. Numerical analysis of Lyapunov exponents confirms that chaotic behaviour emerges throughout the spin-glass region, for any value of the coupling correlations. While negative correlations increase the strength of chaos, positive ones reduce it, with chaos disappearing for reciprocal (i.e. symmetric) couplings, where marginal stability is attained. On the other hand, in finite size non-reciprocal networks, fixed points and limit cycles can arise in the spin-glass region, especially close to the critical line. Finally, we show that chaos is suppressed when the strength of external noise exceeds a certain threshold. Intriguing analogies between chaotic phases in non-equilibrium systems and spin-glass phases in equilibrium are put forward.

Basin entropy as an indicator of a bifurcation in a time-delayed system.

The basin entropy is a measure that quantifies, in a system that has two or more attractors, the predictability of a final state, as a function of the initial conditions. While the basin entropy has been demonstrated on a variety of multistable dynamical systems, to the best of our knowledge, it has not yet been tested in systems with a time delay, whose phase space is infinite dimensional because the initial conditions are functions defined in a time interval [-τ,0], where τ is the delay time. Here, we consider a simple time-delayed system consisting of a bistable system with a linear delayed feedback term. We show that the basin entropy captures relevant properties of the basins of attraction of the two coexisting attractors. Moreover, we show that the basin entropy can give an indication of the proximity of a Hopf bifurcation, but fails to capture the proximity of a pitchfork bifurcation. The Hopf bifurcation is detected because before the fixed points become unstable, a oscillatory, limit-cycle behavior appears that coexists with the fixed points. The new limit cycle modifies the structure of the basins of attraction, and this change is captured by basin entropy that reaches a maximum before the Hopf bifurcation. In contrast, the pitchfork bifurcation is not detected because the basins of attraction do not change as the bifurcation is approached. Our results suggest that the basin entropy can yield useful insights into the long-term predictability of time-delayed systems, which often have coexisting attractors.

Stability Analysis for the Phytoplankton-Zooplankton Model with Depletion of Dissolved Oxygen and Strong Allee Effects

The photosynthetic activity of phytoplankton in the seas is responsible for an estimated 50-80% of the world's oxygen generation. Both phytoplankton and zooplankton require some of this synthesized oxygen for cellular respiration. This study aims to better understand how the oxygen-phytoplankton dynamics are altered due to the Allee effect in phytoplankton development, particularly when considering the time-dependent oxygen generation rate. The dynamic analysis of the model is dedicated to finding the possible equilibrium points. The analysis reveals that three equilibrium points can be obtained. The stability study demonstrates that one of the equilibrium points is always stable. The remaining equilibrium points are stable under specific conditions. We also identify bifurcations originating from these equilibrium points, including transcritical, pitchfork, and Hopf bifurcation. We derive conditions for stable limit cycles (supercritical Hopf bifurcation) and, in some cases, establish the non-existence of solutions. Numerical simulations are performed to validate our theoretical findings. Furthermore, it is noted that the Allee threshold for the phytoplankton population ( significantly influences the overall dynamics of the system. When , the population of plankton is at risk of extinction. On the other hand, when , the population of zooplankton is at risk of extinction. When , the solution reaches a stable condition of coexistence. Conversely, when , the solution exhibits periodic attractor behavior.

Distributed Network of Coupled Oscillators With Multiple Limit Cycles

Distributed Network of Coupled Oscillators With Multiple Limit Cycles

Three solid electrolyte interphases of aqueous zinc batteries

With the increasing demand for enhanced safety performance of rechargeable batteries, water-based zinc-ion batteries have witnessed rapid development due to their exceptional safety characteristics. And, zinc batteries have been widely used in a diverse of various fields such as electric vehicles and electric bicycles. However, the dendrite issue and limited cycle life of the negative electrode pose significant challenges that hinder further advancements in this technology, which greatly limits the development of zinc batteries. For this reason, extensive research has been conducted on various solid electrolyte interphases related to the battery anode. By comprehensively understanding and comparing three materials of solid electrolyte interphase (SEI) utilizing similar controlled negative ion techniques, it can validate the accuracy of this theoretical approach. The utilization of these three materials results in a substantial improvement in battery performance compared to the original battery, particularly in terms of service life; however, each material also exhibits distinct advantages and drawbacks.

Numerical Analysis of Free Play-Induced Aeroelastic Phenomena: A Numerical Approach With Adaptive Step Size Control

This study presents a detailed numerical analysis of nonlinear aeroelastic behavior in a two degree of freedom (DOF) model, focusing on plunge and pitch motions and employing the continuation method (CM) with an adaptive step size control algorithm. The research incorporates free-play nonlinearity at the plunge hinge, a common structural nonlinearity in aeronautics that can induce detrimental limit cycle oscillations (LCOs) during flight. By examining three scenarios—linear response, unhindered plunge motion, and nonlinear stiffness behavior—the study assesses the effects of free play on flutter and LCO phenomena, including discontinuity-induced bifurcations like grazing bifurcation. Additionally, the study explores parameter variation for nonlinear flutter analysis, revealing the dynamics of grazing bifurcation and its impact on LCO behavior. The research also demonstrates the method’s superior accuracy in flutter speed estimation and mode-switching identification, despite higher computational demands. The findings underscore the diminishing influence of nonlinear free-play behavior on LCO amplitude, providing insights with significant implications for aeroelastic design and aircraft safety.