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This paper is on the motion of a neutrally buoyant but circular slip particle in a clockwise double-lid-driven square cavity. The slip flow at the particle surface is implemented by the lattice Boltzmann method with corrected slip boundary schemes. The effects of slip length (Ls), initial particle position, Reynolds number (Re), and particle size (D) are studied on the migration of the slip particle. The motion of the circular slip particle is dominated by the centrifugal and boundary-repulsion forces. The results show that the cavity center is the unique fixed point, and once the slip particle initially deviates from the cavity center, it is always stabilized at the same limit cycle. With the increase in slip length, the limit cycle of the circular slip particle is closer to the cavity boundaries, which brings a stronger centrifugal force to balance the increased boundary-confinement effect. As the slip length, Ls, exceeds 0.02D, the limit cycle forms more quickly than the circular no-slip particle. When Re increases to within 1000, the limit cycle is squashed along the leading diagonal of the cavity and pushed toward the boundaries; however, when Re increases beyond 1000, two developing secondary vortices confine the limit cycle to shrink toward the cavity center. With the increase in particle size, the enhanced boundary confinements lead to the shrinkage of the limit cycle toward the cavity center. [ABSTRACT FROM AUTHOR]

This paper devotes to the study of the classical Abel equation d x d t = g (t) x 3 + f (t) x 2 , where g (t) and f (t) are trigonometric polynomials of degree m ≥ 1. We are interested in the problem that whether there is a uniform upper bound for the number of limit cycles of the equation with respect to m , which is known as the famous Smale-Pugh problem. In this work we generalize an idea from the recent paper (Yu, Chen and Liu, Disc. Cont. Dyn. Syst. Ser. B, 2023) and give a new criterion to estimate the maximum multiplicity of limit cycles of the above Abel equations. By virtue of this criterion and the previous results given by Álvarez et al. and Bravo et al., we completely solve the simplest case of the Smale-Pugh problem, i.e., the case when g (t) and f (t) are linear trigonometric, and obtain that the maximum number of limit cycles, is three, which gives a positive answer to the sixth of 33 open problem proposed by Gasull in paper (Gasull, SeMA J., 2021). [ABSTRACT FROM AUTHOR]

This paper aims to determine the maximum number of limit cycles that can bifurcate from the linear center x ˙ = y {\dot{x}=y} , y ˙ = - x {\dot{y}=-x} , when we perturb it inside a class of piecewise smooth differential systems, which are separated by the straight line y = 0 {y=0} . Using the averaging theory of first order for piecewise smooth differential systems, we prove that if the degree of the polynomial that appears in the perturbation term is equal to n, then the maximum number of limit cycles bifurcating from the period annulus around the linear center is at most n or n - 1 {n-1} , and this number can be reached. [ABSTRACT FROM AUTHOR]

In this paper, a modified Leslie-type predator–prey system with Holling-type III functional response is formulated to investigate the dynamics in the presence of special Allee effect induced by fear factors. First, we calculate the first three focal values by the method of successor function to ensure that the system has an unstable weak focus of multiplicity 3. A focus- or center-type degenerate Bogdanov–Takens singularity of codimension 3 and a weak focus of multiplicity 1 or 2 together with a cusp of codimension 2 are derived for the system. Then multiple bifurcations are explored. It is shown that the system undergoes Hopf bifurcation of codimension 3 and hence three limit cycles are generated. There exists another large stable limit cycle enclosing these three limit cycles by Poincaré–Bendixson theorem. We demonstrate that a degenerate focus-type Bogdanov–Takens bifurcation of codimension 3 can occur by calculating the universal unfolding. It is also shown that a subcritical, supercritical or degenerate Hopf bifurcation occurs together with a Bogdanov–Takens bifurcation. Finally, numerical simulations are used to support the theoretical results. The analysis results reveal that the special Allee effect is beneficial for the persistence and diversity of ecosystem. [ABSTRACT FROM AUTHOR]

This study investigates the wind‐induced flutter phenomenon of long‐span suspension bridges. A mathematical model is established using the energy method, considering the nonlinear aeroelastic behavior of a bridge deck constrained by vertical suspenders under cubic nonlinear damping. The aerodynamic self‐excited force is obtained using the Scanlan flutter theory. The critical state of flutter and the relationship between the amplitude and parameters of the limit cycle oscillation (LCO) are obtained by solving the nonlinear flutter equation using the average perturbation method. The Hopf bifurcation is confirmed to occur in the critical state of the nonlinear motion model using the Jacobian matrix. The results are verified using the Runge–Kutta numerical method, and it is found that the nonlinear response of the flutter LCO is in general agreement with the numerical solution. The study also shows that the degenerate Hopf bifurcation appears at the critical state under the condition of lower cubic damping coefficient, and the nonlinear flutter bifurcation changes from a degenerate Hopf bifurcation to a supercritical Hopf bifurcation as the cubic damping coefficient grows. [ABSTRACT FROM AUTHOR]

In this paper, we consider the bifurcation of small-amplitude limit cycles near the origin in perturbed pendulum systems of the form x ˙ = y , y ˙ = − sin ⁡ (x) + ε Q (x , y) , where Q (x , y) is a smooth or piecewise smooth polynomial in the triple (sin ⁡ (x) , cos ⁡ (x) , y) with free coefficients. We obtain sharp upper bounds on the number of positive zeros of its associated first order Melnikov function near h = 0 for Q (x , y) being smooth and piecewise smooth with the discontinuity at y = 0 or x = 0 , respectively. [ABSTRACT FROM AUTHOR]

This study provides a comprehensive analysis of the dynamics of a three-level vertical food chain model, specifically focusing on the interactions between vegetation, herbivores, and predators in a Snowshoe hare-Canadian lynx system. By simplifying the model through dimensional analysis, we determine conditions for equilibrium existence and identify various types of bifurcations, including Saddle-Node and Hopf bifurcations. Additionally, the study explores codimension-two bifurcations such as Bogdanov–Takens (BT) and zero-Hopf bifurcations. Coefficient formulas of normal forms are derived through the use of center manifold reduction and normal form theory. The study also presents an approximation of homoclinic orbits near a BT bifurcation of the system by computing explicit asymptotics based on regular perturbation methods. Utilizing the MATLAB package MATCONT, a family of limit cycles and their associated bifurcations are computed, including limit point cycles, period-doubling bifurcations, cusp points of cycles, fold-flip bifurcations, and various resonance bifurcations (R1, R2, R3, and R4). The biological implications of the findings are discussed in detail, highlighting how the identified bifurcations and dynamics can impact the population dynamics of vegetation, herbivores, and predators in real-world ecosystems. Numerical experiments validate the theoretical results and provide further support for the conclusions. [ABSTRACT FROM AUTHOR]

In this article, the integrability and linearizability of a class of cubic Z 2 -equivariant systems x ˙ = - 1 2 x - a 21 y + 1 2 x 3 + a 21 x 2 y + a 12 x y 2 + a 03 y 3 , y ˙ = (- q - b 21) y + b 21 x 2 y + b 12 x y 2 + b 03 y 3 , are studied. For any positive integer q, we obtain the first three saddle quantities of the above systems by theoretical analysis. Moreover, for any positive integer q, we derive the necessary and sufficient conditions for the linearizability of the above systems under some assumptions. [ABSTRACT FROM AUTHOR]

The goal of this paper is to study the number of sliding limit cycles of regularized piecewise linear visible–invisible twofolds using the notion of slow divergence integral. We focus on limit cycles produced by canard cycles located in the half-plane with an invisible fold point. We prove that the integral has at most 1 zero counting multiplicity (when it is not identically zero). This will imply that the canard cycles can produce at most 2 limit cycles. Moreover, we detect regions in the parameter space with 2 limit cycles. [ABSTRACT FROM AUTHOR]

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In this paper, we investigate the number of limit cycles bifurcating from a kind of quadratic Hamiltonian system having a homoclinic loop under perturbations of piecewise smooth polynomials with degree $ n $. It is proved that at most $ 10n+[\frac {n-1}2]+7 (n\geq 1) $ limit cycles bifurcating from the periodic annulus if the first order Melnikov function is not identically zero, and it is also proved that there exist piecewise smooth polynomials with degree $ n $ such that the corresponding system has at least $ n+[\frac{n+1}{2}] $ limit cycles. [ABSTRACT FROM AUTHOR]

Electrically assisted turbocharging (EAT) technology is an important technical means to effectively solve the problems of insufficient intake air and poor transient response of traditional turbocharged engines at low speed or acceleration, and to realize the electrification of marine internal combustion engine power system. In the EAT design stage, the original rotor model of a marine turbocharger was established and verified. Subsequently, the EAT rotor dynamics design and analysis were carried out, and the influence of the unbalanced phase and external clearance on rotor vibration was discussed. The results show that when the external clearance is small, extensive sub-synchronous vibration occurs and bifurcation occurs at high speed. When the external clearance is large, the sub-synchronous vibration component almost disappears but high synchronous vibration occurs at a high speed. At the same time, the shaft is significantly affected by the gravity load at low speeds, and the rotor makes a balanced motion on the critical limit cycle at high speed. In order to minimize the vibration and noise of the same type of EAT, it is recommended to control the outer clearance of the EAT bearing in the range of 0.30–0.36mm. The research content provides a reference for the dynamic design and analysis of the EAT, which effectively helps the development of the original engine of the ship electric auxiliary turbocharger and improves the operation stability of the EAT. [ABSTRACT FROM AUTHOR]

In this paper, a single-species logistic model with both fear effect-type feedback control and additive Allee effect is developed and investigated using the new coronavirus as a feedback control variable. When the system introduces additive Allee effect and fear effect-type feedback control, more complicated dynamical behavior is obtained. The system can undergo transcritical bifurcation, saddle-node bifurcation, degenerate Hopf bifurcation and Bogdanov–Takens bifurcation. By numerical simulations, the system exhibits homoclinic bifurcation and saddle-node bifurcation of limit cycles as parameters are altered. Remarkably, it is the first time that two limit cycles have been discovered in a single-species logistic model with the Allee effect. Further, stronger Allee effect or stronger fear effect can lead to the extinction of the species population. [ABSTRACT FROM AUTHOR]

This paper presents a novel viewpoint on nonlinear dust-acoustic waves (DAWs) in an unmagnetized collisionless plasma containing regularized κ distributed electrons and ions (ei) and negative dust grain. The nonlinear oscillatory system based on hybridization of the Van der Pol–Mathieu equation (VdPME) is derived by a new technique. By bifurcation analysis of the planar dynamical system (DS), the effects of parameters with the assistance of phase planes and time series of VdPME are studied. After analyzing the equation to identify the resonance region, a fourth-order Runge–Kutta method is used to solve it numerically. We explained the behavior of DA periodic, stable limit cycle, and chaotic limit cycle wave solutions with different parameters. These types of numerical solutions are illustrated in two-dimensional and three-dimensional graphics by changing the rate at which charged dust grain is produced α, as well as waste β and comparing the results with those of earlier research. A novel bifurcation analysis of VdPE and VdPME is obtained with the effects of the cut-off factor δ of regularized κ-distribution (RKD) distributed ei, the superthermality of ei particles κe,i, the ratio of ion to electron temperature σ, and the ratio of dust to electron density ρ illustrated.  It is noticed that the DAW shows promotion in width and amplitude as the frequency γ increases and it becomes rarefaction as the cut-off parameter δ increases. On the contrary, it becomes compressive under the impact of superthermality κe and κi. The obtained conclusions may help explain and comprehend a variety of applications in experimental plasma containing highly energy regularized κ dispersed ei, as well as in the interstellar medium. [ABSTRACT FROM AUTHOR]

In this paper, the effect of time delay is investigated on the system dynamics of a glucose-insulin model incorporating obesity. Treating the time delay as a bifurcation parameter, the stability switching on the positive equilibrium with global bifurcation is obtained. With the method of normal forms and central manifold theory, the direction and stability of limit cycles arising from Hopf bifurcation are analyzed. Using the method of multiple time scales, the normal form associated with non-resonant double Hopf bifurcation is derived. Moreover, the bifurcations are classified in the two-dimensional parameter plane near the critical point, and numerical simulations are presented to demonstrate the applicability of the theoretical results. Our results indicate that time delay in the glucose-insulin model can not only induce Hopf bifurcation and double Hopf bifurcation but also generate multiple stable periodic solutions. These results may help to understand the dynamical mechanism of glucose-insulin metabolic regulation systems, and to design control strategies for regulating and mitigating the occurrence of related diseases. [ABSTRACT FROM AUTHOR]

In this paper, we apply the second‐order averaging theory for obtaining an explicit expression of the small amplitude periodic solution that bifurcates from a zero‐Hopf equilibrium point of a tritrophic food chain model. This model considers logistic growth rate for the lowest trophic level, Holling functional responses type III for the middle and for the highest level. We first prove that this model has a zero‐Hopf equilibrium point, after we show that from this equilibrium bifurcates a small limit cycle, and finally, we provide the explicit expression of the first two terms in the power series of this limit cycle. These differential systems for which the equilibrium point is non hyperbolic are not easy to study, in particular, if the equilibrium is zero‐Hopf. As far as we know, this is the first time that the averaging theory has been used to exhibit the first and second terms of the power series expansion of the analytical expression of a limit cycle that bifurcates from a zero‐Hopf equilibrium in the food chain models. [ABSTRACT FROM AUTHOR]

The article delves into the trapping and clustering of inertial particles near a rotating vortex pair, using numerical simulations to analyze the phenomenon in a flow governed by Lamb–Oseen vortices. The inclusion of the Basset–Boussinesq history force is found to be crucial in enhancing particle trapping and clustering due to their finite density. The study's findings have implications for various physical applications, such as planetesimal formation and phytoplankton aggregation, shedding light on the dynamics of inertial particles in rotating vortex pairs and emphasizing the significance of the Basset–Boussinesq–History force in understanding solid-liquid systems like microplastics in the ocean. [Extracted from the article]

Exceptional points (EPs)—singularities in the parameter space of non-Hermitian systems where two nearby eigenmodes coalesce—feature unique properties with applications such as sensitivity enhancement and chiral emission. Existing realizations of EP lasers operate with static populations in the gain medium. By analyzing the full-wave Maxwell–Bloch equations, here we show that in a laser operating sufficiently close to an EP, the nonlinear gain will spontaneously induce a multi-spectral multi-modal instability above a pump threshold, which initiates an oscillating population inversion and generates a frequency comb. The efficiency of comb generation is enhanced by both the spectral degeneracy and the spatial coalescence of modes near an EP. Such an "EP comb" has a widely tunable repetition rate, self-starts without external modulators or a continuous-wave pump, and can be realized with an ultra-compact footprint. We develop an exact solution of the Maxwell–Bloch equations with an oscillating inversion, describing all spatiotemporal properties of the EP comb as a limit cycle. We numerically illustrate this phenomenon in a 5-μm-long gain-loss coupled AlGaAs cavity and adjust the EP comb repetition rate from 20 to 27 GHz. This work provides a rigorous spatiotemporal description of the rich laser behaviors that arise from the interplay between the non-Hermiticity, nonlinearity, and dynamics of a gain medium. An ab initio laser theory is developed to show that exceptional-point lasers can self-generate frequency combs through dynamic population inversion. Notably, the repetition rate of these EP combs is independent of the cavity's free spectral range. [ABSTRACT FROM AUTHOR]

Limit-cycle oscillations are studied for ion temperature gradient turbulence, which, in the absence of large diamagnetic (mean) shear flows, saturates through energy transfer from unstable modes to large-scale stable modes via zonal-flow intermediary modes. Oscillations of zonal flow and turbulence levels are strongly constrained by the reactive, largely non-dissipative character of the zonal flows. Since existing predator–prey models for observed oscillations in experiments do not include energy transfer through zonal flows to stable modes, low-order fluid models with this physics are constructed and investigated. A simple three-wave truncation produces low-amplitude zonal flows that slowly oscillate around a zero mean, with turbulence oscillations between coupled wavenumbers that exceed linear frequencies by orders of magnitude. This inconsistency with experimental observations is caused by the weak non-linear drive of zonal flows in three-wave systems and the lack of multiple-wavenumber turbulent interactions. A more comprehensive model that preserves multiple wavenumber interactions within the context of conservative zonal-flow-mediated energy transfer to stable modes accurately reflects observed dynamics when the phase between stable and unstable modes is occasionally randomized. [ABSTRACT FROM AUTHOR]

This paper is concerned with the limit cycle problem of a cubic reversible Hamiltonian system under perturbation of polynomials of degree n with a switching line x = 0. The upper and lower bounds of the number of limit cycles are obtained using the first order Melnikov function and its expansion. The method for calculating the Melnikov function relies upon some iterative formulas, which differs from other approaches. [ABSTRACT FROM AUTHOR]

For the Leslie–Gower predator–prey model with Michaelis–Menten type prey harvesting, the known results are on the saddle-node bifurcation and the Hopf bifurcation of codimensions 1, the Bogdanov–Takens bifurcations of codimensions 2 and 3, and on the cyclicity of singular slow–fast cycles. Here, we focus on the global dynamics of the model in the slow–fast setting and obtain much richer dynamical phenomena than the existing ones, such as global stability of an equilibrium; an unstable canard cycle exploding to a homoclinic loop; coexistence of a stable canard cycle and an inner unstable homoclinic loop; and, consequently, coexistence of two canard cycles: a canard explosion via canard cycles without a head, canard cycles with a short head and a beard and a relaxation oscillation with a short beard. This last one should be a new dynamical phenomenon. Numerical simulations are provided to illustrate these theoretical results. [ABSTRACT FROM AUTHOR]

The classical Hastings–Powell model is well known to exhibit chaotic dynamics in a three-species food chain. Chaotic dynamics appear through period-doubling bifurcation of stable coexistence limit cycle around an unstable coexisting equilibrium point. A specific choice of parameter value leads to a situation, where the chaotic attractor disappears through a collision with an unstable limit cycle originated due to subcritical Hopf-bifurcation around the coexistence equilibrium. As a consequence, the top predator goes to extinction. The main objective of this work is to explore the structural sensitivity of this phenomenon by replacing the Holling-type II functional responses with Ivlev functional responses. In this work, we have shown the existence of two Hopf-bifurcation thresholds and numerically verified the existence of an unstable limit cycle. The model with Ivlev functional responses does not indicate any possibility of extinction of the top predator due to any collision of chaotic attractor with the unstable limit cycle for the chosen range of parameter values. Moreover, the model with Ivlev functional responses depicts an interesting scenario of bistable oscillatory coexistence. [ABSTRACT FROM AUTHOR]

A Leslie–Gower predator–prey model with nonlinear harvesting and a generalist predator is considered in this paper. It is shown that the degenerate positive equilibrium of the system is a cusp of codimension up to 4, and the system admits the cusp-type degenerate Bogdanov–Takens bifurcation of codimension 4. Moreover, the system has a weak focus of at least order 3 and can undergo degenerate Hopf bifurcation of codimension 3. We verify, through numerical simulations, that the system admits three different stable states, such as a stable fixed point and three limit cycles (the middle one is unstable), or two stable fixed points and two limit cycles. Our results reveal that nonlinear harvesting and a generalist predator can lead to richer dynamics and bifurcations (such as three limit cycles or tristability); specifically, harvesting can cause the extinction of prey, but a generalist predator provides some protection for the predator in the absence of prey. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the existence and uniqueness of isolated periodic traveling wave solutions of a family of nonlinear reaction–convection–diffusion equations using the monotonicity of the ratio of Abelian integrals. A numerical study is also presented. [ABSTRACT FROM AUTHOR]

This paper proposes a novel finite‐time limit cycle controller for DC to AC conversion in power boost converters. The stable limit cycle is categorized as a positive limit set. Achieving the control law, based on the concepts of set stability and finite‐time stability, is a challenging problem. To fulfill the control objective, a novel sliding manifold is suggested concerning the structure of the wanted limit cycle which facilitates conditions for the development of finite‐time stability for the limit cycle. By utilizing the sliding mode control method, a control strategy is proposed that guarantees the desired stable limit cycle is created in the phase plane of the closed‐loop system and the phase trajectories reach it in a finite time and remain on it for all future times. This ensures the finite‐time generation of the biased sinusoidal oscillations in the output of the power boost converter from a DC input source. The simulation results of a practical boost converter have validated the effectiveness and feasibility of the presented algorithm. [ABSTRACT FROM AUTHOR]

The industrial standard in the design and development process of NVH(Noise Vibration Harshness) characteristic of brakes is the application of Finite Element(FE) models with a high number of degrees of freedom in the range of one or several millions. Nevertheless, parallel experimental investigations are still indispensable. On the other hand, minimal models with, due to the inclusion of the self-excitation process, at least two degrees of freedom are well known to be capable to explain qualitatively phenomena as instability of the desired non-vibrating solution or limit cycle oscillation but are in general very inaccurate in predicting the dynamics of a specific real brake. This is because the underlying physical assumptions are already too restrictive and model parameters (especially those referring to nonlinearities) are widely unknown. To overcome this problem, the data-driven modeling approach SINDy(Sparse Identification of Nonlinear Dynamics) is applied to identify appropriate nonlinear functions for a brake squeal minimal model. A problem thereby is the limited database. It turns out that the naive implementation of the method yielding the lowest possible residuum does not necessarily provide physically meaningful models and results, respectively. Instead, a constrained model that incorporates physical knowledge is used to robustly identify parameters and reproduce realistic dynamic behavior. Thereby, several appropriate models with coexisting limit cycles and stationary equilibrium are identified. In particular, it was found that the angular position of the brake drum has a significant influence on the model parameters and therefore must be taken into account in a model with long-term validity. [ABSTRACT FROM AUTHOR]

Dry low emissions (DLE) systems are well-known to be susceptible to thermoacoustic instabilities. In particular, transverse, spinning modes of high frequency may appear, and lead to severe damage in a matter of seconds. The thermoacoustic response of an engine is usually specific to the combustor geometry, operating conditions and difficult to reproduce at the lab-scale. In this work, details of high frequency dynamics (HFD) observed during the early development phase of a new DLE system are provided, where a multipeaked spectrum was noticed during testing. Beginning with an analysis of the measured pressure spectra from three different concepts, an analytical model of the clockwise and anticlockwise transverse waves was fitted to the experimental data using a nonlinear curve fitting approach to produce a simple yet useful understanding of the phenomena. A flamelet-based large eddy simulation (LES) of the entire combustion system was used to complement this analysis and confirm the mode shapes using dynamic mode decomposition (DMD). Both approaches independently identified a spinning second-order mode as the dominant one in the high frequency regime. The LES indicates the coupling of a distortion of swirl profile with a precessing vortex core as a possible cause for the onset of instability. With regard to modeling sensitivities, it is shown that subgrid scale combustion modeling has a strong impact on predicted amplitudes. Ultimately, a thickened-flame model with a modified efficiency function provided consistent results. [ABSTRACT FROM AUTHOR]

In this paper, a nonlinear mathematical model for tuberculosis transmission, which incorporates multiple saturated exogenous reinfections, is proposed and explored. The existence of disease-free and endemic steady states is investigated. Disease-free equilibrium (DFE) is locally asymptotically stable (LAS) but not globally asymptotically stable (GAS) when the basic reproduction number, ℛ 0 < 1. However, it is GAS only when there is no exogenous reinfection. The local asymptotic stability and global asymptotic stability of the unique endemic equilibrium point (EEP) are established under certain conditions when ℛ 0 > 1. Further, the EEP is GAS when ℛ 0 > 1 , provided there is no exogenous reinfection. When ℛ 0 is below unity, the presence of multiple endemic equilibria is found which leads to backward bifurcation. It is demonstrated that the system encounters a Hopf-bifurcation when the transmission rate β crosses a critical value, resulting in the formation of limit cycles, i.e. periodic solutions bifurcate around the EEP when β passes a critical value. The stability and direction of Hopf-bifurcation are also studied. The results of the analytical work are validated through numerical simulations. A numerical simulation illustrates that EEP losses its stability via Hopf-bifurcation for specific parameters. However, when the bifurcation parameter β is increased further, the EEP regains its stability. In addition, Hopf-bifurcation occurs due to exogenous reinfection rates p and . Thus, our model shows some important nonlinear dynamical behaviors. [ABSTRACT FROM AUTHOR]

Position-predefined velocity control in the presence of parametric uncertainties as well as the ability to reject external disturbances simultaneously, has been one of the challenges in the area of fish-like swimming robots. To overcome the abovementioned problems, in this article, after identifying parameters of the Chapylgin sleigh model (CSM), which is a low-dimensional time-varying nonholonomic system with limit cycle dynamics, a chattering-free robust backstepping controller (RBSC) and a disturbance observer (DO) are proposed for autonomous position control with external disturbance rejection capability in the presence of uncertainties in velocity space for an under-actuated swimming robot with an internal reaction wheel. The uniform boundedness of the whole system, including RBSC and DO, is theoretically performed by Lyapunov methods. Because of the inherent limit cycle dynamics in the reduced velocity space of the swimming robot, pure usage of the observer results in fluctuating estimations of disturbances. Thus, the moving average technique is employed to enhance the accuracy of the estimated disturbance signals. The simulations are performed for various trajectory tracking tasks, and the results demonstrate the high maneuverability and robustness of the proposed schemes in the perturbed conditions. In addition, computational fluid dynamic validation simulations confirm the reliability of the proposed CSM-based feedback controller in the presence of high-dimensional fish-fluid interaction with the vortex wake of a hydrofoil. [ABSTRACT FROM AUTHOR]

The harmonic balance (HB) method is efficient in predicting the limit cycle oscillations for many dynamic systems including very high dimensional systems in fluid dynamics. Recently, some variations and extensions of the HB method have been developed and employed in the computation of large systems. This study focuses on a new development of the HB method. The new approach, the hybrid harmonic balance method (HHB), is developed based on the ideas in the conventional HB approach and the high dimensional harmonic balance (HDHB) approach. To demonstrate the applications and advantages, the new method is employed on a prototypical dynamic system in comparison with the results from both the HB and HDHB approaches. It is theoretically proved that when twice the number of harmonics in HB are included in the HHB derivation the results from HHB achieve the accuracy of the HB method for the cubic nonlinearity. Numerical simulations reveal that HHB combines the advantages of both the conventional HB and HDHB methods: Ease of implementation for very high dimensional systems regardless of the complexity of the nonlinearities and generation of meaningful solutions while the spurious solutions (from HDHB) are diminished. This new method is more accurate when compared with the HDHB method and much more computationally efficient when compared with the conventional HB method. [ABSTRACT FROM AUTHOR]

This paper focuses on the research of the algorithm for higher order Melnikov functions for piecewise Hamiltonian systems under piecewise perturbations in any order ϵ. Then the formula is applied to the any order ϵ linear perturbations of a general center-center type piecewise linear system. The maximum number of zeros for any order Melnikov functions is 2 and it can be reached. It is worth mentioning that the algorithm can also be applied to smooth case. [ABSTRACT FROM AUTHOR]

Planar differential systems whose angular velocity is constant are called rigid or uniform differential systems. The first rigid system goes back to the pendulum clock of Christiaan Huygens in 1656; since then, the interest for the rigid systems has been growing. Thus, at this moment, in MathSciNet there are 108 articles with the words rigid systems or uniform systems in their titles. Here, we study the dynamics of the planar rigid polynomial differential systems with homogeneous nonlinearities of arbitrary degree. More precisely, we characterize the existence and non-existence of limit cycles in this class of rigid systems, and we determine the local phase portraits of their finite and infinite equilibrium points in the Poincaré disc. Finally, we classify the global phase portraits in the Poincaré disc of the rigid polynomial differential systems of degree two, and of one class of rigid polynomial differential systems with cubic homogeneous nonlinearities that can exhibit one limit cycle. [ABSTRACT FROM AUTHOR]

In this paper, we estimate the number of limit cycles bifurcating from the periodic orbits of the weight-homogeneous polynomial centers of weight-degree 3, when they are perturbed inside the class of piecewise smooth polynomial systems of degree n. By using the first-order averaging method, we give the upper and lower bounds on the maximal number of limit cycles which can bifurcate from the period annuluses. Our results indicate that the nonsmooth systems can have more limit cycles than the smooth ones. [ABSTRACT FROM AUTHOR]

Recently, certain biological field experiments and research findings have revealed that predators not only reduce prey populations through direct predation (lethal effect) but also indirectly affect the growth rate of prey by inducing fear; these nonlethal effects can be carried over seasons or generations. In this paper, we proposed an autonomous and a nonautonomous Leslie–Gower model incorporating some biological factors such as fear and its carry‐over effects, prey refuge, and nonlinear predator harvesting. In the autonomous model, first, we examined the well‐posedness, positivity solutions, and their boundedness. Also, it is shown that new equilibrium points emerge and disappear with the change in intrinsic growth rate of predator. Further, we compute and analyzed the local and global stability at the interior equilibrium points. Furthermore, Hopf bifurcation and direction and stability of limit cycle at interior equilibrium points are established. Moreover, the sensitivity analysis of the biological parameters is carried out numerically with two statistical methods via Latin hypercube sampling and partial rank correlation coefficients. It was found that at the small values of fear factor, carry‐over effect, and refuge behavior of prey parameters, the autonomous system exhibits oscillatory behavior, whereas for small values of prey birth rate and harvesting effort, the system remains stable. However, when intrinsic growth rate of predator increases from a low value to a higher value, the system shows transition from stability to instability and back to stability. We also explore the effects of seasonal variations in biological parameters in the nonautonomous model. Additionally, we investigate the theoretical existence of positive periodic solutions using the continuation theorem. The influences of seasonal variations in some parameters such as prey's birth rate, fear, and its carry‐over effects; refuge behavior of prey; intrinsic growth rate of predator; and harvesting effort are then numerically investigated. The results reveal the emergence of positive periodic solutions and bursting patterns in the seasonally forced model. [ABSTRACT FROM AUTHOR]

In this paper, we study Poincaré, Hopf and homoclinic bifurcations of limit cycles for planar near-Hamiltonian systems. Our main results establish Hopf and homoclinic bifurcation theories by higher order Melnikov functions, obtaining conditions on upper bounds and lower bounds of the maximum number of limit cycles. As an application, we concern a cubic near-Hamiltonian system, and study Hopf and homoclinic bifurcations in detail, finding more limit cycles than [26]. [ABSTRACT FROM AUTHOR]

Direct yaw moment control (DYC) and differential drive-assist steering (DDAS) for distributed-drive vehicles are both realized by allocating the in-wheel motor torque. To address the interference caused by overlapping control objectives, this paper proposes a multilayer control strategy that integrates DYC and DDAS, consisting of an upper controller, a coordinated decision layer, and a torque distribution layer. The upper controller, designed based on the vehicle's dynamic characteristics, incorporates an adaptive fuzzy control DYC system and a dual PID control DDAS system. The coordinated decision layer is developed utilizing a phase-plane dynamic weighting method, delineating region boundaries by applying the double-line and limit cycle methods. The torque distribution strategy is formulated considering motor peak torque and road adhesion conditions. Multi-condition joint simulation experiments indicate that the proposed multilayer control strategy, integrating the advantages of DYC and DDAS, reduces peak steering wheel torque by approximately 10%, peak yaw rate by around 25%, peak sideslip angle by roughly 29%, and peak sideslip angle rate by about 19%, significantly improving driving stability and maneuvering flexibility. [ABSTRACT FROM AUTHOR]

Numerical simulations of a T-tail flutter in subsonic and transonic flow regimes were conducted using a 3D Navier-Stokes code which considers the in-plane motion of the horizontal tail plane (HTP). In our subsonic flutter simulations, we confirmed that the flutter velocity of the T-tail is dependent upon the mean angle of attack of the HTP, and also confirmed that the results show a satisfactory agreement with those predicted using a lifting-surface theory. Thereby we were able to verify the reliability and accuracy of the 3D Navier-Stokes code. In transonic simulations, the sharp transonic-dip phenomena of the dynamic pressure of flutter and limit cycle oscillation (LCO) were predicted. It can be shown that the characteristics of LCO and flutter are considerably different depending on Mach number and the angle of attack of the HTP. As a possible cause of these phenomena, we examined the behavior of the shock wave generated on the upper or lower surface of the HTP, which varies strongly depending on the Mach number and the angle of attack of the HTP. [ABSTRACT FROM AUTHOR]

In contrast to the various well‐known two‐scroll hidden Lorenz‐like attractors, little seems to be known about three‐scroll hidden conservative chaotic flows in the 3D Lorenz‐like system family whose Kaplan–Yorke dimension is approximately equal to three$three$. This technical note first proposes a three‐scroll chaotic system, i.e., ẋ=a(y−x)+dxz$\dot{x} = a(y - x) + dxz$, ẏ=kx+fy−xz$\dot{y} = kx + fy - xz$, ż=−ex2+xy+c$\dot{z} = -ex^{2} + xy + c$, which generates three‐scroll hidden chaotic flows without equilibrium points when (a,d,k,f,e,c)=(1,0.4,0.7,0.5,0.6,172)$(a, d, k, f, e, c)=(1,0.4,0.7, 0.5, 0.6, \frac{1}{72})$. Moreover, its Kaplan–Yorke dimension is DλLE=2.999419$D_{\lambda _{LE}}=2.999419$. Then, for c=0$c=0$ and e(a+fd)=(a−kd)$e(a + fd) = (a - kd)$, this system has another type of singularly degenerate heteroclinic cycle consisting of two different equilibria in the lines of semi‐hyperbolic equilibria Ez$E_{z}$ and Ex$E_{x}$, i.e., Ez=(0,0,z)$E_{z} = (0,0,z)$ and Ex=(x,ex,k+ef)$E_{x} = (x,ex,k+ef)$, with z,x∈R$z, x \in \mathbb {R}$. With a small perturbation of c>0$c > 0$, the kind of three‐scroll hidden conservative chaotic flows described above are created. Finally, five cycles are obtained when the proposed system undergoes Hopf bifurcation. [ABSTRACT FROM AUTHOR]

The Koopman operator framework holds promise for spectral analysis of nonlinear dynamical systems based on linear operators. Eigenvalues and eigenfunctions of the Koopman operator, the so-called Koopman eigenvalues and Koopman eigenfunctions, respectively, mirror global properties of the system's flow. In this paper, we perform the Koopman analysis of the singularly perturbed van der Pol system. First, we show the spectral signature depending on singular perturbation: how two Koopman principal eigenvalues are ordered and what distinct shapes emerge in their associated Koopman eigenfunctions. Second, we discuss the singular limit of the Koopman operator, which is derived through the concatenation of Koopman operators for the fast and slow subsystems. From the spectral properties of the Koopman operator for the singularly perturbed system and the singular limit, we suggest that the Koopman eigenfunctions inherit geometric properties of the singularly perturbed system. These results are applicable to general planar singularly perturbed systems with stable limit cycles. [ABSTRACT FROM AUTHOR]

Fear prompts prey to adopt risk-averse behaviors, such as reduced foraging activity, increased vigilance, and avoidance of areas with high predator presence, which affects its reproduction. In a real scenario, a population requires a minimum density to avoid extinction, known as an Allee threshold. In light of these biological factors, we propose a predator–prey model with (i) a fear effect in a prey population, (ii) an Allee effect in a predator population, and (iii) a non-constant attack rate that modifies the functional response. We ensured the non-negativity and boundedness of the solutions and examined the local and global stability status for each existing steady state solutions. We investigated some deep dynamical properties of the system by varying different parameters, such as cost of fear in prey and strength of the Allee effect in predators and their mortality rate. In codimension one bifurcations, we observed saddle node, Hopf, homoclinic, and coalescence of two limit cycles. Additionally, codimension two bifurcations were observed, including Bautin and Bogdanov Takens bifurcations. To provide a clearer understanding of these bifurcations, we conducted biparametric analysis involving the fear and Allee parameters, as well as the fear parameter and predator mortality rate. Our investigation shows that cost of fear and strength of Allee strongly influences the survival status of the predator. Furthermore, bistability and tristability reveal that the survival and extinction of predator are dependent on the initial population level. Numerical simulations and graphical illustrations are provided to support and validate our theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, two classes of near-Hamiltonian systems with a nilpotent center are considered: the coexistence of algebraic limit cycles and small limit cycles. For the first class of systems, there exist 2 n + 1 limit cycles, which include an algebraic limit cycle and 2 n small limit cycles. For the second class of systems, there exist n 2 + 3 n + 2 2 limit cycles, including an algebraic limit cycle and n 2 + 3 n 2 small limit cycles. [ABSTRACT FROM AUTHOR]

In order to reveal the influence of initial excitation on the bifurcation phenomenon of bridge decks, a new perspective of flow characteristics is developed based on the computational fluid dynamics numerical simulation method. Then, the bifurcation mechanism of vortex-induced vibration (VIV) response and nonlinear flutter response of the H-shaped section is investigated. The results show that when the wind speed is 2 m/s, under a small torsional excitation of 0.5°, the flow field of the H-shaped section will develop into the vortex shedding mode of the vertical vibration, resulting in vertical VIV. However, while under a large excitation of 6°, the flow field will directly transform into the vortex shedding mode of the torsional vibration, resulting in torsional VIV. Therefore, the bifurcation phenomenon of the VIV response is observed. When the wind speed is 4 m/s, the H-shaped section exhibits a nonlinear flutter limit cycle oscillation under a large excitation of 8°, but its response can be ignored under a small excitation of 0.5°. This phenomenon is attributed to the significant change in the transition of the vortex shedding mode from a small amplitude to a stable large amplitude, and the flow field lacks enough energy to complete the transition of the vortex shedding mode, resulting in the bifurcation phenomenon of the nonlinear flutter response. When the wind speed is 3.0 m/s, the large excitation will change the vortex shedding frequency of the new H-shaped section, resulting in the torsional VIV. [ABSTRACT FROM AUTHOR]

The intrinsic physical relevance of higher-order self-excited force (SEF) components has received limited attention, and there is a dearth of formulas that adequately analyze the influence of SEF components on the post-flutter characteristics. Based on Taylor formulas and the principle of independence, semi-empirical polynomial SEF models are developed and validated. The energy input efficiency and role of each order SEF component are examined using the proposed models. By introducing the principle of energy equivalence and approximate average power, theoretical formulas designed to calculate the post-flutter characteristics are established. Finally, the applicability and robustness of the SEF models and theoretical formulas are discussed. Results show that the proposed models can obtain independent higher-order SEF components, which is conducive to the correct analysis of the SEF driving mechanisms. The theoretical formulas can accurately reconstruct the time-varying curves of the flutter characteristics, and the terms in the formulas can explicitly calculate and analyze the mechanism of each SEF model element. It is observed that the higher-order SEF components have a significant impact on the accurate reconstruction of SEFs while barely affecting the system energy. Moreover, the limit cycle oscillation generation mechanisms of the investigated two rectangular cylinders are different, but the variation of the flutter characteristics with time remain the same. [ABSTRACT FROM AUTHOR]

The detection of limit cycles of differential equations poses a challenge due to the type of the nonlinear system, the regime of interest, and the broader context of applicable models. Consequently, attempts to solve Hilbert's sixteenth problem on the maximum number of limit cycles of polynomial differential equations have been uniformly unsuccessful due to failing results and their lack of consistency. Here, the answer to this problem is finally obtained through information geometry, in which the Riemannian metrical structure of the parameter space of differential equations is investigated with the aid of the Fisher information metric and its scalar curvature R. We find that the total number of divergences of | R | to infinity provides the maximum number of limit cycles of differential equations. Additionally, we demonstrate that real polynomial systems of degree n ≥ 2 have the maximum number of 2 (n − 1) (4 (n − 1) − 2) limit cycles. The research findings highlight the effectiveness of geometric methods in analyzing complex systems and offer valuable insights across information theory, applied mathematics, and nonlinear dynamics. These insights may pave the way for advancements in differential equations, presenting exciting opportunities for future developments. [ABSTRACT FROM AUTHOR]

This paper presents a modular approach to motion planning with provable stability guarantees for robots that move through changing environments via periodic locomotion behaviors. We focus on dynamic walkers as a paradigm for such systems, although the tools developed in this paper can be used to support general compositional approaches to robot motion planning with dynamic movement primitives (DMPs). By formulating the planning process as a switching system with multiple equilibria (SSME), we prove that the system's evolution remains within explicitly characterized trapping regions in the state space under suitable constraints on the frequency of switching among the DMPs. These conditions encapsulate the low-level stability limitations in a form that can be easily communicated to the planner. Furthermore, we show how the available primitives can be safely composed online in a receding horizon manner to enable the robot to react to moving obstacles. The proposed framework can be applied in a wide class of 3D bipedal walking models, and offers a modular approach for integrating readily available low-level locomotion control and high-level planning methods. [ABSTRACT FROM AUTHOR]

• The adaptive nonlinear damping is integrated into lateral suspension to improve hunting stability of high-speed train. • Three typical hunting bifurcations with different wheel/rail equivalent conicity functions are analyzed. • An adaptive constraint-following control method is employed to implement nonlinear damping via active suspension system. • The proposed method can significantly improve both carbody and bogie hunting stability. This study employs an active secondary suspension with adaptive nonlinear damping to enhance the hunting stability of high-speed trains. Adjusting passive suspension parameters to optimize ride comfort and hunting stability simultaneously in varied extreme operational conditions poses a significant challenge for high-speed trains. This research integrates high-order displacement-dependent nonlinear damping into the secondary lateral suspension, drawing on insights from field test data. We analyze three typical hunting motion bifurcations: carbody hunting and both subcritical and supercritical bifurcations of bogie hunting. Theoretical analysis demonstrates that (a) the adaptive nonlinear damping narrows the unstable speed range and reduces the amplitude of the limit cycle in carbody hunting, (b) while it does not increase the critical speed for supercritical bogie hunting bifurcation, it substantially reduces the amplitude of the limit cycle, and (c) it increases the nonlinear critical speed for subcritical bogie hunting bifurcation, with the potential to alter the bifurcation from subcritical to supercritical, which holds considerable practical significance. Implementing such nonlinear damping directly through passive structure is challenging. Therefore, in view of this underactuated control problem, the constraint-following control is applied in the active suspension system to inherit the benefits of adaptive nonlinear damping. The simulation results show that the proposed active suspension system can effectively suppress the abnormal vibration caused by hunting motions. [ABSTRACT FROM AUTHOR]

• Extended Goodwin model integrates endogenous technological change and labor supply. • Reinterpretation of induced innovation in terms of mechanization. • Application of 3D-Hopf bifurcation theorem reveals limit cycle dynamics. • Employment rate's effect on productivity as a bifurcation parameter defines cycle stability. • Numerical simulation for 10 OECD economies and sensitivity analysis. This paper extends the Goodwin model of distributive cycles by incorporating the simultaneous endogeneity of technical change and labor supply within a classical-Marxian framework. It reinterprets induced innovation, suggesting that firms optimize mechanization to maximize cost reduction, obtaining a micro-founded relationship between mechanization and the wage share. Additionally, it assumes a positive relationship between labor supply and the employment rate. The resulting three-dimensional dynamical system includes wage share, employment rate, and capital-output ratio as state variables. The Hopf bifurcation theorem reveals the emergence of limit cycles as the employment rate's effect on labor productivity (reserve-army-creation effect) approximates a critical value from below. Numerical simulations for 10 OECD countries illustrate the cyclical nature of the model and its consistency with empirical patterns. Furthermore, a sensitivity analysis explores the effect of parameters variations, emphasizing the social dimensions of productivity and labor supply as critical factors defining the stability of distributive cycles. [ABSTRACT FROM AUTHOR]

Consideration of generalist predators leads to relatively complex dynamics due to alternative food sources. Here, we propose and analyze a prey-predator model with a generalist predator. The availability of alternative food sources for the predator and a density-dependent growth rate induces not only bistability and tristability, but also more complicated dynamical behaviors. We have studied the possible number and geometric configurations of positive equilibria in detail. A systematic bifurcation analysis has revealed the existence of the degenerate Bogdanov-Takens bifurcation of codimension four and degenerate Hopf bifurcation of codimension three. We found that degenerate local bifurcations with a higher codimension are responsible for three limit cycles. Derivation of the analytical conditions for three limit cycles for a suitable range of parameters is a crucial finding of this work. [ABSTRACT FROM AUTHOR]

We study the maximum number of limit cycles that bifurcate from the periodic annulus of the isochronous cubic centre of discontinuous and continuous piecewise differential systems with three zones formed by the discontinuity set Σ = { (x , y) ∈ R 2 : (y = 0) ∨ (x = 0 ∧ y ≥ 0) } . More precisely, we consider the following perturbed systems x ˙ = - y + x 2 y + ϵ p i (x , y) , y ˙ = x + x y 2 + ϵ q i (x , y) , i = 1 , 2 , 3 , where p i and q i are polynomials of degree m. Using the averaging theory of first order, we prove that for m = 1 , 2 , 3 , at most 3, 9 and 15 limit cycles bifurcate from the periodic annulus of the isochronous cubic centre in the discontinuous case, respectively. While for the continuous case, it can appear 1, 5 and 6 limit cycles from the periodic orbits of these centres, respectively. Furthermore, we extend our study when p i and q i are homogeneous polynomials with 1 ≤ m ≤ 3 , obtaining respectively for m = 1 , 2 , 3 at most one, seven and at least twelve limit cycles, which bifurcate from the periodic orbits of the isochronous cubic centre. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the maximum number of limit cycles that can bifurcate from the periodic annulus of the linear center for discontinuous piecewise quadratic polynomial differential systems with four zones separated by two nonlinear curves y = ± x 2 . By analyzing the first order averaged function, we prove that at most 7 crossing limit cycles can produce from periodic annulus of the linear center, and the upper bound is reached. In addition, under particular conditions, we obtain that at least 8 crossing limit cycles can bifurcate from periodic annulus in this class of systems by using the second order averaged function. Our results show that multiple switching curves increase the number of crossing limit cycles in comparison with the case where there is only a single switching curve. [ABSTRACT FROM AUTHOR]