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Piecewise differential systems in the plane have been extensively studied in the last two decades, due to the vast application of these systems to describe natural phenomena. The knowledge of the existence or not of periodic solutions, in particular, of the limit cycles, is very important for understanding the dynamics of the differential systems. In this article we studied the maximum number of limit cycles of discontinuous piecewise differential systems, formed by two Hamiltonians systems of degree one with three distinct lines of discontinuity either a circle, or a parabola, or a hyperbola. [ABSTRACT FROM AUTHOR]

The Lyapunov exponent is a measure of the sensitivity of a dynamic system to any changes and disturbances. It is therefore applicable in many fields of science, including physics, chemistry, economics, psychology, biology, medicine, and technology. Therefore, there is a need to have effective methods for determining the value of this exponent. In particular, it concerns the definition of its sign. The positive value of the Lyapunov exponent confirms the sensitivity of the system, especially when the system is chaotic. A negative value indicates the stability of the dynamic system being tested. The numerical value of the Lyapunov exponent indicates the degree of sensitivity of the dynamic system under study. Although the mathematical definition of the Lyapunov exponent is clear and simple, in practice determining its value is not a trivial matter, especially for continuous models. This paper presents a method for determining the Lyapunov exponent for continuous models. This method is based on the monodromy matrix. The work, for example, presents research on the following models of physical systems: the Van der Pol model with external forcing, the nonlinear mathematical pendulum model with external forcing and the Lorenz weather model. [ABSTRACT FROM AUTHOR]

The aim of this paper is to prove Llibre–Valls's conjecture for the Brusselator system. By transforming the Brusselator system into Liénard equation and applying the limit cycle theory of Liénard equation, we give a positive answer to this conjecture. Numerical simulations are also given to illustrate our theoretical results. • Positive answer of Llibre–Valls's conjecture for the Brusselator system. • Developed a new technique to prove the nonexistence of limit cycle for Liénard equation. • The exact number of limit cycles of Brusselator system. [ABSTRACT FROM AUTHOR]

We have explored the entanglement dynamics in a mechanically coupled double-cavity enhanced by two-level atomic ensembles. The entanglement between different components can be readily generated and transferred by appropriately choosing the parameters of two atomic ensembles, so that the maximal optomechanical entanglements for stable fixed points of the classical nonlinear counterpart of the system are ultimately achieved. Moreover, the entanglement generation scheme is robust with respect to the ambient temperature, in the sense that the maximal optomechanical entanglement is not completely lost up to T = 6 K. We further studied the time evolution of the entanglement in the parameter regimes where the fixed points are no longer stable, and observed that around the classical limit cycles, the optomechanical quantum entanglement also shows time-varying behavior about large average values. Our work has extended the usual time-invariant entanglement generation for stable fixed-point states to entanglement dynamics for more versatile steady states, and the proposed hybrid optomechanical system may serve as a flexible platform for quantum precision measurement as well as exploring the boundary between classical and quantum mechanics. • Stable optomechanical entanglements can be reached by adjusting suitable parameters. • Quantum entanglements on the boundary oscillate about higher average values. • In the unstable region, the entanglement is recovered by including atomic ensembles. • The scheme is readily extended to asymmetric cases for richer entanglement dynamics. [ABSTRACT FROM AUTHOR]

In this paper, we give a simple relation between the coefficients appearing in the expansions of n + 2 (n ∈ Z + , n ≥ 2) Melnikov functions near a compound cycle C (n) , which can be used to simplify some computations. We further give some conditions for a general near-Hamiltonian system to have limit cycles as many as possible near C (n) . Based on this, for a quintic Hamiltonian system with a compound cycle C (2) we prove that it can produce at least 7 2 (n − 2) + 1 2 (1 + (− 1) n) limit cycles near C (2) under polynomial perturbation of degree n (n ≥ 2). • Relationship among the first order Melnikov functions. • Limit cycle bifurcations near a compound loop. • A sharp lower bound of the maximum number of limit cycles of a polynomial system. [ABSTRACT FROM AUTHOR]

In this paper, we present a complete parametric analysis on a nonlinear predator–prey system with the generalized Holling type III functional response and a strong Allee effect. We apply the hierarchical parametric analysis to derive explicit conditions for the existence and stability of equilibrium solutions in a 5-dimensional parameter space. Specifically, a detailed study on Hopf bifurcation is given to show bifurcation of multiple limit cycles, and complex dynamics of multistable states including bistable, tristable and tetrastable phenomena. We also conduct simulations and provide a biological interpretation of the multistable states under various conditions. • A predator–prey generalized Holling type III functional response and a strong Allee effect is investigated. • The hierarchical parametric analysis is applied to study the existence and stability of equilibrium solutions in a 5-dimensional parameter space. • A detailed study on Hopf bifurcation is given to show bifurcation of multiple limit cycles. • Complex dynamics of multistable states including bistable, tristable and tetrastable phenomena are explored. [ABSTRACT FROM AUTHOR]

In this paper, a Gause type predator–prey system with constant-yield prey harvesting and monotone ascending functional response is proposed and investigated. We focus on the influence of the harvesting rate on the predator–prey system. First, equilibria corresponding to different situations are investigated, as well as the stability analysis. Then bifurcations are explored at nonhyperbolic equilibria, and we give the conditions for the occurrence of two saddle–node bifurcations by analyzing the emergence, coincidence and annihilation of equilibria. We calculate the Lyapunov number and focal values to determine the stability and the quantity of limit cycles generated by supercritical, subcritical and degenerate Hopf bifurcations. Furthermore, the system is unfolded to explore the repelling and attracting Bogdanov–Takens bifurcations by perturbing two bifurcation parameters near the cusp. It is shown that there exists one limit cycle, or one homoclinic loop, or two limit cycles for different parameter values. Therefore, the system is susceptible to both the constant-yield prey harvesting and initial values of the species. Finally, we run numerical simulations to verify the theoretical analysis. [ABSTRACT FROM AUTHOR]

We study a relay feedback system (RFS) having an ideal relay element and a linear, time-invariant, second-order plant. The relay element is modeled as an ideal on–off switch. And the plant is modeled using a transfer function that as follows: first, is Hurwitz stable, second, is proper, third, has a positive real zero, andfourth,  has a positive dc gain. We analyze this RFS using a state-space description, with closed-form expressions for the state trajectory from one switching time to the next. We prove that the state transformation from one switching time to the next, first, has a Schur stable linearization, and first,  is a contraction mapping. Then using the Banach contraction mapping theorem, we prove that all trajectories of this RFS converge asymptotically to a unique limit cycle. This limit cycle is symmetric, and is unimodal as it has exactly two relay switches per period. This result helps understand the behavior of the relay autotuning method, when applied to second-order plants with no time delay. We also treat the case where the plant either has no finite zero, or has exactly one zero that is negative. [ABSTRACT FROM AUTHOR]

• Nonlinear liénard type fractional feedback self-excitation is considered. • The control is general in nature and takes van der Pol and Rayleigh type feedback as its special cases. • Limit cycle of any desired frequency and amplitude can be generated for minimum control cost. • Dynamics of generation of limit cycle is explained with equivalent integer-order model of the proposed control. • Theoretical results are verified by simulations and experiments. Anti-control of self-excited oscillation in mechanical and micro-mechanical systems is an important research problem due to its potential applications. In this paper, a novel fractional-order Liénard type nonlinear feedback is proposed to generate a limit cycle of desired frequency and amplitude in a single-degree-of-freedom spring-mass-damper mechanical oscillator. The feedback comprises two different fractional-order terms which are associated with both linear and nonlinear parts of the feedback and thus making it more general, with the van der Pol and Rayleigh type feedback as special cases. The analytical relations for steady-state amplitude and frequency of oscillation with the system and controller parameters are obtained by performing the nonlinear analysis with the method of two-time scale. Bifurcations of amplitude and frequency of oscillation with the fractional orders are studied in details. For any desired frequency and amplitude of oscillation, the controller parameters are obtained for minimum control cost. The analytical results are verified by numerical simulations performed in MATLAB SIMULINK and experiment. An equivalent integer-order model of the proposed fractional-order feedback system is developed to decipher the dynamics behind the generation of the limit cycle at different desired frequencies and amplitudes. [ABSTRACT FROM AUTHOR]

In this article, we study the formation problem for a group of mobile agents in a plane, in which the agents are required to maintain a distribution pattern, as well as to rotate around or remain static relative to a static/moving target. The prescribed distribution pattern is a class of general formations where the distances between neighboring agents or the distances between each agent and the target do not need to be equal. Each agent is modeled as a double integrator and can merely perceive the relative information of the target and its neighbors, and the acceleration of the target. In order to solve the formation problem, a limit-cycle-based controller design is delivered. We divide the overall control objective into two subobjectives, where the first is target circling that each agent keeps its own desired distance to the static/moving target as well as rotating around or remaining static relative to the target as expected, and the second is distribution adjustment that each agent maintains the desired distance to its neighbors. Then, we propose a controller comprised of two parts, where a limit cycle oscillator named a converging part is designed to deal with the first subobjective, while a layout part is introduced to address the second subobjective. One key merit of the controller is that it can be implemented by each agent in its local frame so that only local information is utilized without knowing global information. Theoretical analysis of the convergence to the desired formation, of which the agents are required to be evenly distributed on a circle around the target, is provided for the multiagent system under the proposed controller. Numerical simulations are given to validate the effectiveness of the proposed controller for the cases of general formations, and to show that no collision between agents ever takes place throughout the system's evolution. [ABSTRACT FROM AUTHOR]

• A design of an explicit analytical expression of Poincaré maps for non-autonomous periodically forced nonlinear systems is achieved. • A linearization of the nonlinear dynamics around a desired period-m limit cycle is performed. • We design three different time-piecewise-constant control laws to stabilize the limit cycle. • An application to the chaotic Duffing oscillator is realized to stabilize period-1 and period-2 limit cycles. In this paper, a design of Poincaré maps and time–piecewise–constant state–feedback control laws for the stabilization of limit cycles in periodically–forced, non–autonomous, nonlinear dynamical systems is achieved. Our methodology is based mainly on the linearization of the nonlinear dynamics around a desired period– m unstable limit cycle. Thus, this strategy permits to construct an explicit mathematical expression of a controlled Poincaré map from the structure of several local maps. An expression of the generalized controlled Poincaré map is also developed. To make comparisons, we design three different time–piecewise–constant control laws: an mT –piecewise–constant control law, a T –piecewise–constant control law and a T n –piecewise–constant control law. As an illustrative application, we adopt the chaotic Duffing oscillator. By applying the designed piecewise–constant control laws to the Duffing oscillator, the system is stabilized on its desired period– m limit cycle and hence the chaotic motion is controlled. [ABSTRACT FROM AUTHOR]

A study of data on the Zika outbreak in Brazil in 2015–2016 provides knowledge that Zika infection can trigger brain disorders such as Guillain–Barré Syndrome in adults and microcephaly in newborns. Zika infection is a vector-borne disease most commonly transmitted to humans through an infected Aedes mosquito bite, which is also the primary vector for dengue. Thus, many cases of these two diseases were recorded in the same population. In this study, we formulated a Zika–dengue coinfection model that describes the combined dynamics of Zika and dengue, involving the birth of microcephaly-malformed infants from pregnant women infected with the Zika virus. The analysis of the model states that three equilibrium points can be calculated explicitly, namely a disease-free equilibrium point and two boundary endemic equilibrium points. In addition, there is one implicit coinfection equilibrium point. The basic reproduction number is obtained along with the stability of disease-free equilibrium. Condition for the instability of the two boundary equilibria is obtained. This condition allows for a bifurcation point in the numerical exploration of the coinfection equilibrium point and its stability properties. The numerical simulation shows that the coinfection point appears when the two boundary equilibria are unstable. In addition, two Hopf bifurcation points are found with stable limit cycles between them. [ABSTRACT FROM AUTHOR]

In this paper, the number and distributions of limit cycles bifurcating from a double homoclinic loop and a double heteroclinic loop of piecewise smooth systems with three zones are considered. By introducing a suitable Poincaré map near the double homoclinic loop, three criteria are derived to judge its inner and outer stability. Then through stability-changing method, bifurcation theorems of limit cycles near the double homoclinic and double heteroclinic loops for non-symmetric and symmetric piecewise near-Hamiltonian systems are established. A piecewise linear Z 2 -equivariant system is presented as an application and five limit cycles are obtained, three of which are alien limit cycles. • Criteria for the inner and outer stability of a 3-piecewise double homoclinic loop. • Double homoclinic bifurcation for non-symmetric and symmetric piecewise systems. • Double heteroclinic bifurcation for non-symmetric and symmetric piecewise systems. • Obtaining five limit cycles for the first time for 3-piecewise linear system. [ABSTRACT FROM AUTHOR]

The nonlinear characteristics of the flow-induced vibration of an elastically-supported cylinder-plate assembly are investigated using numerical simulations of the flow past the assembly at a low-Reynolds number R e = 100. The beating phenomenon in the time series of the transverse (lift) force coefficient for both vortex-induced vibration and galloping is studied. The dynamical characteristics of the beating are elucidated using local attributes (local frequency), global attributes (periodicity and symmetry), and the properties of the energy transfer (per unit time) between the cylinder-plate assembly and the surrounding fluid. From this analysis, five distinct types of beating for a cylinder-plate assembly are identified. The nonlinear dynamical characteristics is investigated using a number of methodologies such as the power spectral density, the phase-plane portrait and the Poincaré section. The analysis implies the existence of a close relationship between the multiple characteristic frequencies in the power spectrum and the beating observed in the time variations. Three types of nonlinear oscillations in a cylinder-plate assembly have been identified and characterized in terms of the nature of their limit cycles in the phase plane and the point set distribution in the Poincaré section. The wake mode of a cylinder-plate assembly during beating is found to be invariably asymmetric with highly irregular vortex shapes, e.g., a right-angled vortex designated as S A and a crescent-shaped vortex designated as S B —multiplets (groups) of vortices involving either S A or S B with two elliptically-shaped vortices are closely associated with the observed beating. [ABSTRACT FROM AUTHOR]

In this paper, we study the relation between the coefficients in the expansions of two Melnikov functions near a heteroclinic loop with nilpotent cusps. Based on this relation, we give a condition of obtaining limit cycles near the heteroclinic loop. Further, we present a method to compute more coefficients in the expansions of two Melnikov functions near the heteroclinic loop. As an application, we consider a class of Liénard systems and study the number of limit cycles bifurcated from a heteroclinic loop and an elementary center. • For a heteroclinic loop with n nilpotent cusps, we give the expansions of two Melnikov functions near it. • We give the relation between the coefficients in the expansions of two Melnikov functions. • We present a method to compute the coefficients of the terms (h 2 , h 3 , ......) appearing in the expansions of two Melnikov functions. [ABSTRACT FROM AUTHOR]

• We perform a recurrence analysis of solutions from the generalized 9D Lorenz model. • Our analysis is effective in revealing two kinds of attractor coexistence. • The first kind of coexistence includes chaotic and steady-state solutions. • The second kind of coexistence contains limit cycle and steady-state solutions. • We show the potential of detecting non-chaotic processes for better predictability. Based on recent studies using high-dimensional Lorenz models (LMs), a revised view on the nature of weather has been proposed as follows: the entirety of weather is a superset that consists of both chaotic and non-chaotic processes. We suggest that better predictability may be obtained for non-chaotic processes if they can be identified in advance. In this study, to achieve the goal, we generate recurrence plots (RPs) for classifying various types of solutions obtained using simplified and full versions of the generalized Lorenz model (GLM) with various M modes, including M = 3 , 5 , 7 , and 9. We first perform recurrence analyses of the following solutions: (1) a periodic solution and quasi-periodic solutions containing multiple incommensurate frequencies; (2) a temporal transition from an unstable solution to a limit cycle solution, which is an isolated closed orbit; and (3) the coexistence of two types of solutions such as a steady-state and a chaotic solution or a steady-state and limit cycle solution. Various types of solutions that coexist depend only on the initial conditions (ICs). Additionally, to effectively detect the dependence of the coexistence on ICs, we further complete the following: (1) We derive a new system, referred to as version 2 (V2), by decomposing a total field into a basic state and a perturbation. The V2 system is capable of depicting the linear and nonlinear evolution of perturbations that deviate from the basic state represented by a non-trivial critical point solution. (2) We perform ensemble runs with various ICs distributed over a hypersphere centered at a non-trivial critical point. (3) We produce recurrence plots for the ensemble runs using various radii for the hypersphere. The feasibility of applying the above method for analyzing African Easterly Waves (AEWs) is discussed near the end. [ABSTRACT FROM AUTHOR]

Abstract Phase portraits are an invaluable tool in studying differential systems. Most of known results about global phase portraits are related to the smooth differential systems. This paper deals with a class of planar continuous piecewise linear Liénard differential systems with three zones separated by two vertical lines without symmetry. We provide the topological classification of the phase portraits in the Poincaré disc for systems having a unique singular point located in the middle zone. [ABSTRACT FROM AUTHOR]

Most of the proposed controllers for switched affine systems are able to drive the system trajectory to a sufficiently small neighborhood of a desired state. However, the trajectory behavior in this neighborhood is not generally considered despite the fact that the system performance may be judged by its steady operation as the case of power converters. This note investigates the local stabilization of a desired limit cycle in switched affine systems using the hybrid Poincaré map approach. To this end, interesting algebraic properties of the Jacobian of the hybrid Poincaré map are firstly discussed and used for the controller design to achieve asymptotic stability conditions of the limit cycle. A dc–dc four-level power converter is considered as an illustrative example to highlight the developed results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the impact of delayed tax revenues on the outcomes of fiscal policy. The analytical framework is the Goodwin growth cycle model, which is founded on the Volterra predator-prey equations. We study the dynamic behavior of the system analytically proving the existence of Hopf bifurcations, which may be supercritical and subcritical. In the numerical simulations, which follows the qualitative analysis, it is shown that, given the degree of competition in the markets, fiscal policy purposes may become consistent with their real outcomes only if policy makers are able to control the delay in the structure of the tax system. Nevertheless, there are in the system elements out of the control of the policy makers. These elements imply the possibility to make partially ineffectual the stabilization policy, because of the risk to overcome the minimum public expenditure able to stabilize the system. [ABSTRACT FROM AUTHOR]

We study feedback control for a discrete-time integrator with unitary delay in the presence of quantization both in the control action and in the measurement of the controlled variable. In some applications the quantization effects can be neglected, but when high precision is needed, they have to be explicitly accounted for in control design. In this paper, we propose a switched control solution for minimizing the effect of quantization of both the control and controlled variables for the considered system, that is quite common in the computing systems domain, for example, in thread scheduling, clock synchronization, and resource allocation. We show that the switched solution outperforms the one without switching, designed by neglecting quantization, and analyze necessary and sufficient conditions for the controlled system to exhibit periodic solutions in the presence of an additive constant disturbance affecting the control input. Simulation results provide evidence of the effectiveness of the approach. [ABSTRACT FROM AUTHOR]

In this paper, we study the bifurcation of limit cycles in two special near-Hamiltonian polynomial planer systems which their corresponding Hamiltonian systems have a heteroclinic loop connecting a hyperbolic saddle and a cusp of order two. In these systems, we will compute the asymptotic expansions of corresponding first order Melnikov functions near the loop and the center to analyze the number of limit cycles. Moreover, in the first system, by using the Chebychev criterion, we study the Poincaré bifurcation. [ABSTRACT FROM AUTHOR]

We study Poincaré bifurcation for a planar piecewise near-Hamiltonian system with two regions separated by a non-regular separation line, which is formed by two rays starting at the origin and such that the angle between them is α  ∈ (0, π ). The unperturbed system is a piecewise linear system having a periodic annulus between the origin and a homoclinic loop around the origin for all α  ∈ (0, π ). We give an estimation of the maximal number of the limit cycles which bifurcate from the periodic annulus mentioned above under n -th degree polynomial perturbations. Compared with the results in [13], where a planar piecewise linear Hamiltonian system with a straight separation line was perturbed by n -th degree polynomials, one more limit cycle is found. Moreover, based on our Lemma 2.5 we improve the upper bounds on the maximal number of zeros of the first order Melnikov functions derived in [19]. [ABSTRACT FROM AUTHOR]

In this paper, a four-dimensional model of flexible prices with the central bank's stabilization policy, describing the development of the firms' private debt, the output, the expected rate of inflation and the rate of interest is analyzed. Questions concerning the existence of limit cycles around its normal equilibrium point are investigated. The bifurcation equation is found. The formulae for the calculation of its coefficients are gained. A numerical example is presented by means of numerical simulations. [ABSTRACT FROM AUTHOR]

Dynamics of Goodwin’s accelerator business cycle model is reconsidered. The model is characterized by a nonlinear accelerator and an investment time delay. The role of the nonlinearity for the birth of persistent oscillations is fully discussed in the existing literature. On the other hand, not much of the role of the delay has yet been revealed. The purpose of this paper is to show that the delay really matters. In the original framework of Goodwin [6] , it is first demonstrated that there is a threshold value of the delay: limit cycles arise for smaller values than the threshold and so do sawtooth oscillations for larger values. In the extended framework in which a consumption or saving delay, in addition to the investment delay, is introduced, three main results are demonstrated under assumption of the identical length of investment and consumption delays. The dynamics with consumption delay is basically the same as that of the single delay model. Second, in the case of saving delay, the steady state can coexist with the stable and unstable limit cycles in the stable case. Third, in the unstable case, there is an interval of delay in which the limit cycle or the sawtooth oscillation emerges depending on the choice of the constant initial function. [ABSTRACT FROM AUTHOR]

In this article, we first suggest that the attractor of Brownian motor model is one of the reasons for the directed transport phenomenon of Brownian particle. We take the classical Smoluchowski–Feynman (SF) ratchet model as an example to investigate the relationship between limit cycles and directed transport phenomenon of the Brownian particle. We study the existence and variation rule of limit cycles of SF ratchet model at changing parameters through mathematical methods. The influences of these parameters on the directed transport phenomenon of a Brownian particle are then analyzed through numerical simulations. Reasonable mathematical explanations for the directed transport phenomenon of Brownian particle in SF ratchet model are also formulated on the basis of the existence and variation rule of the limit cycles and numerical simulations. These mathematical explanations provide a theoretical basis for applying these theories in physics, biology, chemistry, and engineering. [ABSTRACT FROM AUTHOR]

In this paper, we study the bifurcation and stability of a ratio-dependent predator-prey model with nonconstant predator harvesting rate. The analysis is carried out both analytically and numerically. We determine stability and dynamical behaviours of the equilibria of this system and characterize codimension 1 and codimension 2 bifurcations of the system analytically. Our bifurcation analysis indicates that the system exhibits numerous types of bifurcation phenomena, including Fold, Hopf, Cusp, and Bogdanov–Takens bifurcations. We use the numerical software MATCONT, to compute curves of equilibria and to compute several bifurcation curves. We especially approximate a family of limit cycles emanating from a Hopf point. Our results generalize and improve some known results and show that the model has more rich dynamics than the ratio-dependent predator-prey model without harvesting rate. [ABSTRACT FROM AUTHOR]

We study the circle formation problem for a group of anonymous mobile agents in a plane, in which the agents are required to converge onto a circle with a preset target as the center, as well as to maintain the desired relative positions when rotating around the target at the same speed. Each agent is modeled as a kinematic point and can merely perceive the relative positions of the target and its limited neighbors. In order to solve the circle formation problem, a limit-cycle-based decoupled-design approach is delivered. We divide the overall control objective into two subobjectives, where the first is target circling that all agents converge onto the circle around the target, and the second is spacing adjustment that each agent maintains the desired distance from its neighbors. Then, we propose to use a controller comprised of converging part and layout part to deal with these two subobjectives, respectively. The former part is based on a limit-cycle oscillator using only the relative position from the target, and the latter is designed by also perceiving the relative position from the agent's neighbors. An important feature of the controller is that it guarantees that no collision between agents ever takes place throughout the system's evolution. Another feature is that some of the parameters in the proposed controller have explicit physical meanings related to the agents’ rotating motion around the target, so that they can be set more reasonable and easily in real applications. Numerical simulations are given to show the effectiveness and performance of the proposed circle formation controller. [ABSTRACT FROM PUBLISHER]

Calcium ions are important in cell process, which control cell functions. Many models on calcium oscillation have been proposed. Most of existing literature analyzed calcium oscillations using numerical methods, and found rich dynamical behaviours. In this paper, we explore a further study on an established three-store model, which contains endoplasmic reticulum (ER), mitochondria and calcium binding proteins. We conduct bifurcation analysis to identify two Hopf bifurcations, and apply normal form theory to study their stability and show that one of them is supercritical while the other is subcritical. Further, we transform the model into a slow-fast system, and then apply the geometrical singular perturbation theory to investigate the mechanism of generating slow-fast motions. The study reveals that the mechanism of generating the slow-fast oscillating behaviour in the three-store calcium model for certain parameter values is due to the relative fast change in the free calcium in cytosol, and relative slow changes in the free calcium in mitochondria and in the bounded Ca 2 + binding sites on the cytosolic proteins. A further parametric study may provide some useful information for controlling harmful effect, by adjusting the amount of calcium in a human body. Numerical simulations are present to demonstrate the correct analytical predictions. [ABSTRACT FROM AUTHOR]

In this paper we unveil the existence of hidden chaotic sets in a simplified Hopfield neural network with three neurons. It is shown that beside two stable cycles, the system presents hidden chaotic attractors and also hidden chaotic transients which, after a relatively long life-time, fall into regular motions along the stable cycles. [ABSTRACT FROM AUTHOR]

In this paper, we study bifurcation of limit cycles from a homoclinic loop connecting a saddle of tangent type for a quadratic Hamiltonian system perturbed by n th degree polynomials, n = 1 , 2 , ... , 13. The main tool is the asymptotic expansion of the related Abelian integral near the homoclinic loop, and the maximal number of independent coefficients gives exact number of limit cycles. Our aim is to obtain more limit cycles by exploring more coefficients in the asymptotic expansion. However, it is usually very difficult to obtain the coefficients of the terms with degree greater than or equal to 2 in the asymptotic expansion. To overcome the difficulty, we derive two auxiliary systems and investigate the expansions for the related Abelian integral. The coefficients of lower degree terms in the new asymptotic expansions are equivalent to those of higher degree terms in the original asymptotic expansion. We obtain n − 1 − n − 2 4 limit cycles near the non-regular homoclinic loop and n − 2 4 limit cycles near the center, and it totally has at least n − 1 limit cycles, when n ∈ { 1 , 2 , ... , 13 }. The cyclicity of period annulus is also estimated by the first order Melnikov functions for n = 3. • A complete discussion for the maximal number of limit cycles near the non-regular homoclinic loop and the center for a quadratic reversible Hamiltonian system perturbed by n -th degree polynomials (n = 1, 2, ...,13) is given. • The maximal number of limit cycles near the non-regular homoclinic loop is at least n − 1 − n − 2 4 and near the center is n − 2 4 when n ∈ { 1 , 2 , ... , 13 }. The obtained results are novel. • The upper bound on the number of limit cycles of the perturbed quadratic reversible Hamiltonian system with n = 3 is given. This is also a new discover. [ABSTRACT FROM AUTHOR]

This paper aims to study impact limit cycles in the planar piecewise linear hybrid systems formed by center type vector fields and reset maps on the impact surfaces. Motivated by Llibre & Teixeira, 2018, where an open problem was posed: Piecewise linear differential systems with only centers can create limit cycles? We answer this problem for piecewise linear hybrid systems separated by one or two parallel straight lines. By using Poincaré map and first integral, we present an estimate of the maximum number of two-zone and three-zone impact limit cycles. When the hybrid systems are separated by a unique straight line, they can have at most 1 two-zone impact limit cycle. When they are separated by two parallel straight lines, we show that such hybrid systems can have at most 2 three-zone impact limit cycles. Furthermore, we employ some numerical examples to illustrate our main results and show that this upper bound can indeed be reached. • The mathematical theory is related to Hilbert's 16th problem. • Hybrid system can be widely detected in various practical fields. • This paper studies the maximum number of impact limit cycles in a class of piecewise linear hybrid system. • We employ some numerical examples to show that this upper bound can indeed be reached. [ABSTRACT FROM AUTHOR]

Two examples of collapse of a Neimark–Sacker torus in the Rössler system are studied. Their stability is evaluated by measuring the rotation number, defined as the long-term average value of the angular phase shift that occurs in the Poincaré map on each pass, and by theoretically calculating the same angular shift within the context of a normalized cubic model. The parameters of the cubic model are evaluated using perturbation theory applied to a limit cycle, which involves first evaluating the response after a fixed time, the known period of the limit cycle, and then calculating the additional effect due to variations in time of flight to reach a fixed plane perpendicular to velocity. For the first example of collapse, the above two methods yield the same result, namely that the collapse is characterized by a constant rotation number as we move away from the bifurcation, leading to a torus that usually coalesces into discrete branches and then collapses due to turning points. This mechanism allows us to define a simple criterion that the parameters of the cubic model must satisfy, which is related to stability. The second example of collapse is more complex: the normalized cubic model suggests correctly that the collapse will be associated with the calculated radius of the torus becoming complex, which leads to a second criterion for the cubic parameters. However, the predicted point of collapse is significantly premature. The actual torus continues to survive until the rotation number approaches zero, leading to a single wave packet of infinite duration that travels between the two known equilibrium points of the system. • Stability of a Neimark–Sacker torus can be analyzed using the angular phase shift in a phase portrait. • The phase shift can be measured experimentally as an average. • The phase shift can be calculated theoretically using a cubic model and perturbation theory. • For a "Resonant" collapse into discrete branches, the total phase shift is constant. • For a "Non-Cubic" collapse, the solution becomes complex, or turns into a single wave packet. [ABSTRACT FROM AUTHOR]

This paper proposes an original interval-based method to compute an outer approximation of all invariant sets (such as limit cycles) of a continuous-time nonlinear dynamic system, which are included inside a prior set of the state space. Contrary to all other existing approaches, our method has the following properties: first, it is guaranteed (a solution cannot be lost); second, it is applicable to a large class of systems without any specific assumption such as the knowledge of a Lyapunov function or any partial linearity; and third, there is no need to integrate the system. [ABSTRACT FROM AUTHOR]

In this paper an asymmetric planar continuous piecewise linear differential system with three zones x ˙ = y − F ( x ) , y ˙ = − g ( x ) is considered. The aim of this paper gives a completely study of limit cycles when this system satisfies such conditions and the uniqueness equilibrium does not lie in the central region. When ( x − x 0 ) g ( x ) > 0 for ∀ x ≠ x 0 and y = F ( x ) is a Z -shaped curve, it owns at most two limit cycles, which exist between a linear Hopf bifurcation surface and a double limit cycle bifurcation surface. Moreover, we prove the conjectures proposed by Ponce et al. [27]. When the uniqueness equilibrium lies in the central region, this system has exactly one limit cycles by others. Finally, some numerical examples are demonstrated. [ABSTRACT FROM AUTHOR]

Based on the study of scholars at home and abroad, this paper improves the three-dimensional IS-LM model in macroeconomics, analyzes the equilibrium point of the system and stability conditions, focuses on the parameters and complex dynamic characteristics when Hopf bifurcation occurs in the three-dimensional IS-LM macroeconomics system. In order to analyze the stability of limit cycles when Hopf bifurcation occurs, this paper further introduces the first Lyapunov coefficient to judge the limit cycles, i.e. from a practical view of the business cycle. Numerical simulation results show that within the range of most of the parameters, the limit cycle of 3D IS-LM macroeconomics is stable, that is, the business cycle is stable; with the increase of the parameters, limit cycles becomes unstable, and the value range of the parameters in this situation is small. The research results of this paper have good guide significance for the analysis of macroeconomics system. [ABSTRACT FROM AUTHOR]

Generation of low jitter, high frequency clock from a low frequency reference clock using classical analog phase-locked loops (PLLs) requires large loop filter capacitor and power hungry oscillator. Digital PLLs can help reduce area but their jitter performance is severely degraded by quantization error. Specifically, their deterministic jitter (DJ), which is proportional to the loop update rate becomes prohibitively large at low reference clock frequencies. We propose a scrambling TDC (STDC) to improve DJ performance and a cascaded architecture with digital multiplying delay locked loop as the first stage and hybrid analog/digital PLL as the second stage to achieve low random jitter in a power efficient manner. Fabricated in a 90 nm CMOS process, the prototype frequency synthesizer consumes 4.76 mW power from a 1.0 V supply and generates 160 MHz and 2.56 GHz output clocks from a 1.25 MHz crystal reference frequency. The long-term absolute jitter of the 160 MHz digital MDLL and 2.56 GHz digital PLL outputs are 2.4 $\rm{{ps}_{rms}}$ and 4.18 $\rm{{ps}_{rms}}$ , while the peak-to-peak jitter are 22.1 ps and 35.2 ps, respectively. The proposed frequency synthesizer occupies an active die area of 0.16mm2 and achieves power efficiency of 1.86 mW/GHz. [ABSTRACT FROM PUBLISHER]

Time delay is an essential ingredient of spatio-temporal predator–prey models since the reproduction of the predator population after predating the prey will not be instantaneous, but is mediated by a constant time lag accounting for the gestation of predators. In this paper we study a predator–prey reaction–diffusion system with time delay, where a stability analysis involving Hopf bifurcations with respect to the delay parameter and simulations produced by a new numerical method reveal how this delay affects the formation of spatial patterns in the distribution of the species. In particular, it turns out that when the carrying capacity of the prey is large and whenever the delay exceeds a critical value, the reaction–diffusion system admits a limit cycle due to the Hopf bifurcation. This limit cycle induces the spatio-temporal pattern. The proposed discretization consists of a finite volume element (FVE) method combined with a Runge–Kutta scheme. [ABSTRACT FROM AUTHOR]

This paper proposes a mathematical model that describes the interaction of prey and predators, assuming logistic growth for both species, harvesting and alternative food for predators and functional response of the Holling II predator. When performing a qualitative analysis to determine conditions in the parameters that allow the possible extinction or preservation of prey and/or predators, a modification of the initial model is made considering that the consumption of prey by predators is restricted if the amount of prey is less than a critical value, whose dynamics is formulated by a planar Filippov system. The study of the discontinuous model is carried out by bifurcation analysis in relation to two parameters: harvesting of predators and critical value of prey. • A model describing the behavior of both species at low and high densities is proposed. • The Filippov method is used to describe the dynamics in the discontinuity region. • A bifurcation study is performed to show the solutions of the discontinuous model. • This study determines conditions that allow the coexistence of both species. [ABSTRACT FROM AUTHOR]

Although switching systems have been investigated intensively, there are few results about limit cycles bifurcated from switching systems with degenerate singular point. In this paper, a method to compute focal values for degenerate critical point of switching systems was proposed. Furthermore, we studied a quartic system in order to illustrate the efficiency of our method. [ABSTRACT FROM AUTHOR]

Eco-evolutionary dynamics is crucial to understand how individuals' behaviors and the surrounding environment interplay with each other. Typically, it is assumed that individuals update their behaviors via linear imitation function, i.e., the replicator dynamics. It has been proved that there cannot be limit circles in such eco-evolutionary dynamics. It suggests that eco-evolutionary dynamics alone is not sufficient to explain the widely observed fluctuating behavior in both the biological and social systems. We extrapolate from the linear imitation function to general imitation function, which can be non-linear. It is shown that the general imitation does not change the internal equilibrium and its local stability. It however leads to limit cycles, which are never present in classical eco-evolutionary dynamics. Moreover, the average cooperation level and average environment state agree with the first and the second component of the internal fixed point. Furthermore, we estimate the location of the emergent limit cycle. On the one hand, our results provide an alternative mechanism to make cooperators and defectors coexist in a periodic way in the "tragedy of the commons" and indicates the global eco-evolutionary dynamics is sensitive to the imitation function. Our work indicates that the way of imitation, i.e., the imitation function, is crucial in eco-evolutionary dynamics, which is not true for evolutionary dynamics with a static environment. • The existence of limit circles in eco-evolutionary dynamics with general imitation functions is proved and the location of the limit circle is estimated. • The non-linearity of the imitation function leads to the stable limit cycles, which represent the coexistence of cooperators and defectors in an oscillatory way. • The results are surprising that the global eco-evolutionary dynamics can be greatly altered by the imitation function alone. [ABSTRACT FROM AUTHOR]

The algebraic criterion for Abelian integral was posed in (Grau et al. Trans Amer Math Soc 2011) and (Mañosas et al. J Differ Equat 2011) to bound the number of limit cycles bifurcating from the center of polynomial Hamiltonian system. Thisapproach reduces the estimation to the number of the limit cycle bifurcating from the center to solve the associated semi-algebraic systems (the system consists of polynomial equations, inequations and polynomial inequalities). In this paper, a systematic procedure with interval analysis has been explored to solve the SASs. In this application, we proved a hyperelliptic Hamiltonian system of degree five with a pair of conjugate complex critical points that could give rise to at most six limit cycles at finite plane under perturbations ɛ ( a + b x + c x 3 + x 4 ) y ∂ ∂ x . Moreover we comment the results of some related works that are not reliable by using numerical approximation. [ABSTRACT FROM AUTHOR]

Vortex-induced vibration of a three-dimensional flexible tube is one of the key problems to be considered in many engineering situations. This paper aims to investigate the nonlinear dynamic behaviors and response characteristics of a three-dimensional tube under turbulent flow. The three-dimensional unsteady, viscous, incompressible Navier–Stokes equation and LES turbulence model are solved with the finite volume approach, and the dynamic equilibrium equations are discretized by the finite element theory. A three-dimensional fully coupled numerical model for vortex-induced vibration of flexible tube is proposed. The model realized the fluid–structure interaction with solving the fluid flow and the structure vibration simultaneously. Based on this model, Response regimes, trajectory, phase difference, fluid force coefficient and vortex shedding frequency are obtained. The nonlinear phenomena of lock-in, phase-switch are captured successfully. Meanwhile, the limit cycle, bifurcation of lift coefficient and displacement are analyzed using phase portrait and Poincare section. The results reveal that, a quasi-upper branch occurs in the present fluid-flexible tube coupling system with high mass-damping and low mass ratio. There is no bifurcation of lift coefficient and lateral displacement occurred in the three-dimensional flexible tube submitted to uniform turbulent flow. [ABSTRACT FROM AUTHOR]

The influence of seeker’s disturbance rejection rate on the stability of on-off guidance loop is analyzed. Firstly, a unified dynamics model for gyro-stabilized seeker and rate-gyro seeker is established by using equivalent transformation principle of loop. And then, equivalent Lure’ model of on-off guidance loop with the isolation parasitic loop is built to analyze the stability of the on-off control system. It is concluded from comparisons that the describing function method is superior to Popov theory and simulation method in the aspects of system conservation and stability trend predication. So, the describing function method is utilized to analyze the stability of the parasitic loop. Furthermore, the influence of cable torque and friction torque on the stable range of parasitic loop under parameters of different vehicles and modulators is obtained. Finally, the lag compensation method is put forward for unstable parasitic loop to improve system’s stability. Simulation results show that the limit cycle is not appeared again in the guidance loop under the vehicle’s attitude disturbance. It avoids pulsed thruster repeatly switching. And, when the vehicle maneuvers to a track target, it saves fuel and assures the guidance accuracy simultaneously. [ABSTRACT FROM AUTHOR]

This paper presents simple but accurate expressions for the noise components caused by clock jitter, in the output signal of self-oscillating sigma delta modulators (SOSDM). Contrary to conventional continuous time sigma delta modulators (CTSDM), the SOSDM's loop contains a strong oscillation, whose attribution to the system's jitter caused noise has not previously been explored. In this paper, the SOSDM system is modeled, and the effect of the self oscillation, the input signal and the quantization noise on the jitter caused noise in the output signal, is calculated. Results are confirmed by system level simulations. [ABSTRACT FROM AUTHOR]

The paper proposes a technique to accurately and efficiently locate periodic steady-state solutions of electric power systems. This technique is based on an enhanced version of the time-domain shooting method (TDSM) and the probe-insertion technique (PIT). The latter has been successfully applied to low-power electronic circuits but it is innovative for the study of electromechanical steady-state periodic behavior of power systems. With this aim, the paper discusses the inherent criticalities of the conventional formulation of power system models (PSMs). Then, a novel formulation is proposed to accommodate the hypotheses and mathematical requirements of the TDSM and PIT. The effectiveness and numerical efficiency of the proposed model and technique are discussed through two case studies based on the IEEE 14-bus and WSCC 9-bus systems. [ABSTRACT FROM AUTHOR]

Limit cycle bifurcations for a class of perturbed planar piecewise smooth systems with 4 switching lines are investigated. The expressions of the first order Melnikov function are established when the unperturbed system has a compound global center, a compound homoclinic loop, a compound 2-polycycle, a compound 3-polycycle or a compound 4-polycycle, respectively. Using Melnikov’s method, we obtain lower bounds of the maximal number of limit cycles for the above five different cases. Further, we derive upper bounds of the number of limit cycles for the later four different cases. Finally, we give a numerical example to verify the theory results. [ABSTRACT FROM AUTHOR]

Real number modeling of analog circuits in hardware description languages (HDLs) has become more common as a part of mixed-signal SoC validation. We propose two methods that both improve the fidelity and simulation speed, and make the event-driven, piecewise linear (PWL) analog functional models easier to write. First we use the accuracy set by users to dynamically determine when a new output segment should be emitted, which is computed without any iteration. This capability allows designers to trade accuracy for simulation speed of analog models without any time-consuming model calibration/error estimation, and creates models which generate events only when needed to maintain output accuracy. We next extend this method to eliminate limit-cycle oscillations that occur when simulating circuits with continuous-time feedback in a discrete-time event simulator. Handling this feedback efficiently allows the user to create the system model from simpler component models. The performance of this modeling approach is demonstrated on various analog filter models, an operational amplifier, and a high-speed, wireline transceiver system, and is 3.1 $\times$ faster than an optimally-chosen, fixed-time step simulation for the transceiver. [ABSTRACT FROM PUBLISHER]

In this paper, an existing method is modified for computing the focal values and period constants of switching systems associated with elementary singular points. In particular, a quadratic switching system is considered to illustrate the computational efficiency of this method. Further, with this method, a cubic switching system is constructed to show existence of 15 limit cycles, which is the best result so far obtained for cubic switching systems. [ABSTRACT FROM AUTHOR]

This paper deals with the problem of creating stable oscillations in the class of nonlinear sandwich systems with dead-zone nonlinearity and unknown disturbances by creating the stable limit cycle. Due to the presence of the non-smooth dead-zone nonlinearity in the middle of the separate subsystems in the sandwich systems' structure, the control synthesis of such systems is complex. In this paper, to deal with the complexities, the control design process is implemented in two steps and a robust adaptive controller is presented by utilizing the sliding mode control (SMC) method. To achieve the control purpose through the SMC, a novel sliding surface is introduced regarding the structure of the admissible limit cycle, which is associated with some characteristics such as frequency, shape, and the amplitude of the desired sustained output oscillations. Simultaneously, the adaptive scheme is employed through the controller design process to extract the proper adaptive laws to estimate the upper bounds of unknown terms. The proposed method guarantees that the admissible limit cycle is generated in the phase plane of the controlled system and consequently verifies that the desired sustained oscillation (with the prescribed properties) is produced at the system's output. It also ensures that all variables in the resulting closed-loop system keep bounded. Finally, to show the effectiveness of the presented method, the given approach is applied to a practical system, and simulation results are given. • The sustained oscillation is created in the nonlinear sandwich system with dead-zone via the limit cycle control approach. • A novel sliding surface is introduced based on the structure of the desired limit cycle. • An adaptive controller is designed by using a novel SMC strategy. • Adaptive laws are extracted to cope with the unknown external disturbances. • The efficiency of the proposed method is verified via simulating a practical system. [ABSTRACT FROM AUTHOR]

While the limit cycles of the discontinuous piecewise differential systems formed by two linear differential systems separated by one straight line have been studied intensively, and up to now there are examples of these systems with at most 3 limit cycles. There are almost no works studying the limit cycles of the discontinuous piecewise differential systems formed by one linear differential system and a quadratic polynomial differential system separated by one straight line. In this paper using the averaging theory up to seven order we prove that the discontinuous piecewise differential systems formed by a linear focus or center and a quadratic weak focus or center separated by one straight line can have 8 limit cycles. More precisely, at every order of the averaging theory from order one to order seven we provide the maximum number of limit cycles that can be obtained using the averaging theory. Primary 34C05, 34A34. • The purpose of this paper is the study of a piecewise differential system formed by a linear focus or center and a quadratic weak focus separated by the straight-line x = 0. • Study the maximum number of limit cycles of this discontinuous piecewise differential system. • Using averaging theory we prove that this piecewise differential system can have at most eight limit cycles. [ABSTRACT FROM AUTHOR]