Identification of momentum life cycle stage of stock price

In this article, we study the continuous and discontinuous planar piecewise differential systems formed only by linear centers separated by one or two parallel straight lines. When these piecewise differential systems are continuous, they have no limit cycles. Also if they are discontinuous separated by a unique straight line, they do not have limit cycles. But when the piecewise differential systems are discontinuous separated two parallel straight lines, we show that they can have at most one limit cycle, and that there exist such systems with one limit cycle.     

Piecewise linear differential systems without equilibria produce limit cycles?

In this article, we study the planar piecewise differential systems formed by two linear differential systems separated by a straight line, such that both linear differential have no equilibria, neither real nor virtual. When the piecewise differential system is continuous, we show that the system has no limit cycles. But when the piecewise differential system is discontinuous, we show that it can have at most one limit cycle.     

Limit cycles in a family of discontinuous piecewise linear differential systems with two zones in the plane

In this paper we study the existence of limit cycles in a one-parameter family of discontinuous piecewise linear differential systems with two zones in the plane. It is characterized for all the parameter values the number of non-sliding limit cycles of the family studied, detecting a rather non-generic bifurcation leading to the simultaneous generation of three limit cycles.     

The solution of the second part of the 16th Hilbert problem for nine families of discontinuous piecewise differential systems

We provide the maximum number of limit cycles of some classes of discontinuous piecewise differential systems formed by two differential systems separated by a straight line, when these differential systems are linear centers or three families of cubic isochronous centers, giving rise to ten different classes of discontinuous piecewise differential systems. These maximum number of limit cycles vary from 0, 1, 2, 3, 5, 7 and 12 depending on the chosen class. For nine of these classes, we prove that the corresponding maximum number of limit cycles are reached. In particular, we have solved the extension of the second part of the 16th Hilbert problem to these classes of discontinuous piecewise differential systems. The main tool used for proving these results is based on the first integrals of the systems which form the discontinuous piecewise differential systems.

     

Analysis and synthesis of oscillator systems described by a perturbed double-well Duffing equation

This paper presents an investigation of limit cycles in oscillator systems described by a perturbed double-well Duffing equation. The analysis of limit cycles is made by the Melnikov theory. Expressing the solutions of the unperturbed Duffing equation by Jacobi elliptic functions allows us to calculate explicitly the Melnikov function, whereupon the final result is a function involving the complete elliptic integrals. The Melnikov function is analyzed with the aid of the Picard–Fuchs and Riccati equations. It has been proved that the considered oscillator system can have two small hyperbolic limit cycles located symmetrically with respect to the y-axis, or one large hyperbolic limit cycle, or two large hyperbolic limit cycles, or one large limit cycle of multiplicity 2. Moreover, we have obtained the conditions under which each of these limit cycles arises. The present work gives the conditions for the arising of limit cycles around the homoclinic trajectory. In this connection, an alternative approach is proposed for obtaining a series expansion of the Melnikov function near the homoclinic trajectory. This approach uses the series expansion of the complete elliptic integrals as the elliptic modulus tends to 1. It is shown that a jumping phenomenon may occur between limit cycles in the analyzed oscillator system. The conditions for the occurrence of this jumping phenomenon are given. A method for the synthesis of an oscillator system with a preliminary assigned limit cycle is also presented in the article. The obtained analytical results are illustrated and confirmed by numerical simulations.     

Residue harmonic balance approach to limit cycles of non-linear jerk equations

A residue harmonic balance is established for accurately determining limit cycles to parity- and time-reversal invariant general non-linear jerk equations with cubic non-linearities. The new technique incorporates the salient features of both methods of harmonic balance and parameter bookkeeping to minimize the total residue. The residue is separated into two parts in each step; one conforms to the present order of approximation and the remaining part for use in the next order. The corrections are governed by a set of linear ordinary differential equations that can be solved easily. Three specific cases of non-linear jerk equations are given to illustrate the validity and efficiency. The approximations to the angular frequency and the limit cycle are obtained and compared. The results show that the approximations obtained are in excellent agreement with the exact solutions for a wide range of initial velocities. The new technique is simple in principle and can be applied to other non-linear oscillating systems.     

An Eigenvalue Method for Calculation of Stability and Limit Cycles in Nonlinear Systems

In the calculation of periodic oscillations of nonlinear systems –so-called limit cycles – approximative and systematic engineeringmethods of linear system analysis are known. The techniques, working inthe frequency domain, perform a quasi-linearization of the nonlinear system,replacing nonlinearities by amplitude-dependent describing functions.Frequently, the resulting equations for the amplitude and frequency ofpresumed limit cycles are solved directly by a graphical procedure in aNyquist plane or by solving the nonlinear equations or a parameteroptimization problem. In this paper, an indirect numerical approach isdescribed which shows that, for a system of nonlinear differentialequations, the eigenvalues of the quasi-linear system simply indicateall limit cycles and, additionally, yield stability regions for thelinearized case. The method is applicable to systems with multiplenonlinearities which may be static or dynamic. It is demonstrated foran example of aircraft nose gear shimmy dynamics in the presence ofdifferent nonlinearities and the results are compared with those fromsimulation.     

Creation–annihilation process of limit cycles in the Rayleigh–Duffing oscillator

The present paper examines the creation?Cannihilation process of limit cycles in the Rayleigh?CDuffing oscillator with negative linear damping and negative linear stiffness. It is obtained by the perturbation method, in which the number of limit cycles in the Rayleigh?CDuffing oscillator varies with the linear damping and stiffness. Numerical simulations are performed in order to confirm the analytically obtained creation?Cannihilation process of limit cycles. Moreover, we compare the process of limit cycles in the Rayleigh?CDuffing oscillator to that of limit cycles in the van der Pol?CDuffing oscillator. The difference in these oscillator is only in nonlinear forces, which causes a qualitative difference in the creation?Cannihilation processes.     

Limit Cycles of Piecewise Smooth Differential Equations on Two Dimensional Torus

In this paper we study the limit cycles of some classes of piecewise smooth vector fields defined in the two dimensional torus. The piecewise smooth vector fields that we consider are composed by linear, Ricatti with constant coefficients and perturbations of these one, which are given in (3). Considering these piecewise smooth vector fields we characterize the global dynamics, studying the upper bound of number of limit cycles, the existence of non-trivial recurrence and a continuum of periodic orbits. We also present a family of piecewise smooth vector fields that posses a finite number of fold points and, for this family we prove that for any 2k number of limit cycles there exists a piecewise smooth vector fields in this family that presents k number of limit cycles and prove that some classes of piecewise smooth vector fields presents a non-trivial recurrence or a continuum of periodic orbits.     

Domains of attraction for multiple limit cycles of coupled van der pol equations by simple cell mapping

One of the most difficult tasks in non-linear analysis is to determine globally various domains of attraction in the state space when there exist more than one asymptotically stable equilibrium states and/or periodic motions. The task is even more demanding if the order of the system is higher than two. In this paper we consider two coupled van der Pol oscillators which admit two asymptotically stable limit cycles. For systems of this kind we show how the method of cell-to-cell mapping can be used to determine the two four-dimensional domains of attraction of the two stable limit cycles in a very effective way. The final results are shown in this paper in the form of a series of graphs which are various two-dimensional sections of the four-dimensional state space.     

The stability of limit cycles in nonlinear systems

In this paper, a necessary condition is first presented for the existence of limit cycles in nonlinear systems, then four theorems are presented for the stability, instability, and semistabilities of limit cycles in second order nonlinear systems. Necessary and sufficient conditions are given in terms of the signs of first and second derivatives of a continuously differentiable positive function at the vicinity of the limit cycle. Two examples considering nonlinear systems with familiar limit cycles are presented to illustrate the theorems.     

The Sliding Bifurcations in Planar Piecewise Smooth Differential Systems

In this paper we are mainly interested in the bifurcation phenomena for a class of planar piecewise smooth differential systems, where a new phenomenon, i.e. sliding heteroclinic bifurcation, is found. Furthermore we will show that the involved systems can present many interesting bifurcation phenomena, such as the (sliding) heteroclinic bifurcation, sliding (homoclinic) cycle bifurcation and semistable limit cycle bifurcation and so on. The system can have two hyperbolic limit cycles, which are bifurcated in one way from a semistable limit cycle, and in another way from a heteroclinic cycle and a sliding cycle. In the proof of our main results, we will use the geometric singular perturbation theory to analyze the dynamics near the sliding region.     

Two-parameter bifurcation analysis of limit cycles of a simplified railway wheelset model

The effect of the nonlinear terms on bifurcation behaviors of limit cycles of a simplified railway wheelset model is investigated. At first, the stable equilibrium state loses its stability via a Hopf bifurcation. The bifurcation curve is divided into a supercritical branch and a subcritical one by a generalized Hopf point, which plays a key role in determining the occurrence of flange contact and derailment of high-speed railway vehicles, and the occurrence of this critical situation is an important decision-making criteria for design parameters. Secondly, bifurcations of limit cycles are discussed by comparing the bifurcation behavior of cycles for two different nonlinear parameters. Unlike local Hopf bifurcation analysis based on a single bifurcation parameter in most papers, global bifurcation analysis of limit cycles based on two bifurcation parameters is investigated, simultaneously. It is shown that changing nonlinear parameter terms can affect bifurcation types of cycles and division of parameter domains. In particular, near the branch points of cycles, two symmetrical limit cycles are created by a pitchfork bifurcation and then two symmetrical cycles both undergo a period-doubling bifurcation to form two stable period-two cycles. Around the resonant points, period orbits can make several turns, whose number of turns corresponds to the ratio of resonance. Thirdly, near the Neimark–Sacker bifurcation of cycles, a stable torus is created by a supercritical Neimark–Sacker bifurcation, which shows that the orbit of the model exhibits modulated oscillations with two frequencies near the limit cycle. These results demonstrate that nonlinear parameter terms can produce very complex global bifurcation phenomena and make obvious effects on possible hunting motions even though a simple railway wheelset model is concerned.

     

The global analysis of higher order nonlinear dynamical systems and the application of cell-to-cell mapping method

In this paper, the general characteristics and the topological consideration of the global behaviors of higher order nonlinear dynamical systems and the characteristics of the application of cell-to-cell mapping method in this analysis are expounded. Specifically, the global analysis of a system of two weakly coupled van der Pol oscillators using cell-to-cell mapping method is presented.The analysis shows that for this system, there exist two stable limit cycles in 4-dimensional state space, and the whole 4-dimensional state space is divided into two almost equal parts which are, respectively, the two asymototically stable domains of attraction of the two periodic motions of the two stable limit cycles. The validities of these conclusions about the global behaviors are also verified by direct long term numerical integration. Thus, it can be seen that the cell-to-cell mapping method for global analysis of fourth order nonlinear dynamical systems is quite effective.     

Switching-induced stable limit cycles

Physical limits place bounds on the divergent behaviour of dynamical systems. The paper explores this situation, providing an example where generator field-voltage limits capture behaviour, giving rise to a stable, though non-smooth, limit cycle. It is shown that shooting methods can be adapted to solve for such non-smooth switching-induced limit cycles. By continuing branches of switching-induced and smooth limit cycles, the paper established the co-existence of equilibria, smooth and non-smooth limit cycles. Furthermore, it is shown that when branches of switching-induced and smooth limit cycles merge, the limit cycles are annihilated at a grazing bifurcation.     

Bifurcation of limit cycles in piecewise-smooth systems with intersecting discontinuity surfaces

This paper deals with bifurcation of limit cycles for perturbed piecewise-smooth systems. Concentrating on the case in which the vector fields are defined in four domains and the discontinuity surfaces are codimension-2 manifolds in the phase space. We present a generalization of the Poincaré map and establish some novel criteria to create a new version of the Melnikov-like function. Naturally, this function is designed corresponding to a system trajectory that interacts with two different discontinuity surfaces. This provides an approach to prove the existence of special type of invariant manifolds enabling the reduction of dynamics of the full system to the two-dimensional surfaces of the invariant cones. It is shown that there exists a novel bifurcation concerning the existence of multiple invariant cones for such system. Further, our results are then used to control the persistence of limit cycles for two- and three-dimensional perturbed systems. The theoretical results of these examples are illustrated by numerical simulations.     

Center-focus problem and limit cycles bifurcations for a class of cubic Kolmogorov model

The problem of limit cycles for the Kolmogorov model is interesting and significant both in theory and applications. In this paper, we investigate the center-focus problems and limit cycles bifurcations for a class of cubic Kolmogorov model with three positive equilibrium points. The sufficient and necessary condition that each positive equilibrium point becomes a center is given. At the same time, we show that each one of point (1,2) and point (2,1) can bifurcate 1 small limit cycles under a certain condition, and 3 limit cycle can occur near (1,1) at the same step. Among the above limit cycles, 4 limit cycles can be stable. The limit cycles bifurcations problem for Kolmogorov model with several positive equilibrium points are hardly seen in published references. Our result is new and interesting.     

The orientation dynamics of small prolate and oblate spheroids in linear shear flows

The present work deals with the stable orientation of oblate and prolate spheroids in general steady linear flows and with the mode of convergence to these stable orientations. The orientation dynamics is governed by the Jeffery equation. The stable orientations are either fixed points or limit cycles in the orientation space. The type of stable orientation depends on whether the eigenvalues of the linear part of Jeffery equation are real or complex. We define prolate and oblate spheroids to be equivalent if the aspect ratio of one is the reciprocal of the other. We show that, in a given flow, equivalent oblate and prolate spheroids possess the same number of fixed points and limit cycles of which only one is stable. If they possess only fixed points, then their corresponding stable fixed points are orthogonal. If they possess one fixed point and one limit cycle each, then the stable fixed point of one is orthogonal to the plane of the limit cycle of the other. The rate of convergence to these attractors is important to consideration of the orientations in time-space varying flow fields. We show that non-normal growth (NNG) of the distance to these attractors may delay the convergence by several characteristic shear time scales. We derive conditions for occurrence of NNG and explicit expressions for the maximal duration of the growth. We consider a specific case of which the vorticity is a stable orientation of prolate spheroids. We analyze the conditions that imply monotonic or, conversely, non-monotonic convergence to this orientation due to NNG. We thereby find the corresponding conditions for convergence of the equivalent oblate spheroids to their attractors, normal to the vorticity. We show that the convergence is monotonic if the vorticity is parallel to the strain tensor’s largest eigenvector, but that NNG occurs if the vorticity is parallel to the strain tensor’s intermediate eigenvector. The NNG duration decreases with increasing vorticity-strain ratio and with the strain intermediate eigenvalue approaching the largest eigenvalue.     

Limit cycles in a sextic Lyapunov system

Employing the inverse integral factor method, the first 13 quasi-Lyapunov constants for the three-order nilpotent critical point of a sextic Lyapunov system are deduced with the help of MATHEMATICS. Furthermore, sufficient and necessary center conditions are obtained, and there are 13 small amplitude limit cycles, which could be bifurcated from the three-order nilpotent critical point. Henceforth, we give a lower bound of limit cycles, which could be bifurcated from the three-order nilpotent critical point of sextic Lyapunov systems. At last, an example is given to show that there exists a sextic system, which has 13 limit cycles.     

On forced oscillations of a simple model for a novel wave energy converter

The dynamics of a simple model for an ocean wave energy converter is discussed. The model for the converter is a hybrid system consisting of a pair of harmonically excited mass–spring–dashpot systems and a set of four state-dependent switching rules. Of particular interest is the response of the model to a wide spectrum of harmonic excitations. Partially because of the piecewise-smooth dynamics of the system, the response is far more interesting than the linear components of the model would suggest. As expected with hybrid systems of this type, it is difficult to establish analytical results, and hence, with the assistance of an extensive series of numerical integrations, an atlas of qualitative results on the limit cycles and other forms of bounded oscillations exhibited by the system is presented. In addition, the presence of unstable limit cycles, the stabilization of the unforced system using low-frequency excitation, the peculiar nature of the response of the system to high-frequency excitation, and the implications of these results on the energy harvesting capabilities of the wave energy converter are discussed.