Holomorphic Flows on Simply Connected Regions Have No Limit Cycles

The dynamical system or flow = f(z), where f is holomorphic on C, is considered. The behavior of the flow at critical points coincides with the behavior of the linearization when the critical points are non-degenerate: there is no center-focus dichotomy. Periodic orbits about a center have the same period and form an open subset. The flow has no limit cycles in simply connected regions. The advance mapping is holomorphic where the flow is complete. The structure of the separatrices bounding the orbits surrounding a center is determined. Some examples are given including the following: if a quartic polynomial system has four distinct centers, then they are collinear.     

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.     

Separatrices and basins of stability from time series data: an application to biodynamics

An approach is presented for identifying separatrices in state space generated from noisy time series data sets which are representative of those generated from experiments. We demonstrate how these separatrices can be found using Lagrangian coherent structures (LCSs), ridges in the state space distribution of the maximum finite-time Lyapunov exponent. As opposed to the current approach which requires a vector field in the state space at each instant of time, this method can be performed using only trajectories reconstructed from time series. As such, this paper forms a bridge connecting methods for evaluating time series data with methods used to evaluate LCSs in vector fields. The methods are applied to a problem in musculoskeletal biomechanics, considered as an exemplar of a class of experimental systems that contain separatrices. In this case, the separatrix reveals a basin of stability for a balancing task, outside of which is a zone of failure. We demonstrate that LCSs calculated from only trajectory data, which samples only portions of the state space, align well with LCSs found using a known vector field. In general, we believe this method provides a fruitful approach for extracting information from noisy experimental data regarding boundaries between qualitatively different kinds of behavior.     

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.     

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.     

Bifurcation of a modified railway wheelset model with nonlinear equivalent conicity and wheel–rail force

This paper presents the bifurcation behaviors of a modified railway wheelset model to explore its instability mechanisms of hunting motion. Equivalent conicity data measured from China high-speed railway vehicle are used to modify the wheelset model. Firstly, the relationships between longitudinal stiffness, lateral stiffness, equivalent conicity and critical speed are taken into account by calculating the real parts of the eigenvalues of the Jacobian matrix and Hurwitz criterion for the corresponding linear model. Secondly, measured equivalent conicity data are fitted by a nonlinear function of the lateral displacement rather than are considered as a constant as usual. Nonlinear wheel–rail force function is used to describe the wheel–rail contact force. Based on these modifications, a modified railway wheelset model with nonlinear equivalent conicity and wheel–rail force is set up, and then, some instability mechanisms of China high-speed train vehicle are investigated based on Hopf bifurcation, fold (limit point) bifurcation of cycles, cusp bifurcation of cycles, Neimark–Sacker bifurcation of cycles and 1:1 resonance. In particular, fold bifurcation of cycles can produce a vast effect on the hunting motion of the modified wheelset model. One of the main reasons leading to hunting motion is due to the fold bifurcation structure of cycles, in which stable limit cycles and unstable limit cycles may coincide, and multiple nested limit cycles appear on a side of fold bifurcation curve of cycles. Unstable hunting motion mainly depends on the coexistence of equilibria and limit cycles and their positions; if the most outward limit cycle is stable, then the motion of high-speed vehicle should be safe in a reasonable range. Otherwise, if the initial values are chosen near the most outward unstable limit cycle or the system is perturbed by noises, the high-speed vehicle will take place unstable hunting motion and even lead to serious train derailment events. Therefore, in order to control hunting motions, it may be the easiest way in theory to guarantee the coexistence of the inner stable equilibrium and the most outward stable limit cycle in a wheelset system.

     

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.     

Bifurcation of limit cycles in fluid film bearings

Rotors supported by journal bearings may become unstable due to self-excited vibrations when a critical rotor speed is exceeded. Linearised analysis is usually used to determine the stability boundaries. Non-linear bifurcation theory or numerical integration is required to predict stable or unstable periodic oscillations close to the critical speed. In this paper, a dynamic model of a short journal bearing is used to analyse the bifurcation of the steady state equilibrium point of the journal centre. Numerical continuation is applied to determine stable or unstable limit cycles bifurcating from the equilibrium point at the critical speed. Under certain working conditions, limit cycles themselves are shown to disappear beyond a certain rotor speed and to exhibit a fold bifurcation giving birth to unstable limit cycles surrounding the stable supercritical limit cycles. Numerical integration of the system of equations is used to support the results obtained by numerical continuation. Numerical simulation permitted a partial validation of the analytical investigation.     

A Perturbation-Incremental Method for Strongly Nonlinear Autonomous Oscillators with Many Degrees of Freedom

The perturbation-incremental method is extended to determine thebifurcations and limit cycles of strongly nonlinear autonomousoscillators with many degrees of freedom. Coupled van der Poloscillators and coupled Rayleigh oscillators are taken as numericalexamples. Limit cycles of the oscillators can be calculated to anydesired degree of accuracy. The stabilities of limit cycles are alsodiscussed.     

Existence, number, and stability of limit cycles in weakly dissipative, strongly nonlinear oscillators

Oscillators control many functions of electronic devices, but are subject to uncontrollable perturbations induced by the environment. As a consequence, the influence of perturbations on oscillators is a question of both theoretical and practical importance. In this paper, a method based on Abelian integrals is applied to determine the emergence of limit cycles from centers, in strongly nonlinear oscillators subject to weak dissipative perturbations. It is shown how Abelian integrals can be used to determine which terms of the perturbation are influent. An upper bound to the number of limit cycles is given as a function of the degree of a polynomial perturbation, and the stability of the emerging limit cycles is discussed. Formulas to determine numerically the exact number of limit cycles, their stability, shape and position are given.     

An analogue rotated vector field of polynomial system

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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.     

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.     

Codimension-two bifurcation analysis in two-dimensional Hindmarsh–Rose model

In this paper, we analyze the codimension-2 bifurcations of equilibria of a two-dimensional Hindmarsh–Rose model. By using the bifurcation methods and techniques, we give a rigorous mathematical analysis of Bautin bifurcation. The main result is that no more than two limit cycles can be bifurcated from the equilibrium via Hopf bifurcation; sufficient conditions for the existence of one or two limit cycles are obtained. This paper also shows that the model undergoes a Bogdanov–Takens bifurcation which includes a saddle-node bifurcation, an Andronov–Hopf bifurcation, and a homoclinic bifurcation. In some case, the globally asymptotical stability is discussed.     

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.     

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.

     

具有多个极限环非线性动力系统的解析近似

应用一种新的解析方法------同伦分析法,研究了一种具有多个 极限环的Rayleigh振子问题. 与所有其他传统方法不同,该方法不依赖于小参数, 且提供了一个简便的途径以确保级数解的收敛, 因此,特别适用于强非线性问题. 将同伦分析法与平均法以及四阶的龙格库塔方法(数值解)做了比较. 结果 表明,平均法在强非线性情况失效, 四阶的龙格库塔法不能找到非稳定的极限环,而同伦分析法不仅适用于强非线性情 况,而且给出了非稳定的极限环.     

Perturbation methods for nonlinear autonomous discrete-time dynamical systems

Two perturbation methods for nonlinear autonomous discrete-time dynamical systems are presented. They generalize the classical Lindstedt-Poincaré and multiple scale perturbation methods that are valid for continuous-time systems. The Lindstedt-Poincaré method allows determination of the periodic or almost-periodic orbits of the nonlinear system (limit cycles), while the multiple scale method also permits analysis of the transient state and the stability of the limit cycles. An application to the discrete Van der Pol equation is also presented, for which the asymptotic solution is shown to be in excellent agreement with the exact (numerical) solution. It is demonstrated that, when the sampling step tends to zero the asymptotic transient and steady-state discrete-time solutions correctly tend to the asymptotic continuous-time solutions.     

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.     

A Perturbation Method for Nonlinear Two-Dimensional Maps

A new perturbation method for a weakly nonlinear two-dimensional discrete-time dynamical system is presented. The proposed technique generalizes the asymptotic perturbation method that is valid for continuous-time systems and detects periodic or almost-periodic orbits and their stability. Two equations for the amplitude and the phase of solutions are derived and their fixed points correspond to limit cycles for the starting nonlinear map. The method is applied to various nonlinear (autonomous or not) two-dimensional maps. For the autonomous maps we derive the conditions for the appearance of a supercritical Hopf bifurcation and predict the characteristics of the corresponding limit cycle. For the nonautonomous maps we show amplitude-response and frequency-response curves. Under appropriate conditions, we demonstrate the occurrence of saddle-node bifurcations of cycles and of jumps and hysteresis effects in the system response (cusp catastrophe). Modulated motion can be observed for very low values of the external excitation and an infinite-period bifurcation occurs if the external excitation increases. Analytic approximate solutions are in good agreement with numerically obtained solutions.