Bifurcation analysis in a limit cycle oscillator with delayed feedback

A kind of limit cycle oscillator with delayed feedback is considered. Firstly, the linear stability is investigated. According to the analysis results, the bifurcation diagram is drawn in the parameter plane. It is found that there are stability switches for time delay, and Hopf bifurcations when time delay crosses through some critical values. Then the direction and stability of the Hopf bifurcation are determined, using the normal form method and the center manifold theorem. Finally, some numerical simulations are carried out to illustrate the results found.     

On limit cycle approximations in the van der Pol oscillator

This paper applies bifurcation analysis to the well-known van der Pol oscillator to obtain approximations of its periodic solutions in the nearly sinusoidal regime. A frequency domain method based on harmonic balance approximations is used for small values of the bifurcation parameter. Moreover, a comparison with some other frequency domain approaches is also given. Finally, a total harmonic distortion is computed using the information provided by the frequency domain approach.     

A homotopy analysis method for limit cycle of the van der Pol oscillator with delayed amplitude limiting

In this work, a powerful analytical method, called Liao’s homotopy analysis method is used to study the limit cycle of a two-dimensional nonlinear dynamical system, namely the van der Pol oscillator with delayed amplitude limiting. It is shown that the solutions are valid for a wide range of variation of the system parameters. Comparison of the obtained solutions with those achieved by numerical solutions and by other perturbation techniques shows that the utilized method is effective and convenient to solve this type of problems with the desired order of approximation.     

Three-dimensional micromechanical analysis of cracks in a ceramic composite

The finite element method is used to perform a three-dimensional micromechanical analysis of a crack in a fiber-reinforced ceramic for cases when the crack length is of the same order of magnitude as the fiber spacing. Three stages in the growth of the crack as it progresses past a set of fibers are considered. The stress intensity factor is calculated under mode-I loading for various crack lengths and the reinforcing effect of the fibers is determined. The results of this study are compared to the results of two-dimensional analyses in order to assess the validity of the two-dimensional approximation.     

On Lienard oscillator models with a pregiven set of limit cycles

Oscillators, of the Lienard type, with an arbitrary predetermined (in number and locations) family of limit cycles are synthesized using the polynomial of the least possible degree as a nonlinear dissipative characteristic. Some new facts regarding the interplay between the set of the generating amplitudes (in the sense of quasi-linear theory) and the zeros of the nonlinear characteristic are established. On this ground a simple procedure for a concrete hardware realization of the oscillators is described. Part of the results is based on intensive numerical experiments.     

Chaotifying a continuous-time system near a stable limit cycle

This paper studies the chaotification problem of driving a continuous-time system chaotic near its stable limit cycle. The controller is designed to ensure the controlled orbit be bounded and, meanwhile, have positive Lyapunov exponents. A numerical example is given to illustrate the effectiveness of the proposed chaotification algorithm.     

Semiclassical approximation for a nonlinear oscillator that is stochastic in the classical limit

L. V. Kirenski Institute of Physics, Siberian Branch, USSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 77, No. 2, pp. 277–284, November, 1988.     

一类三次系统极限环的惟一性

讨论三次系统x=x(A0+A1x+A2y+A3xy-A4y2)y=y(x-1)的极限环问题.得到了该系统不存在极限环和存在惟一极限环的条件.     

Alien limit cycles near a Hamiltonian 2-saddle cycle

It is known that perturbations from a Hamiltonian 2-saddle cycle Γ can produce limit cycles that are not covered by the Abelian integral, even when it is generic. These limit cycles are called alien limit cycles. This phenomenon cannot appear in the case that Γ is a periodic orbit, a non-degenerate singularity, or a saddle loop. In this Note, we present a way to study this phenomenon in a particular unfolding of a Hamiltonian 2-saddle cycle, keeping one connection unbroken at the bifurcation. To cite this article: M. Caubergh et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

一类微分系统的极限环分布情况

用定性分析和数值判定方法,对一类微分系统x=y,y=x(l-bx2) (α-cx2)y(其中l＞0,b＞0,c≠0)的极限环分布情况进行了研究,得出该系统有3个极限环,并且给出了该系统所有极限环的分布情况.     

Asymptotic stability of forced oscillations emanating from a limit cycle

Classical conditions for asymptotic stability of periodic solutions bifurcating from a limit cycle rely on the sign of the derivative of the associated bifurcation function at a zero. In this paper we show that, for analytic systems, this result is of topological nature. This means that it is enough to impose a change of sign at the zero, without any assumption on the succesive derivatives.     

Uniqueness of limit cycle in an immune system

A generalized predator–prey model in immune response is proposed. The conditions that guarantee the uniqueness of the limit cycle in the model are studied. These conditions provide another tool to study the nonlinear oscillatory phenomena in immune response.     

A limit cycle in a non-conservative sytem in 1:2 resonance

Systems with two degrees of freedom under the action of non-conservative forces and small linear viscous friction forces with complete dissipation are considered. A limit cycle that exists under specific conditions in the vicinity of an isolated equilibrium of the system is constructed using asymptotic methods in the case of 1:2 resonance. A criterion for asymptotic stability of this cycle is obtained to within equality-type relations. An estimate of the region of attraction of the limit cycle in a truncated system is given. Oscillations of a two-link rod system on a plane in 1:2 resonance are investigated. ©2011     

Perturbation methods in group analysis

In this paper we describe the theory of approximate groups of transformations which lets us constructively find approximate symmetries of equations with a small parameter. For evolution equations we introduce the concept of formal symmetries and Bäcklund transformations. The theory is applied to a large class of wave and evolution equations.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 34, pp. 85–147, 1989.     

Perturbation analysis of a magnetohydrodynamic boundary layer flow

This paper presents a perturbation analysis study of the flow of an electrically conducting power-law fluid in the presence of a uniform transverse magnetic field over a stretching sheet. The perturbation solutions for small and large values of the mixed convection parameter are explored. The asymptotic behavior of the solutions was examined for different values of the power-law index and the magnetic parameter.     

Melnikov function and limit cycle bifurcation from a nilpotent center

We investigate a general near-Hamiltonian system on the plane whose unperturbed system has a nilpotent center. Our main purpose is to give an algorithm for calculating the first coefficients of the expansion of the first order Melnikov function. We also give an application by using the method and obtain the number of limit cycles of a cubic system.