Determination Of Limit Cycles Research Articles

A real autonomous quadratic system of three differential equations with an infinite number of limit cycles

In this paper, we consider the problem of construction of real autonomous quadratic systems of three differential equations with the nonlocal existence of an infinite number of limit cycles. This means that an infinite number of limit cycles, emerging from the focus due to the Andronov – Hopf bifurcation, can exist in the phase space not only in the vicinity of the focus and not only for parameter values close to the bifurcation value. To solve this problem we use the method of determination of limit cycles as the curves of intersection of an invariant plane with a family of invariant elliptic paraboloids. Then the study of the limit cycles of the constructed system of the third order is reduced to the study of the corresponding system of the second order on each of the invariant elliptic paraboloids. The proof of the nonlocal existence of the limit cycle and the investigation of its stability for such a second-order system is carried out by constructing a topographic system of Poincaré functions or by transforming to polar coordinates.

Study on the stability and bifurcations of limit cycles in higher-dimensional nonlinear autonomous systems

A semi-analytical procedure for studying stability and bifurcationsof limit cycles in higher-dimensional nonlinear autonomous dynamicalsystems is developed. This procedure is based mainly on theincremental harmonic balance (IHB) method. It is composed of threekey steps, namely, the determination of limit cycles by IHB method,the calculation of transition matrix by precise integration (PI)algorithm and the discrimination of limit cycle stability by Floquettheory. As an application, the procedure is used to investigate thedynamics of the limit cycle of a three-dimensional nonlinearautonomous system. The symmetry-breaking bifurcation, the first andthe second period-doubling bifurcations of the limit cycle areidentified. The critical parameter values corresponding to thesebifurcations are calculated. The phase portraits and bifurcationpoints agree well with those of direct numerical integrations byusing Runge-Kutta method.

Determination of Limit Cycle by Hamiltonian Approach for Strongly Nonlinear Oscillators

Determination of Limit Cycle by Hamiltonian Approach for Strongly Nonlinear Oscillators

Determination of limit cycles by iterated homotopy perturbation method for nonlinear oscillators with strong nonlinearity

He's Homotopy Perturbation Method which reduced to an Iterative Scheme is applied to nonlinear oscillators with strong nonlinearity. With the method, the iteration scheme provides excellent approximations to the solutions even though the iteration can only be done to the first stage.

Determination of limit cycle by He's parameter-expanding method for strongly nonlinear oscillators

This paper applies He's parameter-expanding method (PEM) to determine the limit cycles of strongly nonlinear oscillators by which one iteration leads to an accurate solution. Comparison of the obtained results with those of the exact solution shows that the method is very effective and convenient and quite accurate to both linear and nonlinear physics and engineering problems.

Determination of limit cycles by a modified straightforward expansion for nonlinear oscillators

This paper applies He’s modified straightforward expansion method to determine the limit cycles of various nonlinear physical systems. The obtained approximate solutions are uniformly valid for both weakly and strongly nonlinear equations.

Self-excited systems: Analytical determination of limit cycles

Abstract The determination of amplitude and period of limit cycles is a crucial question in non-linear mechanics. Recently, a van der Pol oscillator containing a periodic potential has been considered as a suitable self-excited system in tribological studies. Analytical expressions for the amplitude and the period of limit cycles for such modified van der Pol oscillator are calculated making use of the He’s variational method and the Krylov–Bogoliubov–Mitropolsky (KBM) method. The analytical results are then compared to numerical correspondent results, showing the goodness of the two methods in the range of application.

Determination of Limit Cycles Using Both the Slope of Correlation Integral and Dominant Lyapunov Methods

A method for nonlinear analysis of instabilities in boiling water reactors (BWRs) is presented. Both the Dominant Lyapunov Exponent method and the Slope of the Correlation Integral (SOCI) method are used to analyze the average power reactor monitor (APRM) signals from a BWR. The main advantage of using the two methods in a complementary manner is that doing so results in an enhancement of the capability to analyze noisy systems, such as the APRM signals in a BWR. Previously, such nonlinear analysis had been performed using independently either the Dominant Lyapunov Exponent Method or the SOCI method. These two methods are sensitive to noise in a signal and normally require large amounts of data for a reliable analysis.This proposed system for nonlinear analysis is composed first of a home-developed computer program called “SLOPE,” which is based on the SOCI method. Then, the signal analysis is also performed by the “LENNS” code, which is used to obtain the dominant Lyapunov exponent. Since only the dominant Lyapunov exponent is computed, there is no need to acquire large amounts of data; thus, computational processing time is greatly reduced, even in the case of noisy data.The system was used to analyze BWR signals containing stationary and nonstationary limit cycles. It was found that this method satisfactorily calculates the limit cycles, extracting useful information from noisy signals.

Determination of limit cycles for two-dimensional dynamical systems.

We consider in this paper two-dimensional dynamical systems of the form x\ifmmode \dot{}\else \.{}\fi{}=P(x,y),y\ifmmode \dot{}\else \.{}\fi{}=Q(x,y), where P and Q are analytic functions. We introduce a method for finding the limit cycles of the system. This method consists of searching for a power series solution of the equation P(\ensuremath{\partial}V/\ensuremath{\partial}x)+Q(\ensuremath{\partial}V/\ensuremath{\partial}y)=[(\ensuremath{\partial}P/\ensuremath{\partial}x)+(\ensuremath{\partial}Q/\ensuremath{\partial}y)]V. The limit cycles are determined from the condition V(x,y)=0.

A New Approach to Pulse Frequency Modulated Control

A straight-forward analytical model for PFM controllers is proposed basing on the principles of conventional phase angle modulation. Given in a closed analytical form using the theory of the so-called serraphile functions by Turban, this model simplifies stability analysis, determination of limit cycles, discussion of controllability and particularly the analytical design of PFM control systems. Because of the ability of pulse efficiency reduction the implementation of this model allows to diminish or even avoid limit cycles in some cases. By approximation of the nonlinear model by an equivalent linear element, analysis and design of PFM control systems can be considerably simplified since linear control theory becomes applicable.

非線形離散値フィードバックシステムにおけるリミットサイクルの予測と決定

非線形離散値フィードバックシステムにおけるリミットサイクルの予測と決定

The determination of limit-cycles in nonlinear sampled data systems

The determination of limit-cycles in nonlinear sampled data systems

A method for the evaluation of limit cycles

A method is presented for the determination of limit cycles in feedback systems having one or more non-linear elements. The non-linearities are approximated using a quantized characteristic, that is, a parallel combination of relays, and a previously known result for the exact computation of limit cycles in relay systems, is used. Examples are given where it is seen that very accurate results can be obtained with relatively few quantization steps.

Analytic determination of limit cycle for a class of third-order non-linear system

The purpose of this paper is to show a new criterion on the stability of the limit cycle for a general third-order non-linear system : that the linear-part transfer function can be either stable or unstable and the non-linear element has a real describing function. The presented method is purely analytic. Combining with an analytic method for predicting the amplitudes and frequencies of limit cycles, an analytic process for determining the stable limit cycles of a general third-order non-linear system is completed. Many examples are given.

Modified application of describing functions in the determination of limit cycles for relay systems

A modified method for the study of limit cycles in relay systems using a describing-function approach is presented. Results of applying the method in three systems are quoted. The method is appropriate for situations in which the normal describing function approach fails to predict a limit cycle.

On the determination of limit cycles in relay control systems with additional saturation type non-linear elements†

Exact and approximate methods of determination of limit cycles in a relay control system including an actuator with PI transfer function and limited output are given. No restrictions are imposed on the order of the plant transfer function. The stability of the limit cycles may be checked using one of the well-known methods, using the describing function concept. With the help of an example it is indicated that the methods suggested give reliable results in eases where the use of the conventional describing function fails.