Limit Cycles In Nonlinear Systems Research Articles

Generation of Limit Cycles in Nonlinear Systems: Machine Leaning Based Type-3 Fuzzy Control

A limit cycle is a cyclic path of oscillation on which the states of a nonlinear system settle. A considerable number of practical systems such as robots, converters, and heartbeat require generating the sustainable oscillatory behaviours. The objective is to design a controller to generate limit cycle with specific behaviours. In this paper, a novel fuzzy control (FC) strategy is introduced to create the limit-cycle for nonlinear complex dynamics with unknown uncertainties. The suggested controller benefits from new interval type-3 fuzzy logic, allowing the control synthesis to improve the quality of closed-loop response and robust performance. The adaptively learned backstepping controller based on FC is employed to analyze the convergence and robustness. Various simulations are proposed to ensure the efficiency of the fuzzy-based control law and adaption rules.

A comparison of methods for approximating periodic limit cycles in nonlinear systems with joint friction

AbstractIn this contribution, approximation methods for calculating periodic limit cycles of self‐excited dynamical systems with frictional damping are studied. In industrial application, e.g. the analysis of brake squeal, the so‐called Energy‐Balance has attracted a lot of interest [3], while rather classical Harmonic Balance methods are commonly used in non‐linear analysis. Focussing on the effect of both smooth and non‐smooth non‐linearities, limit cycles were calculated for two minimal models, which are commonly used when dealing with NVH or brake squeal analysis. Moreover these two methods are compared in terms of accuracy and numerical effort.

Limit Cycles in Nonlinear Systems with Fractional Order Plants

In recent years, there has been considerable interest in the study of feedback systems containing processes whose dynamics are best described by fractional order derivatives. Various situations have been cited for describing heat flow and aspects of bioengineering, where such models are believed to be superior. In many situations these feedback systems are not linear and information on their stability and the possibility of the existence of limit cycles is required. This paper presents new results for determining limit cycles using the approximate describing function method and an exact method when the nonlinearity is a relay characteristic.

A Brief Survey of Stochastic Resonance and Its Application to Control

Abstract Noise is generally considered as nuisance in engineering. Hence engineers routinely attempt to filter out noise. But quite remarkably a counter intuitive phenomenon was found in 1980s that changed the perception of noise, giving birth to a phenomenon called Stochastic resonance (SR). In this paper, we review various phenomena that are benefited by noise. Quite obviously, we extend those ideas to Control Systems design. We provide 3 examples: a randomized algorithm for a linear state feedback control law, (ii) minimizing limit cycles in nonlinear systems, and (iii) antilock braking system (ABS) design. Finally, we present some of our arguments in favour of stochastic resonance as a potential tool for system design in a broader perspective.

Numerical investigation into the existence of limit cycles in two-dimensional predator–prey systems

There has been a surge of interest in developing and analysing models of interacting species in ecosystems, with specific interest in investigating the existence of limit cycles in systems describing the dynamics of these species. The original Lotka–Volterra model does not possess any limit cycles. In recent years this model has been modified to take disturbances into consideration and allow populations to return to their original numbers. By introducing logistic growth and a Holling Type II functional response to the traditional Lotka–Volterra-type models, it has been proven analytically that a unique, stable limit cycle exists. These proofs make use of Dulac functions, Liénard equations and invariant regions, relying on theory developed by Poincaré, Poincaré-Bendixson, Dulac and Liénard, and are generally perceived as difficult. Computer algebra systems are ideally suited to apply numerical methods to confirm or refute the analytical findings with respect to the existence of limit cycles in non-linear systems. In this paper a class of predator–prey models of a Gause type is used as the vehicle to illustrate the use of a simple, yet novel numerical algorithm. This algorithm confirms graphically the existence of at least one limit cycle that has analytically been proven to exist. Furthermore, adapted versions of the proposed algorithm may be applied to dynamic systems where it is difficult, if not impossible, to prove analytically the existence of limit cycles.

Limit cycles in nonlinear systems of stabilization

Limit cycles in nonlinear systems of stabilization

Behavior of Limit Cycles in Nonlinear Systems

Behavior of Limit Cycles in Nonlinear Systems

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.

Analysis of stability in quadratic mean of the limit cycles of nonlinear stochastic systems

Consideration was given to the exponential stability in quadratic mean of the stochastically perturbed limit cycles of the nonlinear systems. An approach was developed using the spectral theory of positive operators for the stability analysis. Within the framework of this approach, a positive operator of stochastic stability is assigned to the limit cycle. The spectral radius of this operator characterizes stability of the limit cycle. An iterative numerical method was proposed for calculation of the spectral radius of the stochastic stability operator, and a theorem about its convergence was proved. The constructive potentialities of the results obtained were demonstrated by the example of bifurcational analysis of the stochastic Ressler system at transition to chaos by multiple duplication of the limit cycle period.

Stochastic sensitivity of 3D-cycles

The limit cycles of nonlinear systems under the small stochastic disturbances are considered. The random trajectories of forced system leave the deterministic cycle and form some stochastic bundle around it. The probabilistic description of this bundle near cycle based on stochastic sensitivity function (SSF) is suggested. The SSF is a covariance matrix for periodic solution of linear stochastic first approximation system. This matrix is a solution of the boundary problem for linear matrix differential equation. For 3D-cycles this matrix differential equation on the basis of singular expansion is reduced to the system of three scalar equations only. The possibilities of SSF to describe some peculiarities of stochastically forced Roessler model are demonstrated.

Tensor operators and limit cycles in non-linear systems

The theory of tensor operators is used to obtain criteria for the existence and nonexistence of limit cycles in non-linear systems

A new approach to the prediction of limit cycles

Abstract The describing function method is commonly used for the prediction of limit cycles in non-linear systems. This method is based upon the representation of signals by a Fourier series, and assigning a gain to the non-linear component derived from its response to a sinusoidal input, In this paper we show that a different approach is feasible which in some respects is the dual of the describing function method. The signals are analysed into a Walsh scries, and a gain is assigned to the linear component of the system derived from its response to a square, wave. In many instances the method is likely to give more accurate predictions of limit cycling.

Frequency criteria for the absence of limit cycles in nonlinear systems

A method is derived herein that provides a geometric interpretation of frequency regions in the Nyquist plane over which simple oscillations cannot occur. These criteria answer a question of system behavior when Popov-type criteria are violated.

Bifurcation theorems and limit cycles in nonlinear systems: part II

Duffing's equation was first given in 1918. Recently CHAOS was found and studied by using analog and digital computers. The present paper reports the chaotic behavior for a wide range of parameters and limit cycle in Duffing's equation. Some discussion on the chaotic behavior and its benefit is included.

Bifurcation theorems and limit cycles in nonlinear systems—I.

Two bifurcation theorems are established concerning the qualitative change in the integral curves of the hard-excitation type nonlinear systems at a point of bifurcation (or a branch point) where different regions meet. Two classes of this type (Type B) are considered. These exhibit limit cycles which do not contract to the origin, unlike the soft-excitation type nonlinear systems (Type A) reported by Jonnada and Weygandt. A Type B, Class 1, system is exemplified by the well-known van der Pol equation. A third-order example (B-1 system) is also given. Type B, Class 2 systems exhibit an unstable limit cycle. Second- and third-order examples are given for B-2 systems.

A Computer-Oriented, Parameter-Space Approach to the Synthesis of Nonlinear Control Systems

A systematic synthesis procedure is developed for ensuring an absence of limit cycles in nonlinear systems. Compensator parameter plots, subject to specified constraints, are generated in parameter planes. These plots can be combined to yield design surfaces. The proposed approach is oriented for digital computation but can be adapted easily to a paper and pencil equivalent. The procedure is illustrated by two design examples.