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Publication# A General Family of Morphed Nonlinear Phase Oscillators with Arbitrary Limit Cycle Shape

Abstract

We present a general family of nonlinear phase oscillators which can exhibit arbitrary limit cycle shapes and infinitely large basins of attraction. This general family is the superset of familiar control methods like PD-control over a periodic reference, and rhythmic Dynamical Movement Primitives. The general methodology is based on morphing the limit cycle of an existing phase oscillator with phase-based scaling functions to obtain a desired limit cycle behavior. The introduced methodology can be represented as first, second, or n-th order dynamical systems. The elegance of the formulation provides the possibility to define explicit arbitrary convergence behavior for simple cases. We analyze the stability properties of the methodology with the Poincare-Bendixson theorem and the Contraction Theory, and use numerical simulations to show the properties of some oscillators that are a subset of this general family. (C) 2013 Elsevier B.V. All rights reserved.

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Limit cycle

In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity or as time approaches negative infinity. Such behavior is exhibited in some nonlinear systems. Limit cycles have been used to model the behavior of many real-world oscillatory systems. The study of limit cycles was initiated by Henri Poincaré (1854–1912).

Attractor

In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain close even if slightly disturbed. In finite-dimensional systems, the evolving variable may be represented algebraically as an n-dimensional vector. The attractor is a region in n-dimensional space.

Dynamical system

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