SPECIFIED GLOBAL POINCARE–BENDIXSON ANNULUS WITH THE LIMIT CYCLE OF THE RAYLEIGH SYSTEM

In the work of A. Grin and K. Schneider [1] two algebraic transversal ovals, which form the Poincare–Bendixson annulus 𝐴(𝜆), are analytically constructed. This annulus contains the unique limit cycle of the Rayleigh equation ¨ 𝑥+𝜆( ˙ 𝑥2/3−1) ˙ 𝑥+𝑥 = 0 for all values of the parameter 𝜆 > 0. In the constructed annulus 𝐴(𝜆) inner boundary consists of zero-level set of the Dulac–Cherkas function and outer boundary represents the diffeomorphic image corresponding boundary of such an annulus for the van der Pol system. In this paper new methods of construction two Dulac–Cherkas functions are worked out. With help of these functions a better inner boundary of the Poincare–Bendixson annulus 𝐴(𝜆) depending on the parameter is found. Also, for the Rayleigh system, a procedure for direct finding a polynomial whose zero-level set contains a transversal oval used as the outer boundary 𝐴(𝜆) is proposed. Then we determine the interval for 𝜆 where the better outer boundary of the annulus is a closed contour, which composed from two arcs of the constructed oval and two arcs of non-closed curves belonging to zerolevel set of one from Dulac–Cherkas functions. Thus, finally the specified global Poincare–Bendixson annulus for the limit cycle of the Rayleigh system is presented.