TY - JOUR

T1 - The Songling system has exactly four limit cycles

AU - Galias, Zbigniew

AU - Tucker, Warwick

N1 - Funding Information:
This work was supported in part by the AGH University of Science and Technology.
Publisher Copyright:
© 2021 The Authors
Copyright:
Copyright 2021 Elsevier B.V., All rights reserved.

PY - 2022/2/15

Y1 - 2022/2/15

N2 - Determining how many limit cycles a planar polynomial system of differential equations can have is a remarkably hard problem. One of the main difficulties is that the limit cycles can reside within areas of vastly different scales. This makes numerical explorations very hard to perform, requiring high precision computations, where the necessary precision is not known in advance. Using rigorous computations, we can dynamically determine the required precision, and localize all limit cycles of a given system. We prove that the Songling system of planar, quadratic polynomial differential equations has exactly four limit cycles. Furthermore, we give precise bounds for the positions of these limit cycles using rigorous computational methods based on interval arithmetic. The techniques presented here are applicable to the much wider class of real-analytic planar differential equations.

AB - Determining how many limit cycles a planar polynomial system of differential equations can have is a remarkably hard problem. One of the main difficulties is that the limit cycles can reside within areas of vastly different scales. This makes numerical explorations very hard to perform, requiring high precision computations, where the necessary precision is not known in advance. Using rigorous computations, we can dynamically determine the required precision, and localize all limit cycles of a given system. We prove that the Songling system of planar, quadratic polynomial differential equations has exactly four limit cycles. Furthermore, we give precise bounds for the positions of these limit cycles using rigorous computational methods based on interval arithmetic. The techniques presented here are applicable to the much wider class of real-analytic planar differential equations.

KW - Hilbert 16th problem

KW - Interval arithmetic

KW - Limit cycle

KW - Planar polynomial vector fields

UR - http://www.scopus.com/inward/record.url?scp=85117088374&partnerID=8YFLogxK

U2 - 10.1016/j.amc.2021.126691

DO - 10.1016/j.amc.2021.126691

M3 - Article

AN - SCOPUS:85117088374

SN - 0096-3003

VL - 415

JO - Applied Mathematics and Computation

JF - Applied Mathematics and Computation

M1 - 126691

ER -