On the uniqueness and expression of limit cycles in planar polynomial differential system via monotone iterative technique

ABSTRACT

This paper investigates the limit cycles for planar differential system $\dot{x}=\lambda x-y+P_n(x,y),\dot{y}=x+\lambda y+Q_n(x,y),$where $P_n(x,y),Q_n(x,y)$ are homogeneous polynomials of degree n. Using the method of upper and lower solutions we show that this system has a unique limit cycle under some suitable conditions. Moreover, by employing the monotone iterative technique, we provide a computational method to obtain the expression, or the asymptotical one, of this limit cycle in the polar form. The error estimation of the asymptotical solutions is also provided.

Keywords:

• Limit cycles
• expression
• uniqueness
• monotone iterative technique
• lower and upper solutions

2010 Mathematics Subject Classifications:

• 34A45
• 34B05
• 34C07

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The first author is supported by the NNSF of China [No. 11771101], the major research program of colleges and universities in Guangdong Province [No. 2017KZDX M054], and The Science and Technology Program of Guangzhou, China [No. 201805010001].