Cyclicity of $ (1,3) $-switching FF type equilibria

Xingwu Chen; Jaume Llibre; Weinian Zhang

doi:10.3934/dcdsb.2019153

Article Contents

2019, Volume 24, Issue 12: 6541-6552. Doi: 10.3934/dcdsb.2019153

This issue Previous Article Stability in measure for uncertain heat equations Next Article Nonlinear decomposition principle and fundamental matrix solutions for dynamic compartmental systems

Cyclicity of $ (1,3) $-switching FF type equilibria

1.
Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
2.
Department de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain

^* Corresponding author: Weinian Zhang
^* Corresponding author: Weinian Zhang

Received: January 2019

Early access: July 2019

Published: September 2019

The first author is supported by NSFC #11871355. The second author has been partially supported by the Ministerio de Economía, Industria y Competitividad, Agencia Estatal de Investigación grants MTM2016-77278-P (FEDER) and MDM-2014-0445, the Agència de Gestió d'Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911. The third author is supported by NSFC #11726623 and #11771307

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Abstract

Hilbert's 16th Problem suggests a concern to the cyclicity of planar polynomial differential systems, but it is known that a key step to the answer is finding the cyclicity of center-focus equilibria of polynomial differential systems (even of order 2 or 3). Correspondingly, the same question for polynomial discontinuous differential systems is also interesting. Recently, it was proved that the cyclicity of $ (1, 2) $-switching FF type equilibria is at least 5. In this paper we prove that the cyclicity of $ (1, 3) $-switching FF type equilibria with homogeneous cubic nonlinearities is at least 3.

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Mathematics Subject Classification: Primary: 34C07, 34C23, 34C37, 34K18.

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Table 1. Numbers of positive simple zeros of $ f_1, ..., f_5 $

$ \#Z_+(f_1) $	$ f_1\!\!\equiv \!0 $	$ \#Z_+(f_2) $	$ f_2\!\!\equiv\! 0 $	$ \#Z_+(f_3) $	$ f_3\!\!\equiv\! 0 $	$ \#Z_+(f_4) $	$ f_4\equiv 0 $	$ \#Z_+(f_5) $
$ 0 $	$ \lambda_{11}\!\!=\!0 $	1	$ C_2 $	1	$ C_3 $	$ 2 $	$ C_{41} $	$ 2 $
							$ C_{42} $	$ 2 $
							$ C_{43} $	$ 2 $

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Table 2. Number of positive simple zeros of $ f_6 $

condition for $ f_4\equiv 0 $	condition for $ f_5\equiv 0 $	$ \#Z_+(f_6) $
$ C_{41} $	$ C_{411} $	$ 3 $
$ C_{42} $	$ C_{421} $	$ 2 $
	$ C_{422} $	$ 2 $
	$ C_{423} $	$ 2 $
$ C_{43} $	$ C_{431} $	$ 3 $
	$ C_{432} $	$ 2 $
	$ C_{433} $	$ 3 $

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References

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