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

  • * Corresponding author: Weinian Zhang

    * Corresponding author: Weinian Zhang

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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  • 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.

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

    Citation:

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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 $
     | Show Table
    DownLoad: CSV

    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 $
     | Show Table
    DownLoad: CSV
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