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2023, Volume 22, Issue 3: 970-982. Doi: 10.3934/cpaa.2023016

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Existence of at most two limit cycles for some non-autonomous differential equations

Armengol Gasull^1,2, and
Yulin Zhao^3, ,

1.
Dept. de Matemàtiques, Universitat Autònoma de Barcelona, Edifici C 08193 Cerdanyola del Vallès (Barcelona), Spain
2.
Centre de Recerca Matemàtica, Edifici Cc, Campus de Bellaterra, 08193 Cerdanyola del Vallès (Barcelona), Spain
3.
School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai Campus, Zhuhai, 519082, China

^*Corresponding author: Yulin Zhao
^*Corresponding author: Yulin Zhao

Received: December 2022

Early access: January 2023

Published: March 2023

The first author is supported by Spanish State Research Agency, through the projects PID2019-104658GB-I00 grant and the Severo Ochoa and María de Maeztu Program for Centers and Units of Excellence in R&D (CEX2020-001084-M) and by the grant 2017-SGR-1617 from AGAUR, Generalitat de Catalunya. The second author is supported by the NNSF of China (No. 11971495), Guangdong-Hong Kong-Macau Applied Math Center (No. 2020B1515310014) and Guangdong Basic and Applied Basic Research Foundation (No. 2022A1515012105)

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Abstract

It is know that the non-autonomous differential equations $ {\rm d}x/{\rm d} t = a(t)+b(t)|x| $, where $ a(t) $ and $ b(t) $ are 1-periodic maps of class $ \mathcal{C}^1 $, have no upper bound for their number of limit cycles (isolated solutions satisfying $ x(0) = x(1) $). We prove that if either $ a(t) $ or $ b(t) $ does not change sign, then their maximum number of limit cycles is two, taking into account their multiplicities, and that this upper bound is sharp. We also study all possible configurations of limit cycles. Our result is similar to other ones known for Abel type periodic differential equations although the proofs are quite different.

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Mathematics Subject Classification: Primary: 34C25; Secondary: 34A34, 34C07, 37C27.

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Figure 1. For a given differential equation (1.4), example of the values $ x_i $ and $ \tau_i. $ Also, for a particular solution $ u(t,0,x), $ with $ n = 2 $ and $ r = 4, $ we show the values $ t_1(x) $ and $ t_2(x). $ Notice that for $ x $ in each of the intervals $ J_i $ the number of zeroes of $ u(t,0,x) $ is constant. In particular, for $ x\in J_3, $ this number is $ k_3 = 2. $.

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