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On cyclicity in discontinuous piecewise linear near-Hamiltonian differential systems with three zones having a saddle in the central one

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Abstract

We obtain lower bounds for the maximum number of limit cycles bifurcating from periodic annuli of discontinuous planar piecewise linear Hamiltonian differential systems with three zones separated by two parallel straight lines, assuming that the linear differential subsystem in the region between the two straight lines, called of central subsystem, has a saddle at a point equidistant from these lines. (Obviously, the other subsystems have saddles or centers.) We prove that at least six limit cycles bifurcate from the periodic annuli of these kind of piecewise Hamiltonian differential systems, by linear perturbations. Normal forms and Melnikov functions, defined in two and three zones, are the main techniques used in the proof of the results.

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Data Availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

The first author is partially supported by São Paulo Research Foundation (FAPESP) grant 2023/04061-6. The first, third and fourth authors are also supported by FAPESP grant 2019/10269-3. The second author was supported by Brazilian Federal Agency for Support and Evaluation of Graduate Education (CAPES) grant 88882.434343/2019-01. The third author is partially supported by FAPESP grants 2022/09633-5 and 2018/13481-0 and by National Council for Scientific and Technological Development (CNPq) grants 438975/2018-9 and 309110/2021-1. The fourth author is also partially supported by FAPESP grant 2022/04040-6. The fifth author is partially supported by Pronex/FAPEG/CNPq grant 2017 10 26 7000 508 CAPES grant 88881.068462/2014-01 and CNPq grants 402060/2022-9 and 308652/2022-3. Moreover, this article was possible thanks to the scholarship granted from the CAPES, in the scope of the Program CAPES-Print, process number 88887.310463/2018-00, International Cooperation Project number 88881.310741/2018-01.

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Authors and Affiliations

  1. Instituto de Biociências Letras e Ciências Exatas, Universidade Estadual Paulista (UNESP), R. Cristovão Colombo, 2265, S. J. Rio Preto, SP, 15054-000, Brazil

    Claudio Pessoa & Márcio Gouveia

  2. Departamento de Matemática, Universidade Estadual de Campinas (UNICAMP), R. Sérgio Buarque de Holanda, 651, Campinas, SP, 13083-859, Brazil

    Douglas Novaes

  3. Instituto de Matemática e Estatística, Universidade Federal de Goiâs (UFG), R. Jacarandá, Goiânia, GO, 74690-900, Brazil

    Rodrigo Euzébio

  4. Instituto de Matemática e Computação, Universidade Federal de Itajubá (UNIFEI), Avenida BPS, 1303, Pinheirinho, Itajubá, MG, 37500-903, Brazil

    Ronisio Ribeiro

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  1. Claudio Pessoa
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  2. Ronisio Ribeiro
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  5. Rodrigo Euzébio
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Appendices

Appendix A

The coefficients list of the Melnikov functions for the case CSS is the following:

$$\begin{aligned}{} & {} C_{0}^0 = 2 (p_{00} + p_{10}) + \frac{2b_{\scriptscriptstyle R}}{b_{\scriptscriptstyle L}}(r_{10}-r_{00})+b_{\scriptscriptstyle R}\\{} & {} \quad \bigg (2 v_{01}-2 u_{10}+\frac{1}{\omega _{\scriptscriptstyle LC}}(r_{10} + s_{01}) \\{} & {} \quad ( \pi \mu _2 (1 + \tau _{\scriptscriptstyle LC}^2)-\tau _{\scriptscriptstyle LC})+\frac{\tau _{\scriptscriptstyle RS}}{\omega _{\scriptscriptstyle RS}}(p_{10} + q_{01})\bigg )\\{} & {} \quad +\frac{(-1)^{\mu _2}b_{\scriptscriptstyle R}}{2\omega _{\scriptscriptstyle LC}}(r_{10} + s_{01})\\{} & {} \quad (\tau _{\scriptscriptstyle LC}^2+1)\arccos \bigg (\frac{\tau _{\scriptscriptstyle LC}^2-1}{\tau _{\scriptscriptstyle LC}^2+1}\bigg ) -\frac{b_{\scriptscriptstyle R}}{2\omega _{\scriptscriptstyle RS}}\\{} & {} \quad (p_{10} + q_{01}) (\tau _{\scriptscriptstyle RS}^2-1)\log \bigg (\frac{\tau _{\scriptscriptstyle RS}+1}{\tau _{\scriptscriptstyle RS}-1}\bigg ),\\{} & {} \quad C_{0}^1 = 2 (p_{00} + p_{10}) + \frac{(-1)^{\mu _2}b_{\scriptscriptstyle R}}{\omega _{\scriptscriptstyle LC}}(r_{10} + s_{01})\\{} & {} \quad \arccos \bigg (\frac{\tau _{\scriptscriptstyle LC}^2-1}{\tau _{\scriptscriptstyle LC}^2+1}\bigg ) + b_{\scriptscriptstyle R}\bigg (\frac{2}{b_{\scriptscriptstyle L}}( r_{10}\\{} & {} \quad -r_{00})+\frac{1}{\omega _{\scriptscriptstyle LC}} (r_{10} + s_{01}) \\{} & {} \quad (2 \pi \mu _2 + ( (-1)^{\mu _2}-1 ) \tau _{\scriptscriptstyle LC})+u_{10} (\log (4)-2 ) \\{} & {} \quad + v_{01} (2 + \log (4))\bigg )+\frac{b_{\scriptscriptstyle R}}{\omega _{\scriptscriptstyle RS}}(p_{10} + q_{01})\log \bigg (\frac{\tau _{\scriptscriptstyle RS}+1}{\tau _{\scriptscriptstyle RS}-1}\bigg ), \end{aligned}$$
$$\begin{aligned} \begin{aligned}&C_{0}^2 = \frac{b_{\scriptscriptstyle R}}{2\omega _{\scriptscriptstyle LC}\omega _{\scriptscriptstyle RS}}\bigg (2 \bigg (\pi (r_{10} + s_{01}) \mu _2 \omega _{\scriptscriptstyle RS} + (u_{10} + v_{01})\\&\quad \omega _{\scriptscriptstyle LC} \omega _{\scriptscriptstyle RS}+\frac{(-1)^{\mu _2}\tau _{\scriptscriptstyle LC}\omega _{\scriptscriptstyle RS}}{\tau _{\scriptscriptstyle LC}^2+1}\\&\quad (r_{10} + s_{01})+ \frac{\tau _{\scriptscriptstyle RS}\omega _{\scriptscriptstyle LC}}{\tau _{\scriptscriptstyle RS}^2-1}(p_{10} + q_{01})\bigg )\\&\quad +(-1)^{\mu _2} (r_{10} + s_{01}) \omega _{\scriptscriptstyle RS}\\&\quad \arccos \bigg (\frac{\tau _{\scriptscriptstyle LC}^2-1}{\tau _{\scriptscriptstyle LC}^2+1}\bigg )+(u_{10} + v_{01}) \omega _{\scriptscriptstyle LC} \omega _{\scriptscriptstyle RS}\\&\quad \log (4) + (p_{10} + q_{01}) \omega _{\scriptscriptstyle LC}\\&\quad \log \bigg (\frac{\tau _{\scriptscriptstyle RS}+1}{\tau _{\scriptscriptstyle RS}-1}\bigg )\bigg ),\\&\quad C_{0}^3 = \frac{b_{\scriptscriptstyle R}}{12}\bigg (3 (u_{10} + v_{01})+\frac{8(-1)^{\mu _2}\tau _{\scriptscriptstyle LC}^3}{\omega _{\scriptscriptstyle LC}(\tau _{\scriptscriptstyle LC}^2+1)^2}\\&\quad (r_{10} + s_{01})+\frac{8\tau _{\scriptscriptstyle RS}^3}{\omega _{\scriptscriptstyle RS}(\tau _{\scriptscriptstyle RS}^2-1)^2}\\&\quad (p_{10} + q_{01})\bigg ),\\&\quad C_{1}^0 = 2( p_{00} + p_{10}) + b_{\scriptscriptstyle R} ( v_{01} - 2 u_{00} - u_{10}) \\&\quad + \frac{b_{\scriptscriptstyle R}\tau _{\scriptscriptstyle RS}}{\omega _{\scriptscriptstyle LC}}(p_{10} + q_{01})-\frac{b_{\scriptscriptstyle R}}{2\omega _{\scriptscriptstyle RS}}\\&\quad (p_{10} + q_{01})(\tau _{\scriptscriptstyle RS}^2-1)\log \bigg (\frac{\tau _{\scriptscriptstyle RS}+1}{\tau _{\scriptscriptstyle RS}-1}\bigg ),\\&\quad C_{1}^1 = 2( p_{00} + p_{10}) + b_{\scriptscriptstyle R} \Big ( v_{01} - 2 u_{00} - u_{10}\\&\quad +(u_{10} + v_{01}) \log (2)\Big )+\frac{b_{\scriptscriptstyle R}}{\omega _{\scriptscriptstyle RS}}\\&\quad (p_{10} + q_{01})\log \bigg (\frac{\tau _{\scriptscriptstyle RS}+1}{\tau _{\scriptscriptstyle RS}-1}\bigg ),\\&\quad C_{1}^2 = \frac{b_{\scriptscriptstyle R}}{2}\bigg (\frac{2\tau _{\scriptscriptstyle RS}}{\omega _{\scriptscriptstyle RS}(\tau _{\scriptscriptstyle RS}^2-1)}(p_{10} + q_{01})\\&\quad +(u_{10} + v_{01}) (1 + \log (2))+\frac{1}{\omega _{\scriptscriptstyle RS}}\\&\quad (p_{10} + q_{01})\log \bigg (\frac{\tau _{\scriptscriptstyle RS}+1}{\tau _{\scriptscriptstyle RS}-1}\bigg )\bigg ),\\ \end{aligned} \end{aligned}$$
$$\begin{aligned} \begin{aligned}&C_{2}^0 = \frac{2}{b_{\scriptscriptstyle L}}( r_{10}-r_{00})+2 u_{00} - u_{10} \\&\quad + v_{01} + \frac{1}{\omega _{\scriptscriptstyle LC}}(r_{10} + s_{01}) ( \pi \mu _2 (1 + \tau _{\scriptscriptstyle LC}^2)\\&\quad -\tau _{\scriptscriptstyle LC} )\frac{(-1)^{\mu _2}(\tau _{\scriptscriptstyle LC}^2+1)}{2\omega _{\scriptscriptstyle LC}}(r_{10} + s_{01})\\&\quad \arccos \bigg (\frac{\tau _{\scriptscriptstyle LC}^2-1}{\tau _{\scriptscriptstyle LC}^2+1}\bigg ),\\&\quad C_{2}^1 = \frac{2}{b_{\scriptscriptstyle L}}( r_{10}-r_{00})+2 u_{00} - u_{10} \\&\quad + v_{01} + \frac{1}{\omega _{\scriptscriptstyle LC}}(r_{10} + s_{01})(2 \pi \mu _2 + ( (-1)^{\mu _2}\quad \quad \quad \quad \\&\quad -1) \tau _{\scriptscriptstyle LC})+(u_{10} + v_{01}) \log (2) +\\&\quad \frac{(-1)^{\mu _2}}{\omega _{\scriptscriptstyle LC}}(r_{10} + s_{01})\arccos \bigg (\frac{\tau _{\scriptscriptstyle LC}^2-1}{\tau _{\scriptscriptstyle LC}^2+1}\bigg ),\\&\quad C_{2}^2 = \frac{1}{2}\bigg (\frac{2}{\omega _{\scriptscriptstyle LC}}\bigg (\pi \mu _2+\frac{(-1)^{\mu _2}\tau _{\scriptscriptstyle LC}}{\tau _{\scriptscriptstyle LC}^2+1}\bigg )\\&\quad (r_{10} + s_{01})+\frac{(-1)^{\mu _2}}{\omega _{\scriptscriptstyle LC}}(r_{10} + s_{01})\\&\quad \arccos \bigg (\frac{\tau _{\scriptscriptstyle LC}^2-1}{\tau _{\scriptscriptstyle LC}^2+1}\bigg )+(u_{10} + v_{01}) (1 + \log (2))\bigg ). \end{aligned} \end{aligned}$$

Appendix B

The coefficients list of the Melnikov functions for the case CSC is the following:

$$\begin{aligned}{} & {} C_{0}^0 = \frac{1}{2}\bigg (4 (p_{00} + p_{10}) + 2 b_{\scriptscriptstyle R}\\{} & {} \quad \bigg (\frac{2}{b_{\scriptscriptstyle L}}( r_{10}-r_{00})-2 (u_{10} +v_{01})-\frac{1}{\omega _{\scriptscriptstyle LC}}(r_{10} + s_{01}) \\{} & {} \quad ( \pi \mu _2 (1 + \tau _{\scriptscriptstyle LC}^2)-\tau _{\scriptscriptstyle LC})\bigg )+\frac{2 b_{\scriptscriptstyle R}}{\omega _{\scriptscriptstyle RC}} \\{} & {} \quad (p_{10} + q_{01}) ( \pi \mu _1 (1 + \tau _{\scriptscriptstyle RC}^2)-\tau _{\scriptscriptstyle RC} )\\{} & {} \quad +\frac{(-1)^{\mu _2}b_{\scriptscriptstyle R}}{\omega _{\scriptscriptstyle LC}}(r_{10} + s_{01})(\tau _{\scriptscriptstyle LC}^2+1)\\{} & {} \quad \arccos \bigg (\frac{\tau _{\scriptscriptstyle LC}^2-1}{\tau _{\scriptscriptstyle LC}^2+1}\bigg )+\frac{(-1)^{\mu _1}b_{\scriptscriptstyle R}}{\omega _{\scriptscriptstyle RC}}(p_{10} \\{} & {} \quad + q_{01})(\tau _{\scriptscriptstyle RC}^2+1)\arccos \bigg (\frac{\tau _{\scriptscriptstyle RC}^2-1}{\tau _{\scriptscriptstyle RC}^2+1}\bigg )\bigg ),\\{} & {} \quad C_{0}^1 = \frac{1}{b_{\scriptscriptstyle L}\omega _{\scriptscriptstyle LC}\omega _{\scriptscriptstyle RC}}\bigg (\omega _{\scriptscriptstyle LC}\omega _{\scriptscriptstyle RC}\\{} & {} \quad (2 b_{\scriptscriptstyle L} (p_{00} + p_{10}) + 2 b_{\scriptscriptstyle R} (r_{10}-r_{00}) + 2b_{\scriptscriptstyle L}b_{\scriptscriptstyle R} (v_{01} \\{} & {} \quad -u_{10}))+b_{\scriptscriptstyle L} b_{\scriptscriptstyle R} \bigg (2 \pi ( r_{10} + s_{01}) \mu _2 \omega _{\scriptscriptstyle RC} \\{} & {} \quad + (-r_{10} - s_{01} + (-1)^{\mu _2} (r_{10} \\{} & {} \quad + s_{01})) \tau _{\scriptscriptstyle LC} \omega _{\scriptscriptstyle RC} + p_{10} \omega _{\scriptscriptstyle LC} \\{} & {} \quad (2 \pi \mu _1+(-1 + (-1)^{\mu _1}) \tau _{\scriptscriptstyle LC}) + q_{01} (2 \pi \mu _1 \omega _{\scriptscriptstyle LC} \\{} & {} \quad - \omega _{\scriptscriptstyle LC} \tau _{\scriptscriptstyle RC} + (-1)^{\mu _1} \omega _{\scriptscriptstyle LC} \tau _{\scriptscriptstyle RC})\bigg )\\{} & {} \quad +b_{\scriptscriptstyle L} b_{\scriptscriptstyle R} \bigg ((-1)^{\mu _2}(r_{10} + s_{01})\omega _{\scriptscriptstyle RC}\\{} & {} \quad \arccos \bigg (\frac{\tau _{\scriptscriptstyle LC}^2-1}{\tau _{\scriptscriptstyle LC}^2+1}\bigg )+(-1)^{\mu _1} (p_{10} + q_{01}) \omega _{\scriptscriptstyle LC}\\{} & {} \quad \arccos \bigg (\frac{\tau _{\scriptscriptstyle RC}^2-1}{\tau _{\scriptscriptstyle RC}^2+1}\bigg ) +(u_{10} \\{} & {} \quad + v_{01}) \omega _{\scriptscriptstyle LC} \omega _{\scriptscriptstyle RC} \log (4)\bigg )\bigg ),\\ \end{aligned}$$
$$\begin{aligned} \begin{aligned}&C_{0}^2 = \frac{b_{\scriptscriptstyle L}}{2}\bigg (2 u_{10} + 2\bigg (v_{01}+\frac{1}{\omega _{\scriptscriptstyle RC}}\\&\quad \bigg (\frac{1}{\omega _{\scriptscriptstyle LC}}\bigg (\pi (p_{10} + q_{01}) \mu _1 \omega _{\scriptscriptstyle LC} + \pi (r_{10} + s_{01})\\&\quad \mu _2 \omega _{\scriptscriptstyle RC} +\frac{(-1)^{\mu _2}}{\tau _{\scriptscriptstyle LC}^2+1}\tau _{\scriptscriptstyle LC} \omega _{\scriptscriptstyle LC}(r_{10} + s_{01}) \bigg )\\&\quad +\frac{(-1)^{\mu _1}\tau _{\scriptscriptstyle RC}}{\tau _{\scriptscriptstyle RC}^2+1}(p_{10} + q_{01})\bigg )\bigg )\\&\quad +\frac{(-1)^{\mu _2}}{\omega _{\scriptscriptstyle LC}}(r_{10} + s_{01})\arccos \bigg (\frac{\tau _{\scriptscriptstyle LC}^2-1}{\tau _{\scriptscriptstyle LC}^2+1}\bigg )\\&\quad +\frac{(-1)^{\mu _1}}{\omega _{\scriptscriptstyle RC}}(p_{10} + q_{01})\\&\quad \arccos \bigg (\frac{\tau _{\scriptscriptstyle RC}^2-1}{\tau _{\scriptscriptstyle RC}^2+1}\bigg )+(u_{10} + v_{01}) \log (4)\bigg ),\\&\quad C_{0}^3 = \frac{b_{\scriptscriptstyle R}}{12}\bigg (3 (u_{10} + v_{01})+\frac{8(-1)^{\mu _2}\tau _{\scriptscriptstyle LC}^3}{\omega _{\scriptscriptstyle LC}(\tau _{\scriptscriptstyle LC}^2+1)^2}\\&\quad (r_{10} + s_{01})+\frac{8(-1)^{\mu _1}\tau _{\scriptscriptstyle RC}^3}{\omega _{\scriptscriptstyle RC}(\tau _{\scriptscriptstyle RC}^2-1)^2}\\&\quad (p_{10} + q_{01})\bigg ),\\&\quad C_{1}^0 = \frac{1}{2\omega _{\scriptscriptstyle RC}}\bigg (4 (p_{00} + p_{10}) \omega _{\scriptscriptstyle RC}\\&\quad - 2 b_{\scriptscriptstyle R} (2 u_{00} + u_{10} - v_{01}) \omega _{\scriptscriptstyle RC} + 2 b_{\scriptscriptstyle R} (p_{10} + q_{01}) \\&\quad ( \pi \mu _1 (1 + \tau _{\scriptscriptstyle RC}^2)-\tau _{\scriptscriptstyle RC})+ (-1)^{\mu _1} b_{\scriptscriptstyle R}\\&\quad (p_{10} + q_{01}) (1 + \tau _{\scriptscriptstyle RC}^2)\\&\quad \arccos \bigg (\frac{\tau _{\scriptscriptstyle RC}^2-1}{\tau _{\scriptscriptstyle RC}^2+1}\bigg )\bigg ), \end{aligned} \end{aligned}$$
$$\begin{aligned} \begin{aligned}&C_{1}^1 = \frac{1}{\omega _{\scriptscriptstyle RC}}\bigg (2 (p_{00} + p_{10}) \omega _{\scriptscriptstyle RC}\\&\quad + b_{\scriptscriptstyle R} (p_{10} + q_{01}) (2 \pi \mu _1 + ((-1)^{\mu _1}-1) \tau _{\scriptscriptstyle RC}) \\&\quad + (-1)^{\mu _1} b_{\scriptscriptstyle R}(p_{10} + q_{01})\\&\quad \arccos \bigg (\frac{\tau _{\scriptscriptstyle RC}^2-1}{\tau _{\scriptscriptstyle RC}^2+1}\bigg )+b_{\scriptscriptstyle R} \omega _{\scriptscriptstyle RC} ( v_{01}-2 u_{00} - u_{10} \\&\quad + (u_{10} + v_{01}) \log (2))\bigg ),\\&\quad C_{1}^2 = \frac{b_{\scriptscriptstyle R}}{2\omega _{\scriptscriptstyle RC}(\tau _{\scriptscriptstyle RC}^2+1)}\bigg ((-1)^{\mu _1} (p_{10} + q_{01})\\&\quad \bigg (2 \tau _{\scriptscriptstyle RC} + (1 + \tau _{\scriptscriptstyle RC}^2)\arccos \bigg (\frac{\tau _{\scriptscriptstyle RC}^2-1}{\tau _{\scriptscriptstyle RC}^2+1}\bigg )\bigg )\\&\quad + (1 + \tau _{\scriptscriptstyle RC}^2) \bigg (2 \pi (p_{10} + q_{01}) \mu _1 + (u_{10} + v_{01}) \\&\quad \omega _{\scriptscriptstyle RC} (1 + \log (2))\bigg )\bigg ),\\&\quad C_{2}^0 = \frac{2}{b_{\scriptscriptstyle L}}(r_{10}-r_{00})+2 u_{00} - u_{10} + v_{01} \\&\quad +\frac{1}{\omega _{\scriptscriptstyle LC}}(r_{10} + s_{01}) ( \pi \mu _2 (1 + \tau _{\scriptscriptstyle LC}^2)-\tau _{\scriptscriptstyle LC})\\&\quad +\frac{(-1)^{\mu _2}}{2\omega _{\scriptscriptstyle LC}} (r_{10} + s_{01}) (\tau _{\scriptscriptstyle LC}^2+1)\arccos \bigg (\frac{\tau _{\scriptscriptstyle LC}^2-1}{\tau _{\scriptscriptstyle LC}^2+1}\bigg ),\\&\quad C_{2}^1 = \frac{2}{b_{\scriptscriptstyle L}}(r_{10}-r_{00})+2 u_{00} - u_{10} + v_{01} \\&\quad +\frac{1}{\omega _{\scriptscriptstyle LC}}(r_{10} + s_{01}) (2 \pi \mu _2 + ((-1)^{\mu _2}-1)\\&\quad \tau _{\scriptscriptstyle LC})+\frac{(-1)^{\mu _2}}{\omega _{\scriptscriptstyle LC}}(r_{10} + s_{01}) \arccos \bigg (\frac{\tau _{\scriptscriptstyle LC}^2-1}{\tau _{\scriptscriptstyle LC}^2+1}\bigg )\\&\quad +(u_{10} + v_{01}) \log (2),\\&\quad C_{2}^2 = \frac{1}{2}\bigg (\frac{2}{\omega _{\scriptscriptstyle LC}}(r_{10} + s_{01})\bigg (\pi \mu _2+\frac{(-1)^{\mu _2}\tau _{\scriptscriptstyle LC}}{\tau _{\scriptscriptstyle LC}^2+1}\bigg )\\&\quad +\frac{(-1)^{\mu _2}}{\omega _{\scriptscriptstyle LC}}(r_{10} + s_{01})\\&\quad \arccos \bigg (\frac{\tau _{\scriptscriptstyle LC}^2-1}{\tau _{\scriptscriptstyle LC}^2+1}\bigg )+(u_{10} + v_{01}) (1 + \log (2))\bigg ). \end{aligned} \end{aligned}$$

Conclusion

In this paper, we study the number and position of crossing limit cycles of discontinuous planar piecewise linear Hamiltonian differential systems with three zones separated by two parallel straight lines, assuming that the linear differential subsystem have a saddle in the central one. By estimating the number of simultaneous simple zeros of the first-order Melnikov functions associated with the piecewise near-Hamiltonian differential systems, we prove that at least six crossing limit cycles can bifurcate from the periodic annuli of these kind of piecewise systems, after affine linear perturbations.

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Pessoa, C., Ribeiro, R., Novaes, D. et al. On cyclicity in discontinuous piecewise linear near-Hamiltonian differential systems with three zones having a saddle in the central one. Nonlinear Dyn 111, 21153–21175 (2023). https://doi.org/10.1007/s11071-023-08931-8

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