ea2e595b89075fbaacf5afa48e3f3126 Mat_a2024v12n3p464.pdf 56ef88679b4fed7f97acc277dfb1d98a4e81a634 Mat_a2024v12n3p464.pdf 99cacb1a6a902f5869d9a441e2333974a655cb5d7aa52805fd3c31091a005e06 Mat_a2024v12n3p464.pdf Title: The Solution of the Extended 16th Hilbert Problem for Some Classes of Piecewise Differential Systems Subject: The limit cycles have a main role in understanding the dynamics of planar differential systems, but their study is generally challenging. In the last few years, there has been a growing interest in researching the limit cycles of certain classes of piecewise differential systems due to their wide uses in modeling many natural phenomena. In this paper, we provide the upper bounds for the maximum number of crossing limit cycles of certain classes of discontinuous piecewise differential systems (simply PDS) separated by a straight line and consequently formed by two differential systems. A linear plus cubic polynomial forms six families of Hamiltonian nilpotent centers. First, we study the crossing limit cycles of the PDS formed by a linear center and one arbitrary of the six Hamiltonian nilpotent centers. These six classes of PDS have at most one crossing limit cycle, and there are systems in each class with precisely one limit cycle. Second, we study the crossing limit cycles of the PDS formed by two of the six Hamiltonian nilpotent centers. There are systems in each of these 21 classes of PDS that have exactly four crossing limit cycles. Keywords: discontinuous piecewise differential system; Hamiltonian nilpotent center; cubic polynomial differential system; limit cycle; vector field Author: Louiza Baymout, Rebiha Benterki and Jaume Llibre Creator: LaTeX with hyperref Producer: pdfTeX-1.40.25 CreationDate: Thu Feb 1 09:59:44 2024 CET ModDate: Thu Feb 1 10:04:24 2024 CET Custom Metadata: no Metadata Stream: no Tagged: no UserProperties: no Suspects: no Form: none JavaScript: no Pages: 29 Encrypted: no Page size: 595.276 x 841.89 pts (A4) Page rot: 0 File size: 775478 bytes Optimized: no PDF version: 1.7 name type encoding emb sub uni object ID ------------------------------------ ----------------- ---------------- --- --- --- --------- NPSJEU+CMEX10 Type 1 Builtin yes yes yes 10 0 WEWMPO+URWPalladioL-Bold Type 1 Custom yes yes yes 15 0 GLYPWH+URWPalladioL-Roma Type 1 Custom yes yes yes 21 0 JBKDYN+URWPalladioL-Ital Type 1 Custom yes yes yes 26 0 GIGFZE+CMR10 Type 1 Builtin yes yes yes 31 0 SZFIBJ+CMSY10 Type 1 Builtin 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Destination: section.5 Item: Title: Numerical Examples of Theorem 3 Destination: section.6 Item: Title: Numerical Examples of Theorem 4 Destination: section.7 Item: Title: Numerical Examples of Theorem 5 Destination: section.8 Item: Title: Numerical Examples of Theorem 6 Destination: section.9 Item: Title: Numerical Examples of Theorem 7 Destination: section.10 Item: Title: Numerical Examples of Theorem 8 Destination: section.11 Item: Title: References Destination: section.12 Info: Title: The Solution of the Extended 16th Hilbert Problem for Some Classes of Piecewise Differential Systems Author: Louiza Baymout, Rebiha Benterki and Jaume Llibre Subject: The limit cycles have a main role in understanding the dynamics of planar differential systems, but their study is generally challenging. In the last few years, there has been a growing interest in researching the limit cycles of certain classes of piecewise differential systems due to their wide uses in modeling many natural phenomena. In this paper, we provide the upper bounds for the maximum number of crossing limit cycles of certain classes of discontinuous piecewise differential systems (simply PDS) separated by a straight line and consequently formed by two differential systems. A linear plus cubic polynomial forms six families of Hamiltonian nilpotent centers. First, we study the crossing limit cycles of the PDS formed by a linear center and one arbitrary of the six Hamiltonian nilpotent centers. These six classes of PDS have at most one crossing limit cycle, and there are systems in each class with precisely one limit cycle. Second, we study the crossing limit cycles of the PDS formed by two of the six Hamiltonian nilpotent centers. There are systems in each of these 21 classes of PDS that have exactly four crossing limit cycles. 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