Existence of piecewise linear differential systems with exactly n limit cycles for all n ∈ ℕ

Jaume Llibre; Enrique Ponce; Xiang Zhang

doi:https://doi.org/10.1016/S0362-546X(03)00122-6

Existence of piecewise linear differential systems with exactly n limit cycles for all n ∈ ℕ

Jaume Llibre, Enrique Ponce, Xiang Zhang

Department of Mathematics

Research output: Contribution to journal › Article › Research › peer-review

14 Citations (Scopus)

Abstract

In this paper, we prove that the piecewise linear differential system ẋ = - y - εφ(x), ẏ = x with ε ≠ 0 and φ an odd piecewise linear periodic function of period 4, has exactly n limit cycles in the strip |x| ≤ 2(n+1). Consequently, there are piecewise linear differential systems having infinitely many limit cycles in the real plane. We also provide examples of piecewise linear differential systems having exactly n limit cycles for all n ∈ ℕ. © 2003 Elsevier Science Ltd. All rights reserved.

Original language	English
Pages (from-to)	977-994
Journal	Nonlinear Analysis, Theory, Methods and Applications
Volume	54
DOIs	https://doi.org/10.1016/S0362-546X(03)00122-6
Publication status	Published - 1 Aug 2003

Keywords

Limit cycles
Nonlinear oscillations
Piecewise linear control system

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abstract = "In this paper, we prove that the piecewise linear differential system ẋ = - y - εφ(x), ẏ = x with ε ≠ 0 and φ an odd piecewise linear periodic function of period 4, has exactly n limit cycles in the strip |x| ≤ 2(n+1). Consequently, there are piecewise linear differential systems having infinitely many limit cycles in the real plane. We also provide examples of piecewise linear differential systems having exactly n limit cycles for all n ∈ ℕ. {\textcopyright} 2003 Elsevier Science Ltd. All rights reserved.",

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author = "Jaume Llibre and Enrique Ponce and Xiang Zhang",

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AU - Ponce, Enrique

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N2 - In this paper, we prove that the piecewise linear differential system ẋ = - y - εφ(x), ẏ = x with ε ≠ 0 and φ an odd piecewise linear periodic function of period 4, has exactly n limit cycles in the strip |x| ≤ 2(n+1). Consequently, there are piecewise linear differential systems having infinitely many limit cycles in the real plane. We also provide examples of piecewise linear differential systems having exactly n limit cycles for all n ∈ ℕ. © 2003 Elsevier Science Ltd. All rights reserved.

AB - In this paper, we prove that the piecewise linear differential system ẋ = - y - εφ(x), ẏ = x with ε ≠ 0 and φ an odd piecewise linear periodic function of period 4, has exactly n limit cycles in the strip |x| ≤ 2(n+1). Consequently, there are piecewise linear differential systems having infinitely many limit cycles in the real plane. We also provide examples of piecewise linear differential systems having exactly n limit cycles for all n ∈ ℕ. © 2003 Elsevier Science Ltd. All rights reserved.

KW - Limit cycles

KW - Nonlinear oscillations

KW - Piecewise linear control system

U2 - https://doi.org/10.1016/S0362-546X(03)00122-6

DO - https://doi.org/10.1016/S0362-546X(03)00122-6

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