Simple non-autonomous differential equations with many limit cycles

Bartomeu Coll; Armengol Gasull; Rafel Prohens

Simple non-autonomous differential equations with many limit cycles

Bartomeu Coll, Armengol Gasull, Rafel Prohens

Department of Mathematics

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

2 Citations (Scopus)

Abstract

Consider the family of differential equations on the cylinder, dx/dt = a(t) + b(t) x , where x, t ∈ ℝ, and a, b are real, 1-periodic and smooth functions. The solutions satisfying x(0) = x(1) are called periodic orbits of the equation. The periodic orbits that are isolated in the set of all the periodic orbits are usually called limit cycles. We give a proof, which is self contained, that there is no upper bound for the number of limit cycles of the above type of equations.

Original language	English
Pages (from-to)	29-34
Journal	Communications on Applied Nonlinear Analysis
Volume	15
Issue number	1
Publication status	Published - 1 Jan 2008

Keywords

Bifurcations
Limit cycles
Non-smooth differential equations

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title = "Simple non-autonomous differential equations with many limit cycles",

abstract = "Consider the family of differential equations on the cylinder, dx/dt = a(t) + b(t) x , where x, t ∈ ℝ, and a, b are real, 1-periodic and smooth functions. The solutions satisfying x(0) = x(1) are called periodic orbits of the equation. The periodic orbits that are isolated in the set of all the periodic orbits are usually called limit cycles. We give a proof, which is self contained, that there is no upper bound for the number of limit cycles of the above type of equations.",

keywords = "Bifurcations, Limit cycles, Non-smooth differential equations",

author = "Bartomeu Coll and Armengol Gasull and Rafel Prohens",

year = "2008",

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T1 - Simple non-autonomous differential equations with many limit cycles

AU - Coll, Bartomeu

AU - Gasull, Armengol

AU - Prohens, Rafel

PY - 2008/1/1

Y1 - 2008/1/1

N2 - Consider the family of differential equations on the cylinder, dx/dt = a(t) + b(t) x , where x, t ∈ ℝ, and a, b are real, 1-periodic and smooth functions. The solutions satisfying x(0) = x(1) are called periodic orbits of the equation. The periodic orbits that are isolated in the set of all the periodic orbits are usually called limit cycles. We give a proof, which is self contained, that there is no upper bound for the number of limit cycles of the above type of equations.

AB - Consider the family of differential equations on the cylinder, dx/dt = a(t) + b(t) x , where x, t ∈ ℝ, and a, b are real, 1-periodic and smooth functions. The solutions satisfying x(0) = x(1) are called periodic orbits of the equation. The periodic orbits that are isolated in the set of all the periodic orbits are usually called limit cycles. We give a proof, which is self contained, that there is no upper bound for the number of limit cycles of the above type of equations.

KW - Bifurcations

KW - Limit cycles

KW - Non-smooth differential equations

M3 - Article

VL - 15

SP - 29

EP - 34

JO - Communications on Applied Nonlinear Analysis

JF - Communications on Applied Nonlinear Analysis

SN - 1074-133X

IS - 1

ER -