Structural stability of polynomial second order differential equations with periodic coefficients

Adolfo W. Guzmán

Structural stability of polynomial second order differential equations with periodic coefficients

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

Abstract

This work characterizes the structurally stable second order differential equations of the form x″ = _iⁿ=0 a _i(x)(x′)ⁱ where a_i : ℜ → ℜ are C^r periodic functions. These equations have naturally the cylinder M = S₁ × ℜ as the phase space and are associated to the vector fields X(f) = ya/a_x+f(x, y)a/a_y, where f(x, y) = _iⁿ=0 a_i(x)yⁱa/a_y. We apply a compactification to M as well as to X(f) to study the behavior at infinity. For n ≥ 1, we define a set ∑ⁿ of X(f) that is open and dense and characterizes the class of structural differential equations as above.

Original language	English
Pages (from-to)	1-28
Number of pages	28
Journal	Electronic Journal of Differential Equations
Volume	2004
State	Published - 9 Aug 2004
Externally published	Yes

Keywords

Compactification
Second order differential equation
Singularity at infinity
Structural stability

Cite this

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title = "Structural stability of polynomial second order differential equations with periodic coefficients",

abstract = "This work characterizes the structurally stable second order differential equations of the form x″ = in=0 a i(x)(x′)i where ai : ℜ → ℜ are Cr periodic functions. These equations have naturally the cylinder M = S1 × ℜ as the phase space and are associated to the vector fields X(f) = ya/ax+f(x, y)a/ay, where f(x, y) = in=0 ai(x)yia/ay. We apply a compactification to M as well as to X(f) to study the behavior at infinity. For n ≥ 1, we define a set ∑n of X(f) that is open and dense and characterizes the class of structural differential equations as above.",

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year = "2004",

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N2 - This work characterizes the structurally stable second order differential equations of the form x″ = in=0 a i(x)(x′)i where ai : ℜ → ℜ are Cr periodic functions. These equations have naturally the cylinder M = S1 × ℜ as the phase space and are associated to the vector fields X(f) = ya/ax+f(x, y)a/ay, where f(x, y) = in=0 ai(x)yia/ay. We apply a compactification to M as well as to X(f) to study the behavior at infinity. For n ≥ 1, we define a set ∑n of X(f) that is open and dense and characterizes the class of structural differential equations as above.

AB - This work characterizes the structurally stable second order differential equations of the form x″ = in=0 a i(x)(x′)i where ai : ℜ → ℜ are Cr periodic functions. These equations have naturally the cylinder M = S1 × ℜ as the phase space and are associated to the vector fields X(f) = ya/ax+f(x, y)a/ay, where f(x, y) = in=0 ai(x)yia/ay. We apply a compactification to M as well as to X(f) to study the behavior at infinity. For n ≥ 1, we define a set ∑n of X(f) that is open and dense and characterizes the class of structural differential equations as above.

KW - Compactification

KW - Second order differential equation

KW - Singularity at infinity

KW - Structural stability

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SN - 1072-6691

VL - 2004

SP - 1

EP - 28

JO - Electronic Journal of Differential Equations

JF - Electronic Journal of Differential Equations

ER -

Structural stability of polynomial second order differential equations with periodic coefficients

Abstract

Keywords

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