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# CANARD CYCLES AND HOMOCLINIC BIFURCATION IN A 3 PARAMETER FAMILY OF VECTOR FIELDS ON THE PLANE

Paulo Ricardo da Silva
Publicacions Matemàtiques
Vol. 43, No. 1 (1999), pp. 163-189
Stable URL: http://www.jstor.org/stable/43736654
Page Count: 27
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## Abstract

Let the 3-parameter family of vector fields given by (A) $y\frac{\partial }{{\partial x}} + \left[ {{x^2} + \mu + y\left( {{\nu _0} + {\nu _1}x + {x^3}} \right)}\right]\frac{\partial }{{\partial y}}$ with (x, y, μ, ν₀, ν₁) ∈ R² × R³ ([DRS1]). We prove that if μ →—∞ then (A) is C⁰ -equivalent to (B) $\left[ {y - \left( {bx + c{x^2} - 4{x^3} + {x^4}} \right)} \right]\frac{\partial } {{\partial x}} + \varepsilon \left( {{x^2} - 2x} \right)\frac{\partial } {{\partial y}}$ for ε ↓ 0, b, c ∈ R. We prove that there exists a Hopf bifurcation of codimension 1 when b = 0 and also that, if b = 0, c = 1 2 and ε > 0 then there exists a Hopf bifurcation of codimension 2. We study the "Canard Phenomenon" and the homoclinic bifurcation in the family (B). We show that when ε ↓ 0, b = 0 and c = 12 the attracting limit cycle, which appears in a Hopf bifurcation of codimension 2, stays with "small size" and changes to a "big size" very quickly, in a sense made precise here.

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