Periodic orbits for a class of reversible quadratic vector field on R3 ✩

For a class of reversible quadratic vector fields on R3 we study the periodic orbits that bifurcate from a heteroclinic loop having two singular points at infinity connected by an invariant straight line in the finite part and another straight line at infinity in the local chart U2. More specifically, we prove that for all n ∈ N, there exists εn > 0 such that the reversible quadratic polynomial differential system ˙x = a0 +a1y + a3y2 +a4y2 +ε a2x2 +a3xz , ˙y = b1z +b3yz +εb2xy, ˙z = c1y +c4z2 +εc2xz in R3, with a0 < 0, b1c1 < 0, a2 < 0, b2 < a2, a4 > 0, c2 < a2 and b3 /∈ {c4, 4c4}, for ε ∈ (0, εn) has at least n periodic orbits near the heteroclinic loop. © 2007 Elsevier Inc. All rights reserved