On the bifurcation of double homoclinic loops of a cubic system

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A cubic symmetric system x ̇ = y + ε [ a 1 x + ( a 2 + a 3 + 1 ) x 3 + ( b 1 − 3 ) x 2 y + ( a 2 − 3 a 3 + 1 ) x y 2 + ( b 1 + 1 ) y 3 ] , y ̇ = x − x 3 + ε [ a 1 y + ( b 1 + 1 ) x 3 + ( a 2 + 3 a 3 − 1 ) x 2 y + ( b 1 − 3 ) x y 2 + ( a 2 − a 3 − 1 ) y 3 ] is considered. By computing the focus quantities and saddle quantities, we get the quantities which determine the stability of the double homoclinic loops appearing. Then combining Hopf bifurcation and double homoclinic loop bifurcation, we prove that seven limit cycles can bifurcated from the double homoclinic loops in the above cubic system. As far as we know, this result on the number of limit cycles bifurcated from double homoclinic loops of the cubic system is new.