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We study the classical H\'enon family $f_{a,b}:(x,y)\mapsto(1-ax^2+y,bx)$,$0<a<2$, $0<b<1$, and prove that given an integer $k\geq 1$, there is a set ofparameters $E_k$ of positive two-dimensional Lebesgue measure so that$f_{a,b}$, for $(a,b)\in E_k$, has at least $k$ attractive periodic orbits andone strange attractor. A corresponding statement also holds for theH\'enon-like families. The final main result of the paper is the existence,within the classical H\'enon family, of a positive Lebesgue measure set ofparameters whose corresponding maps have two coexisting strange attractors.

Coexistence phenomena in the H\'enon family