Birth of limit cycles for a class of continuous and discontinuous differential systems in (d+2)-dimension

The orbits of the reversible differential system $\dot{x}=-y$, $\dot{y}=x$, $\dot{z}=0$, with $x,y \in R$ and $z\in R^d$, are periodic with the exception of the equilibrium points $(0,0, z)$. We compute the maximum number of limit cycles which bifurcate from the periodic orbits of the system $\dot{x}=-y$, $\dot{y}=x$, $\dot{z}=0$, using the averaging theory of first order, when this system is perturbed, first inside the class of all polynomial differential systems of degree $n$, and second inside the class of all discontinuous piecewise polynomial differential systems of degree $n$ with two pieces, one in $y> 0$ and the other in $y<0$. In the first case this maximum number is $n^d(n-1)/2$, and in the second is $n^{d+1}$.

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