  We study chiral symmetry breaking in the Nambu--Jona-Lasinio model
regularized in proper-time in arbitrary space-time dimensions through an
iterative procedure by writing the gap equation in the form of a discrete
dynamical system with the coupling constant as the control parameter.
Expectedly, we obtain the critical coupling for chiral symmetry breaking when a
nontrivial solution bifurcates away from the trivial one and becomes an
attractor. By increasing further the value of the coupling constant, we observe
a second bifurcation where the dynamical solution is no longer an attractor,
and observation that holds true in all space-time dimensions. In the
super-strong coupling regime, the system becomes chaotic.
