A 3n − 3 lower bound for limit cycles in continuous piecewise-linear Liénard systems with n corners

We study the maximal number H(n) of limit cycles of planar Liénard systems ẋ = F(x) − y, ẏ = x in which F is continuous and piecewise linear with exactly n corners. For every n ≥ 3 we construct a nonempty open set of such systems, all with slopes in (−2, 2), having at least 3n − 3 hyperbolic limit cycles. Hence H(n) ≥ 3n − 3. For n ≥ 4 this improves the lower bound 3n − 4 of Cai, Chen, Jin and Lu by one; for n = 3 it gives a second proof of H(3) ≥ 6. The construction starts from a three-corner profile with three limit cycles whose normalized return displacement has a zero of order four at infinity. Three slopes unfold this zero into three new cycles. Because F is continuous, restoring the fourth-order zero requires a new corner at a position fixed by the unfolding scale. We show that the new cycles do not reach this corner, so the step can be repeated with the same expansion order and three slope controls. For a seed and a threshold r₀ fixed independently of n, a scaled return estimate shows moreover that the open set can be chosen so that exactly 3n − 6 periodic orbits cross the negative y-axis at distance greater than r₀ from the origin.