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

For every n ≥ 3 we construct a nonempty open set of continuous piecewise-affine Liénard systems ẋ = F(x) − y, ẏ = x with exactly n corners and all slopes in (−2, 2), each having at least 3n − 3 hyperbolic limit cycles. Hence H(n) ≥ 3n − 3. For n ≥ 4 this improves by one the lower bound 3n − 4 of Cai, Chen, Jin and Lu; for n = 3 it gives a new proof of the known bound H(3) ≥ 6. The construction starts from a three-corner profile with three limit cycles at finite amplitude whose normalized displacement function vanishes to order four at infinity. Changing three slopes creates three new cycles at a large scale, and one added corner, placed at that scale, restores the fourth-order vanishing. Each step therefore uses expansion order four and three controls, independently of n. The same estimates show that, above a marker threshold independent of n, the constructed systems have exactly 3n − 6 periodic orbits; the number of periodic orbits below this threshold is not determined. The unfolding and restoration steps also apply over an arbitrary admissible core with q corners, which is kept unchanged on a fixed strip. Ten additional corners prepare the core; then K unfoldings and K − 1 restorations realize any prescribed finite sequence of K exterior groups of one or three hyperbolic cycles, with exactly q + K + 9 corners and an exact count above a marker threshold that depends only on the core. This corner budget serves to preserve the core and is not the one used for the 3n − 3 bound. For a local analytic family of cores, one exterior parameter produces two nondegenerate saddle-node bifurcations of the terminal group, giving a one–three–one cycle diagram; after a reparametrization of this parameter depending analytically on the core parameters, the two fold levels are constant.