Border-collision bifurcations from stable fixed points to any number of coexisting chaotic attractors

The uploaded paper proves that, near a border-collision bifurcation, one can obtain a transition from a stable fixed point (for µ<0) to many coexisting chaotic attractors (for µ>0), and gives an explicit count N[k,n] = (1/(k n)) ∑_{d|a} φ(d) k^{n/d} with a the largest divisor of n coprime to k. Theorem 2.2 states this precisely and provides the formula (2.6) for N[k,n] . The proof proceeds via: (i) scaling to a two-parameter simple form g and a C^1–closeness lemma (Lemma 3.1) ; (ii) an asymptotically stable fixed point for µ<0 (Lemma 4.1) ; (iii) the n-th iterate conjugate to a product of skew tent maps (Lemma 5.1) ; (iv) choosing parameters in the Sk regions where the skew tent map has a k-band chaotic attractor (Section 6) ; (v) fattening intervals Ji so that h(Ji)⊂int(Ji+1) (Lemma 7.1) ; (vi) forming boxes Φv and a combinatorial map ψ(v)=(v2,…,vn,v1+1) so that g(Φv)⊂int(Φψ(v)) (Section 8) ; (vii) counting orbits via Burnside’s lemma to get (2.6) (Proposition 9.1 and its derivation) ; and (viii) proving expansion of f^{kn} on each trapping region (Lemma 10.1) and collating the result (Proof of Theorem 2.2) . The candidate’s solution reproduces this strategy closely: reduction to a two-parameter form; stable fixed point for µ<0; product-of-skew-tent structure for the n-th iterate; fattened intervals and Cartesian-product boxes; the ψ/Θ shift+increment combinatorics; Burnside counting yielding the same N[k,n]; and piecewise-C^r expansion for f^{kn}. Minor discrepancies are cosmetic: the model mentions a near-identity linear scaling S (the paper uses a µ-scaling), and it attributes “slopes >1” to the skew tent already at the level of the n-th iterate (where the paper obtains uniform expansion after the additional k-iterate, i.e., for g^{kn}). These do not affect the core argument. Hence, both the paper and the model’s solution are correct and substantially the same.