Quasiperiodic Forcing by Infinite Fourier Series of the Verhulst Logistic Model Exact Solutions, Stability Analysis, and the Emergence of a Strange Nonchaotic Attractor under Specific Circumstances

We study the logistic equation with a quasiperiodically forced carrying capacity generated by a real-analytic torus hull. Reciprocal linearisation gives an explicit pullback convolution and an exact fibre contraction rate. If the carrying capacity remains bounded away from zero, the attracting invariant graph is continuous, excluding strange nonchaotic behaviour. At a unique nondegenerate zero, an analytic Morse normal form yields the exact shallow-grazing frequency and a two-sided, error-controlled relation between the depth of an isolated passage and the resulting population bottleneck. The reciprocal invariant graph is finite almost everywhere but essentially unbounded; its population counterpart is bounded and upper semicontinuous, positive almost everywhere, and zero on a dense Haar-null set. Direct verification of the sampled skew product therefore gives a pinched strange nonchaotic attractor with sampled per-unit-time fibre Lyapunov exponent -r.