The relationship of topological insulators and superconductors and the field of nonlinear dynamics is widely unexplored. To address this subject, we adopt the linear coupling geometry of the Su-Schrieffer-Heeger model, a paradigmatic example for a topological insulator, and render it nonlinearly in the context of superconducting circuits. As a consequence, the system exhibits topologically enforced bifurcations as a function of the topological control parameter, which finally gives rise to chaotic dynamics, separating phases that exhibit clear topological features.