Failure of finite-jet determination for small-time local controllability

Small-time local controllability asks whether a system can reach a neighborhood of an equilibrium in arbitrarily short time. We give a negative answer to the finite-jet determination question for real-analytic systems with bounded measurable controls. Starting from a previously constructed five-dimensional polynomial system with two controls, we show that adding a positive polynomial term of arbitrarily high order destroys this property. The original system is fixed independently of the prescribed jet order. The proof compares a logarithmic bound for closed trajectories with the positive integral cost of an auxiliary coordinate's excursion.