A driven damped nearly harmonic oscillator with a small an-harmonicity, a cubic term in the force, is known as the Duffing oscillator. The Duffing oscillator shows various interesting features of non-linear response such as bistability and hysteresis. Several features of the Duffing instability have been recently measured using superconducting qubits and nano-mechanical resonators. Linear Paul traps can be well approximated as harmonic but have a small an-harmonicity due to their deviation from a real quadruple shape. We study the steady state of motion of a single trapped Sr⁺ ion subject- to a near-resonance drive and dissipation in a linear Paul trap with a small an-harmonicity. The driving force is applied by an oscillating voltage on the trap end-caps. Dissipation is a result of laser cooling. We measure both the oscillations amplitude and phase and find a good agreement with the Duffing oscillator model. In our experiment dissipation can be conveniently controlled by varying the intensity and detuning of the cooling laser. When the cooling laser is close to resonance the standard Duffing model has to be extended to account for non-linearity in the dissipative force as well.