Yes, it is possible to have an oscillator where the restoring force is non-linear, even for small amplitudes. Oscillators with non-linear restoring forces are known as nonlinear oscillators. In these systems, the relationship between the displacement of the oscillator from its equilibrium position and the restoring force is not linear.
In a simple harmonic oscillator, the restoring force is directly proportional to the displacement from the equilibrium position and acts in the opposite direction. This linear relationship between displacement and restoring force results in simple harmonic motion.
However, in a nonlinear oscillator, the restoring force is not directly proportional to the displacement. Instead, it can be a function of the displacement, its derivatives (velocity, acceleration), or other factors. This nonlinearity can lead to various interesting phenomena, such as the dependence of the oscillation frequency on the amplitude, the occurrence of multiple stable states, or even chaotic behavior.
Nonlinear oscillators can be found in many physical systems, such as pendulums, electrical circuits, mechanical systems, and even biological systems. The behavior of these oscillators can be analyzed using mathematical techniques, such as perturbation theory or numerical simulations.
It's important to note that in the case of small amplitudes, the nonlinearity may have a negligible effect, and the system can still behave approximately as a simple harmonic oscillator. However, as the amplitude increases, the nonlinearity becomes more significant and can lead to deviations from simple harmonic motion.