

At the first glance, nonlinear resonance seems not to be so much different from the (linear) resonance of a harmonic oscillator. But both, the dependency of the eigenfrequency of a nonlinear oscillator on the amplitude and the nonharmoniticity of the oscillation lead to a behavior that is impossible in harmonic oscillators, namely, the foldover effect and superharmonic resonance, respectively. Both effects are especially important in the case of weak damping.
The nonlinear resonance line can be approximated considerably well (see the green line in the figure) by the following
heuristic approach: Take the resonance line of the harmonic oscillator and replace
the eigenfrequency leads to because the width and the height of the resonance line are proportional to and 
If you try to find them for the pendulum driven by a periodic force, you have to choose relatively large driving amplitudes and low values for the damping constant. Nevertheless, you will not find superharmonic resonance at half the eigenfrequency of the pendulum! Why? The reason is symmetry. The undriven pendulum is invariant under reflection symmetry. That is, if you change the sign of , the equation of motion remains unchanged. Due to this symmetry, the nonlinear oscillation of the pendulum does not contain frequencies which are even multiples of the fundamental frequency. Thus, the first superharmonic resonance occurs near one third of the fundamental frequency. Again, the foldover effect occurs. 
QUESTIONS worth to think about: 
