To obtain a description of the motion of the pendulum we need to solve this equation with the initial condition that at t=0, the pendulum starts with zero speed and a given amplitude . Although, the exact solution of this problem is known, simplification can be obtained for small amplitude. For , we can write . Then this equation of motion is linear with the well known harmonic solution.
(1.2)
For large displacements equation (1.1) is no more linear. In this simple case the exact solutions of eq (1.1) are well known in terms of Jacobi elliptic functions. For larger values of the initial amplitude the motion remains periodic but not a simple cosine function and the period depends on the initial amplitude. If we resolve this motion into a number of sinusoidal functions we find that the motion contains many harmonic frequencies. It is important to note that there is no threshold in the problem i.e. there is no value of initial amplitude below which these corrections are zero, but of course they are small. Now, we know that if two pendula are close to each other, oscillations can transfer from one to another.
What happens if the frequency (as obtained in the linear approximation) of one is thrice that of another?
Our description above suggests that a pendulum with resonant frequency can excite one at frequency and this transfer will depend on the amplitude of the first pendulum. Thus we see that such a simple system actually turns out to be nonlinear and in the process of linearization we miss out on many interesting phenomena e.g., the possibility of exciting this system resonantly at a harmonic frequency, the nonsinusoidal nature of oscillations and the dependence of the frequency on the initial amplitude.
. Although, the exact solution of this problem is known, simplification can be obtained for small amplitude. For 
, we can write 
. Then this equation of motion is linear with the well known harmonic solution.
can excite one at frequency 
and this transfer will depend on the amplitude of the first pendulum. Thus we see that such a simple system actually turns out to be nonlinear and in the process of linearization we miss out on many interesting phenomena e.g., the possibility of exciting this system resonantly at a harmonic frequency, the nonsinusoidal nature of oscillations and the dependence of the frequency on the initial amplitude.