Coupled spring-mass system with non-linear spring

The spring forces described thus far assumed the forces to be proportional to displacement. If the spring has non-linearities in it, then the restoring force is no longer just proportional to displacement but is a squared or a cubic function. As noted by Fay & Graham, the restoring spring force can be modeled as:

The ODE's then become:

The following values were used for the simulation:

The initial values used were:

 

The patterns for this set of ODE’s are plotted below. What started as a relatively “simple” 2-DOF system has resulted in a fairly intricate displacement-web with a slight non-linearization of the restoring forces.