The Painlevé analysis is applied to the anharmonic oscillator equation$$\ddot x + d\dot x + Ax + Bx^2 + Cx^3 = 0$$. The following three integrable cases are identified: (i)C=0,d²=25A/6,A>0,B arbitrary, (ii)d²=9A/2,B=0,A>0,C arbitrary and (iii)d²=−9A/4,C=2B²/(9A),A<0,C<0,B arbitrary. The first two integrable choices are already reported in the literature. For the third integrable case the general solution is found involving elliptic function with exponential amplitude and argument.

Integrability and chaotic behaviour in a two-coupled Duffing oscillators are studied. The coupling is nonlinear. Painlevé test is performed to identify integrable cases of damped- and force-free system. Exact analytical solutions are given for the integrable cases. Effect of external periodic forces for (i) single well with infinite height potential, (ii) potential with a hump at the centre and (iii) single well with finite height hump potential are numerically investigated. Occurrence of multiple attractors and period doubling cascades of coexisting attractors is presented.