Abstract Linear and weakly non-linear analyses of convection in a second-order fluid is investigated. The Rivlin–Ericksen constitutive equation is considered to give viscoelastic correction to the momentum equation. The linear and non-linear analyses are, respectively, based on the normal mode technique and truncated representation of Fourier series. The linear theory reveals that the critical eigenvalue is independent of viscoelastic effects and the principle of exchange of stabilities holds. An autonomous system of differential equations representing cellular convection arising in the non-linear study is solved numerically. The non-linear analysis reveals that finite amplitudes have random behaviour. The effect of viscoelasticity on the non-linear solutions is analysed by considering different projections in the phase-space. Also, the transient behaviour concerning the variations of the Nusselt number with time has been investigated. The onset of chaotic motion is also discussed in this paper.