Brinkmann [1] has shown that conformally related distinct Ricci flat solutions are pp-waves. Brinkmann’s result has been generalized to include the conformally invariant source terms. It has been shown that [4] if g_ik and $$\overline g $$_ik (=w⁻²g_ik; w: a scalar function), are distinct metrics having the same Einstein tensor, G_ik=$$\overline G $$_ik, then both represent (generalized) pp-waves and w_i is a null covariantly constant vector of g_ik. Thus pp-waves are the only candidates which yield conformally related nontrivial solutions of G_ik=T_ik=$$\overline G $$_ik, with T_ik being conformally invariant source.

In this paper the functional form of the conformal factor for the conformally related pp-waves/generalized pp-waves has been obtained. It has been shown that the most general pp-wave, conformally related to ds²=−2du[dv−mdy=Hdu]+P⁻²[dy²+dz²], turns out to be (au+b)⁻² ds², where a,b are constants. Only in the special case when m=0, H=1, and P=P(y, z), the conformal factor is (au+b)⁻² or (a(u+v)+b)⁻².

In the present paper the effect of delay on chaos in fractional-order Chen system is investigated. It is observed that inclusion of delay changes chaotic behaviour to limit cycles or stable systems.

Recently, the discrete fractional calculus (DFC) is receiving attention due to its potential applications in the mathematical modelling of real-world phenomena with memory effects. In the present paper, the chaotic behaviour of fractional difference equations for the tent map, Gauss map and 2x(mod 1) map are studied numerically. We analyse the chaotic behaviour of these fractional difference equations and compare them with their integer counterparts. It is observed that fractional difference equations for the Gauss and tent maps are more stable compared to their integer-order version.