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We present an analysis on the behaviour of the Galactic bulge and the LargeMagellanic Cloud (LMC) $\delta$ Scuti stars in terms of period-colour andamplitude-colour (PCAC) relations at maximum, mean and minimum light. Thepublicly available Optical Gravitational Lensing Experiment-IV (OGLE-IV) lightcurves for Galactic bulge and OGLE-III light curves for LMC $\delta$ Scutistars are exploited for the analysis. It has been found that the Galactic bulge$\delta$ Scuti stars obey flat PC relations at maximum/mean/minimum light whilethe LMC $\delta$ Scutis have sloped/sloped/flat PC relations atmaximum/mean/minimum light. Both the Galactic bulge and the LMC $\delta$ Scutishave sloped/flat/sloped AC relations at maximum/mean/minimum. These relationsalso show that Galactic $\delta$ Scutis are hotter as compared to their LMCcounterparts. The period-amplitude (PA) relations for $\delta$ Scutis exhibitdifferent behaviour in the Galactic bulge and the LMC. The LMC variables arefound to have higher amplitudes at a given period. The amplitude of theGalactic bulge $\delta$ Scuti shows a bimodal distribution which can bemodelled using a two-component Gaussian Mixture Model: one component with alower amplitude and another with a higher amplitude. The observed behaviour ofthe $\delta$ Scuti PCAC relations can be explained using the theory of theinteraction of hydrogen ionization front (HIF) and stellar photosphere as wellas the PA diagram. We use MESA-RSP to calculate theoretical non-linearhydrodynamical pulsation models for $\delta$ Scuti stars with inputmetallicities of $Z=0.02$ and $Z=0.008$ appropriate for the Galactic bulge andLMC, respectively. The observed PCAC relations and theoretical calculationssupport the HIF-photosphere interaction theory.

Period-Colour and Amplitude-Colour relations for OGLE-$\delta$ Scuti stars in the Galactic Bulge and LMC