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Using $e^+e^-$ annihilation data corresponding to an integrated luminosity of2.93 $\rm fb^{-1}$ taken at the center-of-mass energy $\sqrt{s}=3.773$~GeV withthe BESIII detector, a joint amplitude analysis is performed on the decays$D^0\to\pi^+\pi^-\pi^+\pi^-$ and $D^0\to\pi^+\pi^-\pi^0\pi^0$(non-$\eta$). Thefit fractions of individual components are obtained, and large interferencesamong the dominant components of $D^{0}\to a_{1}(1260)\pi$,$D^{0}\to\pi(1300)\pi$, $D^{0}\to\rho(770)\rho(770)$ and$D^{0}\to2(\pi\pi)_{S}$ are found in both channels. With the obtained amplitudemodel, the $CP$-even fractions of $D^0\to \pi^+\pi^-\pi^+\pi^-$ and$D^0\to\pi^+\pi^-\pi^0\pi^0$(non-$\eta$) are determined to be $(75.2\pm1.1_{\rmstat.}\pm1.5_{\rm syst.})\%$ and $(68.9\pm1.5_{\rm stat.}\pm 2.4_{\rmsyst.})\%$, respectively. The branching fractions of $D^0\to\pi^+\pi^-\pi^+\pi^-$ and $D^0\to\pi^+\pi^-\pi^0\pi^0$(non-$\eta$) are measuredto be $(0.688\pm0.010_{\rm stat.}\pm 0.010_{\rm syst.})\%$ and$(0.951\pm0.025_{\rm stat.}\pm 0.021_{\rm syst.})\%$, respectively. Theamplitude analysis provides an important model for binning strategy in themeasurements of the strong phase parameters of $D^0 \to 4\pi$ when used todetermine the CKM angle $\gamma (\phi_{3})$ via the $B^{-}\to D K^{-}$ decay.

Amplitude Analysis of $D^0\to\pi^+\pi^-\pi^+\pi^-$ and $\pi^+\pi^-\pi^0\pi^0$