\begin{document}$ e^+e^- $\end{document} annihilation data corresponding to an integrated luminosity of 2.93 \begin{document}$ \rm fb^{-1} $\end{document} taken at the center-of-mass energy \begin{document}$ \sqrt{s}=3.773 $\end{document} GeV with the BESIII detector, a joint amplitude analysis is performed on the decays \begin{document}$ D^0\to\pi^+\pi^-\pi^+\pi^- $\end{document} and \begin{document}$ D^0\to\pi^+\pi^-\pi^0\pi^0 $\end{document}(non-η). The fit fractions of individual components are obtained, and large interferences among the dominant components of the decays \begin{document}$ D^{0}\to a_{1}(1260)\pi $\end{document}, \begin{document}$ D^{0}\to\pi(1300)\pi $\end{document}, \begin{document}$ D^{0}\to\rho(770)\rho(770) $\end{document}, and \begin{document}$ D^{0}\to2(\pi\pi)_{S} $\end{document} are observed in both channels. With the obtained amplitude model, the \begin{document}$ CP $\end{document}-even fractions of \begin{document}$ D^0\to \pi^+\pi^-\pi^+\pi^- $\end{document} and \begin{document}$ D^0\to\pi^+\pi^-\pi^0\pi^0 $\end{document}(non-η) are determined to be \begin{document}$ (75.2\; \pm\; 1.1_{\rm stat.}\; \pm\; 1.5_{\rm syst.})\ $\end{document}% and \begin{document}$(68.9\; \pm\; 1.5_{\rm stat.}\; \pm\; 2.4_{\rm syst.})$\end{document}%, respectively. The branching fractions of \begin{document}$ D^0\to \pi^+\pi^-\pi^+\pi^- $\end{document} and \begin{document}$ D^0\to\pi^+\pi^-\pi^0\pi^0 $\end{document}(non-η) are measured to be \begin{document}$(0.688\; \pm\; 0.010_{\rm stat.}\; \pm\; 0.010_{\rm syst.})$\end{document}% and \begin{document}$(0.951\; \pm\; 0.025_{\rm stat.}\; \pm\; 0.021_{\rm syst.})$\end{document}%, respectively. The amplitude analysis provides an important model for the binning strategy in measuring the strong phase parameters of \begin{document}$ D^0 \to 4\pi $\end{document} when used to determine the CKM angle \begin{document}$ \gamma (\phi_{3}) $\end{document} via the \begin{document}$ B^{-}\to D K^{-} $\end{document} decay."> Amplitude analysis of the decays <inline-formula><tex-math id="M1">\begin{document}${\boldsymbol D^{\bf 0}\bf\to\pi^+\pi^-\pi^+\pi^-} $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/f2ef68bb-a513-43d1-821f-ff75bdc93831/CPC-2023-0533_M1.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/f2ef68bb-a513-43d1-821f-ff75bdc93831/CPC-2023-0533_M1.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}${\boldsymbol D^{\bf 0}\bf\to\pi^+\pi^-\pi^0\pi^0} $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/f2ef68bb-a513-43d1-821f-ff75bdc93831/CPC-2023-0533_M2.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/f2ef68bb-a513-43d1-821f-ff75bdc93831/CPC-2023-0533_M2.png"/></alternatives></inline-formula> -
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