We prove that a pair of heterodimensional cycles can be born at the bifurcations of a pair of Shilnikov loops (homoclinic loops to a saddle-focus equilibrium) having a one-dimensional unstable manifold in a volume-hyperbolic flow with a $\mathbb{Z}_2$ symmetry. We also show that these heterodimensional cycles can belong to a chain-transitive attractor of the system along with persistent homoclinic tangency.
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Figure 1. The dashed curves represent the two homoclinic loops and the solid vertical lines represent leaves of the foliation $\mathcal{F}_0$. The coincidence condition for system $X$ is that, for any point of $\Gamma^+$ lying in a leaf $l$, there exists one point of $\Gamma^-$ that also lies in $l$
Figure 3. As shown in figure (a), we can create an infinite sequence of index-2 point $Q_k^-$ accumulating on $M^-$ while keeping the intersection $W^u(P)\cap W^{ss}(M^-)$ by changing $\mu, \rho$ and $\nu$ together. In figure (b), the intersection $W^u(P)\cap W^s(Q_{k_0}^-)$ is created by changing $\nu$)
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