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So far, it is still unknown whether all the closed characteristics on a symmetric compact star-shaped hypersurface \(\Sigma \) in \(\mathbf{R}^{2n}\) are symmetric. In order to understand behaviors of such orbits, in this paper we establish first two new resonance identities for symmetric closed characteristics on symmetric compact star-shaped hypersurface \(\Sigma \) in \(\mathbf{R}^{2n}\) when there exist only finitely many geometrically distinct symmetric closed characteristics on \(\Sigma \), which extend the identity established by Liu and Long (J Differ Equ 255:2952–2980, 2013) of 2013 for symmetric strictly convex hypersurfaces. Then as an application of these identities and the identities established by Liu et al. (J Funct Anal 166:5598–5638, 2014) for all closed characteristics on the same hypersurface, we prove that if there exist exactly two geometrically distinct closed characteristics on a symmetric compact star-shaped hypersuface in \(\mathbf{R}^4\), then both of them must be elliptic.

Resonance identities and stability of symmetric closed characteristics on symmetric compact star-shaped hypersurfaces