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The gravitational interaction between two objects on similar orbits caneffect noticeable changes in the orbital evolution even if the ratio of theirmasses to that of the central body is vanishingly small. Christou (2005)observed an occasional resonant lock in the differential node $\Delta \Omega$between two members in the Himalia irregular satellite group of Jupiter in the$N$-body simulations (corresponding mass ratio $\sim 10^{-9}$). Using asemianalytical approach, we have reproduced this phenomenon. We alsodemonstrate the existence of two additional types of resonance, involving angledifferences $\Delta\omega$ and $\Delta (\Omega+\varpi)$ between two groupmembers. These resonances cause secular oscillations in eccentricity and/orinclination on timescales $\sim$ 1 Myr. We locate these resonances in $(a,e,i)$space and analyse their topological structure. In subsequent $N$-bodysimulations, we confirm these three resonances and find a fourth one involving$\Delta \varpi$. In addition, we study the occurrence rates and the stabilityof the four resonances from a statistical perspective by integrating 1000 testparticles for 100 Myr. We find $\sim 10-30$ librators for each of theresonances. Particularly, the nodal resonance found by Christou is the moststable: 2 particles are observed to stay in libration for the entireintegration.

Secular resonances between bodies on close orbits: a case study of the Himalia prograde group of jovian irregular satellites