Sure thing, let’s pull up a map of the resonance zones. First we’ll label the mean‑motion resonances—like 2:1, 3:2, 4:3—and note their locations in semi‑major axis. Then we can plot the width of each resonance using the standard formula Δa ≈ a (μ/3)¹/³, where μ is the planet‑star mass ratio. Once we have those widths, we’ll check for overlap by seeing where Δa₁ + Δa₂ exceeds the distance between the resonances; that’s where chaos sets in. If you want the math laid out, just tell me which system we’re looking at and I’ll crunch the numbers.

Let’s pick TRAPPIST‑1. It has seven Earth‑size planets, many in tight resonances. For each planet we calculate the resonant width using Δa ≈ a (μ/3)¹/³. Because the planets are so close, the widths overlap a lot. The 4:3, 3:2, and 2:1 resonances crowd the inner system, so the chaos border lies just beyond planet f. If we run the formula for each pair, we’ll see that the chaotic zone extends roughly from 0.02 AU out to 0.1 AU, basically covering planets b through e. The math tells us the overlap is enough that a small perturbation can push an orbit into a chaotic path, especially for the innermost bodies. Want me to pull up the exact numbers for each resonance?

That’s a solid run‑through, thanks for the numbers. With those widths the resonance overlap region really does look like a thick “traffic jam” from b to h. A tiny change in the mass ratio or a slight shift in orbital spacing could open up a small corridor of stability or, conversely, tighten the chaos. If you’re curious, we could run a quick N‑body simulation just to see how a 1 % tweak in μ affects the long‑term evolution—no need for fancy math, just a few dozen steps and you’ll see the orbits wobble in and out of the chaotic zone. Let me know if you want to dive into that.