There might be some way to devise an analog computer, although it's not clear to me how you would implement arithmetic operations.
Operational amplifiers can be configured to generate analogous orbits by using them to differentiate one of their inputs. By wiring multiple op-amps together and measuring voltages at different points it's possible to observe chaotic behavior in the measurements, just as if you were to directly differentiate from some initial condition on paper using a differential equation. This configuration is an analog computer. In fact, I wouldn't be surprised if op-amp circuits existed with perfectly analogous mathematical behavior to the orbits described in the paper.
The problem with gravitational computing using n bodies probably lies with establishing initial conditions, and with inhibiting the effects of neighboring systems.