How to catch a instability in an ODE using Julia?

You set kappa = 2.2e7, which is a Float64 value (53 binary digits, internally). You can display it with any number of digits you want, using e.g the Printf standard library:

julia> using Printf

julia> @printf("%.2g", kappa)
2.2e+07
julia> @printf("%.1g", kappa)
2e+07

See also