A Pink noise filter is many half octave LPF's cascaded for each decade required (many)
A Brown noise filter is a single Octave 1st order LP filter or an Integrator, which has twice the slope of Pink noise and easy, but not what you want.
I prefer the digital approach of a maximal length pseudo random sequence generator for white noise.
Chaos is not always pink, although I have generated chaos in SMD SMPS buck to boost regulator coils by accident and the tiny SMD coil sounded like running water in the lab.
They used to make ML-PRSG chips but had a 2 second cycle for audio and you need a wider range spanning 5 decades ! or 50 dB in SNR of noise power as signal.
The 1/2 of the lowest frequency will be the length of the shift register (consecutive 1 or 0's) and clock rate or fmin=1/2* fc/N. for N series shift register. Harmonics of pulses are dependant on duty cycle , D and 1/N in the special case of 50% D with odd harmonics only.
So for extra margin and maximally flat spectrum per octave, let's choose a big sequence
These are special prime numbers for XOR feedback. There is only one illegal state either all 1's or 0's if using reset, so an inverted input if there is an even number of XOR inputs or no inverter if using an odd number of FB inputs.
let's use for n outputs from a 32 bit shift register n= 1 to 32, use 2,3,5,7,32 or if you prefer Q0-Q31, n=1,2,4,6,31 with the equivalent of a 5 input XOR gate to D input, thus after reset, D=1 and next Q0 goes from 0 to 1 and the random sequence begins with up to 32 consecutive 0's and 31 comsecutive 1's and a random spread in between. Output can be from the Q0 or any output.
Here is a 16 bit example, which only starts if preset to all 1's. you will need 4 XOR gates and reset to 0's. you can stop any time by gating the clock at random to get a random start, when enabled, or get a predictable sequence start with power on or manual reset.
**broken link removed** that duplicated my experience except only 4 decades with 4 stage filter, you may want a 5 stage filter.
Each filter is -3dB at breakpoint instead of -6dB thus called "half-order linear filter." For White to pink filtering.
The ML-PRSG 32 bit I described fits in box 32bit PRSG for pseudo random sequence generator.
The digital way is most stable in terms of spectral density over time, temperature.
The analog zener diode method is most linear but needs heat regulation for stable input and high gain.
Both need Pink noise filters. Your choice.
If you are wanting to discriminate controlled chaos with biological chaos, consider two as S & N and decide what is more important. High SNR? or smoothest slope on 3dB/decade of noise distribution or ease of controlling minimum frequency by clock rate.
You can also use this
fourier web synthesizer to create any wave or random noise and see spectrum and hear it. Then shift the frequencies lower in software like Audacity, if you have a DC response amplifier.