This might work for you... just a quad op-amp and some passives:
https://www.researchgate.net/publication/309351711_A_simple_chaotic_circuit_with_a_light-emitting_diode
Article also has an example implementation. Might also require a bi-polar supply -- haven't read through it completely yet.
The download was free, so I'm attaching the PDF.
I have always been interested in chaos (the mathematical kind, not political!) so I found this article kind of fascinating. I wired up the circuit given in the pdf and actually got it to work! For certain values of "work"...
Using two LM358N dual op-amps, six 10k resistors, three 10nF caps and a bipolar +/- 15VDC supply (from an Analog Devices little power brick boost converter thing), a couple of LEDs in parallel (one red, one white) and careful setting of the power supply brick's supply I am able to get a very nice random flicker in the LEDs... but at low brightness levels. So I added a lowside switch off the anode end of the LEDs to drive a third LED at full brightness. Anode of circuit LED to Gate of IRF3205, 100k pulldown to 0v rail on mosfet Gate, bright LED cathode to Drain, anode of LED to 1k resistor to +15 rail, mosfet Source to 0V rail. Now the flicker in the first LEDs can be dim-to-invisible, while the mosfet is being driven nicely to give a brilliant amplified flicker in the LED. I've had the circuit running all night and it sort of cycles through mostly bright flicker down to mostly dimmer over spans of 5 or 10 minutes. I have not scoped the circuit yet but I can tell that there is a constant oscillation at relatively high frequency (visual high, not RF high!) with the flickering superimposed onto that as moments when the op-amp outputs swing all the way to the rails. Today if I can take a break from mowing lawns I'll break out the scope and see what is going on. I don't know how to draw the cute fractal strange attractors shown in the pdf though.
The circuit is exquisitely sensitive to input voltage but once the input is set so it flickers nicely, it remains stable. As I said I've had it running for hours. I've got it on a nice tight breadboard but I may go ahead and build it down onto a PCB just for grins. The pdf shows it built rather sloppily on a breadboard with 4 obviously unmatched single-chip opamps, so it seems that there is plenty of room left for experimentation. My build is slightly sensitive to stray capacitance from touching one of the caps.
I also tried the circuit with LM339 quad comparator and LM324 quad opamp and could not get it to work with those quad chips.
ETA: the signal driving the original LED is AC, so the LED can glow with either insertion polarity, and also I put a Schottky diode in series with the IRF3205's Gate. Experiment with polarities of the LEDs and this Gate diode to find the best combination.