How can Chua's circuit work without having a power supply?
The power comes from Chua's Diode.
It is not a passive device. It actively provides power as you can see quickly if you look at its i-v-curve:


i-v curve of Chua's Diode (image from Wikipedia)

The i-v-curve of Chua's diode occupies the 2nd and 4th quadrant. A passive device would occupy only 1st and 3rd Quadrant:



See section "Types of I–V curves" in Wikipedia article Current–voltage characteristic.