I re-drew the diagram in an attempt to understand the secondary's operation. Sometimes re-drawing to the simplest layout helps to figure things out. I suppose it doesn't really matter, but I always start with the input on the left, so that the current flow is to the right. (Insert favorite "Australia is upside down joke here.")
Notice that CY1 is physically bigger than the other two caps. Part of this is due, of course, to it's higher voltage rating, but also because, I suspect, it has a larger capacitance. Over time (several charging cycles), this will be hold a bigger surprise for the poor bug. So, the first two caps get fully charged on the first alternation (after turn on or zapping), but, if the third cap is larger, it will take several steps to charge it fully.
Perhaps you could supply us with the missing capacitance ratings of the caps.
Notice how crappy-looking the soldering is; it looks like there might be some cold solder joints in there.
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If you are going to try building some voltage multipliers, be very careful of how much storage you build into to it (capacitance). Even with a puny circuit like this one, over time it can charge up the largest of caps and kick you through the goal posts of life. Using some dry (waxed? varnished?) wooden sticks with metal probes (paper clips?) and a well-insulated resistor and suitably-rated wires (check!), you can build a discharging apparatus inexpensively. It also is a good idea to place a permanent resistance across the output that is high enough not to load down the output, but low enough to discharge the output cap after the power is off for a few seconds. (But see below about voltage ratings of the resistors.)
Another cheap safety device is a NE-2 neon bulb and series resistor across the output that will let you see when it is discharged, or, more importantly, when it is not. These can be salvaged from old appliances such as electric blankets, etc., but you will need to up the resistance to limit the current when operating at higher than line voltage; add additional resistors in series with the original one, because each resistor (1/2 or 1/4 watt) should not be subjected to more than 500 volts or so. So, if you were getting really frisky and putting out 5KV, you would need at least 10 resistors in a string for the voltage rating, but the more the better. For resistance value, use Ohm's law to calculate the current drawn when the voltage across the resistor is line/mains minus 65 (the lamp voltage). Then use that current to calculate the needed resistance for the output voltage minus 65. See
http://www.giangrandi.ch/electronics/neon/glowui.html for more about this lamp. (Note how they are using an R as a shunt to read the lamp current with a scope.)
Finally, make sure you have a signed will on file and read
this.
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With regard to your lack of a scope, try getting in touch with the amateur radio operators in your area. Someone might have an old (tube?) scope squirrelled away that is "too good to throw away, but too old to use." Even something like an old kit 1MHz scope would be fine for this work with the proper voltage-rated probes. (Watch Dave's recent video on scope grounding first!)
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PS: Would somebody please tell me why my drawing came out so small (before clicking it) as compared to the others? It was of a nice enough size when created in MS Paint and would be a pain if I had to make it say, ten times bigger. Because I can't see the data of your photos, I have to assume you can't see the data for my drawing, so here it is: 33.61 kB, 519x432. What was the data for your schematics?