This arrangement has been extensively studied in various forms, and is considered to be a standard arrangement for the testing of 'Flux Capacitance'.

To understand the principals i will set forth, we must first examine the situation where the coils are 'open', no current flowing through the wires. And thus we are dealing with purely the magnetic component of the device. There are a few primary assumptions made. The size and thickness of the fluxpaths are equal. meaning that both rectangular cores are the same. Both magnets are assumed to be of equal intensity (strength). And as noted in this specific set-up, the Bridge cores are smaller (shorter and thinner) than the fluxpath cores.

Now i will define the points noted in the diagram below, the next post will describe the magnetic action between these points.

A0(north) to A0(south) and B0(north) to B0(south) are the each of te magnets, respectively.

Points A1, B1   and A2, B2 are the core-tap points.

the lines (z) in the center, are indicators to arbitrary "poles" that occur within the core material

The lower image, is the Bridge, and area (X) is the viarable flux capacitor that is created through the length of the bridge.

If we add a second magnet on the opposite side of the fluxpath A, we eliminate the arbitrary pole (z) that would appear in its place, and instead we have two points, in the same location as the other two (z) points, but on the opposite side of the Bridges. The functionality of the Bridge-Capacitor remains the same, however the magnetic intensity through the core A is doubled. This shifts point (x) to approx 3/4's of the way across the Bridge, towards core B. (charging)
Removing this magnet, relaxes the compressed flux through core B, and point (x) shifts back towards the center, undergoing a few oscillatory bounces while it balances out. (discharging)

[[ These balancing oscillations are very rapid, and low intensity, not having any 'real' effect in terms of electromagnetic induction. Such induction of these oscillatory forces results in a faster equilibrium, it behaves like friction, and dissipates as heat, not electricity. ]]

Now, lets look at what happens when we replace the second magnet with a Coil.
Now we have, a Variable-Magnet.

we have the ability to adjust the location of the second set of (z) points, thus control the flux intensity through core A. This gives us control over the magnetic intensity (strength) of the second magnet.
The location of the second (z) points should never approach or cross the Bridge. You can go slightly over 2x the strength of the primary magnets, varies depending on the core material and dimensions, permeability, satturation, ect. If the arbitrary poles cross the bridge, the bridge capacitance circuit becomes identicle to the 1-magnet analysis in the post above, using the variable magnet as the source.
In addition, you have a second set of flux-capacitors that form between the Bridge and the permanent magnet. This completely changes the magnetic schematic of the device. As i will show in an upcomming picture.

If you keep the secondary (z) poles on the side of the bridge with the variable magnet, (as indicated in green in this pic below), the capacitance is the result of the bridge-tap. The distance of point (X) from the center of the bridge, inversley indicates the magnetic-potential between the two flux-paths A & B respectively. An increase in magnetic intensity in the variable magnet results in an increase in the flux intensity through core A, caused by the opposing magnetic fields, which shifts the locations of the 4 (z) points towards the Bridges. This increases the flux intensity both north and south at points A1 and A2 respectively.
Point (X) through both bridges, shifts towards core B, and in turn increases the flux intensity through core B. This causes a corresponding shift in the two (z) points near core B's magnet.
The third (z) point in core B does not change in location because it's balanced, however the magnetic intensity (north and south) on either side of the center (z) point increases with the flux increase through the entire core B.

When the variable magnet is turned off, the 2 additional (z) points in core A dissapear and the centered (z) point reappears in the center of the coil-region. (assuming its a balanced coil positioned centered on the z point to begin with)
The flux intensity collapses (drops) and the flux-capacitor discharges through the Bridges. Core B returns to its normal flux intensity, and the two cores balance out to an equilibrium state, with points (X) in the center of the Bridges.

the addition of the Bridge-Coil, in this particular device,
adds a square-component to the Bridge Capacitor.
essentially, another variable magnet, but in this case, the (z) point in the center, is varied with magnetic intensity. This complicates the mathematics of the flux capacitor, but is minor, in terms of changes in capacitance. This second variable magnet drives an imbalanced capacitance circuit through core B. The bridge can no longer be treated as a single (or two identicle series) capacitor.

Instead, what you have is a capacitance between the bridge point B1 and the two (z) points at the top of core B. one bewteen the magnet, and one between the coil.
in series with another capacitance between the south bridge (B2) and bottom two (z) points. I'll call these capacitors CB1 and CB2

the flux capacitance of CB1 is the square of the change in flux intensity between the Bridge Coil, and the Coil on core B.

the flux capacitance of CB2 is the square of the change in flux intensity between the (X) point in the south core
    and the Coil on core B.
These are two entirely different numbers and i'll explain why.

At the top of the bridge, the addition of the bridge magnet acts as a capacitor between the permanent magnet, and the induced magnetic flux from the coil on core B.

At the bottom of the bridge, there is no bridge magnet, so the changes in magnetic flux are opposed by the flux intensity through core A. This causes a shift in the south (X) point, towards core A.
So the capacitance of the south Bridge is greater then that of the north bridge, but the change in flux intensity is sqrt of the change in the north Bridge.
The B coil, in this sense, is acting as a flux diode, passing flux in one direction from north Bridge to south Bridge. The magnetic current cannot flow in the other direction, because the pressure is greater on the north side.

and so the schematic would be shown as two variable flux capacitors of different value, in series, and if operated post-Bridge, the addition of the parallel flux capacitor.

secondary inputs for individual operation of the A coil and the Bridge coil gives further control over the function of the bridge flux capacitor.