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decreasing from front to back is the cost (and ability to pay) for war. Without the financial capacity to make war, threat goes from left to right smoothly at the back of the box as tension gradually increases without the onset of armed conflict. When effective fighting capacity is cheap and/or already well funded, the country well armed, the increases in threat go from left to right at the front edge of the box and encounter the cliff of catastrophe and war is declared. Cost of, or ability to wage war varies from the front to back, and serves as the splitting factor. Considering prison riots, social tension is the normal factor and alienation (degree of identification with prison authority) is the splitting factor.” Using factial expressions of dogs sketched by the Konrad Lorenz, Christopher Zeeman then of Warwick Mathematics Institute in England, considered countenances reflecting increasing rage as the normal factor, the amount of fear was the splitting factor. Increasing rage at high fear increased smoothly at the back of the box; at low fear, increasing rage falls off the cliff to an animal attack at the front of the box.” He paced as he talked, occasionally looking up to see if I was following him. He continued,

“A light above the box casts a shadow from the roof to the floor, outlining the gradually widening fold created by the transition from the smoothly rising back of the roof to its `S-shaped’ front. This triangle on the x-y causal floor is the region in which the discontinuity in the result surface roof results and is called the bifurcation set. An increasing amount of the causal `normal factor’ is represented from left to right along the `x’ dimension, the results of which change smoothly at the back of the roof but encounter a discontinuous jump up or fall down crossing the inaccessible crevice in the `S’ fold at the front of the roof. Again, the triangular shadow on the floor made by the fold indicates the parameter region in which discontinuous changes in the result surface occur. The reason the parameter that determines the front to back location of the left to right movement of the `normal factor’ is called the `splitting factor’ becomes obvious. Its value determines whether the results induced by increasing amounts of `normal factor’ will be smoothly changing or generate a discontinuous jump. The entire visualizable object is called a cusp catastrophe and it along with higher dimensional parameter region-inspired shapes such as the

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