4.3.9 (Alternative derivation of scaling law) For systems close to a saddlenode bifurcation, the scaling law Thomeneck ~ 001/2) can also be derived as follows. a) Suppose thatx has a characteristic scale 00"\"), where a is unknown for now. Then x = ru, where u ~ 0(1). Similarly, suppose I = rbr, with 1' ~ 0(1). Show that)? = r + x2 is we r =r+r2"u2 thereby trans formed to 0'? b) Assume that all terms in the equation have the same order with respect to r, and thereby derive a = is b = 'i. 4.3.10 (Nongeneric scaling laws) In deriving the square-root scaling law for the time spent passing through a bottleneck, we assumed that)? had a quadratic minimum. This is the generic case, but what if the minimum were of higher order? Suppose that the bottleneck is governed byi=r+x2, wheren > 1 is an integer. Using the method of Exercise 4.3.9, Show that Tbottieneck % orb, and determine b and 0. (It's acceptable to leave c in the form of a definite integral. If you know complex variables and residue theory, you should be able to evaluate c exactly by integrating around the boundary of the pie-slice {z = rem : 0 s 9 s n/n, 0 s r s R} and lettingR > 00,)