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This transformation is called a scaling collapse since it has the effect of collapsing the curves at different scales of measurement on top of each other into one normalised relationship. An Example of Complex Behaviour and Fractal Time Series in Economics 36 Complexity Theory and Network Centric Warfare The scaling relationship in time may be determined theoretically for Brownian motion. It follows from the mean squared analysis that the mean squared displacement, Δ2 , of the Brownian motion X(t), is described by Δ2 ∝ t .

Thus σ 2 Δt is the variance of the proportional change in stock price, S, during time Δt . The instantaneous variance rate for S is σ 2 S 2 . This implies that S can be represented by an Ito process that has an instantaneous drift rate μS and instantaneous variance rate σ 2 S 2 . This can be written: dS = μ Sdt + σ Sdz or dS = μ dt + σ dz S and ⎛ σ2⎞ ⎟dt + σdz . d ln S = ⎜⎜ μ − 2 ⎟⎠ ⎝ The change in ln S between times t and T is thus normally distributed: ⎡⎛ ⎤ σ2⎞ ⎜ ⎟⎟(T − t ), σ T − t ⎥ ln S T − ln S ≈ φ ⎢⎜ μ − 2 ⎠ ⎣⎝ ⎦ where S T is the stock price at a future time T, S is the stock price at the current time, and φ (m, s) denotes a normal distribution with mean m and standard deviation s.

This phenomenon is known as thermal conduction. In this new state that the system has reached in response to a constraint, stability will prevail again and the behaviour will eventually be as simple as at equilibrium. However, by removing the system from equilibrium further and further, through an increase in ΔT , we observe that suddenly, at a value of ΔT that we will call critical, matter begins to perform a bulk movement. Moreover, this movement is far from random; the fluid is structured in a series of small structures known as Benard cells.

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