# American Psychologist

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1 American Psychologist The Complex Dynamics of Wishful Thinking: The Critical Positivity Ratio Nicholas J. L. Brown, Alan D. Sokal, and Harris L. Friedman Online First Publication, July 15, doi: /a CITATION Brown, N. J. L., Sokal, A. D., & Friedman, H. L. (2013, July 15). The Complex Dynamics of Wishful Thinking: The Critical Positivity Ratio. American Psychologist. Advance online publication. doi: /a

4 is always determined, at least in principle, from the initial conditions, even though it may not be given by any simple explicit formula. Example 2: Consider a population x of some species living in a limited territory, with a maximum sustainable population X max. Then a plausible (though of course extremely oversimplified) model for the growth of this population is given by the differential equation Alan D. Sokal additional variables), and only on the value of x at that same moment of time (i.e., not on the values in the past). (DE5) The rate of change of x at time t is exactly F x t, where F is an explicitly specified function. The mathematical model of the phenomenon under study thus consists of Equation 1 together with the specification of the function F. Example 1: Consider a bank account with continuously compounded interest. Then the amount of money in the account (x) increases with time (t) according to the differential equation dx rx (2) dt where r is the interest rate. This has the form of Equation 1 with F x rx. This same equation describes the cooling of a coffee cup; the decay of radioactive atoms; and a vast number of other physical, biological, and social phenomena. (In some of these applications, r is a negative number.) Equation 2 is an example of a linear differential equation, because the function F x rx is linear (i.e., doubling x causes F x to precisely double). Equation 2 happens to have a simple solution, namely x t x 0 e rt, where x 0 is the account balance at time 0, and e is the base of natural logarithms. This formula illustrates an important general principle: The solution of a differential equation is completely determined by the initial conditions, that is, the value(s) of the dependent variable(s) at time 0. It should be stressed that most differential equations do not have simple solutions that can be explicitly written down; rather, the solutions have to be studied numerically, by computer. Nevertheless, the solution at arbitrary time t dx dt rx 1 x X max (3) where r is some positive number, which has the form of Equation 1 with F x rx 1 x X max. Equation 3 is an example of a nonlinear differential equation, because here the function F is nonlinear (i.e., doubling x does not cause F x to precisely double). Of course, population is not, strictly speaking, a continuous variable, since there is no such thing as a fractional person. This would appear to conflict with principle DE1 above. However, if the population x is large (e.g., millions of people), then only a negligible error is made by treating the population as if it were a continuous variable. Several dependent variables. Let us now consider the case in which we have several dependent variables. We assume once again that both the independent variable t and the dependent variables x 1,...,x n can be treated as continuous quantities, and that x 1,...,x n vary smoothly as a function of t. Then a system of (first-order) differential equations for the functions x 1 t,...,x n t is a system of equations of the form dx 1 dt F 1 x 1,..., x n É dx n dt F n x 1,..., x n where F 1,...,F n are specified functions. The model is constituted by the system of Equation 4 together with the specification of the functions F 1,...,F n. Example 3: Lorenz (1963), building on work by Saltzman (1962), introduced the following system of differential equations as a simplified model of convective flow in fluids: dx X Y d (5a) dy rx Y XZ d (5b) dz bz XY d (5c) Here, the independent variable is a dimensionless variable that is proportional to time t. The dependent variables (4) Month 2013 American Psychologist 3

5 Harris L. Friedman X, Y, and Z are also dimensionless and represent various aspects of the fluid s motion and its temperature gradients. (In the solutions studied by Lorenz, X and Y oscillate between positive and negative values, while Z stays positive.) Finally,, b, and r are positive dimensionless parameters that characterize certain properties of the fluid and its flow; in particular, r measures (roughly speaking) the strength of the tendency to develop convection. For future reference, let us explain what is meant by dimensionless. A quantity is called dimensionful if its numerical value depends on an arbitrary choice of units. For instance, lengths (measured in meters or furlongs or...) are dimensionful, as are times and masses. By contrast, a quantity is called dimensionless if its numerical value does not depend on a choice of units. For instance, the ratio of two lengths is dimensionless, because the units cancel out when forming the ratio; likewise for the ratio of two times or of two masses. Physical laws are best expressed in terms of dimensionless quantities, since these are independent of the choice of units; this explains why Lorenz (1963) rescaled his equations as he did. In particular, Lorenz s variable equals t T, where T is a particular characteristic time in the fluid-flow problem. Nonlinear Dynamics and Chaos The mathematical field of nonlinear dynamics popularly known as chaos theory is founded on the observation that simple equations can under some circumstances have extremely complicated solutions. In particular, fairly simple systems of nonlinear differential equations can exhibit sensitive dependence to initial conditions; that is, small changes in the initial conditions can lead to deviations in the subsequent trajectory that grow exponentially over time. Such systems, while deterministic in principle, can be unpredictable in practice beyond a limited window of time; the behavior can appear random even though it is not. We refer the reader to Lorenz (1993), Stewart (1997), and Williams (1997) for sober nontechnical introductions to nonlinear dynamics, and to Strogatz (1994) and Hilborn (2000) for introductions presuming some background in undergraduate mathematics and physics. The Lorenz system, revisited. The Lorenz system (Equations 5a 5c above) illustrates nicely some of the concepts of nonlinear dynamics. First, this system has a fixed point at X Y Z 0: If the system is started there, it stays there forever. (Physically, this fixed point corresponds to a fluid at rest.) For r 1 it turns out that this fixed point is stable: If the system is started near the fixed point, it will move toward the fixed point as time goes on. For r 1 this fixed point is unstable: If started near the fixed point, the system will move away from it. For r 1 there is another pair of fixed points, at X Y b r 1, Z r 1. (Physically, these fixed points correspond to a steady-state convective flow.) It turns out that these fixed points are stable for r r crit and are unstable for r r crit, where r crit b 3 b 1 and we assume b 1. What happens for r r crit? Lorenz (1963) investigated the trajectories numerically and found that they tend to a butterfly-shaped set now known as the Lorenz attractor. This set is a fractal: It is neither two-dimensional (a surface) nor three-dimensional (a volume) but something inbetween. Moreover, the trajectories near the attractor exhibit sensitive dependence to initial conditions (i.e., they are chaotic). 2 One final remark: Mathematicians studying the Lorenz system typically (though not always) fix and b and vary r. Saltzman (1962) chose the illustrative values 10 and b 8 3; then Lorenz (1963) followed him, as have most (though not all) workers ever since. However, there is nothing magical about these values; indeed, any other values within a fairly wide range would produce qualitatively similar behavior (Sparrow, 1982, pp ). When Can Differential Equations Validly Be Applied? When used properly, differential equations constitute a powerful tool for modeling time-dependent phenomena in the natural and social sciences. However, a number of preconditions must be met if such an application is to be valid. In order to apply differential equations to a specific natural or social system, one must first do the following: (VA1) Identify and define precisely the variables that specify the state of the system at a given moment of time. Per principles DE1 and DE2 (above), these variables must be continuous (or at least approximately so), not discrete, 2 Actually, the Lorenz attractor is born at a value r A slightly less than r crit ; for r A r r crit, both the fixed points and the Lorenz attractor are stable (Sparrow, 1982, pp ). 4 Month 2013 American Psychologist

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