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298 chapter 12TABLE 12.1Several Nonlinear Maps to ExploreName f(x) Name f(x)Logistic µx(1 − x) Tent µ(1 − 2 |x − 1/2|)Ecology xe µ(1−x) Quartic µ[1 − (2x − 1) 4 ]Gaussian e bx2 + µ12.7 Other Maps (Exploration)Bifurcations and chaos are characteristic properties of nonlinear systems. Yet systemscan be nonlinear in a number of ways. Table 12.1 lists four maps that generatex i sequences containing bifurcations. The tent map derives its nonlinear dependencefrom the absolute value operator, while the logistic map is a subclass of theecology map. Explore the properties of these other maps and note the similaritiesand differences.12.8 Signals of Chaos: Lyapunov Coefficients ⊙The Lyapunov coefficient λ i provides an analytic measure of whether a system ischaotic [Wolf 85, Ram 00, Will 97]. Physically, the coefficient is a measure of thegrowth rate of the solution near an attractor. For 1-D maps there is only one suchcoefficient, whereas in general there is a coefficient for each direction in space. Theessential assumption is that neighboring paths x n near an attractor have an n (ortime) dependence L ∝ exp(λt). Consequently, orbits that have λ>0 diverge andare chaotic; orbits that have λ =0remain marginally stable, while orbits with λ