Geoscience Reference
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Fig. 12.9 The Lorenz attractor as revealed by the never-repeating trajectory of a single chaotic
orbit (from Motter and Campbell 2013 , Fig. 2). The spheres represent iterations of the Lorenz
equations, calculated using the parameters originally used by Lorenz ( 1963 ). Spheres are colored
according to iteration count. The two lobes of the attractor resemble a butterfly, a coincidence that
helped earn sensitive dependence on initial conditions. Hence, its nickname: “the butterfly effect”
illustrating some aspects of chaotic behavior is the logistic equation that can be
written as dx / dt
x ) where x and t are non-dimensional variables for popu-
lation size and time, respectively. There are no parameters in this equation because
characteristic time and representative population size were defined and used.
The solution of this logistic equation has fixed points at x
¼
x (1
¼
0 and 1, respectively.
The fixed point at x
0 is unstable in that solutions in its immediate vicinity diverge
away from it. On the other hand, the fixed point at x
¼
1 has solutions in its
immediate vicinity that are stable. Introducing, the new variable, x 1 ¼
¼
x
1, and
x 10 e t
where x 10 is assumed to be small but constant. All such solutions “flow” toward
x
neglecting the quadratic term, the logistic equation has the solution x 1 ¼
1 in time. They are not chaotic.
Chaotic solutions evolve in time with exponential sensitivity to the initial condi-
tions. The so-called logistic map arises from the recursive relation x k +1 ¼
¼
a
x k (1
x k )
with iterations for k
. May ( 1976 ) found that the resulting iterations have a
remarkable range of behavior depending on the value of the constant a that is chosen.
Turcotte ( 1997 , Sect. 10.1) discusses in detail that there now are two fixed points at
x
¼
0, 1,
...
0and1- a 1 , respectively. The fixed point at x
¼
¼
0 is stable for 0
<
a
<
1and
unstable for a
>
1. The other fixed point is unstable for 0
<
a
<
1, stable for 1
<
a
<
3,
and unstable for a
¼3 a so-called flip bifurcation occurs. Both singular points
are unstable and the iteration converges on an oscillating limit cycle. At a
> 3. At a
3.449479,
another flip bifurcation occurs and there suddenly are four limit cycles. Writing a 1 ¼
¼
3
and a 2 ¼
) define intervals
with 2 a i limit cycles that satisfy the iterative relation a i+1 - a i ¼
3.449459, it turns out that the constants a i ( i
¼
1, 2,
...
,
1
F 1
( a i
a i1 )
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