
Egwald Mathematics: Nonlinear Dynamics:
The Logistic Map and Chaos
by
Elmer G. Wiens
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introduction  logistic map  fixed points  linearized stability analysis  bifurcation analysis transcritical bifurcation  period doubling bifurcation  second order map  second order bifurcation  fourth order map period doubling cascade  logistic bifurcation diagram  third order map  f^{3} saddlenode bifurcation f^{3} flip bifurcations  f^{3} period doubling cascade  interactive logistic map diagrams  definitions of chaos  references
The Logistic Map and Chaos: Introduction
Introduction
One can use the onedimensional, quadratic, logistic map to demonstrate complex, dynamic phenomena that also occur in chaos theory and higher dimensional discrete time systems.
The Logistic Map
The logistic interative map with parameter r is:
x_{t+1} = f(x_{t}, r) = r * x_{t} * (1 + x_{t}), x_{0} = x0 >= 0. (1)
For r values in excess of 3.57, the orbits x(t, x0) = {x0, x_{1}, x_{2}, ... } depend crucially on the initial condition x0. Slight variations in x0 result in dramatically different orbits, an important characteristic of chaos.
Fixed Points
Fixed points of f satisfy:
f(x, r) = r * x * (1  x) = x. (2)
Thus the fixed points of f are the roots of the quadratic equation:
r * x ^{2}  (r  1) * x = x * (r * x  (r  1)) = 0, (3)
which are:
x_{1} = 0,
x_{2} = (r  1) / r. (4)
The fixed point x_{2} is nonnegative if r >= 1.
Linearized Stability Analysis
One can analyze the local stability of the difference equation (1) by examining the partial derivative of f with respect to x evaluated at each fixed point x^{*}:
λ = f_{x}(x^{*}, r) = r * (1  2 * x^{*}). (5)
where λ is called the multiplier or eigenvalue. Substituting x_{1} and x_{2} into (5) yields:
λ_{1} = r * (1  2 * x_{1}) = r,
λ_{2} = r * (1  2 * x_{2}) = r * (1  2 * (r  1) / r)) = 2  r.
If r > 1 → λ_{1} > 1 → x_{1} = 0 is unstable (repeller);
if 1 < r < 3 → 1 > λ_{2} > 1 → x_{2} = (r  1) / r is stable (attractor).
Bifurcation Analysis
When a change in a parameter results in a qualitative change in the dynamics of a nonlinear process, the process is said to have gone through a bifurcation.
When r = 1 → λ_{1} = λ_{2} = 1 & x_{1} = x_{2} = 0 → (x^{*}, r_{c}) = (0 , 1) is a nonhyperbolic fixed point. In fact, it is a transcritical bifurcation point of the mapping f.
Check the transcritical bifurcation conditions at (x^{*}, r_{c}) = (0, 1)
(TC1) f(0, r_{c}) = 0 for all r
(TC2) f_{x}(0, r_{c}) = λ = 1 → (0, r_{c}) is nonhyperbolic
(TC3) f_{x,r}(x, r) = 1  2 * x → a = f_{x,r}(0, r_{c}) = 1 ≠ 0
(TC4) f_{x,x}(x, r) = 2 * r → b = (1/2) * f_{x,x}(0, r_{c}) = 2 ≠ 0
 (6) 
The transcritical bifurcation conditions are confirmed.
Transcritical Bifurcation: r = 1
The following diagrams display the phase diagrams and solution trajectories of x(t, x0) for the discrete time, nonlinear dynamic process. At r = 1, the process undergoes a transcritical bifurcation with x_{1} switching from attractor to repeller. Meanwhile, the fixed point x_{2} turns positive and becomes an attractor, as can be seen in the diagrams for r = 2.
r = 1 x_{1} = 0 attractor; transcritical bifurcation



r = 2 x_{1} = 0 repeller, x_{2} = 1 /2 attractor



Period Doubling Bifurcation: r = 3
When r = 3 → λ_{2} = 1 & x_{2} = 2 / 3 → (x^{*}, r_{c}) = (2 / 3 , 3) is a nonhyperbolic fixed point. In fact, it is a period doubling (supercritical flip) bifurcation point of the mapping f.
Check the period doubling bifurcation conditions at (x^{*}, r_{c}) = (2 / 3, 3)
f(x, r) = r * x * (1  x)
f_{x}(x, r) = r * (1  2 * x)
f_{x,x}(x, r) = 2 * r

f_{r}(x, r) = x * (1  x)
f_{x,r}(x,r) = (1  2 * x)
f_{x,x,x}(x,r) = 0

(F1) f(x_{2}, r_{c})  x_{2} = 3 * 2 / 3 * (1  (2 / 3))  2 / 3 = 0,
(F2) f_{x}(x_{2}, r_{c}) = 3 * (1  2 * 2 / 3) = 1,
(F3) f_{r} * f_{x,x} + 2 * f_{x,r} = (2 / 9) * (6) + 2 * (1 / 3) = 2 ≠ 0,
(F4) c = 2 * f_{x,x,x}  3 * (f_{x,x})^{2} = 2 * 0  3 * (6)^{2} =  108 ≠ 0.

The period doubling bifurcation conditions are confirmed.
Since c < 0 in expression F4, the fixed point x_{2} flips from being an attractor to being a repeller. Meanwhile, two stable fixed points of the secondorder map f^{2} emerge that bracket the unstable x_{2}. The trajectory x_{t} switches between the two attracting fixed points of f^{2}.
The following diagrams display the phase diagrams and solution trajectories of x(t, x0) for the discrete time, nonlinear dynamic process. At r = 3, the f map undergoes a period doubling (flip) bifurcation with x_{2} switching from attractor to repeller. Meanwhile, two stable fixed points x_{3} and x_{4} of the second order map, f^{2} emerge.
r = 2.8 x_{1} = 0 repeller, x_{2} = 0.64 attractor;



r = 3 x_{1} = 0 repeller, x_{2} = 2 / 3 attractor; flip bifurcation



r = 3.3 x_{1} = 0 repeller, x_{2} = 0.7 repeller



The Second Order Map of f (f^{2})
Associated with the logistic map:
x_{t+1} = f(x_{t}, r) = r * x_{t} * (1 + x_{t}), x_{0} = x0 >= 0.
is the second order map f^{2} given by:
x_{t+2} = f(x_{t+1}, r) = f (f(x_{t}, r), r ) = r * (r * x_{t} * (1 + x_{t})) * (1 + (r * x_{t} (1 + x_{t}))), or
x_{t+2} = f^{2}(x_{t}, r) =
r^{3} * x_{t}^{4} + 2 * r^{3} * x_{t}^{3}  (r^{3} + r^{2}) * x_{t}^{2} + r^{2} * x_{t}
(7)
The partial derivative of f^{2} with respect to x is:
f_{x}^{2}(x, r) = 4 * r^{3} * x^{3} + 6 * r^{3} * x^{2}  2 * (r^{3} + r^{2}) * x + r^{2}. (8)
Fixed points of (7) satisfy:
f^{2}(x, r) =
r^{3} * x^{4} + 2 * r^{3} * x^{3}  (r^{3} + r^{2}) * x^{2} + r^{2} * x = x. (9)
The fixed points of f^{2} are:
x_{1} = 0,
x_{2} = (r  1) / r,
x_{3} = (1 + r + sqrt(r^{2}  2 * r  3)) / (2 * r),
x_{4} = (1 + r  sqrt(r^{2}  2 * r  3)) / (2 * r),

whose multipliers are:
λ_{i}^{2} = f_{x}^{2}(x_{i}, r).
The first two fixed points of f^{2}are also fixed points of f.
The following graphs display the phase diagrams of f in blue and f^{2} in red. The stable fixed points x_{3} and x_{4} of f^{2} emerge as r increases past 3. At r = 3, f^{2} has a fixed point of multiplicity three (ie x_{2} = x_{3} = x_{4} = 2 / 3). For r > 3, x_{3} and x_{4} bracket x_{2}, and establish the period2 cycle seen in the above trajectory diagram for r = 3.3. Eventually, these period2 fixed points become unstable, and undergo flip bifurcations with respect to f^{4}, the period doubling map of f^{2}.
r = 2.8

r = 3 flip bifurcation

r = 3.3




Second Order (f^{2}) Flip Bifurcation : r = 1 + sqrt(6)
When r_{c} = 1 + sqrt(6) →
x_{3} = 0.85 & x_{4} = 0.44 and λ^{2}_{3} = 1 & λ^{2}_{4} = 1 → x_{3} & x_{4} nonhyperbolic fixed points of f^{2}. In fact, they are period doubling (supercritical flip) bifurcation points of the mapping f^{2}.
Check the period doubling bifurcation conditions at (x_{3,4}, r_{c}) for f^{2}:
f^{2}(x, r) = r^{3} * x^{4} + 2 * r^{3} * x^{3} + (r^{3}  r^{2}) * x^{2} + r^{2} * x
f^{2}_{x}(x, r) =4 * r^{3} * x^{3} + 6 * r^{3} * x^{2} + 2 * (r^{3}  r^{2}) * x + r^{2}
f^{2}_{x,x}(x, r) = 12 * r^{3} * x^{2} + 12 * r^{3} * x  2 * r^{3}  2 * r^{2}

f^{2}_{r}(x, r) = 3 * r^{2} * x^{4} + 6 * r^{2} * x^{3} + (3 * r^{2}  2 * r) * x^{2} + 2 * r * x
f^{2}_{x,r}(x,r) = 12 * r^{2} * x^{3} + 18 * x^{2} * r^{2} + 2 * (3 * r^{2}  2 * r) * x + 2 *r
f^{2}_{x,x,x}(x,r) = 24 * r^{3} * x + 12 * r^{3}

(F1) f^{2}(x_{3}, r_{c})  x_{3} = 0.85  0.85 = 0,
(F2) f^{2}_{x}(x_{3}, r_{c}) = 1,
(F3) f^{2}_{r} * f^{2}_{x,x} + 2 * f^{2}_{x,r} = (0.2992) * (43.0679) + 2 * (1.5445) = 9.7980 ≠ 0,
(F4) c = 2 * f^{2}_{x,x,x}  3 * (f^{2}_{x,x})^{2} = 2 * (344.7201)  3 * (43.0679)^{2} =  4875.1 ≠ 0.

(F1) f^{2}(x_{4}, r_{c})  x_{4} = 0.44  0.44 = 0,
(F2) f^{2}_{x}(x_{4}, r_{c}) = 1,
(F3) f^{2}_{r} * f^{2}_{x,x} + 2 * f^{2}_{x,r} = ( 0.4673) * (15.4719) + 2 * ( 1.2839) = 9.7980 ≠ 0,
(F4) c = 2 * f^{2}_{x,x,x}  3 * (f^{2}_{x,x})^{2} = 2 * (59.1446)  3 * ( 15.4719)^{2} = 836.43 ≠ 0.

The period doubling bifurcation conditions are confirmed.
Since c < 0 in the F4 expressions, the fixed points x_{3} and x_{4} flip from attractors to repellers. Meanwhile, four stable fixed points of the fourthorder map f^{4} emerge bracketing the unstable x_{3} and x_{4}.
The following diagrams display the phase diagrams and solution trajectories of x(t, x0) for the discrete time, nonlinear dynamic process. At r = 1 +sqrt(6) = 3.449489742, f^{2} undergoes a period doubling (flip) bifurcation with x_{3} and x_{4} switching from attractors to repellers. The trajectory x_{t} switches between the four attracting fixed points of f^{4}, creating a stable fourcycle.
r = 3.4 x_{3} = 0.84 attractor, x_{4} = 0.45 attractor;



r = 1 + sqrt(6) x_{3} = 0.85 attractor, x_{4} = 0.44 attractor; flip bifurcation



r = 3.5 x_{3} = 0.86 repeller, x_{4} = 0.43 repeller



The Fourth Order Map of f (f^{4})
Associated with the logistic map:
x_{t+1} = f(x_{t}, r) = r * x_{t} * (1 + x_{t}), x_{0} = x0 >= 0.
is the fourth order map f^{4} given by:
x_{t+4} = f^{4}(x_{t}, r) = f ( f ( f ( f (x_{t}, r), r ), r ), r ) = f^{2} ( f^{2} (x_{t}, r), r ) ),
a polynomial in x of degree 16. Consequently, the 4th order map f^{4} has 16 fixed points, most of which will be complex, except the four fixed points f^{4} shares with f^{2}, and the 4 fixed points that emerge at r = 1 + sqrt(6), bracketing x_{3} and x_{4}.
The following graphs display the phase diagrams of f in blue, f^{2} in red, and f^{4} in black.
After the f^{2} period doubling bifurcation, four stable fixed points of the fourth order map, f^{4} emerge, x_{5} and x_{6} bracketing x_{3}, and x_{7} and x_{8} bracketing x_{4}.
r = 3.4

r = 1 + sqrt(6) f^{2} flip bifurcation



r = 3.5

r = 3.54409 f^{4} flip bifurcation



The bifurcation diagram below summarizes the fixed points and bifurcations of f, f^{2}, and f^{4}. The f map experiences a transcritical bifurcation at r = 1, and a period doubling bifurcation at r = 3. In turn, the f^{2} map undergoes a period doubling bifurcation at r = 1 + sqrt(6), while the f^{4} map undergoes a period doubling bifurcation a r = 3.54409. Stable fixed points of f, f^{2}, and f^{4} are in blue, while unstable fixed points are in red.
Period Doubling Cascade
As r increases, period doublings occur as f, f^{2}, f^{4}, f^{8}, . . . bifurcate at r_{1} = 3, r_{2} = 3.449, r_{3} = 3.54409, r_{4} = 3.5644, . . .. These {2^{n}} cycles of {f^{2n}} and {r_{n}} sequences follow the Feigenbaum rule:
δ = (r_{n}  r_{n1}) / (r_{n+1}  r_{n}) → 4.6692 as n → ∞.
In the limit as n → ∞, r_{n} → r^{*} = 3.570.
Logistic Map Bifurcation Diagram
The bifurcation diagram shows the set of stable fixed points, {x^{*}(r)}, as a function of the parameter r for the logistics map:
x_{t+1} = f(x_{t}, r) = r * x_{t} * (1 + x_{t}), x_{0} = x0 >= 0. (10)
For 1 < r < r^{*}, the period doubling cascade of the sequence of maps {f^{2n}} determines the attracting fixed points.
For r^{*} < r <= 4, the band of stable fixed points expands to cover the entire vertical [0, 1] interval at r = 4. For many values of r in the horizontal [r^{*}, 4] interval, the orbit:
{x_{t+1} = f(x_{t} ,r), x_{0} = x0}
exhibits aperiodic — chaotic — behaviour . Also for r in this [r^{*}, 4] interval, periodic windows appear. The widest window at r = p3 = 1 + sqrt(8) is the period 3 window. As r increases from r^{*} to p3, period 6 and period 5 windows also appear.
Depending on the value of the parameter r, orbits of the logistic map appear orderly or chaotic.
The Third Order Map of f, f^{3}
For r < p3, the f^{3} map has two unstable fixed points it shares with the f map. At r = p3 = 1 + sqrt(8), three fixed points emerge, where the graph of the f^{3} map has 3 points of tangency with the y = x line. Here, the f^{3} map undergoes a triple saddlenode bifurcation.
r = 3.8 x_{1} = 0, x_{2} = 0.7368 repellers



r = p3 = 1 + sqrt(8) = 3.8284 saddlenode bifurcation x_{1} = 0, x_{2} = 0.7388 repellers
x_{3} = 0.15999288, x_{4} = 0.51435528 x_{5} = 0.9563178 attractors



SaddleNode Bifurcation of f^{3}: r = 1 + sqrt(8) = 3.8284
The following table verifies the triple saddlenode bifurcation of the f^{3} map at the fixed points x_{3} = 0.15999288, x_{4} = 0.51435528, and x_{5} = 0.9563178.

x_{3}

x_{4}

x_{5}

(SN1) f(x^{*}, r_{c})  x^{*}= 0 → (x^{*}, r_{c}) is a fixed point

0

0

0

(SN2) f_{x}(x^{*}, r_{c}) = 1 → (x^{*}, r_{c}) is nonhyperbolic

1

1

1

(SN3) a = f_{r}(x^{*}, r_{c}) ≠ 0

0.78

2.0

0.223

(SN4) b = (1/2) * f_{x,x}(x^{*}, r_{c}) ≠ 0

88.9

34

310.6

The f^{3} map undergoes Type Three (a < 0, b > 0) saddlenode bifurcations at x_{3} and x_{4}, and a Type Two (a > 0, b < 0) saddlenode bifurcation at x_{4}.
One can see the effects of these bifurcations in the diagram below. At r = 3.839, x_{3}, x_{4}, and x_{5} have each split into a pair of fixed points, one stable and the other unstable.
r = 3.839 x_{1} = 0, x_{2} = 0.7395 repellers
x_{3} = 0.1498883, x_{4} = 0.48917231 x_{5} = 0.95929992 attractors
x_{6} = 0.16904010, x_{7} = 0.53924723, x_{8} = 0.95383662 repellers



Third Order Flip Bifurcation : r = 3.8414991
At r = 3.8414991, x_{3}, x_{4}, and x_{5} flip from attractors to repellers. The f^{3} map undergoes a flip bifurcation. Pairs of stable fixed points of the f^{9} emerge bracketing x_{3}, x_{4}, and x_{5}.
r = 3.8414991 x_{1} = 0, x_{2} = 0.7397 repellers
x_{3} = 0.14872043, x_{4} = 0.48634402 x_{5} = 0.953596008 attractors
triple period doubling bifurcation
x_{6} = 0.169988878, x_{7} = 0.54200732, x_{8} = 0.95965839 repellers



When r_{c} = 3.8414991 →
x_{3} = 0.14872043, x_{4} = 0.48634402 x_{5} = 0.953596008 and λ^{3}_{3,4,5} = 1 → x_{3,4,5} are nonhyperbolic fixed points. In fact, they are period doubling (supercritical flip) bifurcation points of the mapping f^{3}.
Check the period doubling bifurcation conditions at (x_{3,4,5}, r_{c}) for f^{3}:
(F1) f^{3}(x_{3}, r_{c})  x_{3} = 0.1487  0.1487 = 0,
(F2) f^{3}_{x}(x_{3}, r_{c}) = 1,
(F3) f^{3}_{r} * f^{3}_{x,x} + 2 * f^{3}_{x,r} = (0.89) * (199.9) + 2 * ( 7.9) = 162.13 ≠ 0,
(F4) c = 2 * f^{3}_{x,x,x}  3 * (f^{3}_{x,x})^{2} = 2 * (.0013173)  3 * (199.9)^{2} = 117200 ≠ 0.

(F1) f^{3}(x_{4}, r_{c})  x_{4} = 0.4863  0.4863 = 0,
(F2) f^{3}_{x}(x_{4}, r_{c}) = 1,
(F3) f^{3}_{r} * f^{3}_{x,x} + 2 * f^{3}_{x,r} = ( 2.15) * (71.9) + 2 * (3.73) = 162.13 ≠ 0,
(F4) c = 2 * f^{3}_{x,x,x}  3 * (f^{3}_{x,x})^{2} = 2 * (281.1)  3 * (71.9)^{2} = 16091 ≠ 0.

(F1) f^{3}(x_{5}, r_{c})  x_{5} = 0.9536  0.9536 = 0,
(F2) f^{3}_{x}(x_{5}, r_{c}) = 1,
(F3) f^{3}_{r} * f^{3}_{x,x} + 2 * f^{3}_{x,r} = ( 0.27) * (710.2) + 2 * ( 16.25) = 162.13 ≠ 0,
(F4) c = 2 * f^{3}_{x,x,x}  3 * (f^{3}_{x,x})^{2} = 2 * (25671)  3 * (710.2)^{2} = 1461800 ≠ 0.

The period doubling bifurcation conditions are confirmed.
The control of the pairs of stable fixed points of the f^{3} map that bracket the unstable x_{3}, x_{4}, and x_{5} fixed points of f^{3} of the orbits {x_{t}} is evident in the next diagram.
r = 3.845 x_{1} = 0, x_{2} = 0.7399 repellers



The f^{3} Period Doubling Cascade
As r increases, period doublings occur at r_{1} = p3^{1}, r_{2} = p3^{2}, r_{3} = p3^{3}, .... as f^{3}, f^{9}, f^{27}, . . . flip bifurcate.
The orbit diagram around the fixed point x_{3} of f^{3} in the region 3.8414991 < r < 3.857 and 0.13 < x < 0.18 shows the period doubling cascade of the sequence of maps {f^{3n}} determining the attracting fixed points of the specific f^{3n} map for p3^{n} <= r < p3^{n+1}.
In the limit as n → ∞, p3^{n} → p3^{*}.
The intermittent emergence of order and chaos revealed in the above miniature orbit diagram for r in the interval [3.8414991, 3.857] mirrors the pattern of the complete orbit diagram for r in the interval [3, 4] available below.
As r converges to 4, the {x_{t}} orbits become increasingly chaotic, evident in the diagrams for f and f^{3}, and the solution trajectory.
r = 3.9 x_{1} = 0, x_{2} = 0.7436 repellers
x_{3} = 0.132652527, x_{4} = 0.448717753 x_{5} = 0.964743512 repellers
x_{6} = 0.180986006, x_{7} = 0.578097280, x_{8} = 0.951213178 repellers



For r > 4, most trajectories leave the [0, 1] vertical interval.
Interactive Logistic Map Diagrams
Click to pop a new window with some
interactive logistic map diagrams.
Definitions of Chaos
"Stochastic behavour in a deterministic system."
"Chaos is aperiodic longterm behavour in a deterministic system that exhibits sensitive dependence on initial conditions" (Strogatz 323).
Let V be a set. The mapping f: V → V is said to be chaotic on V if:
1. f has sensitive dependence on initial conditions,
2. f is topologically transitive (all open sets in V within the range of f interact under f),
3. periodic points are dense in V. (Devaney 50)
"A chaotic map possesses three ingredients: unpredictability, indecomposability, and an element of regularity "(Devaney 50).
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