DefinitionThe logistic map is defined by

ut+1 = rut(1− ut), 0 < r < 4.

The steady states and the corresponding eigenvalues λ = f ′(u∗) are

u∗1 = 0, λ1 = r ,

u∗2 =r − 1

r, λ2 = 2− r .

Curves for r = 1, 1.5,. . . 3.5.0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

ut

ut+

1

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Stable fixed point r = 2.8, λ = 2− r = −0.8

0 10

1

ut

ut+

1

0 10 200

1

t

ut

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Unstable fixed point r = 3.2, λ = 2− r = −1.2

0 10

1

ut

ut+

1

0 10 200

1

t

ut

Peter Olsson (Ume̊a University) Discrete logistic map September 12, 2019 3 / 1

To understand period doubling

Consider the map from ut to ut+2 defined by

ut+1 = rut(1− ut),

ut+2 = rut+1(1− ut+1).

0 10

1

A

B

C

ut

ut+

2,

ut+

1

Figure for r = 3.2.

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Period doubling, again!

Figure for r = 3.5

0 10

1

A

B

C

ut

ut+

2,

ut+

1

The behaviors at A and C are unstable, f ′ < −1!

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Oscillations with period 4, r = 3.5.

0 10 20 300

1

t

ut

The same reasoning may be applied for ut+4 which gives an oscillation ofperiod eight. . . this period doubling may be continued without limit.

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The path to chaos

For 1 < r ≤ 3 there is a unique solution (r − 1)/r .

For 3 < r ≤ 1 +√

6(≈ 3.45) the system has periodic fluctuationsbetween two values.

For 1 +√

6 < r < 3.54 (approximately) the system has periodicoscillations between four values.

For 3.54 < r < 3.57 the system oscillates between 8, 16, 32, values,etc.

At r ≈ 3.57 is the onset of chaos. We can no longer see anyoscillations of finite period and slight variations in the initial valueyields dramatically different results over time.

Peter Olsson (Ume̊a University) Discrete logistic map September 12, 2019 7 / 1

Period doublings and the onset of chaos

1 2 3 40.0

0.5

1.0

r

ut

Peter Olsson (Ume̊a University) Discrete logistic map September 12, 2019 8 / 1