Dynamics of nonlinear & chaotic systems Lecture 4: Nonlinear maps & periodic orbits)

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Dynamics of nonlinear &

chaotic systems Lecture 4:

Nonlinear maps &

periodic orbits) S. Denisov

Dynamics of nonlinear & chaotic systems Lecture 4: Nonlinear maps & periodic

orbits)

S. Denisov

Theo I, Institut f¨ ur Physik, Universit¨at Augsburg

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Nonlinear maps &

Logistic map: Period-1 orbits

Map: x _n+1 = M(x _n ) = rx _n (1 − x _n ) γ = M ⁰ (x) = r (1 − 2x) x = 0 : γ ₁ = |M ⁰ (0)| = r

x = 1 − 1/r : γ ₂ =

|M ⁰ (1 − 1/r)| = |2 − r|

For 0 ≤ 0 ≤ 1: x = 0 is a stable orbit with the basin of attraction [0, 1]

For 1 ≤ 0 ≤ 3: x = 1 − 1/r is a

stable orbit with the basin of

attraction [0, 1]

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Nonlinear maps &

Logistic map: Period-1 orbits

Map: x _n+1 = rx _n (1 − x _n ) What about r > 3 ?

M ⁰ (x) = r(1 − 2x) x = 0 : unstable x = 1 − 1/r : unstable

Peiod p(p = 2, 3...) - orbits?

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Nonlinear maps &

Logistic map: Period-p orbits?

Map: r = 3.1, x ₀ = 0.1 x ₀ = 0.1, x ₁ = 0.27912...,

x ₂ = 0.62359..., x ₃ = 0.7276...,

...

x ₉ = 0.58566..., x ₁₀ = 0.7522...

x ₁₁ = 0.5777..., x ₁₂ = 0.7562...

Peiod 2 - orbits: x ₁ = 0.5225..., x ₂ = 0.8113...

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Nonlinear maps &

Logistic map: Period-2 orbits

Map for period-2 orbits: x _n+2 = M ² (x _n )

This period 2 - orbit is stable: γ ₁ , γ ₂ < 1!

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Nonlinear maps &

Logistic map: Period-doubling bifurcation

Bifuraction diagram: plot x _n , x _n + 1, ... when n is a larger

number (say, n = 10000)

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Nonlinear maps &

Period-doubling bifurcation: An icon

In laser dynamics (1987)

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Nonlinear maps &

Period-doubling bifurcation: An icon

In solid-state electronics (1991)

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Nonlinear maps &

Logistic map: further increase of r

The orbit is stable no longer: γ ₁ , γ ₂ > 1!

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Nonlinear maps &

Logistic map: further increase of r

Period 3 - orbit?

Map for period 3 - orbits: x _n+3 = M ³ (x _n )

The orbit is unstable

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Nonlinear maps &

Logistic map: further increase of r

Period 4 - orbit?

Map for period 4 - orbits: x _n+4 = M ⁴ (x _n )

The orbit is stable!

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Nonlinear maps &

Logistic map: further increase of r

Bifuraction diagram:

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Nonlinear maps &

Logistic map: Another period-doubling bifurcation

Bifuraction diagram: Cascade of PD bifuractions

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Nonlinear maps &

Logistic map: Cascade of PD bifuractions

Feigenbaum’s theory (1975)

lim _m→∞ _r ^r

^m

^−r

^m−1

m+1

−r −m = δ _F

δ _F = 4.6692016... is Feigenbaum’s constant

lim _m→∞ r _m = 3.57...

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Nonlinear maps &

Logistic map: Cascade of PD bifuractions

Feigenbaum’s constant

δ _F =

4.6692016091029906718532038204662016172581855774757686327456513430 0413433021131473713868974402394801381716598485518981513440862714 2027932522312442988890890859944935463236713411532481714219947455 6443658237932020095610583305754586176522220703854106467494942849 8145339172620056875566595233987560382563722564800409510712838906 1184470277585428541980111344017500242858538249833571552205223608 7250291678860362674527213399057131606875345083433934446103706309 4520191158769724322735898389037949462572512890979489867683346116 2688911656312347446057517953912204556247280709520219819909455858 1946136877445617396074115614074243754435499204869180982648652368 438702799649017397793425134723808737136211601860128186102056...

D. Broadhurst, ”Feigenbaum’s constant to 1018 decimal

places” (1999)

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Nonlinear maps &

Logistic map: what is beyond r _∞ = 3.57...?

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Nonlinear maps &

Logistic map: what is beyond r _∞ = 3.57...?

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Nonlinear maps &

Logistic map: Feigenbaum’s scenario

Repeats itself

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Nonlinear maps &

Logistic map: Feigenbaum’s scenario

Self-similarity

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Nonlinear maps &

Feigenbaum’s scenario: universality

x _n+1 = x _n + δ[1 − exp(x _n )]

Mass transfer in binary stars (2006)

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Nonlinear maps &

Feigenbaum’s scenario: analog model

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Nonlinear maps &

Logistic map: A period-3 orbit?

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Nonlinear maps &

Logistic map: Period-3 orbit’s window

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Nonlinear maps &

Logistic map: Period-3 orbit

Map for period-3 orbits: x _n+3 = M ³ (x _n )

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Nonlinear maps &

Logistic map: Period-3 orbit & tangent bifurcation

Tangent bifurcation

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Nonlinear maps &

Logistic map: Period-3 orbit & Feigenbaum’s scenario

Feigenbaum’s scenario still holds!

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Nonlinear maps &

Logistic map: Two types of bifurcations

Period-doubling: 2, 4, 8, ..., 2 ^m , ...

Tangent: 3, 3 × 2, 3 × 4, ..., 3 × 2 ^m , ...

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Nonlinear maps &

Logistic map: Period - 5 orbit

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Nonlinear maps &

Logistic map: Something about ALL periodic orbits

Sharkovskii’s theorem (1964)

“Period-3 means chaos”, i.e. when the period-3 orbit is

unstable then ALL the periodic orbits are unstable.

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Nonlinear maps &

Dynamics of nonlinear & chaotic systems Lecture 4: Nonlinear maps & periodic orbits)

Dynamics of nonlinear & chaotic systems Lecture 4: Nonlinear maps & periodic

orbits)

S. Denisov

Theo I, Institut f¨ ur Physik, Universit¨at Augsburg

Logistic map: Period-1 orbits

Map: x n+1 = M(x n ) = rx n (1 − x n ) γ = M 0 (x) = r (1 − 2x) x = 0 : γ 1 = |M 0 (0)| = r

x = 1 − 1/r : γ 2 =

|M 0 (1 − 1/r)| = |2 − r|

For 0 ≤ 0 ≤ 1: x = 0 is a stable orbit with the basin of attraction [0, 1]

For 1 ≤ 0 ≤ 3: x = 1 − 1/r is a

stable orbit with the basin of

attraction [0, 1]

Logistic map: Period-1 orbits

Map: x n+1 = rx n (1 − x n ) What about r > 3 ?

M 0 (x) = r(1 − 2x) x = 0 : unstable x = 1 − 1/r : unstable

Peiod p(p = 2, 3...) - orbits?

Logistic map: Period-p orbits?

Map: r = 3.1, x 0 = 0.1 x 0 = 0.1, x 1 = 0.27912...,

x 2 = 0.62359..., x 3 = 0.7276...,

...

x 9 = 0.58566..., x 10 = 0.7522...

x 11 = 0.5777..., x 12 = 0.7562...

Peiod 2 - orbits: x 1 = 0.5225..., x 2 = 0.8113...

Logistic map: Period-2 orbits

Map for period-2 orbits: x n+2 = M 2 (x n )

This period 2 - orbit is stable: γ 1 , γ 2 < 1!

Logistic map: Period-doubling bifurcation

Bifuraction diagram: plot x n , x n + 1, ... when n is a larger

number (say, n = 10000)

Period-doubling bifurcation: An icon

In laser dynamics (1987)

Period-doubling bifurcation: An icon

In solid-state electronics (1991)

Logistic map: further increase of r

The orbit is stable no longer: γ 1 , γ 2 > 1!

Logistic map: further increase of r

Period 3 - orbit?

Map for period 3 - orbits: x n+3 = M 3 (x n )

The orbit is unstable

Logistic map: further increase of r

Period 4 - orbit?

Map for period 4 - orbits: x n+4 = M 4 (x n )

The orbit is stable!

Logistic map: further increase of r

Bifuraction diagram:

Logistic map: Another period-doubling bifurcation

Bifuraction diagram: Cascade of PD bifuractions

Logistic map: Cascade of PD bifuractions

Feigenbaum’s theory (1975)

lim m→∞ r r

−r

−r −m = δ F

δ F = 4.6692016... is Feigenbaum’s constant

lim m→∞ r m = 3.57...

Logistic map: Cascade of PD bifuractions

Feigenbaum’s constant

δ F =

D. Broadhurst, ”Feigenbaum’s constant to 1018 decimal

places” (1999)

Logistic map: what is beyond r ∞ = 3.57...?

Logistic map: what is beyond r ∞ = 3.57...?

Logistic map: Feigenbaum’s scenario

Repeats itself

Logistic map: Feigenbaum’s scenario

Self-similarity

Feigenbaum’s scenario: universality

x n+1 = x n + δ[1 − exp(x n )]

Mass transfer in binary stars (2006)

Feigenbaum’s scenario: analog model

Logistic map: A period-3 orbit?

Logistic map: Period-3 orbit’s window

Logistic map: Period-3 orbit

Map for period-3 orbits: x n+3 = M 3 (x n )

Logistic map: Period-3 orbit & tangent bifurcation

Tangent bifurcation

Logistic map: Period-3 orbit & Feigenbaum’s scenario

Feigenbaum’s scenario still holds!

Logistic map: Two types of bifurcations

Period-doubling: 2, 4, 8, ..., 2 m , ...

Map: x _n+1 = M(x _n ) = rx _n (1 − x _n ) γ = M ⁰ (x) = r (1 − 2x) x = 0 : γ ₁ = |M ⁰ (0)| = r

x = 1 − 1/r : γ ₂ =

|M ⁰ (1 − 1/r)| = |2 − r|

Map: x _n+1 = rx _n (1 − x _n ) What about r > 3 ?

M ⁰ (x) = r(1 − 2x) x = 0 : unstable x = 1 − 1/r : unstable

Map: r = 3.1, x ₀ = 0.1 x ₀ = 0.1, x ₁ = 0.27912...,

x ₂ = 0.62359..., x ₃ = 0.7276...,

x ₉ = 0.58566..., x ₁₀ = 0.7522...

x ₁₁ = 0.5777..., x ₁₂ = 0.7562...

Peiod 2 - orbits: x ₁ = 0.5225..., x ₂ = 0.8113...

Map for period-2 orbits: x _n+2 = M ² (x _n )

This period 2 - orbit is stable: γ ₁ , γ ₂ < 1!

Bifuraction diagram: plot x _n , x _n + 1, ... when n is a larger

The orbit is stable no longer: γ ₁ , γ ₂ > 1!

Map for period 3 - orbits: x _n+3 = M ³ (x _n )

Map for period 4 - orbits: x _n+4 = M ⁴ (x _n )

lim _m→∞ _r ^r

^−r

−r −m = δ _F

δ _F = 4.6692016... is Feigenbaum’s constant

lim _m→∞ r _m = 3.57...

δ _F =

Logistic map: what is beyond r _∞ = 3.57...?

Logistic map: what is beyond r _∞ = 3.57...?

x _n+1 = x _n + δ[1 − exp(x _n )]

Map for period-3 orbits: x _n+3 = M ³ (x _n )

Period-doubling: 2, 4, 8, ..., 2 ^m , ...

Tangent: 3, 3 × 2, 3 × 4, ..., 3 × 2 ^m , ...