Chaos, Complexity & Christianity

Carlos E. PuenteUniversity of California, Davis

Université Interdisciplinaire de ParisAvril 11, 2012

esis

We humans, with a soul,

may learn from recent advancements

regarding natural complexity

in order to find peace…

The hypotenuse: the pathway to peace

Faith lessons from chaotic fig trees

Outline

Pythagorean eorem

a

b

c

A Game for Kids

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A Game for Kids

70% 30%

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A Game for Kids

70% 30%

⋮

A Game for Kids

70% 30%

…

⋮

A Game for Kids

…

⋮game defines a multiplicative cascade

11 1

1 2 11 3 3

1

e Cascade, Twelve Levels Later

intertwined thorns via layers having distinct densities

ultimate support on each of the layers is dust

game generates a multifractal

h=56.69 𝑝=0.7

Another Game for Kids

50% 50%

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Another Game for Kids

50% 50%

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Another Game for Kids

50% 50%

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Another Game for Kids

cascade yields equal disjoint thorns over dust

varying the hole size gives topologically the layers on first game

Moral: the two games are intimately related

Accumulated Masses of Clay

...simply from the dynamics of the games

notches above: etc.

plateaus below: etc.

Accumulated Masses of Clay

cumulative sets have no derivatives and are locally

flat

they have maximal lengths:

also maximal combining the games and adding

randomness

A Veritable Deception

A Veritable Deception

A Veritable Deception

a devil’s staircase…

layers in one-dimensional turbulence as in first

cascade

dissipation: atmospheric, boundary layer, wake of

a cylinder…

eddies

𝑅𝑒=𝑣 ∙𝐿𝜈

𝑝=0.7 universal

(Meneveau and Sreenivasan, 1987)Fully-Developed Turbulence

Our Turbulent Times

inequities

competition

disparities

discriminations

forced equality

fear

selfish postures and actions

of the world under poverty

kids die a day for lack of water

violence and terror…

An Optimal System?

𝑝=0.7 𝑛=20

5, 10, 20 and 40% largest thorns have

57, 70, 84 and 95% of the mass

An Optimal System?

5, 10, 20 and 40% largest thorns have

57, 70, 84 and 95% of the mass

this is quite close to the United States: 59, 71,

84 and 95%

Moral: as both cascades dissipate the energy,

if we ride them, they lead us to bite the dust

𝑝=0.7 𝑛=20

Common-Sense Code for Peace

run cascade in reverse to achieve equilibrium

live at low Reynolds numbers to avoid

violence

cut mountains and fill valleys to restore unity

Common-Sense Code for Peace

run cascade in reverse to achieve equilibrium

live at low Reynolds numbers to avoid

violence

cut mountains and fill valleys to restore unity

1=0.999…

e Unique Geometric Solution

a straight and solid condition without thorns or

dust

the hypotenuse is the path of peace!

Moral: humbly rectify and love everyone to find

truth

𝑑=√2

𝑌=𝑋

50-50 no holes

A Veritable Invitation

A Veritable Invitation

to the Origin…

Equilibrium is an Improbable Point

it is hypocritical to judge if we are “far away” from

the point

a convex surface is found for a finite number of

levels:

balance is found easily aided by gravity

in truth there is darkness between 6 and 9…

0𝑝 1

Our Options

equilibrium

calmness

rectitude

fifty-fifty

shortest

reconciliation

integration,

wholeness

unity

positive,

to the future

turbulence

violence

wickedness

inequity

longest

separation

division,

emptiness

dust

negative,

to the past

Pathways

Two options before us

two pathways ahead,

the one is the longest

the other straight.

We journey directlyor go by the legs,we follow intentlyor end up in pain.

By walking the flatness

or hiking the spikes,

we travel in lightness

or take serious frights.

e incentive is unityand the call proportion,the key is

forgivenessand the goal true

notion.

In wandering wickedness

there is never fruit,but in ample humbleness

one encounters the root.

e Fig Tree & e Bible

Adam and Eve covered with fig leaves (Gn 3:7)

A symbol of Israel together with the vine (Hos

2:14)

King Hezekiah healed by prophet Isaiah (Is

38:21)

Nathanael is seen under the fig tree (Jn 1:47–51)

The parable of the barren fig tree (Lk 13:6–9)

Jesus curses a fruitless fig tree (Mt 21:18–22, Mk

11:12–23)

The lesson from a fig tree (Mk 13:28–31, Lk

21:29–33)

e Logistic Map

...a population of “rabbits” as a function of time

different cases depending on the parameter :

𝑋𝑘+1=𝛼 𝑋𝑘 (1−𝑋𝑘 ) ,𝛼∈ [0,4 ]

e Logistic Map

...a population of “rabbits” as a function of time

different cases depending on the parameter :

𝑋𝑘+1=𝛼 𝑋𝑘 (1−𝑋𝑘 ) ,𝛼∈ [0,4 ]

𝛼=2.8

e Logistic Map

...a population of “rabbits” as a function of time

different cases depending on the parameter :

𝑋𝑘+1=𝛼 𝑋𝑘 (1−𝑋𝑘 ) ,𝛼∈ [0,4 ]

𝛼=2.8

From the Origin to Bifurcations

𝑋∞=0

0<𝛼≤1

𝑌=𝑋

From the Origin to Bifurcations

𝑌=𝑋

𝑋∞=0

0<𝛼≤1 1<𝛼≤3

From the Origin to Bifurcations

𝑋∞=0

period 2

0<𝛼≤1

𝑌=𝑋

1<𝛼≤3

From the Origin to Bifurcations

1<𝛼≤3

period 4

𝑌=𝑋

𝑋∞=0

period 2

0<𝛼≤1

aperiodic

aperiodic

𝑌=𝑋(𝛼>𝛼∞≈3.5699)

Periodicities and Chaos Intertwined

𝑌=𝑋

period 5aperiodic

period 3

(𝛼>𝛼∞≈3.5699)

aperiodic

Periodicities and Chaos Intertwined

e Diagram of Bifurcations

the tree contains periodic behavior, for any natural

number

Dust Galore & Self-Similarity

after the dusty strange attractors are common

in the periodic “buds” there are small copies of the

tree:

Intermediate Bud of Period 3

and similar for all intermediate buds of period 9 and

all others

orns Galore

the chaotic tree contains many

multifractal thorns

they combine imbalances and holes, as

in :

they happen at ends of white bands of buds for all

periods

openings durations

bifurcations happen in an orderly

fashion:

(Feigenbaum, 1978)Universality on the Fig Tree

Universality on Other Chaotic Trees

straight root, “tender branch,” branches, and dust

(“fig leaves”)

𝑓 ( 𝑋 )=𝛼 𝑋 (1−𝑋 )3𝑓 ( 𝑋 )=𝛼 𝑋 (1− 𝑋 3 )

relevant in biology, ecology, chemistry, physics,

economics…

the dynamics of convection are reproduced if 𝛼 is

heat

In the Summit of Chaos

all appears to wander in a dusty “strange

attractor” when :

In the Summit of Chaos

however, unstable extensions of the tree need to be excluded:

all appears to wander in a dusty “strange

attractor” when :

Oscillations on the Summit

pre-images of eventual repetitions every three generations

are not in attractor

similar trees for other points need to be excluded, and

for every period

the ultimate attractor is therefore dust

An Escape to the Origin

there are also few points that improbably return to the root

pre-images of zero rest despite high heat

Plain Common Sense

it is better not to miss the point

to avoid chaos and its infernal turbulence:

come down the tree lowering your heat to find the root

abandon yourself to the threshold, , to reach the Origin

move away from nonlinearities () to escape the dust

A Scientific Prophetic Tree?

an allegorical fig tree without fruit, consistently cursed:

this tree and other trees have tender branches and

many buds

the ax is at the root of the trees, Nathanael is at the

root

we may curse the fig tree as He rebuked the wind

Our Options

order

simplicity

serenity

peace

decreasing

God’s way

below

abandonment

obedience

blessing

resting

Heaven

disorder

complexity

turbulence

chaos

increasing

our way

above

selfishness

rebelliousness

curse

wandering

hell

Feigenbaum’s Parabel

Foliage of disordertrapped in empty

dust,jumps astir forever

enduring subtle thrust.

Crossing of the outset

leaving faithful rootlooming tender

offsetfailing to yield fruit

Cascade of bifurcations,

increasing heat within,

inescapable succession

of branches bent by wind.

Sprouting of dynamics

attracted to the strange,

o infinity remindingat the origin: the

flame.

Could it be, oh my friends

that science offers a rhyme?

For a rotten tree foretells

the very advent of time.

Could it be, oh how plain,

that nature extends a call?

For old parable proclaims

the crux in growing small.