order in chaotic dynamical systemscomez/order in chaos.pdffractal systems introduction for many...

ORDER IN CHAOTIC DYNAMICAL SYSTEMS

Dogan Comez

Math ClubNorth Dakota State University

April 24, 2015

IntroductionLogistic system

The system T (x) = 4x(1 − x)Symbolic dynamics representation

Fractal systems

Introduction

Many physical phenomena can be described mathematically as adynamical system;

i.e., a set of objects X with certain sharedproperties that evolve in time.

Examples of dynamical systems:

simple (or damped) pendulum

solar system

the stock market

population of animals

spread of epidemics, etc.

Some of these systems are highly regular, such as the solar system,or the simple pendulum, and some are irregular.

Dogan Comez ORDER IN CHAOTIC DYNAMICAL SYSTEMS



Fractal systems

Introduction

Many physical phenomena can be described mathematically as adynamical system; i.e., a set of objects X with certain sharedproperties that evolve in time.

Examples of dynamical systems:

simple (or damped) pendulum

solar system

the stock market

population of animals

spread of epidemics, etc.

Some of these systems are highly regular, such as the solar system,or the simple pendulum, and some are irregular.




Fractal systems

Introduction

For many dynamical systems, regardless the simplicity of theirmathematical modeling, the long term evolution appears randomand does not exhibit any discernible regularity or order.

(Chaoticbehavior)

Example: The motion of a magnetic pendulum over a planecontaining two or more attractive magnets.

The magnet over which the pendulum ultimately comes to rest(due to frictional damping) is highly dependent on the startingposition and velocity of the pendulum.

Another (very interesting) example: The Logistic system(representing population behavior).




Fractal systems

Introduction

For many dynamical systems, regardless the simplicity of theirmathematical modeling, the long term evolution appears randomand does not exhibit any discernible regularity or order. (Chaoticbehavior)







Fractal systems

Introduction


Example:

The motion of a magnetic pendulum over a planecontaining two or more attractive magnets.






Fractal systems

Introduction








Fractal systems

Introduction




Another (very interesting) example:

The Logistic system(representing population behavior).




Fractal systems

Introduction








Fractal systems

Logistic system

Set up:

Suppose we are studying the growth or decline of thepopulation of a certain species. For, count the number of speciesat times 0, 1, 2, 3, and so on.

Assumptions:

Population growth/decline is at a rate proportional to theexisting population (r)

There is a maximum capacity (0 < C <∞)

Hence, if Pn represents the population at time n, then thepopulation at time n + 1 is given by Pn+1 = r Pn(1− Pn

C ).

Logistic Equation

If T represents time from one instance to the next, this equationcan be rewritten as T (x) = rx(1− x), where x ∈ [0, 1].




Fractal systems

Logistic system

Set up: Suppose we are studying the growth or decline of thepopulation of a certain species.

For, count the number of speciesat times 0, 1, 2, 3, and so on.

Assumptions:




C ).

Logistic Equation





Fractal systems

Logistic system

Set up: Suppose we are studying the growth or decline of thepopulation of a certain species. For, count the number of speciesat times 0, 1, 2, 3, and so on.

Assumptions:




C ).

Logistic Equation





Fractal systems

Cases of the Logistic system

Let’s see how does this system behave for different values of r .Case 0 < r ≤ 1 :∗

0 1

1

Tx = x(1− x)

For any initial (positive) value of x , the sequence (orbit of x)x , Tx , T 2x , T 3x , . . . converge to 0; hence the population willeventually die whatever the initial number is. NO FUN!




Fractal systems



0 1

1

Tx = x(1− x)

For any initial (positive) value of x , the sequence (orbit of x)x , Tx , T 2x , T 3x , . . . converge to 0;

hence the population willeventually die whatever the initial number is. NO FUN!




Fractal systems



0 1

1

Tx = x(1− x)

For any initial (positive) value of x , the sequence (orbit of x)x , Tx , T 2x , T 3x , . . . converge to 0; hence the population willeventually die whatever the initial number is.

NO FUN!




Fractal systems



0 1

1

Tx = x(1− x)

For any initial (positive) value of x , the sequence (orbit of x)x , Tx , T 2x , T 3x , . . . converge to 0; hence the population willeventually die whatever the initial number is. NO FUN!




Fractal systems


Case 1 < r ≤ 3 :

0 1

1

Tx = 2x(1− x)0 1

1

Tx = 3x(1− x)

For 1 < r ≤ 2,∗ the orbit x , Tx , T 2x , T 3x , . . . of any x 6= 0converge to 1

2 quicklyFor 2 < r ≤ 3, the orbit {T nx} of any x 6= 0 converge to 2

3slowly (fluctuates around 2

3 for a while).Hence, the population will eventually reach the fixed valuer−1r whatever the initial value is. NO FUN!




Fractal systems


Case 1 < r ≤ 3 :

0 1

1

Tx = 2x(1− x)

0 1

1

Tx = 3x(1− x)








Fractal systems


Case 1 < r ≤ 3 :

0 1

1

Tx = 2x(1− x)0 1

1

Tx = 3x(1− x)








Fractal systems


Case 1 < r ≤ 3 :

0 1

1

Tx = 2x(1− x)0 1

1

Tx = 3x(1− x)

For 1 < r ≤ 2,∗

the orbit x , Tx , T 2x , T 3x , . . . of any x 6= 0converge to 1







Fractal systems


Case 1 < r ≤ 3 :

0 1

1

Tx = 2x(1− x)0 1

1

Tx = 3x(1− x)


2 quickly

For 2 < r ≤ 3, the orbit {T nx} of any x 6= 0 converge to 23

slowly (fluctuates around 23 for a while).

Hence, the population will eventually reach the fixed valuer−1r whatever the initial value is. NO FUN!




Fractal systems


Case 1 < r ≤ 3 :

0 1

1

Tx = 2x(1− x)0 1

1

Tx = 3x(1− x)


2 quicklyFor 2 < r ≤ 3,

the orbit {T nx} of any x 6= 0 converge to 23

slowly (fluctuates around 23 for a while).





Fractal systems


Case 1 < r ≤ 3 :

0 1

1

Tx = 2x(1− x)0 1

1

Tx = 3x(1− x)




3 for a while).





Fractal systems


Case 1 < r ≤ 3 :

0 1

1

Tx = 2x(1− x)0 1

1

Tx = 3x(1− x)




3 for a while).Hence, the population will eventually reach the fixed valuer−1r whatever the initial value is.

NO FUN!




Fractal systems


Case 1 < r ≤ 3 :

0 1

1

Tx = 2x(1− x)0 1

1

Tx = 3x(1− x)








Fractal systems

Case r > 3

When r > 3 the population oscillates between two, four, eight, etc.values as r increases.

r = 3.8

It is not obvious which points have which type of oscillatorybehavior. ∗∗

LET’S HAVE SOME FUN!




Fractal systems

Points of interest

Sort out the points according to the nature of their behavior as:

fixed point (Tx = x)

periodic point (there exists an n such that T nx = x),

eventually periodic point, non-periodic point, etc.

The origin is a fixed point of any logistic system.

Case 0 < r ≤ 3

0 < r ≤ 1 : Fixed point: 0. The orbit {T nx} → 0 for every x ;no periodic point of period greater than 1. ORDER!

1 < r ≤ 3 : Two fixed points: 0 and r−1r . {T nx} → r−1

r forall x 6= 0; no periodic point of period greater than 1. ORDER!




Fractal systems

Points of interest






Case 0 < r ≤ 3

0 < r ≤ 1 :

Fixed point: 0. The orbit {T nx} → 0 for every x ;no periodic point of period greater than 1. ORDER!






Fractal systems

Points of interest






Case 0 < r ≤ 3

0 < r ≤ 1 : Fixed point: 0.

The orbit {T nx} → 0 for every x ;no periodic point of period greater than 1. ORDER!






Fractal systems

Points of interest






Case 0 < r ≤ 3

0 < r ≤ 1 : Fixed point: 0. The orbit {T nx} → 0 for every x ;

no periodic point of period greater than 1. ORDER!






Fractal systems

Points of interest






Case 0 < r ≤ 3

0 < r ≤ 1 : Fixed point: 0. The orbit {T nx} → 0 for every x ;no periodic point of period greater than 1.

ORDER!






Fractal systems

Points of interest






Case 0 < r ≤ 3







Fractal systems

Points of interest






Case 0 < r ≤ 3


1 < r ≤ 3 :

Two fixed points: 0 and r−1r . {T nx} → r−1





Fractal systems

Points of interest






Case 0 < r ≤ 3


1 < r ≤ 3 : Two fixed points: 0 and r−1r .

{T nx} → r−1r for

all x 6= 0; no periodic point of period greater than 1. ORDER!




Fractal systems

Points of interest






Case 0 < r ≤ 3



r forall x 6= 0;

no periodic point of period greater than 1. ORDER!




Fractal systems

Points of interest






Case 0 < r ≤ 3



r forall x 6= 0; no periodic point of period greater than 1.

ORDER!




Fractal systems

Points of interest






Case 0 < r ≤ 3







Fractal systems

Points of interest

If r > 3,

we have two fixed points: 0 and r−1r . Furthermore,

3 < r ≤ 1 +√

6 : the orbits of almost all points oscillate betweentwo values (depending on r),

r = 3.3

1 +√

6 < r ≤ 3.54 (approx.) : the orbits of almost all pointsoscillate among four values,r > 3.54 (approx.) : the orbits of almost all points oscillate among8 values, then 16, 32, and so on.




Fractal systems

Points of interest

If r > 3, we have two fixed points: 0 and r−1r .

Furthermore,

3 < r ≤ 1 +√


r = 3.3

1 +√





Fractal systems

Points of interest

If r > 3, we have two fixed points: 0 and r−1r . Furthermore,

3 < r ≤ 1 +√


r = 3.3

1 +√





Fractal systems

Points of interest


3 < r ≤ 1 +√

6 :

the orbits of almost all points oscillate betweentwo values (depending on r),

r = 3.3

1 +√





Fractal systems

Points of interest


3 < r ≤ 1 +√


r = 3.3

1 +√





Fractal systems

Points of interest


3 < r ≤ 1 +√


r = 3.3

1 +√

6 < r ≤ 3.54 (approx.) : the orbits of almost all pointsoscillate among four values,

r > 3.54 (approx.) : the orbits of almost all points oscillate among8 values, then 16, 32, and so on.




Fractal systems

Points of interest


3 < r ≤ 1 +√


r = 3.3

1 +√





Fractal systems

Points of interest

Approx. r = 3.56995 is the end of period-doubling cascade.

Beyond this value, it is impossible to predict if {T nx} isconvergent, or has periodic or oscillatory behavior.

CHAOS!∗∗




Fractal systems

Points of interest

Approx. r = 3.56995 is the end of period-doubling cascade.Beyond this value, it is impossible to predict if {T nx} isconvergent, or has periodic or oscillatory behavior.

CHAOS!∗∗




Fractal systems

Tent map

Tent map

For r = 4, the Logistic system is equivalent to the Tent system:

0 1

1

Tx =

{2x, if 0 ≤ x < 1

2

2(1− x), if 12≤ x ≤ 1

This equivalence is given by the (invertible) map φ(x) = sin2(πx2 ).




Fractal systems

Tent map

Fixed points: 0, 23 .∗

1 −→ 0 (eventually fixed)14 −→ 1

2 −→ 1 −→ 0 (eventually fixed)34 −→ 1


3 (eventually fixed)25 −→ 4

5 −→ 25 (periodic with period 2)

27 −→ 4

7 −→ 67 −→ 2

7 (periodic with period 3)19 −→ 2

9 −→ 49 −→ 8

9 −→ 29 (event. periodic with period 3).

Do you see a pattern here? Are there other periodic points? ∗∗




Fractal systems

Tent map


1

−→ 0 (eventually fixed)14 −→ 1





27 −→ 4

7 −→ 67 −→ 2


9 −→ 49 −→ 8






Fractal systems

Tent map


1 −→ 0

(eventually fixed)14 −→ 1





27 −→ 4

7 −→ 67 −→ 2


9 −→ 49 −→ 8






Fractal systems

Tent map


1 −→ 0 (eventually fixed)

14 −→ 1





27 −→ 4

7 −→ 67 −→ 2


9 −→ 49 −→ 8






Fractal systems

Tent map


1 −→ 0 (eventually fixed)14

−→ 12 −→ 1 −→ 0 (eventually fixed)

34 −→ 1




27 −→ 4

7 −→ 67 −→ 2


9 −→ 49 −→ 8






Fractal systems

Tent map



2

−→ 1 −→ 0 (eventually fixed)34 −→ 1




27 −→ 4

7 −→ 67 −→ 2


9 −→ 49 −→ 8






Fractal systems

Tent map



2 −→ 1





27 −→ 4

7 −→ 67 −→ 2


9 −→ 49 −→ 8






Fractal systems

Tent map



2 −→ 1 −→ 0





27 −→ 4

7 −→ 67 −→ 2


9 −→ 49 −→ 8






Fractal systems

Tent map



2 −→ 1 −→ 0 (eventually fixed)

34 −→ 1




27 −→ 4

7 −→ 67 −→ 2


9 −→ 49 −→ 8






Fractal systems

Tent map



2 −→ 1 −→ 0 (eventually fixed)34

−→ 12 −→ 1 −→ 0 (eventually fixed)

13 −→ 2



27 −→ 4

7 −→ 67 −→ 2


9 −→ 49 −→ 8






Fractal systems

Tent map




2

−→ 1 −→ 0 (eventually fixed)13 −→ 2



27 −→ 4

7 −→ 67 −→ 2


9 −→ 49 −→ 8






Fractal systems

Tent map




2 −→ 1




27 −→ 4

7 −→ 67 −→ 2


9 −→ 49 −→ 8






Fractal systems

Tent map




2 −→ 1 −→ 0




27 −→ 4

7 −→ 67 −→ 2


9 −→ 49 −→ 8






Fractal systems

Tent map




2 −→ 1 −→ 0 (eventually fixed)

13 −→ 2



27 −→ 4

7 −→ 67 −→ 2


9 −→ 49 −→ 8






Fractal systems

Tent map




2 −→ 1 −→ 0 (eventually fixed)13

−→ 23 (eventually fixed)

25 −→ 4


27 −→ 4

7 −→ 67 −→ 2


9 −→ 49 −→ 8






Fractal systems

Tent map





3



27 −→ 4

7 −→ 67 −→ 2


9 −→ 49 −→ 8






Fractal systems

Tent map





3 (eventually fixed)

25 −→ 4


27 −→ 4

7 −→ 67 −→ 2


9 −→ 49 −→ 8






Fractal systems

Tent map





3 (eventually fixed)25

−→ 45 −→ 2


7 −→ 67 −→ 2


9 −→ 49 −→ 8






Fractal systems

Tent map






5

−→ 25 (periodic with period 2)

27 −→ 4

7 −→ 67 −→ 2


9 −→ 49 −→ 8






Fractal systems

Tent map






5 −→ 25

(periodic with period 2)27 −→ 4

7 −→ 67 −→ 2


9 −→ 49 −→ 8






Fractal systems

Tent map







27 −→ 4

7 −→ 67 −→ 2


9 −→ 49 −→ 8






Fractal systems

Tent map







27

−→ 47 −→ 6


19 −→ 2

9 −→ 49 −→ 8






Fractal systems

Tent map







27 −→ 4

7

−→ 67 −→ 2


9 −→ 49 −→ 8






Fractal systems

Tent map







27 −→ 4

7 −→ 67

−→ 27 (periodic with period 3)

19 −→ 2

9 −→ 49 −→ 8






Fractal systems

Tent map







27 −→ 4

7 −→ 67 −→ 2

7

(periodic with period 3)19 −→ 2

9 −→ 49 −→ 8






Fractal systems

Tent map







27 −→ 4

7 −→ 67 −→ 2

7 (periodic with period 3)

19 −→ 2

9 −→ 49 −→ 8






Fractal systems

Tent map







27 −→ 4

7 −→ 67 −→ 2


9 −→ 49 −→ 8

9 −→ 29

(event. periodic with period 3).





Fractal systems

Tent map







27 −→ 4

7 −→ 67 −→ 2


9 −→ 49 −→ 8






Fractal systems

Tent map







27 −→ 4

7 −→ 67 −→ 2


9 −→ 49 −→ 8


Do you see a pattern here?

Are there other periodic points? ∗∗




Fractal systems

Tent map







27 −→ 4

7 −→ 67 −→ 2


9 −→ 49 −→ 8






Fractal systems

Tent map

Fact.1

If 0 < x ≤ 1 is rational, then it is eventually periodic. Furthermore,a point 0 < a

b < 1 is periodic if a is even and b is odd integer. ∗∗

(The set of periodic points of T is dense in [0, 1]!)

Question. Is there an integer N greater than the periods of all theperiodic points?

Theorem (Li and Yorke)

If f : [0, 1]→ [0, 1] is a continuous map and if it has a periodicpoint of period 3, then it has periodic points of any period k.

Hence, the Tent map (equivalently, the Logistic map) has periodicpoints of any period k ≥ 1!




Fractal systems

Tent map

Fact.1

If 0 < x ≤ 1 is rational, then it is eventually periodic.

Furthermore,a point 0 < a










Fractal systems

Tent map

Fact.1

If 0 < x ≤ 1 is rational, then it is eventually periodic. Furthermore,a point 0 < a










Fractal systems

Tent map

Obviously, if x is irrational, then it is not a periodic point;

hencethere are infinitely many (uncountable) non-periodic points.

Question: How do these non-periodic points behave?

Fact.2

If 0 < x < 1 is irrational, then the orbit {T nx} has an elementarbitrarily close to any point y ∈ [0, 1].

That is, the Tent (Logistic) map is topologically transitive (i.e.,has a point whose orbit is dense).




Fractal systems

Tent map

Obviously, if x is irrational, then it is not a periodic point; hencethere are infinitely many (uncountable) non-periodic points.

Question: How do these non-periodic points behave?

Fact.2

If 0 < x < 1 is irrational, then the orbit {T nx} has an elementarbitrarily close to any point y ∈ [0, 1].

That is, the Tent (Logistic) map is topologically transitive (i.e.,has a point whose orbit is dense).




Fractal systems

Tent map

Recall: for any x ∈ R, there exists an irrational (rational) numbery arbitrarily close to x . ∗∗

Hence, for the Tent map, for two distinct points very close to eachother while one is periodic (rational) the other non-periodic(irrational) and gets arbitrarily close to any point in [0, 1].This is exactly the characterization of a chaotic dynamicalsystem; namely, a system that

has a dense collection of periodic points,

is sensitive to the initial condition of the system (so thatinitially nearby points can evolve quickly into very differentstates), and

is topologically transitive. ∗∗




Fractal systems

Tent map


Hence, for the Tent map, for two distinct points very close to eachother while one is periodic (rational) the other non-periodic(irrational) and gets arbitrarily close to any point in [0, 1].

This is exactly the characterization of a chaotic dynamicalsystem; namely, a system that







Fractal systems

Tent map


Hence, for the Tent map, for two distinct points very close to eachother while one is periodic (rational) the other non-periodic(irrational) and gets arbitrarily close to any point in [0, 1].This is exactly the characterization of a chaotic dynamicalsystem;

namely, a system that







Fractal systems

Tent map


Hence, for the Tent map, for two distinct points very close to eachother while one is periodic (rational) the other non-periodic(irrational) and gets arbitrarily close to any point in [0, 1].This is exactly the characterization of a chaotic dynamicalsystem; namely, a system that







Fractal systems

Symbolic dynamics

Let A = [0, 12 ], and B = ( 1

2 , 1]. Give an address to each x ∈ [0, 1] :

T nx ↔ 0 if T nx ∈ A and T nx ↔ 1 ifT nx ∈ B.

Hence, we have ∗

0 = (0, 0, 0, . . . ) since T n(0) ∈ A for n ≥ 0.23 = (1, 1, 1, . . . ) since T n( 2

3 ) ∈ B for n ≥ 0.

1 = (1, 0, 0, 0, . . . ) since T (1) ∈ A for n ≥ 1.12 = (0, 1, 0, 0, . . . ) since T ( 1

2 ) ∈ B; 12 ,T

n( 12 ) ∈ A for n ≥ 2.

14 = (0, 1, 1, 0, 0, 0, . . . ) since T n( 1

4 ) ∈ B if n = 1, 2 andT n( 1

4 ) ∈ A for n ≥ 3.25 = (0, 1, 0, 1, 0, 1, 0, 1, . . . ) since T n( 2

5 ) ∈ A for n = 2k andT n( 2

5 ) ∈ B for n = 2k + 1.27 = (0, 1, 1, 0, 1, 1, 0, 1, 1, . . . )19 = (0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, . . . ) , and so on.




Fractal systems

Symbolic dynamics

Let A = [0, 12 ], and B = ( 1

2 , 1].

Give an address to each x ∈ [0, 1] :


Hence, we have ∗


3 ) ∈ B for n ≥ 0.


2 ) ∈ B; 12 ,T

n( 12 ) ∈ A for n ≥ 2.

14 = (0, 1, 1, 0, 0, 0, . . . ) since T n( 1

4 ) ∈ B if n = 1, 2 andT n( 1

4 ) ∈ A for n ≥ 3.25 = (0, 1, 0, 1, 0, 1, 0, 1, . . . ) since T n( 2

5 ) ∈ A for n = 2k andT n( 2

5 ) ∈ B for n = 2k + 1.27 = (0, 1, 1, 0, 1, 1, 0, 1, 1, . . . )19 = (0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, . . . ) , and so on.




Fractal systems

Symbolic dynamics

Let A = [0, 12 ], and B = ( 1



Hence, we have ∗


3 ) ∈ B for n ≥ 0.


2 ) ∈ B; 12 ,T

n( 12 ) ∈ A for n ≥ 2.

14 = (0, 1, 1, 0, 0, 0, . . . ) since T n( 1

4 ) ∈ B if n = 1, 2 andT n( 1

4 ) ∈ A for n ≥ 3.25 = (0, 1, 0, 1, 0, 1, 0, 1, . . . ) since T n( 2

5 ) ∈ A for n = 2k andT n( 2

5 ) ∈ B for n = 2k + 1.27 = (0, 1, 1, 0, 1, 1, 0, 1, 1, . . . )19 = (0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, . . . ) , and so on.




Fractal systems

Symbolic dynamics

Let A = [0, 12 ], and B = ( 1



Hence, we have ∗

0 = (0, 0, 0, . . . ) since T n(0) ∈ A for n ≥ 0.

23 = (1, 1, 1, . . . ) since T n( 2

3 ) ∈ B for n ≥ 0.


2 ) ∈ B; 12 ,T

n( 12 ) ∈ A for n ≥ 2.

14 = (0, 1, 1, 0, 0, 0, . . . ) since T n( 1

4 ) ∈ B if n = 1, 2 andT n( 1

4 ) ∈ A for n ≥ 3.25 = (0, 1, 0, 1, 0, 1, 0, 1, . . . ) since T n( 2

5 ) ∈ A for n = 2k andT n( 2

5 ) ∈ B for n = 2k + 1.27 = (0, 1, 1, 0, 1, 1, 0, 1, 1, . . . )19 = (0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, . . . ) , and so on.




Fractal systems

Symbolic dynamics

Let A = [0, 12 ], and B = ( 1



Hence, we have ∗


3 ) ∈ B for n ≥ 0.


2 ) ∈ B; 12 ,T

n( 12 ) ∈ A for n ≥ 2.

14 = (0, 1, 1, 0, 0, 0, . . . ) since T n( 1

4 ) ∈ B if n = 1, 2 andT n( 1

4 ) ∈ A for n ≥ 3.25 = (0, 1, 0, 1, 0, 1, 0, 1, . . . ) since T n( 2

5 ) ∈ A for n = 2k andT n( 2

5 ) ∈ B for n = 2k + 1.27 = (0, 1, 1, 0, 1, 1, 0, 1, 1, . . . )19 = (0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, . . . ) , and so on.




Fractal systems

Symbolic dynamics

Let A = [0, 12 ], and B = ( 1



Hence, we have ∗


3 ) ∈ B for n ≥ 0.

1 = (1, 0, 0, 0, . . . ) since T (1) ∈ A for n ≥ 1.

12 = (0, 1, 0, 0, . . . ) since T ( 1

2 ) ∈ B; 12 ,T

n( 12 ) ∈ A for n ≥ 2.

14 = (0, 1, 1, 0, 0, 0, . . . ) since T n( 1

4 ) ∈ B if n = 1, 2 andT n( 1

4 ) ∈ A for n ≥ 3.25 = (0, 1, 0, 1, 0, 1, 0, 1, . . . ) since T n( 2

5 ) ∈ A for n = 2k andT n( 2

5 ) ∈ B for n = 2k + 1.27 = (0, 1, 1, 0, 1, 1, 0, 1, 1, . . . )19 = (0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, . . . ) , and so on.




Fractal systems

Symbolic dynamics

Let A = [0, 12 ], and B = ( 1



Hence, we have ∗


3 ) ∈ B for n ≥ 0.


2 ) ∈ B; 12 ,T

n( 12 ) ∈ A for n ≥ 2.

14 = (0, 1, 1, 0, 0, 0, . . . ) since T n( 1

4 ) ∈ B if n = 1, 2 andT n( 1

4 ) ∈ A for n ≥ 3.25 = (0, 1, 0, 1, 0, 1, 0, 1, . . . ) since T n( 2

5 ) ∈ A for n = 2k andT n( 2

5 ) ∈ B for n = 2k + 1.27 = (0, 1, 1, 0, 1, 1, 0, 1, 1, . . . )19 = (0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, . . . ) , and so on.




Fractal systems

Symbolic dynamics

Let A = [0, 12 ], and B = ( 1



Hence, we have ∗


3 ) ∈ B for n ≥ 0.


2 ) ∈ B; 12 ,T

n( 12 ) ∈ A for n ≥ 2.

14 = (0, 1, 1, 0, 0, 0, . . . ) since T n( 1

4 ) ∈ B if n = 1, 2 andT n( 1

4 ) ∈ A for n ≥ 3.

25 = (0, 1, 0, 1, 0, 1, 0, 1, . . . ) since T n( 2

5 ) ∈ A for n = 2k andT n( 2

5 ) ∈ B for n = 2k + 1.27 = (0, 1, 1, 0, 1, 1, 0, 1, 1, . . . )19 = (0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, . . . ) , and so on.




Fractal systems

Symbolic dynamics

Let A = [0, 12 ], and B = ( 1



Hence, we have ∗


3 ) ∈ B for n ≥ 0.


2 ) ∈ B; 12 ,T

n( 12 ) ∈ A for n ≥ 2.

14 = (0, 1, 1, 0, 0, 0, . . . ) since T n( 1

4 ) ∈ B if n = 1, 2 andT n( 1

4 ) ∈ A for n ≥ 3.25 = (0, 1, 0, 1, 0, 1, 0, 1, . . . ) since T n( 2

5 ) ∈ A for n = 2k andT n( 2

5 ) ∈ B for n = 2k + 1.

27 = (0, 1, 1, 0, 1, 1, 0, 1, 1, . . . )19 = (0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, . . . ) , and so on.




Fractal systems

Symbolic dynamics

Let A = [0, 12 ], and B = ( 1



Hence, we have ∗


3 ) ∈ B for n ≥ 0.


2 ) ∈ B; 12 ,T

n( 12 ) ∈ A for n ≥ 2.

14 = (0, 1, 1, 0, 0, 0, . . . ) since T n( 1

4 ) ∈ B if n = 1, 2 andT n( 1

4 ) ∈ A for n ≥ 3.25 = (0, 1, 0, 1, 0, 1, 0, 1, . . . ) since T n( 2

5 ) ∈ A for n = 2k andT n( 2

5 ) ∈ B for n = 2k + 1.27 = (0, 1, 1, 0, 1, 1, 0, 1, 1, . . . )

19 = (0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, . . . ) , and so on.




Fractal systems

Symbolic dynamics

Let A = [0, 12 ], and B = ( 1



Hence, we have ∗


3 ) ∈ B for n ≥ 0.


2 ) ∈ B; 12 ,T

n( 12 ) ∈ A for n ≥ 2.

14 = (0, 1, 1, 0, 0, 0, . . . ) since T n( 1

4 ) ∈ B if n = 1, 2 andT n( 1

4 ) ∈ A for n ≥ 3.25 = (0, 1, 0, 1, 0, 1, 0, 1, . . . ) since T n( 2

5 ) ∈ A for n = 2k andT n( 2

5 ) ∈ B for n = 2k + 1.27 = (0, 1, 1, 0, 1, 1, 0, 1, 1, . . . )19 = (0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, . . . ) , and so on.




Fractal systems

Symbolic dynamics

There is a 1-1 correspondence between orbits of the Tent map andelements of the set {0, 1}∞={all sequences of 0s and 1s }.

The action of T on [0, 1] is represented on {0, 1}∞ by thesemi-shift map σ as

σ(a0, a1, a2, a3, . . . ) =

{(a1, a2, a3, . . . ) if a0 = 0

(1− a1, 1− a2, 1− a3, . . . ) if a0 = 1.

Therefore, σ(1, 1, 0, 0, 0, . . . ) = (0, 1, 1, 1, . . . ),σ(0, 1, 1, 0, 0, 0, . . . ) = (1, 1, 0, 0, 0, . . . ),σ2(0, 1, 1, 0, 0, 0, . . . ) = (0, 1, 1, 1, . . . ), etc.

The advantage of passing to the system ({0, 1}∞, σ), known assymbolic dynamical system is that: (i) it is much simpler, (ii) itenables us to see many features of the system independently of thespecifics of the underlying set.




Fractal systems

Symbolic dynamics

There is a 1-1 correspondence between orbits of the Tent map andelements of the set {0, 1}∞={all sequences of 0s and 1s }.The action of T on [0, 1] is represented on {0, 1}∞ by thesemi-shift map σ as

σ(a0, a1, a2, a3, . . . ) =

{(a1, a2, a3, . . . ) if a0 = 0

(1− a1, 1− a2, 1− a3, . . . ) if a0 = 1.






Fractal systems

Symbolic dynamics


σ(a0, a1, a2, a3, . . . ) =

{(a1, a2, a3, . . . ) if a0 = 0

(1− a1, 1− a2, 1− a3, . . . ) if a0 = 1.

Therefore, σ(1, 1, 0, 0, 0, . . . ) = (0, 1, 1, 1, . . . )

,σ(0, 1, 1, 0, 0, 0, . . . ) = (1, 1, 0, 0, 0, . . . ),σ2(0, 1, 1, 0, 0, 0, . . . ) = (0, 1, 1, 1, . . . ), etc.





Fractal systems

Symbolic dynamics


σ(a0, a1, a2, a3, . . . ) =

{(a1, a2, a3, . . . ) if a0 = 0

(1− a1, 1− a2, 1− a3, . . . ) if a0 = 1.

Therefore, σ(1, 1, 0, 0, 0, . . . ) = (0, 1, 1, 1, . . . ),σ(0, 1, 1, 0, 0, 0, . . . ) = (1, 1, 0, 0, 0, . . . )

,σ2(0, 1, 1, 0, 0, 0, . . . ) = (0, 1, 1, 1, . . . ), etc.





Fractal systems

Symbolic dynamics


σ(a0, a1, a2, a3, . . . ) =

{(a1, a2, a3, . . . ) if a0 = 0

(1− a1, 1− a2, 1− a3, . . . ) if a0 = 1.






Fractal systems

Symbolic dynamics


σ(a0, a1, a2, a3, . . . ) =

{(a1, a2, a3, . . . ) if a0 = 0

(1− a1, 1− a2, 1− a3, . . . ) if a0 = 1.


The advantage of passing to the system ({0, 1}∞, σ), known assymbolic dynamical system is that:

(i) it is much simpler, (ii) itenables us to see many features of the system independently of thespecifics of the underlying set.




Fractal systems

Symbolic dynamics


σ(a0, a1, a2, a3, . . . ) =

{(a1, a2, a3, . . . ) if a0 = 0

(1− a1, 1− a2, 1− a3, . . . ) if a0 = 1.


The advantage of passing to the system ({0, 1}∞, σ), known assymbolic dynamical system is that: (i) it is much simpler,

(ii) itenables us to see many features of the system independently of thespecifics of the underlying set.




Fractal systems

Symbolic dynamics


σ(a0, a1, a2, a3, . . . ) =

{(a1, a2, a3, . . . ) if a0 = 0

(1− a1, 1− a2, 1− a3, . . . ) if a0 = 1.






Fractal systems

Symbolic dynamics

Periodic points of period n ↔ periodic {1, 0}-sequences oflength n.

Sequences corresponding to the eventually periodic pointsexhibit similar pattern, except that first several terms may notfollow a pattern.

Non-periodic points are represented by sequences having(eventually) non repeating patterns.

A transitive point:

(0 1 00 01 10 11 000 001 010 100 011 101 110 111 0000 0001 . . . ) ∗∗




Fractal systems

Back to order

Recap:

A chaotic system has an intrinsic “order”, in the sense that:

One can identify points in the system which are periodic, andthose which are transitive

Points in certain parts of a chaotic system tend to converge tothese points (the set of these points constitute the attractorof the system).

Attractors can be a single point (r = 1 in Logistic map), afinite set (1 < r < 3.5 in Logistic map) or more complicatedsets (Tent map). When the attractor is not expressible ascountable union or intersection of geometric objects (likepoints, lines, surfaces, etc.), it is called a strange attractor.




Fractal systems

Back to order

Recap:A chaotic system has an intrinsic “order”, in the sense that:







Fractal systems

Back to order


One can identify points in the system which are periodic,

andthose which are transitive






Fractal systems

Back to order








Fractal systems

Back to order




Attractors can be a single point (r = 1 in Logistic map), afinite set (1 < r < 3.5 in Logistic map) or more complicatedsets (Tent map).

When the attractor is not expressible ascountable union or intersection of geometric objects (likepoints, lines, surfaces, etc.), it is called a strange attractor.




Fractal systems

Back to order








Fractal systems

Cantor set

Tent map T : [0, 1]→ [0, 1], given by the equation

0 1

1

Tx =

{3x, if 0 ≤ x < 1

2

3(1− x), if 12≤ x ≤ 1

Actual domain of T is T−1[0, 1] = [0, 13 ] ∪ [ 2

3 , 1].




Fractal systems

Cantor set

Second iteration: the points in the intervals ( 19 ,

29 ) and ( 7

9 ,89 )

are out of the range of T 2.

Dom(T 2) = T−2[0, 1] = [0, 19 ] ∪ [ 2

9 ,13 ] ∪ [ 2

3 ,79 ] ∪ [ 8

9 , 1].

Third iteration: the points in the intervals( 1

27 ,2

27 ), ( 727 ,

827 ), ( 19

27 ,2027 ), and ( 25

27 ,2627 ) are out of the range.

Dom(T 3) = T−3[0, 1] =[0,1

27] ∪ [

2

27, ,

1

9] ∪ [

2

9,

7

27] ∪ [

8

27,

1

3]

∪ [2

3,

19

27] ∪ [

20

27,

7

9] ∪ [

24

27,

25

27] ∪ [

26

27, 1],

and so on . . .




Fractal systems

Cantor set

Second iteration: the points in the intervals ( 19 ,

29 ) and ( 7

9 ,89 )

are out of the range of T 2.Dom(T 2) = T−2[0, 1] = [0, 1

9 ] ∪ [ 29 ,

13 ] ∪ [ 2

3 ,79 ] ∪ [ 8

9 , 1].

Third iteration: the points in the intervals( 1

27 ,2

27 ), ( 727 ,

827 ), ( 19

27 ,2027 ), and ( 25

27 ,2627 ) are out of the range.

Dom(T 3) = T−3[0, 1] =[0,1

27] ∪ [

2

27, ,

1

9] ∪ [

2

9,

7

27] ∪ [

8

27,

1

3]

∪ [2

3,

19

27] ∪ [

20

27,

7

9] ∪ [

24

27,

25

27] ∪ [

26

27, 1],

and so on . . .




Fractal systems

Cantor set

The set of points in [0,1] that is mapped back to [0,1] under alliterations of the Tent map is

⋂∞n=1 T

−n[0, 1] :

which is the standard Cantor set! A fractal!




Fractal systems

Cantor set


⋂∞n=1 T

−n[0, 1] :

which is the standard Cantor set!

A fractal!




Fractal systems

Cantor set


⋂∞n=1 T

−n[0, 1] :

which is the standard Cantor set! A fractal!




Fractal systems

Strange attractors

What happened?

All the points that eventually leave [0, 1] under T are removedfrom the system, what is left is the set of attractors of themap.

Cantor set is the strange attractor of the Tent (Logistic)map

T (x) =

3x if x ∈ [0,1

2)

3(1− x) if x ∈ [1

2, 1].

Many fractals are strange attractors of some dynamicalsystems.




Fractal systems

More fun

What more to do?

Study subsystems of a given dynamical system

Study fractals as dynamical systems and see the “order” in afractal

Study fractals as symbolic systems and study its subsystems(subfractals), develop construction of new fractals out ofexisting one(s), calculate and compare fractal dimensions(Hausdorff, box, packing, etc.) of these fractals

and, continue having FUN.

THANK YOU!




Fractal systems

More fun

What more to do?



Study fractals as symbolic systems and study its subsystems(subfractals),

develop construction of new fractals out ofexisting one(s), calculate and compare fractal dimensions(Hausdorff, box, packing, etc.) of these fractals


THANK YOU!




Fractal systems

More fun

What more to do?



Study fractals as symbolic systems and study its subsystems(subfractals), develop construction of new fractals out ofexisting one(s),

calculate and compare fractal dimensions(Hausdorff, box, packing, etc.) of these fractals


THANK YOU!




Fractal systems

More fun

What more to do?





THANK YOU!




Fractal systems

More fun

What more to do?




and,

continue having FUN.

THANK YOU!




Fractal systems

More fun

What more to do?





THANK YOU!


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