2015 one-dimensional maps

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One-dimensional maps

March 23, 2015 1M. Biey - Dip. di Elettronica e Telecomunicazioni - Politecnico di Torino

“Simple dynamical systems do not necessarily

lead to simple dynamical behavior” .

(R. M. May, 1976)

__________________________R. M. May, Simple mathematical models with very complicated dynamics, Nature, 261, 459, 1976.


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Maps

March 23, 2015 M. Biey - Dip. di Elettronica Telecomunicazioni - Politecnico di Torino 2

The solutions of dynamical models with discrete time,

are characterized by sequences of points, each of whichis determined from an initial point (the initial condition).

Such models are called recurrence relationships,

recurrences, iterations, or simply maps. Their explicit

form is written:

X is a p-dimensional real vector, n denotes the discrete

time, Λ is a parameter vector, X 0 is the vector of initial

conditions.


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Maps (cont.)

Maps arise in various ways: As model of natural phenomena

E.g.: Capital growth: , where I is the annual rate of interest

As tools for analyzing differential equationsE.g.: Forward Euler Integration Formula:

As simple examples of chaos


1( ) ( )

k k k x f x x x hf x

1 (1 )

n n C C I


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Maps (cont.)


Simple examples of one-dimensional maps:

1 0( ), ( 0)

n n x f x x n x

If the vector X has dimension 1, we have one-

dimensional

maps:

f is a smooth function from the real line to itself

and x0 is the initial condition.

The sequence x0, x1, x2, …. is called the

orbit (or trajectory ) starting from x0 .


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Maps (cont.)


A geometric way of thinking:Cobweb construction to

iterate the map graphically:


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Fixed points


DefinitionSuppose x* satisfies f ( x*) = x*.

Then, if xn = x*, it follows that

xn+1 = f ( xn) = f ( x*) = x*

Hence the orbit remains at x* for all futureiterations.

x* is called a fixed (or equilibrium) point .

Fixed points are the intersections of the mapwith the straight line xn+1 = xn

What about the stability of a fixed point?


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Fixed points (cont.)


Linear stability of a fixed point x*

xn = x* + ηn : does ηn grow or decay as n increases?

xn+1 = x* + ηn+1 = f ( x* + ηn) = f ( x*) + f ′ ( x*) ηn + (ηn2

)

ηn+1 f ′ ( x*) ηn λ ηn , with λ = f ′ ( x*) Then

η1 = λ η0 , η2 = λ

η1 = λ2 η0 , ..... , ηn = λ

n η0


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Linear stabilityIf the multiplier λ satisfies

( *) 1 f x

then the fixed point x* is (linearly) stable.

If the fixed point is unstable.

Nothing can be said about the marginal case

( *) 1 f x

( *) 1 f x . In this case, graphical way can help!


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Problem:Consider the map . Determine thestability of the fixed point x* = 0.

1 sin

n n x x

At x* = 0 we have .

Nothing can be said about stability. We use a

geometrical approach:

(0) cos(0) 1 f

x* = 0 is a stable equilibrium point!


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Logistic Map

Logistic (or quadratic) map[*]

:


1 (1 )

n n n x rx x (1)

________________________________

[*] Discrete-time analog of the logistic equation for population growth


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Logistic Map (cont.)

Fixed points:


* 0, (0 4)

1* 1 , ( 1)

x r

x r r

REMARKS:For 0 < r < 1 there is just an equilibrium point at the origin ( x* can notbe negative)

A second equilibrium point appears for r > 1

It is stable for 1< r < 3

For r > 3 a period-2 cycle is born


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Bifurcations


A bifurcation represents the sudden

appearance of a qualitatively different solutionfor a nonlinear system as some parameter is

varied.

r

x

Logistic map

Bifurcations occur in four basic varieties:flip, fold, pitchfork, and transcritical (Rasband 1990)


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Bifurcations (cont.)


In the logistic map, at r = 1 the equilibrium x = 0

becomes unstable and a new equilibrium point appears:that is called a transcritical bifurcation.

At r = 3, x* becomes unstable and a 2-cycle appears.

That is called a flip (or period-doubling) bifurcation

FLIP

TRANSCRITICAL


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In one-dimensional maps a 2-cycle exists if and

only if there are two points p and q such that

f ( p) = q and f (q) = p

Equivalently, p must satisfy

f ( f ( p)) = p

Hence, p is a fixed point of the second-iterate

map f 2( x) = f ( f ( p)) . The same for q.

In conclusion:

p and q are fixed points of the second-iterate

map.


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p and q are evaluated by solving the equation f 2( x) – x = 0:

2 (1 ) 1 (1 ) 0r x x rx x x

The fixed points of the 1-st iterate are solution of theabove equation (in fact, f ( x*) = x*, so also f 2( x*) = x*).

After factoring out x and [ x – (1 – 1/r )], we get a

quadratic equation:

2 1 1 11 1 0x x r r r


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p and q are given by

( 1) ( 3)( 1),

2

r r r p q

r

A 2-cycle exists for all r > 3. For r < 3 the roots arecomplex, which means that a 2-cycle does not exist.

What about its stability?


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For evaluating the stability of the 2-cycle, it is sufficient toevaluate the stability of a fixed point, since p and q are fixed

points of the 2nd-iterate map.

Let us compute the multiplier, say, at p (*):

( ( ( ))) ( ( )) ( ) ( ) ( )x p

d f f x f f p f p f q f p

dx

_____________________________________

(*) Remember that:


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Evaluate the multiplier at p:


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The first values where bifurcations take place:

Source: S. H. Strogatz, Nonlinear dynamics and chaos, Perseus Books, 1994.

What happens if r > r ∞ ?

1 6


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Chaos


Qualitative definition(1):

Chaos is a bounded aperiodic long-term

behavior in a deterministic system that exhibits

sensitive dependence on the initial conditions,

thereby rendering long-term predictionimpossible.

1. Bounded: trajectories remain confined in a finite set

2. Aperiodic long-term behavior: there are trajectories which do not settle

down to fixed points, periodic orbits or quasi periodic orbits as t → ∞.

3. Deterministic: the system has no random or noisy inputs or parameters.4. Sensitive dependence on the initial conditions: nearby trajectories separate

exponentially fast.

_____________________________

(1) S.H. Strogatz, Nonlinear dynamics and chaos, p. 323, Perseus Books Pu.,

1994.


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Lyapunov Exponent in One Dimension


In a chaotic system neighboring orbits should separate

exponentially fast, on average.

Let us compute the separation between two orbitsstarting from two nearby initial conditions: x0 and x0 + ε

To what extent two solutions starting from nearly equal

initial conditions are repelling each others?

1st orbit:2

0 0 0 0, ( ), ( ), , ( )n x f x f x f x

2nd orbit:2

0 0 0 0, ( ), ( ), , ( )n x f x f x f x


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The absolute value of separation after n iterates of the two

maps is

01 2 0

'' ' '( ) ( ) ( )n

n n n x x f f x f x f x

0 0( ) ( )n n

n x f x f x

To a 1st order approximation we have

Then, taking the log

1

0

ln ln '( ) lnn

n i i

x f x


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The average value of the sum is

0( )

0 0( ) ( )

n x n n

n x f x f x e

0( )x is called the Lyapunov exponent (LE) of the orbit.

It represents the mean exponential separation of two orbits that

have near initial conditions. A positive LE is a signature of chaos

1

00

'1( ) ln ( )n

i i

x f x n

So we have

From which we get

0ln ( ) ln

n x n x


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Recall the last equation:

dividing by |ε|, taking the limit for ε → 0, andremembering that we get

0

0

( )

0

( )( )

n n x n n

x

x df x e f x

dx

1

00

0( ) 1 1 1lim ln lim ln ( ) lim ln ( )

n n n

i n n n i

x x

f x f x n n n

Taking the limit as n → ∞, we get the definition of LE:

0( )

0 0( ) ( ) n x n n

n x f x f x e

0( )n x

n x e


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Two fundamental facts [*]

:The LE does not depend on the trajectory. More

precisely, it is the same for all the x0 in the basin of

attraction of a given attractor

The limit for n going to infinity exists

In general, one needs to use a computer to

calculate Lyapunov exponents

[*] V.I. Oseledets, A multiplicative ergodic theorem. Lyapunov characteristic numbers

for dynamical systems, Trans.Moscow Math. Soc. 19(1968), 197-231.

Moscov.Mat.Obsch.19(1968),179-210.

2015 one-dimensional maps

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