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Chaos and Bifurcation

Mathematical Modeling

& Simulation

POINCARÉ Section

An New Representation

For chaotic oscillations, neither the time function x(t) nor the phase diagram v(x) are very illuminating. Quite often deeper systematics appear, if one uses the phase diagram and marks those points that belong to a certain phase of the motor excitation (quasi ‘stroboscopically‘): such a representation is called Poincaré Section.

The diagram on the left shows an oscillation with doubled period and

thus two different Poincaré points. With period quadrupling, there

would be four points and chaotic oscillations would show an infinite

number. Often the Poincaré points assemble on marked trajectories,

the socalled Strange Attractors.

Flows vs. Maps

• System of ordinary first-order

differential equations:

• where t = independent variable
• yi  y 1 , y 2 ,... , yn.
• Its an example of aflow
• Flow – gives rise to continuous evolution of “field lines” in the n-

dimensional space (or “phase space”)

• If the volume in space remains constant with time, the flow is

conservative.

• If the volume in space decreases with time, the flow is dissipative.
• A dampening effect in dynamics like friction

( ) 1 2

( 1 )

2

2 3

2

1

( , , , , )

;

;

;

;

,

n n

n n f t y y y y

y y

dt

d y y

dt

dy y

y y

















definition:

• a set V (e.g. real numbers between 0 and 1); 0 ≤ x ≤ 1
• a map of the kind f:V  V
• Linear maps: xn+1 = axn + b
• a and b are constants
• linear maps are invertible with no ambiguity  no chaos
• Non-linear maps: The logistic map;

One-dimensional maps

f ( x )   x  1  x 

xn  1  f ( xn )

Logistic Equation for the Dynamics of Population

• The logistic equation (the Verhulst model or logistic growth curve) is

a model of population growth first published by Pierre Verhulst

• The model is continuous in time, but a modification of the

continuous equation to a discrete quadratic recurrence equation

known as the logistic map is also widely used.

• The continuous version of the logistic model is described by the

differential equation:

  K

rN K N dt

dN  

 where r (or μ) is a parameter for rate of maximum population

growth and K is the so-called carrying capacity (that is the

maximum sustainable population).

• Dividing both sides by K and defining x = N/K, will give the differential equation: rx (^1 x ) dt

d x  

 It is known as

the logistic

equation and

has solutions

shown in

figure

Details of the logistic map

y=x

f(x) 0 1

1

y=x

f(x) 0 1

1

0 0.5 1

1

0 0.5 1

1

Fixed

point

fixed

point

2-cycle?

chaos?

a) b)

c)

d)

Analyze first a)  b) then b)  c) , …

FIXED POINTS:

Condition for existence: xf = f(xf)

For a Logistic map: xf( 1 – μ + μxf ) = 0 ;

it means that xf = 0 and xf = 1 – 1/μ and μ is greater or equal to One.

Notice: since 0 ≤ x ≤ 1 ; the second fixed point exists only for μ >=1.

STABILITY:

Let us Define the distance of xn from the fixed point xf

dn = xn – xf

Then consider a neighborhood of xf ( Taylor expansion) :

|dn+1| = | xn+1 – xf |= f(xf + dn) – xf = | df/dx |xf |dn|

Then a requirement that |dn+1| < |dn| means that

| df/dx|xf < 1.

Bifurcation : Plot of fixed points vs μ

MATLAB Program for Bifurcation

% Logistics Map % Classic chaos example. Plots semi-stable values of % x(n+1) = rx(n)(1-x(n)) as r increases from 2.8 to 4. clear scale = 100000; % determines the level of rounding maxpoints = 900; % determines maximum values to plot N = 900; % number of "r" values to simulate a =2.8 ; % starting value of "r" b = 4.0; % final value of "r"... anything higher diverges. ymin = 0.0; ymax = 1.0; % y-axis scale rs = linspace(a,b,N); % vector of "r" values M = 1000; % number of iterations of logistics equation % Loop through the "r" values for j = 1:length(rs) r=rs(j); % get current "r" x=zeros(M,1); % allocate memory x(1) = 0.5; % initial condition (can be from 0 to 1) for i = 2:M, % iterate x(i) = rx(i-1)(1-x(i-1)); end % only save those unique, semi-stable values out{j} = unique(round(scale*x(end-maxpoints:end))); End %%% continued on next page - - -

First case study

• scale = 100000; % determines

the level of rounding

• maxpoints = 900; % determines

maximum values to plot

• N = 900; % number of "r" values

to simulate

• a = 2.8; % starting value of "r"
• b = 3.6; % final value of "r“
• ymin = 0.3 ; ymax = 0.92; % y-

axis scale

• M = 1000; % number of iterations

of logistics equation

Xmin = 2. Xmax = 3. Ymin = 0. Ymax = 0.

Parameter, μ

xn

2.8 3.

Input data for first case study:

•μ =1.0; below this

value the population

cannot survive;

•μ =2.0; oscillatory

approach to the

asymptotic value

•μ =3.0; “period” of

the population

doubles

•μ =3.45; “something

else happens”

The Logistic Map

Parameter, μ

xn

Xmin = 2. Xmax = 3. Ymin = 0. Ymax = 0.

2.8 3.

xn+1 = μxn(1 - xn)

Xmin = 2. Xmax = 3. Ymin = 0. Ymax = 0.

The Logistic Map

Parameter, μ

xn

3.

2.9 3.

Period quadruples at 3.

Xmin = 2. Xmax = 3. Ymin = 0. Ymax = 0.

The Logistic Map

• 3.544090 – period of 8
• 3.564407 – period of 16
• 3.568759 – period of 32
• 3.569692 – period of 64
• 3.569946 – period

doubling ends

Parameter, μ

xn

3.535 3.

Parameter, μ

x n

2. 9

Look into it specially in

the range of μ = 3.5 to

Xmin = 3. Xmax = 3. Ymin = 0. Ymax = 0.