Introduction to chaos

Deterministic Chaos
U n i
PHYS 306/638
University of Delaware

Chaos vs. Randomness
• Do not confuse chaotic with random:
Random:
– irreproducible and unpredictable
Chaotic:
– deterministic - same initial conditions lead to same
final state… but the final state is very different for
small changes to initial conditions
– difficult or impossible to make long-term predictions

Clockwork (Newton) vs. Chaotic (Poincaré)
•Suppose the Universe is madeUniverse matter interacting according
of particles of
to Newton laws → this is just a dynamical system governed by a (very
large though) set of differential equations
•Given the starting positions and velocities of all particles, there is a
unique outcome → P. Laplace’s Clockwork Universe (XVIII Century)!

Brief Chaotic History:
Poincaré

Brief Chaotic History:
Lorenz

Chaos in the Brave New World of
• Poincaré created anComputers understand such
original method to
systems, and discovered a very complicated dynamics,but:
"It is so complicated that I cannot even draw the
figure."

An Example… A Pendulum
• starting at 1, 1.001, and 1.000001 rad:

Changing the Driving
Force
• f = 1, 1.07, 1.15, 1.35, 1.45

Chaos in Physics
• Chaos is seen in many physical systems:
– Fluid dynamics (weather patterns),
– some chemical reactions,
– Lasers,
– Particle accelerators, …
• Conditions necessary for chaos:
– system has 3 independent dynamical variables
– the equations of motion are non-linear

Why Nonlinearity and 3D Phase
Space?

Dynamical Systems
• A dynamical system is defined as a deterministic mathematical
prescription for evolving the state of a system forward in time
• Example: A system of N first-order, autonomous ODE
dx1 
= F ( x1 , x2 ,K , xn ) 
dt

dx2
= F ( x1 , x2 ,K , xn )  set of points ( x1 , x2 ,K , xn ) is phase space
 
dt ⇒ 
M  [ x1 (t ), x2 (t ),K , xn (t )] is trajectory or flow


dxN
= F ( x1 , x2 ,K , xn ) 
dt 


N ≥ 3 + nonlinearity ⇒ CHAOS becomes possible!

Damped Driven Pendulum:
Part I
• This system demonstrates features of
chaotic motion:
dθ dθ
2

ml 2 + c + mg sin θ = A cos(ω D t + φ )
dt dt
• Convert equation to a dimensionless form:

dω dθ
+q + sin θ = f 0 cos(ω D t )
dt dt

• Part II
3 dynamic variables: ω, θ, t
• the non-linear term: sin θ
• this system is chaotic only for certain values
of q, f0 , and ωD
• In these examples:
– ωD = 2/3, q = 1/2, and f0 near 1
dx1
dθ   dt = f 0 cos x3 −sin x2 − qx1
x1 =
dt  
 dx2
x2 =θ  ⇒ = x1
x3 = ωD t   dt
 dx3
  dt = ωD


Part III
• to watch the onset of chaos (as f0 is
increased) we look at the motion of the
system in phase space, once transients die
away
• Pay close attention to the period doubling
that precedes the onset of chaos...

f0 = 1.35 f0 = 1.45 f0 = 1.47
f0 = 1.48 f0 = 1.49 f0 = 1.50

Sound waves... f = 1.48
f = 1.49
f = 1.50

2

1
amplitude

0

-1

-2

0.000 0.005 0.010 0.015 0.020 0.025

time (s)

Forget About Solving
Equations!
New Language for Chaos:
• Attractors (Dissipative Chaos)
• KAM torus (Hamiltonian Chaos)
• Poincare sections
• Lyapunov exponents and Kolmogorov entropy
• Fourier spectrum and autocorrelation functions

Poincaré Section:
Examples

Poincaré Section:
Pendulum
• The Poincaré section is a slice of the 3D
phase space at a fixed value of: ωDt mod 2π
• This is analogous to viewing the phase
space development with a strobe light in
phase with the driving force. Periodic
motion results in a single point, period
doubling results in two points...

Poincaré Movie
• To visualize the 3D surface that the chaotic
pendulum follows, a movie can be made in
which each frame consists of a Poincaré
section at a different phase...
• Poincare Map: Continuous time evolution is
replace by a discrete map

Pn +1 = f P ( Pn )

f0 = 1.07

f0 = 1.48

f0 = 1.50

f0 = 1.15
q = 0.25

Attractors
• The surfaces in phase space along which the
pendulum follows (after transient motion
decays) are called attractors
• Examples:
– for a damped undriven pendulum, attractor is
just a point at θ=ω=0. (0D in 2D phase space)
– for an undamped pendulum, attractor is a curve
(1D attractor)

Strange Attractors
• Chaotic attractors of dissipative systems (strange

non-integer dimension
attractors) are fractals

⇒ Our Pendulum: 2 < dim < 3

• The fine structure is quite complex and similar to
the gross structure: self-similarity.

What is Dimension?
• Capacity dimension of a line and square:
N ε N ε
1 L 1 L

2 L/2
4 L/2
4 L/4
8 L/8 16 L/4
22n L/2n
N ( ε ) = L (1 / ε )
d d 2n L/2n

d c = lim log N (ε ) / log(1 / ε )
ε →0

Trivial Example: Point, Line,
Surface, …
l
N (ε ) :
ε

N (ε ) : 2

A
N (ε ) :
ε 2

Non-Trivial Example: Cantor
Set
• The Cantor set is produced as follows:
N ε
1 1

2 1/3

4 1/9

8 1/27
dc = lim log 2n / log 3n
n→∞
d c = log 2 / log 3 < 1

Lyapunov Exponents: Part
I
• The fractional dimension of a chaotic
attractor is a result of the extreme
sensitivity to initial conditions.
• Lyapunov exponents are a measure of the
average rate of divergence of neighbouring
trajectories on an attractor.

II
• Consider a small sphere in phase space…
after a short time the sphere will evolve into
an ellipsoid:

ε eλ1t
ε
ε eλ2t

III
• The average rate of expansion along the
principle axes are the Lyapunov exponents
• Chaos implies that at least one is > 0
• For the pendulum: Σλi = − q (damp coeff.)
– no contraction or expansion along t direction so
that exponent is zero
– can be shown that the dimension of the attractor
is: d = 2 - λ1 / λ2

Dissipative vs Hamiltonian
• Chaos
Attractor: An attractor is a set of states (points in the phase space),
invariant under the dynamics, towards which neighboring states in a given basin
of attraction asymptotically approach in the course of dynamic evolution. An
attractor is defined as the smallest unit which cannot be itself decomposed into
two or more attractors with distinct basins of attraction. This restriction is
necessary since a dynamical system may have multiple attractors, each with its
own basin of attraction.
• Conservative systems do not have attractors, since the motion
is periodic. For dissipative dynamical systems, however, volumes
shrink exponentially so attractors have 0 volume in n-dimensional
phase space.
• Strange Attractors: Bounded regions of phase space
(corresponding to positive Lyapunov characteristic exponents) having zero
measure in the embedding phase space and a fractal dimension. Trajectories
within a strange attractor appear to skip around randomly

Dissipative vs Conservative
Chaos: Lyapunov Exponent
• Properties
For Hamiltonian systems, the Lyapunov exponents exist
in additive inverse pairs, while one of them is always 0.
• In dissipative systems in an arbitrary n-dimensional
phase space, there must always be one Lyapunov
exponent equal to 0, since a perturbation along the path
→ (-, -,
results in no divergence. -, -, ...) fixed point (0-D)
→ (0, -, -, -, ...) limit cycle (1-D)
→ (0, 0, -, -, ...) 2-torus (2-D)
→ (0, 0, 0, -, ...) 3-torus, etc. (3-D, etc.)
→ (+, 0, -, -, ...) strange (chaotic) (2+-D)
→ (+, +, 0, -, ...) hyperchaos, etc. (3+-D)

Logistic Map: Part I
• The logistic map describes a simpler system that
exhibits similar chaotic behavior
• Can be used to model population growth:

xn = µ xn −1 (1 − xn −1 )
• For some values of µ, x tends to a fixed point, for
other values, x oscillates between two points
(period doubling) and for other values, x becomes
chaotic….

Logistic Map: Part II
• To demonstrate… xn = µ xn −1 (1 − xn −1 )
x
n

x n -1

Bifurcation Diagrams: Part
I
• Bifurcation: a change in the number of
solutions to a differential equation when a
parameter is varied
• To observe bifurcatons, plot long term
values of ω, at a fixed value of ωDt mod 2π
as a function of the force term f0

Bifurcation Diagrams: Part
II
• If periodic → single value
• Periodic with two solutions (left or right
moving) → 2 values
• Period doubling → double the number
• The onset of chaos is often seen as a result
of successive period doublings...

Bifurcation of the Logistic
Map

Feigenbaum Number
• The ratio of spacings between consecutive
values of µ at the bifurcations approaches a
universal constant, the Feigenbaum number.
µ k − µ k −1
lim = δ = 4.669201...
k → ∞ µ k +1 − µ k

– This is universal to all differential equations
(within certain limits) and applies to the
pendulum. By using the first few bifurcation
points, one can predict the onset of chaos.

Chaos in PHYS
Determinis
306/638

dω tic   ω = ω − ∆ t ( Ω sin θ + qω − f2
sin t )
= −Ω sin θ − qω + f sin t  
2 n+1 n n n D
dt
D
 
⇒ θ n+1 = θ n + ∆ tω n +1
dθ  
=ω
dt 
 θ 0 = π , ω 0 = 0

 2
•Aperiodic motion confined to strange attractors in the phase space
•Points in Poincare section densely fill some region
∞
x(ω) = ∫ e x(t ) dt ⇒ P (ω) =| x(ω) |
iωt 2
0
∞
C (τ ) = ∫
0
[ ( x(t ) − x ) ⋅ ( x(t +τ ) − x )] dt
•Autocorrelation function drops to zero, while power spectrum goes
into a continuum

Deterministic Chaos in PHYS
306/638

Do Computers in Chaos Studies Make
any Sense? Theorem: Although a
Shadowing
numerically computed chaotic trajectory
diverges exponentially from the true
trajectory with the same initial
coordinates, there exists an errorless
trajectory with a slightly different initial
condition that stays near ("shadows") the
numerically computed one. Therefore, the
fractal structure of chaotic trajectories
seen in computer maps is real.

Introduction to chaos

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Introduction to chaos

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