Lyapunov Exponents
Chaos and TimeSeries Analysis
10/3/00 Lecture #5 in Physics 505
Comments on Homework
#3 (Van der Pol Equation)

Some people only took initial conditions inside the attractor

For b < 0 the attractor becomes a repellor (time reverses)

The driven system can give limit cycles and toruses but not chaos (?)

Can get chaos if you drive the dx/dt equation instead of
dy/dt
Review (last
week)  Dynamical Systems Theory

Types of attractors/repellors:

Equilibrium points (radial, spiral, saddle) 0D

Limit cycles (closed loops) 1D

2Toruses (quasiperiodic surfaces) 2D

NToruses (hypersurfaces) ND

Strange attractors (fractal) Noninteger D
(Attractor dimension < system dimension)

Stability of equilibrium points:

Find equilibrium point: f(x) = 0 ==> x*,
etc.

Calculate partial derivatives f_{x} etc. at equilibrium

Construct the Jacobian matrix J

Find the characteristic equation: det(J  l)
= 0

Solve for the D eigenvalues: l_{1},
l_{2},
...l_{D}

Find the eigenvectors (if needed) from JR = lR

Stable and unstable manifold (inset & outset)

Organize the phase space

Plot position of eigenvalues in complexplane

If any have Re(l) > 0, point is unstable

Index is number of eigenvalues with Re(l)
> 0

Dimension of outset = index

Volume expansion: dV/dt / V = l_{1}
+ l_{2} + l_{3}
+ ...

By convention, l_{1} > l_{2}
> l_{3} > ...

An attractor has dV/dt < 0

Different rules for stability of fixed points for maps

In 1D, X = X_{0}l^{n}
is stable if l < 1

In 2D and higher, stable if all l are inside
unit circle

Bifurcations occur when l touches unit
circle

Examples of chaotic dissipative flows in 3D:

Driven pendulum

dx/dt = v

dv/dt = sin x  bv + A sin wt

A = 0.6, b = 0.05, w = 0.7

Driven nonlinear oscillator (Ueda)

dx/dt = v

dv/dt = x^{3}  bv + A sin
wt

A = 2.5, b = 0.05, w = 0.7

Driven Duffing oscillator

dx/dt = v

dv/dt = x  x^{3}  bv
+ A sin wt

A = 0.7, b = 0.05, w = 0.7

Driven Van der Pol oscillator

dx/dt = v

dv/dt = x + b(1  x^{2})v
+ A sin wt

A = 0.61, b = 1, w = 1.1 (a torus)

Can get chaos with drive in dx/dt equation

Lorenz attractor

dx/dt = p(y  x)

dy/dt = xz + rx  y

dz/dt = xy  bz

p = 10, r = 28, b = 8/3

Rössler attractor

dx/dt = y  z

dy/dt = x + ay

dz/dt = b + z(x  c)

a = b = 0.2, c = 5.7

Simplest dissipative chaotic flow

dx/dt = y

dy/dt = z

dz/dt = x + y^{2}  Az

A = 2.107

Other simple chaotic flows
General Properties of
Lyapunov Exponents

A measure of chaos (how sensitive to initial conditions?)

Lyapunov exponent is a generalization of an eigenvalue

Average the phasespace volume expansion along trajectory

2D example:

Circle of initial conditions evolves into an ellipse

Area of ellipse: A = pd_{1}d_{2}
/ 4

Where d_{1} = d_{0}e^{l}1^{t
}is
the major axis

And d_{2} = d_{0}e^{l}2^{t
}is
the minor axis

Magnitude and direction continually change

We must average along the trajectory

As with eigenvalues, dA/dt / A = l_{1}
+ l_{2}

Note: l is always real (sometimes
base2, not basee)

For chaos we require l_{1}
> 0 (at least one positive LE)

By convention, LEs are ordered from largest to smallest
l_{1} > l_{2} > l_{3} >
...

In general for any dimension:

(hyper)sphere evolves into (hyper)ellipsoid

One Lyapunov exponent per dimension

Units of Lyapunov exponent:

Units of l are inverse seconds for flows

Or inverse iterations for maps

Alternate units: bits/second or bits/iteration

Caution: False indications of chaos

Unbounded orbits can have l_{1}
> 0

Orbits can separate but not exponentially

Can have transient chaos
Lyapunov Exponent for
1D Maps

Suppose X_{n}_{+1} = f(X_{n})

Consider a nearby point X_{n} + dX_{n}

Taylor expand: dX_{n}_{+1}
= df/dX dX_{n}
+ ...

Define e^{l} = dX_{n}_{+1}/dX_{n}
= df/dX (local Lyapunov number)

Local Lyapunov exponent: l =
log df/dX

Can use any base such as log_{e} (ln) or log_{2}

Since df/dX is usually not constant over the orbit,

We average <log df/dX> over many iterations

For example, logistic map:

df/dX = A(1  2X), and

log df/dX is minus infinity at X = 1/2

l(A) has a complicated
shape

There are infinitely many negative spikes

A = 4 gives l = ln(2) (or 1
bit per iteration)
Lyapunov Exponents for
2D Maps

Suppose X_{n}_{+1} = f(X_{n},
Y_{n}),
Y_{n}_{+1}
= g(X_{n},
Y_{n})

Area expansion: A_{n}_{+1} = A_{n}e^{l}1^{+l}2
(as with eigenvalues)

l_{1} + l_{2}
= <log (A_{n}_{+1}/A_{n})> = <log
det J> = <log f_{x}g_{y}  f_{y}g_{x}>

For example, Hénon
map:

X_{n}_{+1} = 1  CX_{n}^{2}
+ BY_{n} [= f(X, Y)]

Y_{n}_{+1} = X_{n}
[= g(X, Y)]

Alternate representation: X_{n}_{+1} =
1  CX_{n}^{2} + BX_{n}_{1}

Note: This reduces to quadratic map for B = 0

Usual parameters for chaos: B = 0.3, C = 1.4

l_{1} + l_{2}
= <log f_{x}g_{y}  f_{y}g_{x}>
= log B = 1.204 (basee)
(or 1.737 bits per iteration in base2)

Contraction is the same everywhere (unusual)

Numerical calculation gives l_{1}
= 0.419 (basee)
(or 0.605 bits per iteration in base2)

Hence l_{2} = 1.204  0.419 = 1.623
(basee)
(or 2.342 bits per iteration in base2)
Lyapunov Exponents for
3D Flows

Sum of LEs: Sl = l_{1}
+ l_{2} + l_{3}
= <trace J> = <f_{x} + g_{y}
+ h_{z}>

Must be negative for an attractor (dissipative system)

This is the divergence of the flow

It is the fractional rate of volume expansion (or contraction)

For a conservative (Hamiltonian) system, sum is zero

For nonpoint attractors, one exponent must = 0
[corresponding to the direction of the flow]

For a chaotic system, one exponent must be positive

Start with any initial condition in the basin of attraction

Iterate until the orbit is on the attractor

Select (almost any) nearby point (separated by d_{0})

Advance both orbits one iteration and calculate new separation d_{1}

Evaluate log d_{1}/d_{0} in any convenient
base

Readjust one orbit so its separation is d_{0} in same
direction as d_{1}

Repeat steps 46 many times and calculate average of step 5

The largest Lyapunov exponent is l_{1}
= <log d_{1}/d_{0}>

If map approximates an ODE, then l_{1}
= <log d_{1}/d_{0}> / h

A positive value of l_{1} indicates
chaos
General character
of exponents in 3D flows:
l_{1} 
l_{2} 
l_{3} 
Attractor 
neg 
neg 
neg 
equilibrium point 
0 
neg 
neg 
limit cycle 
0 
0 
neg 
2torus 
pos 
0 
neg 
strange (chaotic) 

For flows in dimension higher than 3:

(0, 0, 0, , ...) 3torus, etc.

(+, +, 0, , ...) hyperchaos, etc.
KaplanYorke (Lyapunov)
Dimension

Attractor dimension is a geometrical measure of complexity

Random noise is infinite dimensional (infinitely complex)

How do we calculate the dimension of an attractor? (many ways)

Suppose system has dimension N (hence N Lyapunov exponents)

Suppose the first D of these sum to zero

Then the attractor would have dimension D
(in D dimensions there would be neither expansion nor contraction)

In general, find the largest D for which l_{1}
+ l_{2} + ... + l_{D}
> 0
(The integer D is sometimes called the topological dimension)

The attractor dimension would be between D and D + 1

However, we can do better by interpolating:
D_{KY} = D + (l_{1}
+ l_{2} + ... + l_{D})
/ l_{D+1}

The KaplanYorke conjecture is that D_{KY} agrees
with other methods

Multipoint interpolation doesn't work

2D Map Example: Hénon map
(B = 0.3, C = 1.4)

l_{1} = 0.419 and l_{2}
= 1.623

D = 1 and D_{KY} = 1 + l_{1}
/ l_{2} = 1 + 0.419 / 1.623 = 1.258

Agrees with intuition and other calculations

3D Flow Example: Lorenz Attractor (p = 10, r
= 28, b = 8/3)

Numerical calculation gives l_{1}
= 0.906

Since it is a flow, l_{2} = 0

l_{1} + l_{2}
+ l_{3} = <f_{x} +
g_{y}
+ h_{z}> = p  1  b = 13.667

Therefore, l_{3} = 14.572

D = 2 and D_{KY} = 2 + l_{1}
/ l_{3} = 2 + 0.906 / 14.572 =
2.062

Chaotic flows always have D_{KY} > 2
[Chaotic maps can have any dimension]
Precautions

Be sure orbit is bounded and looks chaotic

Be sure orbit has adequately sampled the attractor

Watch for contraction to zero within machine precision

Test with different initial conditions, step size, etc.

Supplement with other tests (Poincaré section, Power spectrum,
etc.)
J. C. Sprott  Physics 505
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