Lyapunov Exponent

# 1 Definitions

As defined in OU MS327:

# 1.1 2D Real Map

$x_{n+1} = f(x_n, y_n) ; y_{n+1} = g(x_n, y_n)$

# 1.2 Jacobian Matrix

$J_n = \begin{pmatrix} \frac{\partial f}{\partial x_n} & \frac{\partial f}{\partial y_n} \\ \frac{\partial g}{\partial x_n} & \frac{\partial g}{\partial y_n} \end{pmatrix}$

$J = \prod_1^n J_n$

# 1.3 First Lyapunov Exponent

$h_1 = \lim_{n \to \infty} \frac{1}{n} \log \left| J u \right|$

# 1.4 Second Lyapunov Exponent

$h_1 + h_2 = \lim_{n \to \infty} \frac{1}{n} \log \left| \det J \right |$

# 2 Properties

Assuming the orbit is bounded and not asymptotically periodic:

$h_1 > 0$ means chaotic
$h_1 + h_2 < 0$ means attractor
chaotic and attractor means strange attractor

# 2.1 Lyapunov Dimension

For strange attractors:

$\dim_L = 1 + \frac{h_1}{|h_2|}$

Otherwise if $h_1 < 0$ , then $\dim_L = 0$ , and if $h_1 + h_2 \ge 0$ then $\dim_L = 2$ .

# 3 Calculation

Using the Burning Ship as an example in the C programming language:

// initial coordinates
double a = ...;
double b = ...;
double x = 0;
double y = 0;

// generic unit vector
double t = 2 * M_PI * rand() / (double) RAND_MAX;
double u = cos(t);
double v = sin(t);

// Lyapunov exponent accumulators
double h1 = 0;
double h1h2 = 0;

int escaped = 0;
for (int n = 1; n < N; ++n)
{
  // check orbit remains bounded
  double x2 = x * x;
  double y2 = y * y;
  if (x2  + y2 > 4)
  {
    escaped = 1;
    break;
  }

  // Burning Ship iteration
  double xn = x2 - y2 + a;
  double xy = 2 * x * y;
  double yn = (xy >= 0 ? xy : -xy) + b;

  // Burning Ship Jacobian matrix
  double Jxx = 2 * xn;
  double Jxy = -2 * yn;
  double Jyx = 2 * (xy >= 0 ? yn : -yn);
  double Jyy = 2 * (xy >= 0 ? xn : -xn);

  // accumulate h1
  double un = Jxx * u + Jxy * v;
  double vn = Jyx * u + Jyy * v;
  double l2 = un * un + vn * vn;
  double l = sqrt(l2);
  h1 += log(l);
  u = un / l;
  v = vn / l;

  // accumulate h1+h2
  double det = Jxx * Jyy - Jxy * Jyx;
  h1h2 += log(det >= 0 ? det : -det);
  x = xn;
  y = yn;
}

// finish
if (escaped == 0)
{
  h1 /= N;
  h1h2 /= N;
  ...
}

# 4 No Strange Attractors

# 4.1 The Mandelbrot Set

The Mandelbrot set map is complex $z \to z^2 + c$ :

$x_{n+1} = x_n^2 - y_n^2 + a$ $y_{n+1} = 2 x_n y_n + b$

$J = \begin{pmatrix} 2x & -2y \\ 2y & 2x \end{pmatrix}$

$J \begin{pmatrix} u \\ v \end{pmatrix} = \begin{pmatrix} 2 x u - 2 y v \\ 2 y u + 2 x v \end{pmatrix}$

$\begin{align} |Ju|^2 &= (2 x u - 2 y v)^2 + (2 y u + 2 x v)^2 \\ &= 4x^2 u^2 + 4y^2 v^2 - 8 x y u v + 4 y^2 u^2 + 4 x^2 v^2 + 8 x y u v \\ &= 4 (x^2 + y^2) (u^2 + v^2) \\ &= 4 (x^2 + y^2) \end{align}$

$|\det J| = 4 (x^2 + y^2)$

$h_1 = \lim_{n \to \infty} \frac{1}{n} \frac{1}{2} \log (4 (x^2 + y^2))$

$h_1 + h_2 = \lim_{n \to \infty} \frac{1}{n} \log (4 (x^2 + y^2))$

That is, $h_1 + h_2 = 2 h_1$ , so they have the same sign.

But strange attractors require them to have opposite signs.

Therefore there are no strange attractors in the Mandelbrot set map.

# 4.2 Complex-Analytic

The Cauchy-Riemann equations mean the Jacobian has the form

$J = \begin{pmatrix} s & -t \\ t & s \end{pmatrix}$

So $|J (u \, v)^T|^2 = s^2 + t^2 = |\det J|$ and the same argument as above applies.

Therefore there are no strange attractors in 1D complex-analytic maps considered as 2D real maps.

# 4.3 The Burning Ship

Case $xy \ge 0$ , as Mandelbrot set:

$|J u|^2 = 4 (x^2 + y^2)$

$\det J = 4 (x^2 + y^2)$

Case $xy < 0$ :

$J = \begin{pmatrix} 2x & -2y \\ -2y & -2x \end{pmatrix}$