# Calculating the Maximal Lyapunov Exponent for a Hamiltonian System

I am consider the following Hamiltonian: $$\mathcal{H} = \frac{1}{2}(\dot{x}^2 + \dot{y}^2) + \frac{1}{2}(x^2 + y^2) + x^2y - \frac{y^3}{3}.$$ The first step I took were to solve the equations of motion which gave me 4 functions: $$x(t)$$, $$\dot{x}(t)$$, $$y(t)$$, $$\dot{y}(t)$$. However in the Wikipedia article on Lyapunov Exponents, their equation for the maximal Lyapunov exponent $$\lambda(x_0) = \lim_{n\to0}\frac{1}{n}\sum_{i=0}^{n-1}\ln{|f'(x_i)|}$$ depends only on one function $$f(x)$$. How does this generalize to my system?

• Commented Feb 23, 2022 at 10:01
• Also, the equation you list is for discrete systems, not EDOs; and the linked Wikipedia entry already generalizes to more dimensions, using the Jacobian matrix of the system. Commented Feb 23, 2022 at 10:09
• Does this answer your question? How to calculate the maximal Lyapunov exponent(s) of a multidimensional system? Commented Feb 23, 2022 at 13:43
• Yeah I looked at the post but I couldn't figure out what the maps f(x_n, y_n) and g(x_n, y_n) would be since after solving Hamilton's equations, I got functions of t. Also would my state vector be the position or a 4d vector also including the velocities?
– Dan
Commented Feb 23, 2022 at 16:24
• Dan, you're right, my bad, that was for maps, not EDOs. Maybe this is more helpful, with its links to papers 1 and 2. Commented Feb 23, 2022 at 16:33