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?
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Related: How to calculate the maximal Lyapunov exponent(s) of a multidimensional system? – stafusa Feb 23 '22 at 10:01
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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. – stafusa Feb 23 '22 at 10:09
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1Does this answer your question? How to calculate the maximal Lyapunov exponent(s) of a multidimensional system? – stafusa Feb 23 '22 at 13:43
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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 Feb 23 '22 at 16:24
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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. – stafusa Feb 23 '22 at 16:33
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I read through the paper and I didn't get how this is useful for my problem. Like they gave a formula of the maximal lyapunov exponent for a differentiable mapping $T$ but what would that be in my case? – Dan Mar 02 '22 at 04:52
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Dan, actually the Benettin paper covers the continuous case as well. If you didn't consider it clear, you might prefer to consult Ott's book, chapter 4.4, which has a more thorough description of the method. – stafusa Mar 02 '22 at 13:23