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It has been shown that Chua's system has a Hamiltonian-Poisson realization (Arieşanu 2013). That is, there exists a Hamiltonian $H=f(x)$ over $x\in \mathbb{R}^3$ and an skew-symmetric matrix $\Pi\in \mathbb{R}^{3,3}$ such that $\dot{x} = \Pi \cdot \nabla{H}$ gives Chua's system. (The matrix elements of $\Pi$ give the Poisson brackets $\Pi_{ij}=\{x_i,x_j\}$ between the coordinates.)

As this system is known to be chaotic and dissipative, I want to know whether other such systems have such a realization. In particular, I want to consider the chaotic Lorenz-63 system. But it's not obvious what the Hamiltonian would have to be in such a case, and without that I can't proceed to finding skew-symmetric matrix $\Pi$. Is such a Hamiltonian known?

For reference, I was also looking at these articles:

But I didn't find in them a closed form of the Hamiltonian. Any help is great.

Semiclassical
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BAYMAX
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    There are two problems. The first is that the Lorentz system is dissipative, i.e. the flow $\phi^t$ is volume contracting on $\mathbb R^3$ ($\det d \phi^t < 1$). Contrast this with a Hamiltonian system, which always preserves the Liouville measure (Lebesgue measure in this case). The second is that the Lorentz system is, as usually stated, odd-dimensional, whereas Hamiltonian systems are always even-dimensional. – A Blumenthal Mar 14 '17 at 19:42
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    @ABlumenthal The first paper in the question uses a definition of Hamilton-Poisson structure. It's not the Hamiltonian system about which you think first when you read this question. So no, that's not a problem here: they did it for Chua system, which is also odd-dimensional and dissipative. – Evgeny Mar 14 '17 at 22:05
  • It would be very nice if we continue the discussion in chat's but I don't know how I can do that ? – BAYMAX Mar 15 '17 at 02:55
  • @ABlumenthal i am unable to find a closed form of Hamiltonian in Lorenz attractor! – BAYMAX Mar 15 '17 at 05:56
  • @Evgeny You have a closed form of Hmailtonian for Lorenz attractor? – BAYMAX Mar 15 '17 at 05:59
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    If I knew, I would write it as an answer :) Have you seen this article: J.-M. Ginoux and B. Rossetto, “Differential geometry and mechanics: applications to chaotic dynamical systems”? They refer to it when they discuss theorems related to Hamilton-Poisson structure, maybe it has methods for finding it? – Evgeny Mar 15 '17 at 07:24
  • @Evgeny Yes,I have seen it , you can join this room for discussion for which I shall be grateful - http://chat.stackexchange.com/rooms/55403/lorenz-attractor-dynamical-systems-chaos-theory – BAYMAX Mar 15 '17 at 07:46

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