# Nature of separatrix in Fokker--Planck Hamiltonian with two degrees of freedom

+ 4 like - 0 dislike
374 views

Background The semiclassical (weak noise, small $D$) limit of the Fokker--Planck equation

$$\frac{\partial P}{\partial t}=D\frac{\partial^2 P}{\partial x^2}-\frac{\partial}{\partial x}(v(x) P)$$

can be described in terms of the classical Hamiltonian

$$H = p^2 +v(x) p$$

where $v(x)$ denotes the drift. This procedure is described in Graham and Tél, Journal of Statistical Physics, 35, 729 (1984), and the resulting Hamiltonian is called the Freidlin--Wentzell (FW) Hamiltonian (although I can't find where in their book Random perturbations of dynamical systems it is discussed).

My problem I am concerned with an example with two degrees of freedom, written in complex notation as

$$H=|p|^2+p(a z+bz^2)+\bar p(\bar a \bar z+\bar b\bar z^2),$$

with Poisson bracket

$$\{f,g\}=\frac{\partial f}{\partial z}\frac{\partial g}{\partial p} -\frac{\partial f}{\partial p}\frac{\partial g}{\partial z} +\frac{\partial f}{\partial \bar z}\frac{\partial g}{\partial \bar p} -\frac{\partial f}{\partial \bar z}\frac{\partial g}{\partial \bar p}.$$

In this case the complex drift velocity $v(z)=az+bz^2$ generates a Möbius transformation on $z$, with fixed points at $z=0,-a/2b$. Making the canonical transformation

$$\xi_1 = z+\frac{\bar p}{a+\bar a},\qquad \xi_2 = \bar p,$$

with $\{\xi_1,\bar \xi_2\}=1$, one finds that the quadratic part of the Hamiltonian ($b=0$) takes the form

$$H_{b=0} = (a+\bar a) \text{Re}\,\xi_1 \bar \xi_2 + i(a-\bar a)\text{Im}\,\xi_1\bar\xi_2,$$

which reveals the fixed point at $z=0$ to be a focus-focus singularity. I believe this is a general feature of the FW Hamiltonian.

$\xi_1=0$, $\xi_2=0$ are the stable and unstable manifolds (which is which depends on the sign of $\text{Re}\, a$) in the vicinity of the origin. $\xi_2=0$ lies in the $p=0$ plane, corresponding to the drift under $v(z)$ that connects the two fixed points, while the $\xi_1=0$ manifold presumably connects to the other fixed point through $p\neq 0$, forming a heteroclinic cycle of order 2.

Can someone say something more about the nature of this separatrix? Is the system integrable (in which case a second conserved quantity would give us the separatrix immediately), or failing that, is the separatrix smooth?

Update The figure below shows some representative trajectories highly suggestive of a smooth separatrix. Here $a=-0.2+5i$ and $b=0.2$. The $z$-axis is $|p|$. Möbius drifts in the $p=0$ plane between the two fixed points are followed by excursions out of the plane. The focus-focus singularities are clearly visible. This post imported from StackExchange MathOverflow at 2014-11-07 11:21 (UTC), posted by SE-user Austen

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOver$\varnothing$lowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.