Claudia VALLS

Departament de Matematica Aplicada i Analisi
UNIVERSITAT DE BARCELONA
Gran Via de les Corts Catalanes, 585
08007 Barcelona 
SPAIN
claudia@maia.ub.es

A generalization of the standard separatrix map in the classical Arnold's example of diffusion with two equal parameters

Abstract : Symplectic maps appear in a natural way as Poincar\'e sections of Hamiltonian systems with three or more three degrees of freedom. Let us consider the system with Hamiltonian

$$\begin{array}{c} \vspace*{2mm} H(\vec{I},x,y,\vec{\phi},\varepsilon)= h_{0}(x,y,\vec{I},\varepsilon)+ \mu \varepsilon h_{1}(x,y,I_{2},\vec{\phi}) = \\ \frac{1}{2} (x^{2} + I_{2}^{2})+I_{3} + \varepsilon (\cos y -1)+ \varepsilon \mu (\cos y -1) (\cos \phi_{2} + \cos \phi_{3}),

\end{array} $$

where

$(x,\vec{I}) \in G \subset \Rset^{3}$, and $(y,\vec{\phi}) \in \Tset^{3}.$ The above Hamiltonian is real analytic in the parameter $\varepsilon$, it is an entire function of $x,\vec{I}$, $y,\vec{\phi}$ and we shall consider $\mu=\varepsilon$.

Taking suitable Poincar\'e sections, the resulting map is given as follows:\\ $(x,I_{2},\theta_{2},\theta_{3}) \stackrel{f}{\rightarrow} (x',I_{2}',\theta_{2}', \theta_{3}'),$ where

$$\begin{array}{rcl}\vspace*{2mm} x' &=& x + \Delta x, \\

\vspace*{2mm} I_{2}' &=& I_{2} + \Delta I_{2}, \\

\theta'_{2} &=& \theta_{2} + c_1 {I'}_2 +

\vspace*{2mm} \frac{I_{2}'}{\sqrt{\varepsilon}} \log \abs{\frac{c_2}{x'}}, \\

\theta'_{3} &=& \theta_{3} + c_1 + \frac{1}{\sqrt{\varepsilon}} \log

\abs{\frac{c_2}{x'}} ,

\end{array}$$

where $\Delta x$ and $\Delta I_2$ are given by the related splitting functions, $c_1=2\log (\pi/y_{0})/\sqrt{\varepsilon},$ $c_2=x_{0} y_{0} \sqrt{\varepsilon}$ and $(x_{0},y_{0})$ are the initial values of $(x,y).$

Moreover, $f$ can be seen as a generalization to higher dimensions of the standard separatrix map defined on the cilinder $\Rset$ x $\Tset.$ It is given by $(x,y)\rightarrow (x',y')$ where

$$\begin{array}{rcl} \vspace*{2mm} y'&=& y +\sin (2 \pi x), \\

x'&=& x + c +d \log \abs{y'} , \end{array}$$

$c$ and $d$ being two parameters.