View source for Zaslavsky web map

==Stochastic web map==

The stochastic web map, known as '''Zaslavsky web map''', occurs
in Hamiltonian systems with <math>N>1</math> satisfying the degeneracy condition <math>Hess H_0=0</math>
(Zaslavsky et al., 1986; Zaslavsky et al., 1991).

The web map is generated by a periodically kicked linear oscillator with the
Hamiltonian:
:<math>\label{mylabel}
H=(1/2)(\dot{x}^2+\omega_0^2x^2)-(\omega_0K/T) \cos x \sum_{m=-\infty}^{\infty}
\delta(t/T-m) 
</math>
 
where <math>T</math> is the period and <math>K</math> is the intensity of the kicks. The corresponding map 
<math>\hat{T_{\alpha}}</math> connects the dimensionless coordinates 
<math>(u=\dot{x}/\omega_0, v=-x)</math> between two successive kicks:
:<math>
u_{n+1}= (u_n+K\sin v_n)\cos \alpha + v_n \sin \alpha  
</math>
<math>\label{mylabel1}
\hat{T_{\alpha}}\ :</math>
 
:<math>
v_{n+1}= -(u_n+K\sin v_n)\sin \alpha + v_n \cos \alpha. 
</math>
The most interesting case occurs when the oscillator and the kicks are in resonance:  <math>\alpha =  \alpha_q 
= 2 \pi /q\ ,</math> <math>q \in  {\mathbb N}\ ,</math> <math>q \geq 3\ .</math> Then the stochastic web
tiles the phase plane <math>(u,v)</math> (see Figs. 1-5).
<table>
  <tr>
    <td>[[Image:Figure1GZ.gif |thumb|270px|Thin stochastic web for <math>q=4\ ,</math> <math>K=1.5</math> is filled by the trajectory of the map <math>\hat{T_{\alpha}}\ .</math> There are invariant curves and isolated from the web stochastic layers inside the cells created by the web.]]
    </td>
    <td>[[Image:Figure2GZ.gif |thumb|270px|Magnification of the stochastic web in Fig.1 shows that area of the web is non-uniformly filled and has very complex pattern with islands and subislands.]]
   </td>
   <td>[[Image:Figure3GZ.gif |thumb|270px|Stochastic web for <math>q=6\ ,</math> <math>K=2</math> by the only trajectory of the map <math>\hat{T_{\alpha}}\ .</math> Nonuniform density of points along the web represents a result of the random walk process along the web for a finite time.]]
   </td>
  </tr>
</table>
<table>
  <tr>
   <td>[[Image:Figure4GZ.gif |frame|left|Animation of the random walk process created by the only trajectory of the map <math>\hat{T_{\alpha}}</math> for <math>q=5\ ,</math> <math>K=0.8\ .</math> The 5-fold symmetry web emerges in the phase space as a result of the random walk.]]
   </td>
   <td>[[Image:Figure5GMZ482.gif |frame||The same as in Fig.4 but for <math>q=8\ ,</math> <math>K=.8108745\ .</math>]]
   </td>
  </tr>
</table>
<table>
  <tr>
   <td>[[Image:Figure6GZ.gif |thumb|270px|"Thick isolines" of the web skeleton Hamiltonian 
<math>H_q(u,v)\ ,</math>  for <math>q=4\ .</math> Different colors correspond to the sets of points 
<math>(u,v)</math> that belong to different layers of  <math>H_q \in (E_m-\Delta E/2, E_m+\Delta E/2)\ .</math>]]
   </td>
   <td>[[Image:Figure7GZ.gif |thumb|270px|The web skeleton , as in Fig.6, but for <math>q=3\ .</math>]]
   </td>
   <td>[[Image:Figure8GZ.gif |thumb|270px|The web skeleton , as in Fig.6, but for <math>q=5\ .</math> The pattern is similar to the 5-fold symmetry 2-dimensional quasicrystal.]]
   </td>
  </tr>
</table>
<table>
  <tr>
   <td>[[Image:Figure9GZ.gif |thumb|270px|The web skeleton , as in Fig.6, but for <math>q=8\ .</math>]]
   </td>
   <td>[[Image:Figure10GMZ482.gif |frame|right|Animation of a particle trajectory along the 4-fold symmetry stochastic web for <math>K=6.349</math> demonstrates anomalous (non-Gaussian) diffusion with Levy flights.]]
   </td>
  </tr>
</table>
Some properties of the stochastic webs are: (a) The stochastic web 
can appear for <math>\alpha</math> rational or fairly close to
a rational for finite <math>K\ ;</math> 
(b)The structure of the web can be characterized by the web skeleton, i.e. by some net of channels filled by any trajectory  that starts within the web. Evidently, the skeleton is an invariant, i.e. its shape doesn't depend on time.
(c)The web structure, i.e. its skeleton, has a symmetry of the crystalline type for 
<math>q \in \{q_c\} \equiv (3,4,6) </math> and of the quasicrystal 
type if <math>q \not\in \{q_c\}\ ;</math>
(d) The stochastic web exists for an arbitrary small <math>K</math> if <math>q \in \{q_c\}</math> and the size of the meshes is independent of  <math>K\ .</math> It is conjectured that for
<math>q \not\in \{q_c\}</math> the web exists for any small <math>K</math> but the smaller is
<math>K\ ,</math> the larger are the web's meshes.