Discrete Dynamics -- The Web Map

The dynamics that is generated by the Hamiltonian (1.17) can obviously be split into two parts, namely the trivial dynamics of a free harmonic oscillator between two successive kicks, and the kick dynamics itself. Because of the stroboscopic nature of the kick it makes sense to consider the dynamic variables $x$ and $p$ at those times only when kicks occur. Let

be the values of $x$ and $p$ immediately before the $n$-th kick. Then the discrete dynamics generated by (1.17) is given by the POINCARÉ map^1.2

or

$$\hspace*{-0.35cm} \fbox{$ \displaystyle \rule[-0.85cm]{0.0... ...+ (p_n+V_0\sin x_n)\cos T. \end{array} % \hspace*{0.1cm} $}$$ (1.9)

Starting at the phase space point $(x_n,p_n)^t$, first the momentum is shifted by $V_0\sin x_n$ due to the kick; then the new point $(x_n,p_n+V_0\sin x_n)^t$ is submitted to harmonic rotation in phase space for a period of time of length $T$. Figure 1.1 gives a graphical account of this dynamics of the map.

In the following I pay particular attention to the resonance cases

$$T_{\rm res} \; = \; \frac{P}{Q} \, \pi \quad \mbox{with} \quad P,Q\in\mathbb{N}.$$ (1.10)

The most important resonances are those for which there are exactly $q$ kicks per period $2\pi$ of the unforced oscillator:

$$T_{\rm res} \; = \; \frac{2\pi}{q} \quad \mbox{with} \quad q\in\mathbb{N}. % \quad q\in % \NN_+,$$ (1.11)

It is just in these resonance cases that a stochastic web develops, as will be seen in section 1.2. In fact, webs with both translational and rotational symmetry can only occur for just a few special values of $q$; this is discussed in subsection 1.2.1. Due to this connection between stochastic webs and the map (1.21) in these particular cases of resonance, maps of this type are commonly called web maps. (See for example [ZSUC91], but note that there a different scaling is used.) In the formulation used here the web map is given by

$$M_q: \quad \left\{ \begin{array}{lcl} x_{n+1} & = & \dis... ... + (p_n+V_0\sin x_n) \cos\frac{2\pi}{q}; \end{array} \right.$$ (1.12)

it depends on the two parameters $q\in\mathbb{N}$ and $V_0\in\mathbb{R}_+$.

I now briefly discuss the web map for some specific values of $q$. The case $q=1$ (i.e. $T=2\pi$) corresponds to cyclotron resonance, where the frequency of the kick coincides with the frequency of the harmonic oscillator:

$$M_1: \quad \left\{ \begin{array}{lcl} x_{n+1} & = & x_n \\ [0.2cm] p_{n+1} & = & p_n+V_0\sin x_n \end{array} \right.$$ (1.13)

This iteration is trivially solved by

$$\begin{array}{lcl} x_n & = & x_0 \; = \; \mbox{const.} \\ [0.2cm] p_n & = & p_0+nV_0\sin x_0 \, . \end{array}$$ (1.14)

In the context of the original model system of a particle moving within an electromagnetic field (subsection 1.1.1), the $x$-component of the particle position at times of kick is subject to uniform acceleration, as shown in figure 1.2.

For $q=2$ ($T=\pi$) the dynamics is similar to that of equation (1.26): the map is given by

$$M_2: \quad \left\{ \begin{array}{lcl} x_{n+1} & = & -x_n... ...[0.2cm] p_{n+1} & = & -p_n-V_0\sin x_n , \end{array} \right.$$ (1.15)

with the explicit solution

$$\begin{array}{lcl} x_n & = & (-1)^n x_0 \\ [0.2cm] p_n & = & (-1)^n \left( p_0+nV_0\sin x_0 \right). \end{array}$$ (1.16)

In this case of half-integer cyclotron resonance the particle's momentum at times of kick increases in quite the same way as for $q=1$, but separately for odd and even $n$; the solution (1.28) can be obtained from equation (1.26) by successively reflecting the orbit points about the origin of phase space.

Because of their evident simplicity -- in particular, the dynamics is confined to one-dimensional lines in phase space and no two-dimensional web structures can arise -- I do not discuss the maps $M_1$ or $M_2$ any further.

For $q\geq 3$ the dynamics becomes much more interesting, and complicated structures in phase space can develop. The most important example is the web map for $q=4$ ($T=\pi /2$), the dynamics of which has been sketched in figure 1.1:

$$M_4: \quad \left\{ \begin{array}{lcl} x_{n+1} & = & p_n + V_0\sin x_n\\ [0.2cm] p_{n+1} & = & -x_n. \end{array} \right.$$ (1.17)

Each kick is followed by a rotation in phase space through a quarter circle. The web maps for $q=3$ and $q=6$ ($T=2\pi /3$ and $T=\pi /3$, respectively) are also of importance but not given here explicitly, because the corresponding formulae cannot be further simplified much beyond the form of equation (1.24).

The structures that evolve in phase space when $M_q$ is iterated for a large number of times are discussed in section 1.2, with emphasis laid on the cases $q=3,4,6$.

Footnotes

... map^1.2

Alternatively, $x$ and $p$ immediately after the kicks could be considered. The resulting POINCARÉ map would be topologically conjugate to $M$ as given by equations (1.20).