The Influence of the Kick Amplitude

In this subsection I demonstrate the influence of the kick amplitude $V_0$ for the case $q=4$; similar results can be obtained for $q=3$ and $q=6$. The following discussion holds for small values of $V_0$. The idea is to derive a separatrix mapping that describes the dynamics of the kicked harmonic oscillator in the vicinity of the separatrices and allows to estimate the width of the channels of diffusive dynamics.

The first step is to identify the ${\mathcal V}$ of equation (1.39c) as a high frequency perturbation to the time averaged Hamiltonian ${\mathcal H}_4$ (1.44). At kick times $t_n=n\pi/2$, $n\in\mathbb{Z}$, one has

$$\sqrt{2J}\sin\left( \vartheta+\frac{k\pi}{2} \right) \; = \; \sqrt{2J}\sin\left( (\vartheta+t_n)+\frac{\pi}{2}(k-n) \right),$$ (1.25)

which gives, with the help of equations (1.35), for every fourth kick ($n=4m$, $m\in\mathbb{Z}$)

$$x\cos\frac{k\pi}{2}+p\sin\frac{k\pi}{2},$$

such that the action-angle variables $\vartheta,J$ have been exchanged for the original $x,p$. ${\mathcal V}$ now takes the form

$${\mathcal V}(x,p,t) \; = \; \frac{V_0}{\pi} \sum_{k=1}^4 \... ...l=1}^\infty \cos\left\{l\left(t-\frac{k\pi}{2}\right)\right\},$$ (1.26)

which holds for every fourth kick time $t_n=t_{4m}$ only. This is a useful feature of equation (1.50), as it facilitates the discussion of the dynamics near one of the four separatrices enclosing each phase space cell, rather than near all four of them: while for the dynamics of the web map $M_4$, corresponding to the full Hamiltonian (1.17), it takes four iterates to return into the neighbourhood of an initial value near a separatrix, the same neighbourhood is reached by a single iteration of $M_4^4$, or equivalently by the discrete dynamics generated by the Hamiltonian ${\mathcal H}_4+{\mathcal V}$, where ${\mathcal V}$ in the form (1.50) is used.

From the last formula it becomes clear that ${\mathcal V}$ can indeed be treated as a high frequency perturbation of the HARPER Hamiltonian ${\mathcal H}_4$: the time dependence of ${\mathcal V}$ is given by cosine terms the largest period of which is $2\pi$, whereas the smallest period of ${\mathcal H}_4$ is $2\pi^2/V_0$ -- as is shown below on page -- and can thus be made as large as desired in the limit $V_0\to 0$.

In [LL73] it is argued that in a first approximation the higher frequency terms of a perturbation expanded as in equation (1.50) can be neglected, although they contribute with roughly the same weight ($\sim V_0$) as the small frequency terms. Therefore I can drop all cosine terms with $l\geq 3$ and get the approximant

$${\mathcal V}(x,p,t) \; \approx \; \frac{2V_0}{\pi} \, (\cos x-\cos p)\cos 2t,$$ (1.27)

where it turns out to be useful to keep the explicit $t$-dependence, although strictly speaking this formula applies for $t=2m\pi$ only.

Now consider a typical orbit of the web map in the stochastic region of the web. For some $n\in\mathbb{N}$ the corresponding orbit point $(x_n,p_n)^t$ will be close to and just below of the midpoint of the HARPER separatrix $p(x)=\pi-x$ with $0<x<\pi$, which is displayed in the upper right quadrant of figure 1.13.^1.7

For notational convenience I define $P_i:=(x_n,p_n)^t$ with some $i\in\mathbb{Z}$. At $P_i$, ${\mathcal H}_4$ takes on a certain value ${\mathcal E}_i\approx 0$ since the separatrices are characterized by ${\mathcal H}_4(x,p)=0$. Let $P_i$ be the initial value (at time $\tau_i$) for the successive application of the fourth iterate of the iteration of the web map (1.29), $M_4^4$. The resulting orbit in the $(x,p)$-plane is quasiperiodic if $P_i$ is not too near to the separatrix (cf. figure 1.7f). First the orbit follows the separatrix anticlockwise, until it comes close enough to the hyperbolic point $(0,\pi)^t$, where the orbit turns left and follows the perpendicular separatrix $p(x)=\pi+x$ with $-\pi<x<0$; this is shown schematically in figure 1.14. At some time $\tau_{i+1}=\tau_i+2j\pi$ (where $j$ is the according number of iterations of $M_4^4$) the orbit reaches the point $P_{i+1}=M_4^{4j}P_i$, which is defined as that iterate of $P_i$ under $M_4^4$ that comes closest to the centre of the second separatrix. This point again gives rise to a certain value of ${\mathcal H}_4$,  ${\mathcal E}_{i+1}$. The desired separatrix mapping is now given by the change of the value of ${\mathcal H}_4$ during one such quarter revolution and by the corresponding (approximate) quarter period:

In order to determine an approximate expression for $\Delta {\mathcal E}$, I first derive an explicit solution of the HARPER dynamics

on the separatrix $p=\pi-x$ for $0<x<\pi$. Such a solution is given by

$$\begin{array}{lcl} x(t) & = & \displaystyle 2\arctan\exp\l... ...(t-\tau_i)}\right)\\ [0.5cm] p(t) & = & \pi-x(t), \end{array}$$ (1.26)

where $\tau_i$ is chosen in such a way that $x(\tau_i)=p(\tau_i)=\pi/2$, corresponding to the midpoint of the separatrix. In figure 1.13 the corresponding trajectory connects the limiting points $(\pi,0)^t$ and $(0,\pi)^t$ of the separatrix, for $t$ going from $-\infty$
to $\infty$.

The rate of change of the value of ${\mathcal H}_4$ during the ${\mathcal V}$-perturbed dynamics can then approximately be calculated as

$$\frac{{\mbox{d}}{\mathcal H}_4}{{\mbox{d}}t} \; \approx \; ... ...}\dot{x} + \frac{\partial {\mathcal H}_4}{\partial p}\dot{p},$$ (1.27)

where the partial derivatives are to be taken from equations (1.53), and $\dot{x},\dot{p}$ stem from the Hamiltonian equations according to

$$H(x,p,t) \; \approx \; \frac{ V_0}{\pi}(\cos x+\cos p) + \frac{2V_0}{\pi}(\cos x-\cos p)\cos 2t,$$ (1.28)

which is used as an approximant to the full Hamiltonian (1.17) at times $t_{4m}$. This gives

$$\frac{{\mbox{d}}{\mathcal H}_4}{{\mbox{d}}t} \; \approx \; -\frac{4V_0^2}{\pi^2} \sin x(t) \sin p(t) \cos 2t,$$ (1.29)

which holds for all $t_n$, $n\in\mathbb{Z}$, again. Using the solution (1.54) for $x(t)$ and $p(t)$, the right hand side of equation (1.57) can be approximated by

$$-\frac{4V_0^2}{\pi^2} \sin^2 \left( 2\arctan \exp\left( -\... ...rac{\cos 2t}{\cosh^2\left( \frac{V_0}{\pi}(t-\tau_i) \right)},$$

such that by integration along the whole separatrix I finally obtain^1.8

$$\Delta {\mathcal E}({\mathcal E}_i,\tau_i;V_0) \; \approx ... ...x \;\, -\frac{8\pi\cos 2\tau_i}{\sinh\frac{\pi^2}{V_0}}. % %$$ (1.31)

Thus, while an exact formula for $\Delta {\mathcal E}$ certainly depends on ${\mathcal E}_i$, in the framework of this first approximation $\Delta {\mathcal E}$ is independent of ${\mathcal E}_i$.

It remains to determine the time interval $\Delta\tau$ in equation (1.52b). Within the cells, i.e. away from the separatrices, the equations of motion of the unperturbed HARPER system can be integrated using Jacobian elliptic functions. With

$$c \; := \; \cos x(t) + \cos p(t),$$ (1.32)

which remains constant on each integral curve of the HARPER system, and

$$\tilde{x}(t) \; := \; \cos x(t)$$ (1.33)

equation (1.53a) can be transformed into

$$\dot{\tilde{x}}^2 \; = \; \left( \frac{V_0}{\pi}\right)^2 \left( 1-\tilde{x}^2 \right) \left( 1-(c-\tilde{x})^2 \right).$$ (1.34)

This is essentially the characteristic differential equation satisfied by the elliptic function ${\mbox{cd}}(t;k)$ [DV73], where the parameter $k$ is determined by the value of c:

$$k(c) \; = \; \frac{2-\vert c\vert}{2+\vert c\vert}. % \frac{2\mp c}{2\pm c} \quad \mbox{for} \quad c\gl 0.$$ (1.35)

With $\vert c\vert<2$ the solution of the characteristic equation gives

$$\begin{array}{lcl} \cos x(t) & = & \displaystyle \frac{c}{... ...ad c\gl 0 \\ [0.5cm] \cos p(t) & = & c-\cos x(t). \end{array}$$ (1.36)

For real arguments, as in the present case, ${\mbox{cd}}(t;k)$ is periodic with period $4K$, $K$ being a complete elliptic integral of the first kind:

$$% K(k) = \Int_0^{\pi/2} \frac{\dop\varphi}{\sqrt{1-k^2\sin^... ...{\pi}{2}} \frac{{\mbox{d}}\varphi}{\sqrt{1-k^2\sin^2\varphi}}.$$ (1.37)

This integral is bounded below by $\pi/2$. (The value $\pi/2$ corresponds to the case $\vert c\vert=2$, characterizing the centres of the phase space cells.) Therefore the period of the HARPER solution $\big(x(t),p(t)\big)^t$,^1.9

$$T_{\mbox{\scriptsize\textsc{Harper}}} \; = \; \frac{8K\big(k... ...aystyle \left(1+\frac{\vert c\vert}{2}\right)\frac{V_0}{\pi}},$$ (1.38)

is not smaller than $2\pi^2/V_0$, which tends to infinity for $V_0\to 0$. This justifies the above treatment of ${\mathcal V}$ as a high frequency perturbation of ${\mathcal H}_4$ (see pages f).

Approximate expressions for the integral (1.65) can be found in [AS72] and yield here, in the vicinity of a separatrix:

$$K\big(k(c)\big) \; \approx \; \frac{1}{2} \log \frac{8}{\vert c\vert} \quad \mbox{for} \quad c\to 0.$$ (1.39)

Taking into account that for the separatrix mapping only a quarter of a full revolution has to be considered, equations (1.66) and (1.67) finally give the desired expression for $\Delta\tau$,

$$\Delta\tau({\mathcal E}_i,\tau_i;V_0) \; \approx \; \frac{\pi}{V_0} \log\frac{8V_0}{\pi\vert{\mathcal E}_{i+1}\vert};$$ (1.40)

when solving the upcoming equation (1.71) it turns out that it is technically more convenient to use ${\mathcal E}_{i+1}$ here as a replacement for $c$ rather than ${\mathcal E}_i$. The ${\mathcal E}_i$- and $\tau_i$-dependence in this formula comes from ${\mathcal E}_{i+1}({\mathcal E}_i,\tau_i;V_0)$ -- see equations (1.52a) and (1.59).

With equations (1.59) and (1.68), the separatrix mapping (1.52), approximating the dynamics of $M_4$ near the separatrices, is completely specified. Note that the separatrix mapping can also be obtained in a different way: up to terms of order $V_0^2$, the fourth iterate of the web map (1.29) is given by

$${x \choose p} \; \longmapsto \; {x-2V_0\sin p \choose p+2V_0\sin x} \; = \; M_4^4{x \choose p} + {\cal O}(V_0^2).$$ (1.41)

This map can be generated using the kicked HARPER Hamiltonian

$$\tilde{H}(x,p,t) \; = \; \frac{V_0}{\pi} \left\{ \cos x + \... ...ty}^\infty \delta \left( \frac{t}{2\pi}-n \right) \right\},$$ (1.42)

which finds its application in solid state physics, for example in the theory of electrons in certain magnetic fields [KSD92,Dan95,FGKP95], alongside its unkicked counterpart, the HARPER Hamiltonian ${\mathcal H}_4$. Submitting $\tilde{H}$ to manipulations similar to those of the present subsection, formulae (1.59) and (1.68) can be derived once again.

Using the separatrix mapping, I now can proceed to the estimation of the width of the channels of diffusive dynamics. As a criterion for the border between regular dynamics within the meshes and stochastic dynamics in the channels,

$$% \max_{\tau_n\in\RR} \max_{\tau_i} \left\vert \frac{\partial\tau_{i+1}}{\partial\tau_i}-1 \right\vert \; \approx \; 1$$ (1.43)

may be used, since this expression characterizes the region of phase space where the value of $\tau_i$ begins to change significantly under iteration of the separatrix mapping. Solving relation (1.71) for $\vert{\mathcal E}_{i+1}\vert$ and renaming it $\vert{\mathcal E}_{\mbox{\scriptsize border}}\vert$, I get

$$\vert{\mathcal E}_{\mbox{\scriptsize border}}\vert \; \appro... ...\frac{16\pi^2}{V_0 \sinh\frac{\pi^2}{V_0}}, % e^{-\pi^2/V_0}$$ (1.44)

such that with equation (1.44) I finally obtain for the width $w$ of the channels:

$$% w(V_0) \approx \frac{32\sqrt{2}\pi^3}{V_0^2} e^{-\pi^2/V_... ...0^2} e^{-\frac{\pi^2}{V_0}} \quad \mbox{for} \quad V_0 \to 0.$$ (1.45)

This expression scales with $V_0^{-2} e^{-\frac{\pi^2}{V_0}}$ and, not surprisingly, tends to zero in the limit $V_0\to 0$.

Figure 1.15 illustrates this behaviour: for several values of $V_0$ stochastic webs are obtained numerically by iterating the web map $M_4$ for a large number of times; then the channel widths of these webs are measured -- by determining the largest distance of any two points of the web which approximately lie on the line $p=x$ and near the point $(\pi/2,\pi/2)^t$ -- and compared with the numbers given by the approximating formula (1.73). The agreement between the analytical formula
and the numerical data is reasonably good for $V_0 {\protect\begin{array}{c} <\protect\\ [-0.3cm]\sim \protect\end{array}} 1$
and improves for $V_0\to 0$. Figure 1.15 also indicates that for $V_0 {\protect\begin{array}{c} <\protect\\ [-0.3cm]\sim \protect\end{array}} 0.3$
the precision of the computer algorithm does not suffice any more to produce accurate numerical results, because in this parameter range the web gets very thin -- the width of the channels shrinks to less than $10^{-9}$ here. Similar results hold for the other values of $q\in\tilde{{\mathcal Q}}$.

In this subsection I have discussed how the kick strength $V_0$ of the kicked harmonic oscillator determines the shape of its phase portrait. In particular it has been shown that the periodic kicking acts in a way which is typical for perturbed systems: starting from an integrable system at $V_0=0$ (here: the harmonic oscillator) the kicking (with $V_0\neq 0$) renders the system nonintegrable, and the area of the phase space region of irregular, chaotic dynamics grows with increasing perturbation parameter $V_0$.

Having investigated the shape and the size of the channels of irregular motion in this subsection and the previous one, in the next subsection I turn to the discussion of the most characteristic dynamical aspect of the irregular motion.

Footnotes

...HarperDynamics.^1.7

Any other separatrix $p(x)=\pm x+(2k+1)\pi$ with $l\pi<x<(l+1)\pi$ and $k,l\in\mathbb{Z}$ could be considered as well. It depends on the choice of $k$ and $l$ whether the rotation is clockwise or anticlockwise. The cell centred around the origin of the $(x,p)$-plane exhibits anticlockwise rotation, and neighbouring cells have opposite directions of revolution. See figure 1.13.

... obtain^1.8

More explicitly, $\Delta {\mathcal E}$ has to be calculated in two steps: first one integrates along the separatrix $p=\pi-x$ from $x=\pi/2$ to $x=0$; then the perpendicular separatrix $p=\pi+x$ is followed from $x=0$ to $x=-\pi/2$. A closer investigation shows that this procedure can be replaced by integrating along $p=\pi-x$ from $x=\pi$ up to $x=0$; this corresponds to the time integral from $-\infty$ to $\infty$ using the on-separatrix solution (1.54), as in equation (1.59).

...,^1.9

Note that the period of $(x(t),p(t))^t$ is twice the period of $\cos x(t)$, $\cos p(t)$.