Control of Hamiltonian chaos as a possible tool to control anomalous transport in fusion plasmas

It is shown that a relevant control of Hamiltonian chaos is possible through suitable small perturbations whose form can be explicitly computed. In particular, it is possible to control (reduce) the chaotic diffusion in the phase space of a Hamiltonian system with 1.5 degrees of freedom which models the diffusion of charged test particles in a turbulent electric field across the confining magnetic field in controlled thermonuclear fusion devices. Though still far from practical applications, this result suggests that some strategy to control turbulent transport in magnetized plasmas, in particular, tokamaks, is conceivable. The robustness of the control is investigated in terms of a departure from the optimum magnitude, of a varying cutoff at large wave vectors, and of random errors on the phases of the modes. In all three cases, there is a significant region of maximum efficiency in the vicinity of the optimum control term.

Figure 1

Contour plot of $V(x,y,t)$ given by Eq. (3.3) for $t=0$, $a=1$, and $N=25$.

Figure 2

Contour plot of $f_2$ given by Eq. (3.8) for $a=1$ and $N=25$.

Figure 3

Contour plot of $f_3$ given by Eq. (3.9) for $t=0$, $a=1$, and $N=25$.

Figure 4

Poincaré surface of section of a trajectory obtained for Hamiltonian (3.3) using a generic initial condition assuming $a=0.8$.

Figure 5

Poincaré surface of section of a trajectory obtained using a generic initial condition as in Fig. 1 and adding the control term (3.8) to Hamiltonian (3.3) with $a=0.8$.

Figure 6

Mean square displacement $⟨r^2\left(t\right)⟩$ vs time $t$ obtained for Hamiltonian (3.3) with three different values of $a=0.7,0.8,0.9$.

Figure 7

Diffusion coefficient $D$ vs $a$ obtained for Hamiltonian (3.3) (open squares) and Hamiltonian (3.3) plus control term (3.8) (full circles).

Figure 8

Values of the Lyapunov indicator $\nu ({\mathbf{x}}_{\mathbf{0}},T)$ as a function of time $T$ for three trajectories obtained for Hamiltonian (3.3) with $a=0.4$ for the following initial conditions: one strongly chaotic (a) ${\mathbf{x}}_{\mathbf{0}}=(0.865,0.39)$, one weakly chaotic (b) ${\mathbf{x}}_{\mathbf{0}}=(0.8766,0.39)$, and one quasiperiodic (c) ${\mathbf{x}}_0=(0.895,0.39)$.

Figure 9

Poincaré sections of the three trajectories of Fig. 8.

Figure 10

Lyapunov indicator map $\nu ({\mathbf{x}}_0,T=200)$ for Hamiltonian (3.3) for $a=0.4$.

Figure 11

Lyapunov indicator map $\nu ({\mathbf{x}}_0,T=200)$ for Hamiltonian (3.3) for $a=0.4$ with the control term $f_2$ given by Eq. (3.8).

Figure 12

Poincaré sections of the trajectories of (a) Hamiltonian (3.3) and (b) Hamiltonian (3.3) with control term $f_2$ given by Eq. (3.8), with initial conditions ${\mathbf{x}}_0=(0.085,0.385)$.

Figure 13

Histograms of the Lyapunov indicators at $T=200$ for $a=0.4$ with and without control term computed in Figs. 10, 11.

Figure 14

Histograms of the Lyapunov indicators at $T=200$ for $a=0.6$ without control term (upper panel), with control term $f_2$ (middle panel), and with control term $f_2+f_3$ (lower panel).

Figure 15

PDF of the magnitude of the horizontal step size for Hamiltonian (3.3) with $a=0.7$ without the control term (open squares), with the control term $f_2$ (full circles), and with a truncated control term with 12 modes (open circles).

Figure 16

PDF of the horizontal step sizes of the trajectories with small Lyapunov indicator (smaller than 7, full squares) and with large Lyapunov indicator (larger than 7, open squares) for Hamiltonian (3.3) with $a=0.4$.

Figure 17

Diffusion coefficient $D$ vs the magnitude of the control term (3.8) for $a=0.7$. The horizontal dashed line corresponds to the value of $D$ without control term. The dash-dotted line is a piecewise linear interpolation.

Figure 18

Contour plot of the truncation of $f_2(x,y)$ for $a=0.7$ containing the 12 Fourier modes of highest amplitude.

Figure 19

Diffusion coefficient $D$ vs the number of Fourier modes $n$ in the truncation of the control term $f_2$ for $a=0.7$. The dashed line corresponds to the case without control term, the solid line corresponds to the value of the diffusion with the full control term $f_2$, and the dash-dotted line corresponds to a power law interpolation $(\propto n^{-1∕2})$.

Figure 20

Diffusion coefficient $D$ vs the phase error $\gamma$ in the expression (3.8) of $f_2$ for $a=0.7$. The dashed line corresponds to the case without control term, the solid line corresponds to the case with the full control term, and the dash-dotted line corresponds to a quadratic interpolation.

Figure 21

Global picture of the control: The bold curved segment of curve represents a set of integrable Hamiltonians around $H_0$. The gray circles are the domains of existence of a given invariant torus around an integrable Hamiltonian. The gray arrows represent two ways of controlling the Hamiltonian $H_0+\epsilon V$: the first one of order $\epsilon$ and the second one of order ${\epsilon }^2$. The arrows from $H_c$ to $H_c^{\prime }$ and from $H_c^{\prime }$ to $H_0+{\epsilon }^{\prime }V$ represent close to identity canonical transformations.