Topology of magnetic field lines: Chaos and bifurcations emerging from two-action systems

Nonlinear dynamics of magnetic field lines generated by simple electric current elements are investigated. In general, the magnetic field lines show behavior similar to that of the Hamiltonian systems; in fact, they can be generally transformed into Hamiltonian systems with 1.5 degrees of freedom, obey the Kolmogorov-Arnold-Moser (KAM) theorem, and generate chaotic trajectories. In the case where unperturbed systems are described by two action (slow) and one angle (fast) variables, however, it is found that the periodic orbits of the unperturbed systems vanish for arbitrarily small symmetry-breaking perturbations (a breakdown of the KAM theorem) and drifting or periodic trajectories appear. The mechanism of this phenomenon is investigated analytically by weak nonlinear stability analysis. It is also shown numerically that scattering processes of the perturbed system exhibit typical features of chaotic dynamical systems.

Figure 1

(a) The wire configuration of the single coil system. (b) and (c) The rotation number and the Poincaré section $\varphi =\pi /2$ in the one-action case. Parameters are set as $I_C=0.1,I_L=1.0$, and $B_x=0.005$. (d) For the two-action case, the Poincaré section $z=0$ is shown for six orbits that start at points $(-r_0/\sqrt{2},r_0/\sqrt{2},0)(0.38⩽r_0⩽0.455)$. The parameters values are $I_C=0.1,I_L=0.0$, and $B_x=0.001$. (e) A typical orbit in the invariant subspace $\varphi =0$. The parameter values are the same as in (d) except that $B=0.02$. (Inset) An orbit of the approximated equations (4) and (5).

Figure 2

For the single coil system with a uniform magnetic field, the two-dimensional scattering map $(y_f,z_f)=S(y_i,z_i)$ is displayed (see the text). The parameters are set as $I_C=0.1,I_L=0.0$, and $B_x=0.005$ (two-action case). The values of the final coordinates $y_f$ (a) and $z_f$ (b) are shown in grayscale; the horizontal and vertical axes are the initial coordinates $y_i$ and $z_i$, respectively. In (c) and (d), the final coordinates $y_f$ (c) and $z_f$ (d) are shown as functions of $y_i$, while the value of $z_i$ is fixed to $0.4$.

Figure 3

(a) The wire configuration of the parallel coil system. A hyperbolic fixed point is indicated by an arrow. (b) The Poincaré surface $\varphi =\pi /2$ in the one-action case. The parameters are set as $I_C=I_L=1.0$ and $B_x=0.002$. (c)–(e) For two-action cases with uniform magnetic fields ${\mathbf{B}}_{\mathbf{p}}=(B_x,0,0)$. The current intensities are set as $I_C=0.1$ and $I_L=0$. (c) A Poincaré section $z=0$ for six orbits that start from the points $(-r_0/\sqrt{2},r_0/\sqrt{2},0)(0.38⩽r_0⩽0.455)$. The strength of the uniform magnetic field is set as $B_x=0.0002$. (d) A projection of a typical orbit with positive $y$ values onto the $\mathit{xz}$ plane for $B_x=0.02$. (e) The hyperbolic fixed point, and stable and unstable manifolds on the invariant subspace $\varphi =0$ for $B_x=0.02$. The arrows indicate the directions of the stable and unstable manifolds. (f)–(h) For two-action cases with nonuniform magnetic fields ${\mathbf{B}}_{\mathbf{p}}=((z-a/2)B_x,0,0)$. (f) A Poincaré section $z=0$ for six orbits that start from the points $(-r_0/\sqrt{2},r_0/\sqrt{2},0)(0.38⩽r_0⩽0.455)$. The strength of the uniform magnetic field is set as $B_x=0.005$. (g) A projection of a typical orbit with positive $y$ values onto the $\mathit{xz}$ plane for $B_x=0.02$. (h) The hyperbolic fixed point, and stable and unstable manifolds on the invariant subspace $\varphi =0$ for $B_x=0.05$.