Jumps of adiabatic invariant at the separatrix of a degenerate saddle point

We consider a slow-fast Hamiltonian system with two degrees of freedom. One degree of freedom corresponds to slow variables, and the other one corresponds to fast variables. A characteristic ratio of the rates of change of slow and fast variables is a small parameter κ. For every fixed value of the slow variables, in the phase portrait of the fast variables there are a saddle point and separatrices passing through it. When the slow variables change, phase points may cross the separatrices. The action variable of the fast motion is an adiabatic invariant of the full system as long as a trajectory is far from the separatrices: value of the adiabatic invariant is conserved with an accuracy of order of κ on time intervals of order of 1/κ. A passage through a narrow neighborhood of the separatrices results in a jump of the adiabatic invariant. We consider a case when the saddle point is degenerate. We derive an asymptotic formula for the jump of the adiabatic invariant which turns out to be a value of order of κ(3/4) (in the case of a non-degenarate saddle point a similar jump is known to be a value of order of κ). Accumulation of these jumps after many consecutive separatrix crossings leads to the "diffusion" of the adiabatic invariant and chaotic dynamics. We verify the analytical expression for the jump of the adiabatic invariant by numerical simulations. We discuss application of the obtained results to the description of charged particle dynamics in the Earth magnetosphere.

Particle chaos in the Earth's magnetotail

Nonlinear particle dynamics is studied both in current sheets and near neutral lines. The parameter governing particle chaos in a current sheet with a constant normal component, B(n), is kappa=(R(min)/rho(max))(1/2), where R(min) is the minimum field line radius of curvature and rho(max) is the maximum gyroradius. In such a current sheet, motion can be viewed as a combination of a component normal to the current sheet and a tangential component. The parameter kappa represents the ratio of the characteristic time scale of the normal component to the tangential, and thus, particle chaos is maximized for kappa approximately 1. For kappa<<1, the slow motion preserves the action integral of the fast motion, J(z), except near the separatrix, the phase space boundary separating motion that crosses the current sheet midplane from that which does not. Near a linear neutral line, it is found that the parameter b(n), which is the ratio of the characteristic vertical and horizontal field strengths, rather than kappa governs particle chaos. In the limit b(n)<<1, the slow motion again preserves J(z), and J(z) has the same analytic form as in a constant B(n) current sheet. In the limit of b(n)<<1, the structure of x-p(x) phase space is controlled by the stable and unstable manifolds associated with the unstable fixed point orbit at (x,p(x))=(0,0), and this structure lies along a contour of constant J(z).