Geometric investigation of chaos unfolding in Hamiltonian systems

Abstract

In this work we revisit the geometric approach to chaos in Hamiltonian dynamics, by means of the Jacobi-Levi-Civita equation (JLCE). We inspect numerically two low-dimensional dynamical systems; show that, along chaotic orbits, the exponential divergence between nearby trajectories quantified by the JLCE does not unfold in a continuous manner, rather is closer to a multiplicative discrete process: in correspondence of each turning point, where the trajectory bounces away from the boundary of the energetically allowed region, the relative separation increases sharply and abruptly. We highlight through analytical and numerical arguments that the chaotic rather than regular nature of the trajectory is determined by the details of the scattering with the boundary, and interpret these results in terms of parametric resonance theory, and specifically the Mathieu equation.

Figures (7)

• Figure 1: An example of regular trajectory in the HH system. Left-top plot: the black curve is the trajectory in the $(x,y)$ plane computed using Eq. (\ref{['eq:geodesics']}) in the interval from $s= 0$ to $s = 30$. The blue curve is the energy boundary, $E = V= 1/11$. Initial conditions are: $x(0) = 0.2172, y(0) = 0.11205$ (shown as the red dot), $(dx/ds)(0) = -1.63898, (dy/ds)(0) = 2.49126$. Right-top plot shows a part of the time trace ${\cal R}(s)$. Notice that ${\cal R}_{min} = 2/E^2 = 242$. Left-bottom plot is the Poincaré plot, obtained taking the intersections of the trajectory with the line $y = 0, v_y > 0$. The right-bottom plot shows the power spectra of $\sqrt{{\cal R}/2}$. Frequency is normalized to $\omega_0 = \sqrt{{\cal R}_{min}/2}= E^{-1}=11$.
• Figure 2: A chaotic trajectory in the HH system. Initial conditions are the same of the previous figure, but now the energy is $E = 1/7$, ${\cal R}_{min} =98$ .
• Figure 3: The content of the figure is the same of Figs. (\ref{['fig:1']}, \ref{['fig:2']}), relative to a trajectory of the aHH system. Here, $E = 1/20$, ${\cal R}_{min} =800$, and initial conditions are: $x(0) = -0.2172, y(0) = 0.14205, (dx/ds)(0) = -2.90459, (dy/ds)(0) = -7.01845$.
• Figure 4: Upper plot: the red curve is ${\cal R}$ from a chaotic trajectory of the Hénon-Heiles system; the blue curve is the deviation $J_\perp$ (arbitrarily rescaled in order to fit the window) computed from Eq. (\ref{['eq:j2db']}), feeded with ${\cal R}$. The lower plot shows an expanded view of the same curves, over a restricted time span. The energy is $E = 1/6.25$. Initial conditions are: $x(0) = -0.0172, y(0) = 0.24205, (dx/ds)(0) = -2.90459, (dy/ds)(0) = 3.01845$.
• Figure 5: The red curve is ${\cal R}$ from a regular trajectory of the anti-Hénon-Heiles system; the blue curve is the deviation $J_\perp$ (arbitrarily rescaled in order to fit the window) computed from Eq. (\ref{['eq:j2db']}), feeded with ${\cal R}$. The energy is $E = 1/20$. Initial conditions are: $x(0) = 0.12, y(0) = 0.03, (dx/ds)(0) = -2, (dy/ds)(0) = 3$.