Local predictability and nonhyperbolicity through finite Lyapunov exponent distributions in two-degrees-of-freedom Hamiltonian systems

By using finite Lyapunov exponent distributions, we get insight into both the local and global properties of a dynamical flow, including its nonhyperbolic behavior. Several distributions of finite Lyapunov exponents have been computed in two prototypical four-dimensional phase-space Hamiltonian systems. They have been computed calculating the growth rates of a set of orthogonal axes arbitrarily pointed at given intervals. We analyze how such distributions serve or not for tracing the orbit nature and local flow properties such as the unstable dimension variability, as the axes are allowed or not to tend to the largest stretching direction. The relationship between the largest and closest to zero exponent distribution is analyzed. It shows a linear dependency at short intervals, related to the number of degrees of freedom of the system. Finally, the hyperbolicity indexes, associated to the shadowing times, are calculated. They provide interesting information at very local scales, even when there are no Gaussian distributions and the values cannot be regarded as random variables.

Figure 1

Poincaré cross sections with the $y=0$ plane for the studied Hamiltonian and three values of the $ϵ$ parameter. Initial condition is $x=0.03744$, $y=0$, $\dot{x}=0.0480$. (a) $ϵ=4.5$, period-2 orbit (marked with a cross). (b) $ϵ=4.4$, between tori orbit (big lobe). (c) $ϵ=4.6$, chaotic orbit (full filling the phase space). The cross section of another ergodic orbit ($x=0.03$, $y=0$, $\dot{x}=0.04796$) has been also plotted in the first and second cases in order to ease the visualization of the phase portrait.

Figure 2

Finite Lyapunov exponent distributions for the Eq. (3) Hamiltonian and $ϵ=4.5$, 2-periodic orbit. (a) $\Delta t=0.01$, (b) $\Delta t=1$, (c) $\Delta t=7$, (d) $\Delta t=10$, (e) $\Delta t=100$.

Figure 3

Finite Lyapunov exponent distributions for the Eq. (3) Hamiltonian and $ϵ=4.4$, between tori orbit. (a) $\Delta t=0.01$, (b) $\Delta t=1$, (c) $\Delta t=7$, (d) $\Delta t=10$, (e) $\Delta t=100$.

Figure 4

Finite Lyapunov exponent distributions for the Eq. (3) Hamiltonian and $ϵ=4.6$, chaotic orbit. (a) $\Delta t=0.01$, (b) $\Delta t=1$, (c) $\Delta t=7$, (d) $\Delta t=10$, (e) $\Delta t=100$.

Figure 5

Finite Lyapunov exponent distributions for the Hénon-Heiles Hamiltonian. (a) An unstable periodic orbit and $\Delta t=0.01$, (b) a period-5 orbit and $\Delta t=0.01$, (c) a period-5 orbit and $\Delta t=1$, (d) a period-5 orbit and $\Delta t=10$.

Figure 6

Finite time Lyapunov exponent distributions for the Hénon-Heiles Hamiltonian. All panels are for the cycle orbit. (a) $\Delta t=0.01$, (b) $\Delta t=1$, (c) $\Delta t=10$.

Figure 7

Hyperbolicity index, calculated from the distributions of ${\chi }_1$ (upper) and ${\chi }_2$ (lower) finite-time Lyapunov exponents. Equation (3) Hamiltonian values are marked by diamonds, (dotted) period-2, (dashed) between tori orbit, (dashed dotted) chaotic. Hénon-Heiles Hamiltonian values are marked with triangles, (isolated point) UPO, (long dashed) period-5, (solid) cycle.