Chaos and Regularity in the Double Pendulum with Lagrangian Descriptors

Abstract

In this paper we apply the method of Lagrangian descriptors as an indicator to study the chaotic and regular behavior of trajectories in the phase space of the classical double pendulum system. In order to successfully quantify the degree of chaos with this tool, we first derive Hamilton's equations of motion for the problem in non-dimensional form, showing that they can be written compactly using matrix algebra. Once the dynamical equations are obtained, we carry out a parametric study in terms of the system's total energy and the other model parameters (lengths and masses of the pendulums, and gravity), to determine the extent of the chaotic and regular regions in the phase space. Our numerical results show that for a given mass ratio, the maximum chaotic fraction of phase space trajectories is attained when the pendulums have equal lengths. Moreover, we give a characterization of the growth and decay of chaos in the system in terms of the model parameters, and explore the hypothesis that the chaotic fraction follows an exponential law over different energy regimes.

Figures (7)

• Figure 1: Diagram of the double pendulum system, taken from AssencioHam
• Figure 2: Contours of the potential energy function given in \ref{['pot_dpend']} for the double pendulum system with $\alpha = 1$ and $\beta = 2$. We have marked the equilibrium points of Hamilton's equations using magenta squares (saddles), circles (local minimum) and diamonds (local maximum).
• Figure 3: A) Poincaré section in the phase space slice given by Eq. \ref{['psec']} for the system with $\alpha = 1$, $\sigma = 1$ and $\mathcal{H}_0 = 20$. We have marked with an orange point a chaotic initial condition, and with blue a regular one; B) Chaos indicators defined in Eq. \ref{['chaos_inds']} calculated for the initial conditions in panel A).
• Figure 4: Classification of a random ensemble of $10^4$ initial conditions carried out by means of the chaos indicators based on Lagrangian descriptors. The orange dots correspond to chaotic motion, whereas blue dots indicate regular behavior. For this simulation we have selected an energy of $\mathcal{H}_0 = 20$, and the model parameters are set to $\alpha = 1$ and $\sigma = 1$. A) Histogram showing the distribution of the $D^n_{L}$ indicator. We have marked with a vertical red line the threshold that separates chaotic from regular behavior; B) Poincaré section overlaid with the classified initial conditions using the $D^n_{L}$ indicator. The red dot shows an initial condition that has not been correctly classified by the method.; C) and D) Same analysis, but for the $S^n_{L}$ indicator.
• Figure 5: Maximum chaotic fraction of phase space as a function of the model parameters $\alpha = l_1/l_2$ and $\sigma = m_1/m_2$ calculated with the indicator $S^n_L$. The range of energy levels used for the simulations is $[\mathcal{H}_1,\mathcal{H}_4+130]$.