The purpose of this article is to discuss two basic ideas of Henri Poincaré in the theory of dynamical systems. The first one, the recurrence theorem, got at first a lot of attention but most scientists lost interest when finding out that long timescales were involved. We will show that recurrence can be a tool to find complex dynamics in resonance zones of Hamiltonian systems; this is related to the phenomenon of quasi-trapping. To demonstrate the use of recurrence phenomena we will explore the $ 2:2:3 $ Hamiltonian resonance near stable equilibrium. This will involve interaction of low and higher order resonance. A second useful idea is concerned with the characteristic exponents of periodic solutions of dynamical systems. If a periodic solution of a Hamiltonian system has more than two zero characteristic exponents, this points at the existence of an integral of motion besides the energy. We will apply this idea to examples of two and three degrees-of-freedom (dof), the Hénon-Heiles (or Braun's) family and the $ 1:2:2 $ resonance.
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Figure 2. The $ 2:2:3 $ resonance with the actions $ I_1, I_2, I_3 $ in $ 10\, 000 $ timesteps for Hamiltonian (3) starting outside the primary resonance zones; initial conditions $ q_1(0) = 0.3, q_2(0) = 1.2, q_3(0) = 0.5 $, velocities zero, and parameter values $ \varepsilon = 0.1 $ and (5). Left $ I_1, I_2 $ showing strong energy exchanges, the $ q_1 $ and $ q_2 $ normal modes are unstable. Right the action $ I_3 $ showing variations of order $ 0.04 $ (between $ 0.375 $ and $ 0.450 $)
Figure 3. The $ 2:2:3 $ resonance. Left the Euclidean distance $ d $ in $ 10\, 000 $ timesteps of the orbits to their initial conditions in phasespace outside the primary resonance zones and parameter values of fig. 2; recurrence takes many more timesteps. Right the Euclidean distance $ d $ for the same Hamiltonian (3) but starting at $ q_1(0) = 0.3, q_2(0) = 1.2, q_3(0) = 0 $ and velocities zero; we have now $ q_3(t) = \dot{q}_3(t) = 0, t \geq 0 $. The recurrence is different and faster in this case of pure $ q_1, q_2 $ interaction
Figure 5. Recurrence when starting outside primary resonance zone $ M_1 $ and $ M_2 $ in $ 20\, 000 $ timesteps. $ E_1 = 3.06 $ as in fig. 3 with now left $ q_1(0) = 0.3, q_2(0) = 1.2 $; $ q_3(0) = 1.32068 $, the critical value in $ M_1 $ given by eq. (11). The orbits passing through $ M_1 $ encounter the critical case of $ q_3(0) $, recurrence is slow and different from the case of fig. 3. Right the case $ q_1(0) = 0.3, q_2(0) = 1.2 $; $ q_3(0) = 0.9420 $, the critical value in $ M_2 $ given by eq. (12). The recurrence takes much longer which suggests that this location corresponds with unstable periodic solutions
Figure 6. The actions when starting outside primary resonance zone $ M_1 $ and $ M_2 $ in $ 20\, 000 $ timesteps. Each orbit starts outside the primary resonance zones with $ q_1(0) = 0.3 $, $ q_2(0) = 1.2 $ and all velocities zero. In the first two figures (top) we took $ q_3(0) = 1.32068 $, the critical value in $ M_1 $ given by eq. (11). $ I_3(t) $ varies with magnitude 0.06 in according with the error estimate. The instability of the normal modes forces considerable exchange of energy of the first two modes. The next two figures show the instability of $ M_2 $. When passing this primary resonance zone the critical value of $ q_3(0) = 0.9420 $ taken from eq. (12) plays a part
Figure 4. Recurrence when starting in the primary resonance zone $ M_1 $ in $ 1000 $ timesteps. $ E_1 = 3.06 $ as in fig. 3 with now $ q_1(0) = 0.7823, q_2(0) = 0.9581 $; $ q_3(0) = 1.32068 $, the value given by eq. (11) which puts the orbits in the secondary resonance zone. Initial velocities are zero. The top figure left shows the variations of the actions $ I_1, I_2 $, the top figure right shows $ I_3 $ with small variations as predicted. The recurrence $ d $ (below left) is strong and regular as the orbits are caught in the resonance zone $ M_1 $ with near fairly stable dynamics; the picture for 1000 timesteps shows some white segments but hides the fine-structure of recurrence transitions shown when enlarged for 100 timesteps (below right)
Figure 8. Recurrence starting in the primary resonance zone $M_1$ in $10\, 000$ timesteps. $E_1 = 3.06$ (as in fig. 3) with $q_1(0) = 0.7823, q_2(0) = 0.9581$; $q_3(0) = 1.32068$, the value given by eq. (11) which puts the orbits in the secondary resonance zone. Initial velocities are zero. We have chosen $b_1 = 10, b_2, = b_3 = b_4 = 0$ to demonstrate the instability caused by the secondary resonance corresponding with $\chi_2$. The top figure left shows the variations $0.5$ and $0.3$ of the actions $I_1, I_2$, the top figure right shows $I_3$ with variation $0.2$. The corresponding recurrence $d$ is shown in the third figure.
Figure 7. Dynamics when starting in the primary resonance zone $M_2$ in (because of the instability) $10\, 000$ timesteps. $E_1 = 3.06$ as in fig. 3 with now $q_1(0) = 1.1063, q_2(0) = 0.5532$; $q_3(0) = 0.94202$, values putting the orbits in $M_2$. Initial velocities are zero. The top figure left shows strong variations of the actions $I_1, I_2$, the top figure right shows $I_3$; the variations are strong as we start near an unstable secondary resonance. The orbits are leaving the unstable resonance zone $M_2$. Below the corresponding recurrence $d$.
Figure 9. The actions of the $1:2:2$ resonance can be displayed on a simplex where the front plane contains the solutions on an energy manifold. Dots represent periodic solutions. The figure left is based on first order normalization and shows a continuous family of periodic solutions at action value $\tau_1 = 0$. Right shows the simplex obtained by second order normalization; the continuous family breaks up into six unstable periodic solutions. The figures are from [17] and [21].
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Henri Poincaré (1854-1912) at the time when he submitted his prize-essay for the birthday of the Swedish king Oscar Ⅱ.
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Recurrence when starting outside primary resonance zone
The actions when starting outside primary resonance zone
Recurrence when starting in the primary resonance zone
Recurrence starting in the primary resonance zone
Dynamics when starting in the primary resonance zone
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