Classification of equilibrium points and periodic solutions

We classify equilibrium points according to their symmetrical properties in Table 2. There exist four kinds of equilibrium points in Eqs. (7). Figure 2 shows locations of each equilibrium point in (r₁, s₁) and (r₂, s₂) space. The origin (��) is an equilibrium point with full symmetry $\Gamma$. The equilibrium points (��) and (��) are $\sigma_1$ - and $\sigma_2$ -invariant equilibrium points by the definition Eq. (10), respectively. The equilibrium points (� ) has only symmetry operation I₄, thus it is asymmetry. When there is an asymmetrical equilibrium point, the orbit of $\Gamma$ (14) defines immediately three other equilibrium points, see Fig. 2.

By the definition of symmetrical periodic solution, i.e., Eq. (12), we classify periodic solutions observed in Eqs. (7), see Table 3. From this table, we see that in total eight kinds of periodic solutions exist in Eqs. (7) (In Ref. [8] they classified periodic solutions into five types). All of them is sketched on (r₁, r₂) plane in Fig. 3. We will use the name written in each sub-caption here.

Table2: Classification of equilibrium points (abbrev. EP) according to their symmetrical properties.  

type (m) symmetry operation #EP symbol

full symmetry (0) 1 ��

2 ��

2 ��

asymmetry 4 �~

  
Figure2: Classification of equilibrium points according to their symmetrical properties. Symbols are referred to Table 2. Axes (r_i, s_i) represent subspace satisfying $\sigma_i x = x$ where x = (r₁, s₁,  r₂, s₂) called $\sigma_i$-invariant subspace (i=1, 2). The origin is invariant under coordinate transformation by $\sigma_1$, $\sigma_2$ and $\bar{I_4}$.

Table3: Classification of periodic solutions (abbrev. PS) according to their symmetrical properties.  

symmetries #PS comment

1 in-phase

.. 1 anti-phase

.. 2 "shifted" in-phase

.. 2 "shifted" anti-phase

.. 2 "shifted" almost in-phase

.. 2 "shifted" almost anti-phase

.. 2

.. 4 asymmetry

   Figure3: Classification of periodic solutions. Parenthesized number indicates number of periodic solutions corresponding to #PS in Table 3.