Definition of symmetrical properties

[Equation]

Equations (7) can be rewritten as   If there exists a matrix P satisfying   then Eq. (8) is called P-symmetrical equation.
[Equilibrium point]

We say that an equilibrium point e₀ such that

 

Pe₀ = e₀ (10)

is P-invariant equilibrium point.
[Periodic solution]

We assume a periodic solution of Eq. (8) with initial condition as

$$x(t) = \varphi(x_0, t).$$ (11)

If there exists a matrix P and a time $\tau_P$ such that   then we call that the periodic solution $\varphi(x, t)$ is $(P, \tau_P)$- symmetrical periodic solution. The phase difference $\phi$ between the waveforms of each oscillator is defined by: where L is the period of the periodic solution. We call solutions in-phase and anti-phase when $\phi=0$ and $\phi=\pi$, respectively. Thus symmetries of periodic solutions have both a spatial component P and a temporal component $\tau_P$.

Consider Eqs. (7), matrices satisfying Eq. (9) are

where I₂ is $2\times 2$ identity matrix, O is $2\times 2$zero matrix and $\bar{I_2} = -I_2$. The set $\Gamma$: forms an abelian group with the multiplication as shown in Table

1.

  Table1: Group table of $\Gamma$.