Chimera states through invariant manifold theory

Here $\mathcal{O}\left(\cdot \right)$ stands for Landau's symbols for order functions. So, $t=\mathcal{O}\left(\delta \left(\varepsilon \right)\right)$ is: there exists ɛ₀ ⩾ 0 and K ⩾ 0 such that 0 ⩽ t ⩽ K|δ(ɛ)| for 0 ⩽ ɛ ⩽ ɛ₀.

We select initial conditions satisfying the restrictions for system be in M and we add a small random uniform vector with each coordinate drawn from the interval (0, 0.01).

For all simulations involving the BA networks, we integrate the equations using an explicit Runge–Kutta method of order four.

The complete Möbius group consists of all fractional linear transformations $G\left(w\right)=\frac{aw+b}{cw+d}$ of the complex plane such that ad − bc ≠ 0 but we only consider the subgroup that preserves the unit circle. As in related literature we refer to this subgroup as the Möbius group.

Here we mean the topological boundary, that is, $\partial {M}_{\mathrm{a}\mathrm{s}\mathrm{y}\mathrm{n}\mathrm{c}}={\bar{M}}_{\mathrm{a}\mathrm{s}\mathrm{y}\mathrm{n}\mathrm{c}}{\backslash}{M}_{\mathrm{a}\mathrm{s}\mathrm{y}\mathrm{n}\mathrm{c}}$.