No abstract

We report the appearance of three-dimensional (3D) multiheaded chimera states that display cascades of self-organized spatiotemporal patterns of coexisting coherence and incoherence. We demonstrate that the number of incoherent chimera domains can grow additively under appropriate variations of the system parameters generating thereby head-adding cascades of the scroll wave chimeras. The phenomenon is derived for the Kuramoto model of N identical phase oscillators placed in the unit 3D cube with periodic boundary conditions, parameters being the coupling radius r and phase lag α. To obtain the multiheaded chimeras, we perform the so-called 'cloning procedure' as follows: choose a sample single-headed 3D chimera state, make appropriate scale transformation, and put some number of copies of them into the unit cube. After, start numerical simulations with slightly perturbed initial conditions and continue them for a sufficiently long time to confirm or reject the state existence and stability. It is found, by this a way, that multiple scroll wave chimeras including those with incoherent rolls, Hopf links and trefoil knots admit this sort of multiheaded regeneration. On the other hand, multiple 3D chimeras without spiral rotations, like coherent and incoherent balls, tubes, crosses, and layers appear to be unstable and are destroyed rather fast even for arbitrary small initial perturbations.

We report the diversity of scroll wave chimeras in the three-dimensional (3D) Kuramoto model with inertia for N 3 identical phase oscillators placed in a unit 3D cube with periodic boundary conditions. In the considered model with inertia, we have found novel types of patterns which do not exist in a pure system without inertia. In particular, a scroll wave torus-like chimera is obtained under random initial conditions. In contrast to a pure system without inertia, where all chimera states have incoherent inner parts, these states can have partially coherent or fully coherent inner parts in a system with inertia, as exemplified by a scroll wave torus-like chimera. Solitary states exist in the considered model as separate states or can coexist with scroll wave chimeras in the oscillatory space. We also propose a method of construction of 3D images using solitary states as solutions of the 3D Kuramoto model with inertia.