The study of nonlinear systems continues to yield new and fascinating phenomena. Recently, a striking development has emerged in complex networks: bringing together concepts from statistical and nonlinear physics allowed for the discovery of a novel effect, called the chimera state. It yields a nice example of unexpected, counter-intuitive dynamics in which an array of identical oscillators split into two domains: in one the oscillators are coherent and phase locked, in the other incoherent and desynchronized. Chimera state was discovered in 2002 by Kuramoto and Battogtock [1]; its intriguing mythological name was proposed two years later by Abrams and Strogatz in [2]. Currently, this is an area of intense theoretical and experimental research. An accurate understanding of chimera states is expected to bring important insights on many complex spatiotemporal phenomena from brain operation principles to optical and mechanical systems, and even turbulence in hydrodynamics, see e.g. [3]. Another remarkable new discovered pattern corresponds to the situation when only one or a few oscillators split off from the main synchronized cluster and begin to oscillates with different averaged frequency (i.e. Poincare winding number), called the solitary state. In contrary to chimeras, solitary states generically present in networks with Newtonian dynamics, e.g. Kuramoto model with inertia. Arising in a homoclinic bifurcation, solitary states are Lyapunov stable moreover, their total number grows exponentially with the network size. Intriguingly, all together they obey an essential part of the parameter space preserving in both thermodynamic and conservative limits [4]. An important technological example is supplied by power grids, where solitaries mutually co-exist with the desired synchronous behavior; as a result they can provoke a prompt desynchronization (grid blackout) under sudden disturbances, in spite of tremendous controlling efforts [5]. Very recently, both chimeras and solitary states have been reported for a Vicsek-type model of self propelled particles, opening therefore new perspectives for modeling emergent collective behavior ranging from colloidal particles in physics to swarming and flocking in biology [6].

[1] Y.Kuramoto & D.Battogtokh [2002] "Coexistence of coherence and incoherence in nonlocally coupled phase oscillators" Nonlinear Phenom. Complex Syst. 5, 380. [1] Y.Kuramoto & D.Battogtokh [2002] “Coexistence of coherence and incoherence in nonlocally coupled phase oscillators” Nonlinear Phenom. Complex Syst. 5, 380.
[2] D.Abrams & S.Strogatz [2004] “Chimera states for coupled oscillators”, Phys. Rev. Lett. 93, 174102.
[3] L.Lager, B.Penkovsky, and Yu.Maistrenko [2015] “Laser Chimeras as a paradigm for multi-stable patterns in complex systems”, Nat. Communications, 6, 7752.
[4] P.Jaros, D. Dudkovsky, S.Brezetsky, R.Levchenko, T.Kapitaniak, and Yu.Maistrenko [2018] “Solitary states for coupled oscillators with inertia”. CHAOS 28, 011103.
[5] F.Hellman, P.Schultz, P. Jaros, R. Levchenko, T. Kapitaniak, J.Kurths, and Yu.Maistrenko [2019] “Network induced multistability: Lossy coupling and exotic solitary states”. (Nat.Com: under review).
[6] N.Kruk, Yu.Maistrenko, and H.Koeppl. [2018] “Self-propelled chimeras”. Phys.Rev. E 95, 032219.

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