Chimera states in coupled Kuramoto oscillators with inertia

Abstract

The dynamics of two symmetrically coupled populations of rotators is studied for different values of the inertia. The system is characterized by different types of solutions, which all coexist with the fully synchronized state. At small inertia the system is no more chaotic and one observes mainly quasiperiodic chimeras, while the usual (stationary) chimera state is not anymore observable. At large inertia one observes two different kind of chaotic solutions with broken symmetry: the intermittent chaotic chimera, characterized by a synchronized population and a population displaying a turbulent behaviour, and a second state where the two populations are both chaotic but whose dynamics adhere to two different macroscopic attractors. The intermittent chaotic chimeras are characterized by a finite life-time, whose duration increases as a power-law with the system size and the inertia value. Moreover, the chaotic population exhibits clear intermittent behavior, displaying a laminar phase where the two populations tend to synchronize, and a turbulent phase where the macroscopic motion of one population is definitely erratic. In the thermodynamic limit these states survive for infinite time and the laminar regimes tends to disappear, thus giving rise to stationary chaotic solutions with broken symmetry contrary to what observed for chaotic chimeras on a ring geometry.In 2002, simulations of abstract mathematical models revealed the existence of counterintuitive "chimera states", where an oscillator population splits into two parts, with one synchronizing and the other oscillating incoherently, even though the oscillators are identical. Since then, these counterintuitive states have become a relevant subject of investigation for experimental and theoretical scientists active in different fields, as testified by the rapidly increasing number of publications in recent years (for a review see [1, 2]). In this paper we analyze novel chimera states emerging in two symmetrically coupled populations of oscillators with inertia. In particular, the introduction of inertia allows the oscillators to synchronize via the adaptation of their own frequencies, in analogy with the mechanism observed in the firefly Pteroptix malaccae [3]. The modification of the classical Kuramoto model with the addition of an inertial term results in first order synchronization transitions and complex hysteretic phenomena [4][5][6][7][8][9]. Furthermore, networks of rotators have recently found applications in different technological contexts, including disordered arrays of Josephson junctions [10] and electrical power grids [11][12][13][14] and they could also be relevant for micro-electromechanical systems and optomechanical crystals, where chimeras and other partially disordered states likely play an important role with far reaching ramifications.

Chimera at the phase-flip transition of an ensemble of identical nonlinear oscillators

“…Surprisingly spatiotemporal patterns mimicking chimera states were also found in real world systems, which include the unihemispheric sleep of animals [55], the multiple time scales of sleep dynamics [56], and so on. Very recently, chimera states have also been reported in a network of two populations of Kuramoto model with inertia [57,58], which is also a model used in the analysis of power grids [59]. Further investigations on identifying the intricacies involved in the mechanism of the onset of chimera states is of vital importance from the perspective of neuroscience because of the concept of "bumps" of neuronal activity [60,62] associated with it.…”

Spontaneous synchronisation and nonequilibrium statistical mechanics of coupled phase oscillators

“…Some representative ones are Refs. [48,49,50,51,52,53,54,55]. This review was entirely devoted to studies of mean-field interaction between the oscillators, namely, the case where every oscillator interacts with every other with a strength that is the same for every pair, thereby representing an extreme case of long-range interactions.…”

“…Eq. (51). As may be shown [43], the expansion coefficients b n for this case satisfy b 0 (θ, 0) ∼ exp[r st cos θ], b n (θ, 0) = 0 for n > 0, so that only the n = 0 term in the expansion (57) has to be taken into account; then, with Φ 0 (x) ∼ exp(−x 2 /2), the product Φ 0 v/ √ 2T Φ 0 v/ √ 2T appearing in the expansion correctly reproduces the velocity-part of the density…”

Importance sampling of rare events in chaotic systems

“…uniform in the phase-space). The FTLE and its distribution was introduced in the 1980s [27][28][29] and has been used to study turbulent flows [30], Hamiltonian dynamics [31], chimera states [32], characterize dynamical trapping [27,[33][34][35][36], among others [37,38]. The distribution of FTLE is related to the generalized dimensions [39] and often follows a large deviation principle, where t o is the extensive parameter [19,39].…”