Stable concurrent synchronization in dynamic system networks

Abstract

In a network of dynamical systems, concurrent synchronization is a regime where multiple groups of fully synchronized elements coexist. In the brain, concurrent synchronization may occur at several scales, with multiple "rhythms" interacting and functional assemblies combining neural oscillators of many different types. Mathematically, stable concurrent synchronization corresponds to convergence to a flow-invariant linear subspace of the global state space. We derive a general condition for such convergence to occur globally and exponentially. We also show that, under mild conditions, global convergence to a concurrently synchronized regime is preserved under basic system combinations such as negative feedback or hierarchies, so that stable concurrently synchronized aggregates of arbitrary size can be constructed. Robustnesss of stable concurrent synchronization to variations in individual dynamics is also quantified. Simple applications of these results to classical questions in systems neuroscience and robotics are discussed.

A Survey on Aerial Swarm Robotics

“…Such an exponentially-safe and robust synchronization framework can also be used to study the synchronization stability and robustness of networked nonlinear dynamics connected by a synchronization controller or by diffusive communication couplings [27], [73]. One major advantage of incremental stability in a synchronization framework [27], [28], [73] over the passivity formalism is that a hierarchicallycombined structure of dynamic systems, emphasized in this paper, can be handled more easily because of differential contraction analysis without using some implicit motion integrals.…”

“…One major advantage of incremental stability in a synchronization framework [27], [28], [73] over the passivity formalism is that a hierarchicallycombined structure of dynamic systems, emphasized in this paper, can be handled more easily because of differential contraction analysis without using some implicit motion integrals.…”

“…Instead of exchanging communication continuously or over finite intervals of time, as in the previous cases, it is sufficient for stability to exchange information between neighboring agents at discrete instants of time. Conditions for stability in such cases have been found for single integrator dynamics on undirected graphs [75], consensus on balanced digraphs [76], convergence to a trajectory on time-varying graphs [77], and synchronization of general nonlinear dynamics on balanced graphs [28], [73], [78]. These conditions typically depend on the underlying dynamics and also help determine the conditions under which communication must be triggered.…”

Robotic Herding of a Flock of Birds Using an Unmanned Aerial Vehicle

“…Flocking dynamics described by Reynolds' rule are known to be asymptotically stable under fairly weak conditions on the topology of the underlying graph [17], [18], [24], [25]. Stronger results, such as exponential stability, have been found for linear time-varying consensus protocol for single integrator systems [26], in the context of synchronization for second-order Euler-Lagrange systems [20]- [22], and using tools from dynamical systems theory for time-invariant, undirected graph topologies in second-order, two-dimensional (2-D) flocking dynamics [27]. Preliminary results on exponential stability of flocks under tree and star topology constraints are presented in [8].…”

Phase Synchronization Control of Robotic Networks on Periodic Ellipses with Adaptive Network Topologies

“…3 For the nonlinear stability proofs, we use contraction analysis, 16 which has recently been successfully applied to network systems. 3,21,31 Third, the proposed coordinate transformation method and the phase angle shift method facilitate a phase angle shift in any ellipse in 3D space so that the elliptical motions of the networked EL system can be described by the combination of circular and sinusoidal motions in a new coordinate system. We investigate the effectiveness of the proposed methods by simulating swarms of spacecraft rotating and reconfiguring in multiple periodic relative orbits.…”