Complex partial synchronization patterns in networks of delay-coupled neurons

We study the spatio-temporal dynamics of a multiplex network of delay-coupled FitzHugh–Nagumo oscillators with non-local and fractal connectivities. Apart from chimera states, a new regime of coexistence of slow and fast oscillations is found. An analytical explanation for the emergence of such coexisting partial synchronization patterns is given. Furthermore, we propose a control scheme for the number of fast and slow neurons in each layer.DFG, 163436311, SFB 910: Kontrolle selbstorganisierender nichtlinearer Systeme: Theoretische Methoden und Anwendungskonzept

Negative-coupling resonances in pump-coupled lasers

We consider coupled lasers, where the intensity deviations from the steady state, modulate the pump of the other lasers. Most of our results are for two lasers where the coupling constants are of opposite sign. This leads to a Hopf bifurcation to periodic output for weak coupling. As the magnitude of the coupling constants is increased (negatively) we observe novel amplitude effects such as a weak coupling resonance peak and, strong coupling subharmonic resonances and chaos. In the weak coupling regime the output is predicted by a set of slow evolution amplitude equations. Pulsating solutions in the strong coupling limit are described by discrete map derived from the original model.Comment: 29 pages with 8 figures Physica D, in pres

Switching to nonhyperbolic cycles from codimension two bifurcations of equilibria of delay differential equations

In this paper we perform the parameter-dependent center manifold reduction near the generalized Hopf (Bautin), fold-Hopf, Hopf-Hopf and transcritical-Hopf bifurcations in delay differential equations (DDEs). This allows us to initialize the continuation of codimension one equilibria and cycle bifurcations emanating from these codimension two bifurcation points. The normal form coefficients are derived in the functional analytic perturbation framework for dual semigroups (sun-star calculus) using a normalization technique based on the Fredholm alternative. The obtained expressions give explicit formulas which have been implemented in the freely available numerical software package DDE-BifTool. While our theoretical results are proven to apply more generally, the software implementation and examples focus on DDEs with finitely many discrete delays. Together with the continuation capabilities of DDE-BifTool, this provides a powerful tool to study the dynamics near equilibria of such DDEs. The effectiveness is demonstrated on various models

Various Dynamical Regimes, and Transitions from Homogeneous to Inhomogeneous Steady States in Nonlinear Systems with Delays and Diverse Couplings

This dissertation focuses on the effects of distributed delays modeled by \u27weak generic kernels\u27 on the collective behavior of coupled nonlinear systems. These distributed delays are introduced into several well-known periodic oscillators such as coupled Landau-Stuart and Van der Pol systems, as well as coupled chaotic Van der Pol-Rayleigh and Sprott systems, for a variety of couplings including diffusive, cyclic, or dynamic ones. The resulting system is then closed via the \u27linear chain trick\u27 and the linear stability analysis of the system and conditions for Hopf bifurcations that initiate oscillations are investigated. A variety of dynamical regimes and transitions between them result. As an example, in certain cases the delay produces transitions from amplitude death (AD) or oscillation death (OD) regimes to Hopf bifurcation-induced periodic behavior, where typically we observe the delayed limit cycle shrinking or growing as the delay is varied towards or away from the bifurcation point respectively. The conditions for transition between AD parameter regimes and OD parameter regimes are investigated for systems in which OD is possible. Depending on the coupling, these transitions are mediated by pitchfork or transcritical bifurcations. The systems are then investigated numerically, comparing with the predictions from the linear stability analysis and previous work. In several cases the various transitions among AD, OD and periodic domains that we observe are more intricate than the simple AD states, and the rough boundaries of the parameter regimes where they occur, which have been predicted by linear stability analysis and also experimentally verified in earlier work. The final chapter extends these studies by including the effects of periodically amplitude modulated distributed delays in both position and velocity. The existence of quasiperiodic solutions motivates the derivation of a second slow flow, together with a comparison of results and predictions from the second slow flow and the numerical results, as well as using the second slow flow to approximate the radii of the toroidal attractor. Finally, the effects of varying the delay parameter are briefly considered

Dynamics of Patterns

Patterns and nonlinear waves arise in many applications. Mathematical descriptions and analyses draw from a variety of fields such as partial differential equations of various types, differential and difference equations on networks and lattices, multi-particle systems, time-delayed systems, and numerical analysis. This workshop brought together researchers from these diverse areas to bridge existing gaps and to facilitate interaction