Oscillations of delay differential equations

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N-[4-(2-Morpholino­eth­oxy)phen­yl]acetamide monohydrate

In the title compound, C14H20N2O3·H2O, the geometry about the morpholine N atom implies sp 3 hybridization. In the crystal, symmetry-related mol­ecules are linked by inter­molecular N—H⋯O, O—H⋯O and O—H⋯N hydrogen bonds, forming infinite chains along the b axis. The chain structure is further stabilized by intra­molecular C—H⋯O inter­actions

Synchronization in a neuronal feedback loop through asymmetric temporal delays

We consider the effect of asymmetric temporal delays in a system of two coupled Hopfield neurons. For couplings of opposite signs, a limit cycle emerges via a supercritical Hopf bifurcation when the sum of the delays reaches a critical value. We show that the angular frequency of the limit cycle is independent of an asymmetry in the delays. However, the delay asymmetry determines the phase difference between the periodic activities of the two components. Specifically, when the connection with negative coupling has a delay much larger than the delay for the positive coupling, the system approaches in-phase synchrony between the two components. Employing variational perturbation theory (VPT), we achieve an approximate analytical evaluation of the phase shift, in good agreement with numerical results.Comment: 5 pages, 4 figure

Characteristics of a Delayed System with Time-dependent Delay Time

The characteristics of a time-delayed system with time-dependent delay time is investigated. We demonstrate the nonlinearity characteristics of the time-delayed system are significantly changed depending on the properties of time-dependent delay time and especially that the reconstructed phase trajectory of the system is not collapsed into simple manifold, differently from the delayed system with fixed delay time. We discuss the possibility of a phase space reconstruction and its applications.Comment: 4 pages, 6 figures (to be published in Phys. Rev. E

4-[2-(4-Meth­oxy­phen­yl)eth­yl]-3-(thio­phen-2-ylmeth­yl)-1H-1,2,4-triazol-5(4H)-one monohydrate

In the title compound, C16H17N3O2S·H2O, the triazole ring makes a dihedral angle of 34.63 (6)° with the benzene ring. The thio­phene ring is disordered over two orientations [occupancy ratio = 0.634 (4):0.366 (4)] which make dihedral angles of 54.61 (16) and 54.57 (31)° with the triazole ring. Inter­molecular N—H⋯O and O—H⋯O hydrogen bonds stabilize the crystal structure

Exact synchronization bound for coupled time-delay systems

We obtain an exact bound for synchronization in coupled time-delay systems using the generalized Halanay inequality for the general case of time-dependent delay, coupling, and coefficients. Furthermore, we show that the same analysis is applicable to both uni- and bidirectionally coupled time-delay systems with an appropriate evolution equation for their synchronization manifold, which can also be defined for different types of synchronization. The exact synchronization bound assures an exponential stabilization of the synchronization manifold which is crucial for applications. The analytical synchronization bound is independent of the nature of the modulation and can be applied to any time-delay system satisfying a Lipschitz condition. The analytical results are corroborated numerically using the Ikeda system

4-[3-(1H-Imidazol-1-yl)prop­yl]-3-methyl-5-(thio­phen-2-ylmeth­yl)-4H-1,2,4-triazole monohydrate

In the title compound, C14H17N5S·H2O, the triazole ring makes dihedral angles of 48.15 (8) and 84.92 (8)° with the imidazole and thio­phenyl rings, respectively. The water mol­ecule is involved in inter­molecular O—H⋯N hydrogen bonding

Dynamics of FitzHugh-Nagumo excitable systems with delayed coupling

Small lattices of $N$ nearest neighbor coupled excitable FitzHugh-Nagumo systems, with time-delayed coupling are studied, and compared with systems of FitzHugh-Nagumo oscillators with the same delayed coupling. Bifurcations of equilibria in N=2 case are studied analytically, and it is then numerically confirmed that the same bifurcations are relevant for the dynamics in the case $N>2$. Bifurcations found include inverse and direct Hopf and fold limit cycle bifurcations. Typical dynamics for different small time-lags and coupling intensities could be excitable with a single globally stable equilibrium, asymptotic oscillatory with symmetric limit cycle, bi-stable with stable equilibrium and a symmetric limit cycle, and again coherent oscillatory but non-symmetric and phase-shifted. For an intermediate range of time-lags inverse sub-critical Hopf and fold limit cycle bifurcations lead to the phenomenon of oscillator death. The phenomenon does not occur in the case of FitzHugh-Nagumo oscillators with the same type of coupling.Comment: accepted by Phys.Rev.

Delay-enhanced coherent chaotic oscillations in networks with large disorders

We study the effect of coupling delay in a regular network with a ring topology and in a more complex network with an all-to-all (global) topology in the presence of impurities (disorder). We find that the coupling delay is capable of inducing phase-coherent chaotic oscillations in both types of networks, thereby enhancing the spatiotemporal complexity even in the presence of 50% of symmetric disorders of both fixed and random types. Furthermore, the coupling delay increases the robustness of the networks up to 70% of disorders, thereby preventing the network from acquiring periodic oscillations to foster disorder-induced spatiotemporal order. We also confirm the enhancement of coherent chaotic oscillations using snapshots of the phases and values of the associated Kuramoto order parameter. We also explain a possible mechanism for the phenomenon of delay-induced coherent chaotic oscillations despite the presence of large disorders and discuss its applications.Comment: 13 pages, 20 figure