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# Research

My current research interests are in the field of** Dynamical Systems and Applications**.

Theory:

- Systems with time-delays,
- Networks and lattices of coupled systems,
- PDEs with time-delays

Examples of applications:

- Dynamics of optoelectronic systems,
- Neural networks - synchronization, plasticity, dynamics...

Below, I will post some examples in more details.

# Self-organized resistance to noise of neuronal networks with spike-timing dependent plasticity

- Bifurcation diagram for coupling weights. Blue-shaded region: bidirectional coupling is stable. Red-shaded region: unidirectional coupling is stable. Inset: Schematic diagram of the model of two coupled neurons with STDP.
- © Serhiy Yanchuk

One of a fundamental adaptation mechanism of the nervous system is spike time-dependent plasticity (STDP). Depending on the spiking behavior of neural cells, plasticity regulates the coupling between individual cells and controls the network connectivity. Jointly with the Institute of Neuroscience and Medicine - Neuromodulation (Research Center Jülich) we study ensembles of synchronized spiking neurons with adaptive coupling, that are perturbed by an independent random input. For such networks, the phenomenon of self-organized resistance to noise has been reported that is characterized by an increase of the overall coupling and preservation of synchrony in the neural populations with STDP in response to the external noise growth [1]. We studied further theoretically the influence of noise on a microscopic level by considering only two coupled neurons [2], where the underlying mechanism can be studied in more details.

[1] O. V. Popovych, S. Yanchuk, P. A. Tass, *Self-organized noise resistance of oscillatory neural networks with spike timing-dependent plasticity, *Sci. Rep. 3, 2926 (2013)

[2] L. Lücken, O. Popovych, P. Tass, S Yanchuk, *Noise-enhanced coupling between two oscillators with long-term plasticity*, Phys. Rev. E. 93, 032210 (2016), [preprint]

# Reduction of interaction delays in networks

- A small network resembling a connectivity between several regions in the brain. The component-wise timeshift transformation leads to a system with a reduced number of delays (right)
- © Serhiy Yanchuk

Delayed interactions are a common property of coupled natural systems and arise in a variety of different applications. For instance, in neural or laser networks, the signals propagate at finite speed giving rise to delayed connections. In realistic situations the delays are not identical on different connections. We show how it is possible to reduce the number of different delays and simplify the models without loss of information. Our method allows also to determine certain dynamic invariants in delay-coupled networks (e.g. such as round-trip delays along the fundamental semicycles).

[1] L. Lücken, J. P. Pade, K. Knauer, S. Yanchuk, *Reduction of interaction delays in networks, *EPL (Europhys. Lett.) 103, 10006 (2013)

[2] L. Lücken, J.P. Pade, K. Knauer, *Classification of coupled dynamical systems with multiple delays: finding the minimal number of delays*, SIAM J. Appl. Dyn.l Syst. 14, 286 (2015).

# Dissipative solitons and delayed feedback

Localized solutions are important nonlinear phenomena in spatially extended systems. For instance, laser cavity solitons are self-localized peaks of the intensity of light in an optical cavity. In our publication [1], we study how time-delayed feedback influences such structures. In particular, we report a novel destabilization mechanism of localized solutions leading to the appearance of multiple coexistent laser cavity solitons. We show that at large delays apart from the drift and phase instabilities the soliton can exhibit a delay-induced modulational instability that produces a zigzagging motion of the solitons, which are either periodic or chaotic.

[1] D. Puzyrev, A. Vladimirov, S. Gurevich, S. Yanchuk. *Modulational instability and zigzagging of dissipative solitons induced by delayed feedback*. Phys. Rev. A **93**, 041801(R) (2016)

[2] D. Puzyrev, S. Yanchuk, A. Vladimirov, S. Gurevich. *Stability of plane wave solutions in complex Ginzburg-Landau equation with delayed feedback. *SIAM J. Appl. Dyn. Syst. 13, 986 (2014)

# Spiking behavior in loops or chains of neurons (or other oscillatory systems)

[1] V. Klinshov, D. Shchapin, S. Yanchuk, V. Nekorkin, *Jittering waves in rings of pulse oscillators, *Phys. Rev. E 94, 012206 (2016)

[2] M. Kantner, E. Schöll, S. Yanchuk, *Delay-induced patterns in a two-dimensional lattice of coupled oscillators,* Nature Scientific Reports 5, 8522 (2015).

[3] J. P. Pade, L. Lücken, S. Yanchuk, *The dynamical impact of a shortcut in unidirectionally coupled rings of oscillators, *Math. Model. Nat. Phenom. 8, 173-189 (2013)

[4] M. Kantner, S. Yanchuk, *Bifurcation analysis of delay-induced patterns in a ring of Hodgkin-Huxley neurons, *Phil. Trans. Roy. Soc. 371, 20120470 (2013)

[5] O. Popovych, S. Yanchuk and P. Tass, *Delay- and coupling-induced firing patterns in oscillatory neural loops*, Phys. Rev. Lett. 107, 228102 (2011), [PDF]

[6] S. Yanchuk, P. Perlikowski, O. Popovych, P. Tass, *Variability of spatio-temporal patterns in non-homogeneous rings of spiking neurons*, Chaos **21**, 047511 (2011)

[7] P. Perlikowski, S. Yanchuk, M. Wolfrum, A. Stefanski, P. Mosiolek, and T. Kapitaniak, *Routes to complex dynamics in a ring of unidirectionally coupled systems.* Chaos, 20, 013111 (2010)

[8] S. Yanchuk and M. Wolfrum, *Destabilization patterns in chains of coupled oscillators*, Phys. Rev. E 77 (2008) 026212.

# Multi-cluster states in adaptive networks

[1] R. Berner, E. Schöll, S. Yanchuk Multi-clusters in adaptive networks, ArXiv preprint (2018);

[2] D. Kasatkin, S. Yanchuk, E. Schöll, V. Nekorkin, Self-organized emergence of multilayer structure and chimera states in dynamical networks with adaptive couplings. Phys. Rev. E, 96, 062211 (2017).

# Spectrum of systems with long time-delays

Delay differential equations with large delay play an important role in the modeling of many real-world systems, especially for optoelectronic systems with optical feedback or coupling. One of the basic questions in such systems concerns the stability of steady states or periodic solutions and, in particular, spectral properties of the corresponding linearized systems. In [1-2] we investigated the spectrum of steady states and periodic solutions for delay differential equations (DDEs) with one large delay.

We classified rigorously which types of limits this spectrum can have as delay increases. Our results provide simple and sometimes explicit approximation formula for the spectrum, which are useful in applications [3].

[1] M. Lichtner, M. Wolfrum, S. Yanchuk, The Spectrum of Delay Differential Equations with Large Delay}. SIAM J. Mathematical Analysis, 43, 788 (2011);

[2] J.~Sieber, M.~Wolfrum, M.~Lichtner, S.~Yanchuk. On the stability of periodic orbits in delay equations with large delay, Discr. Cont. Dyn. Systems A, 33, 3109 (2013);

[3] S.~Yanchuk, M.~Wolfrum. A multiple time scale approach to the stability of external cavity modes in the Lang–Kobayashi system using the limit of large delay, SIAM J. Appl. Dyn. Systems, 9, 519 (2010)

# General properties of periodic solutions of delay systems

In [1] we describe generic properties of systems with one time delay, which are related to the appearance and stability of periodic solutions. In particular, we show that delay systems generically have families of periodic solutions, which are reappearing for infinitely many delay times. As delay increases, the solution families overlap leading to increasing coexistence of multiple stable as well as unstable solutions.

[1] S. Yanchuk, P. Perlikowski, Delay and periodicity, Phys. Rev. E, 79, 046221 (2009)