Amplitude death in oscillators coupled by asymmetric connection delays with tree graph topology
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THE EUROPEAN PHYSICAL JOURNAL B
Regular Article
Amplitude death in oscillators coupled by asymmetric connection delays with tree graph topology Yuki Okigawa 1 , Yoshiki Sugitani 1,a , and Keiji Konishi 2 1
2
Department of Electrical and Electronic Engineering, Ibaraki University 4-12-1 Nakanarusawa, Hitachi, Ibarak 316-8511, Japan Department of Electrical and Information Systems, Osaka Prefecture University 1-1 Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531, Japan Received 8 January 2020 / Received in final form 23 April 2020 Published online 6 July 2020 c EDP Sciences / Societ`
a Italiana di Fisica / Springer-Verlag GmbH Germany, part of Springer Nature, 2020 Abstract. The present study investigates amplitude death in an oscillator network with asymmetric connection delays and a tree graph topology. A frequency domain analysis reveals that the local steady-state stability is dominated by the sum of the connection delays between connected oscillators. Based on this observation, we show that the local steady-state stability can be reduced to that with symmetric connection delays, as long as the sum of the connection delays between connected oscillators satisfies a certain condition. Numerical simulations verify the analytical results.
1 Introduction Mutual interactions among oscillators cause two major oscillation quenching phenomena: oscillation death (OD) and amplitude death (AD) [1–3]. Oscillation death is the emergence of a stable heterogeneous steady state by interactions, whereas AD is the stabilization of a homogeneous steady state in coupled oscillators. Since AD can eliminate unwanted oscillations in many practical systems, it has been actively investigated not only from an academic perspective [4–18], but also from an engineering point of view [19–24]. The connection delay in the interactions plays a crucial role in the occurrence of AD [25]. Since connection delay is inherent in real-world networks, AD in delayedcoupled oscillators has been intensively investigated both analytically and experimentally [5–12,16–18,20–23]. Most studies on AD consider only symmetric connection delays: τij = τji holds for any i and j, where τij is the connection delay from oscillator j to oscillator i (see Fig. 1). In real-world networks, however, such as neural networks [26], the connection delays are usually not symmetric, and are instead asymmetric (i.e., τij 6= τji ). To the best of our knowledge, there have been few efforts to deal with AD induced with asymmetric connection delays, because such delays complicate the analytical treatment of the stability. Amplitude death induced with asymmetric connection delays has been numerically observed [26,27]. Although these studies [26,27] confirm the occurrence of AD in various network topologies, they a
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do not provide a sufficient analytical discussion. A previous study [28] considered two oscillators with recurrent coupling of asymmetric connection delays (i.e., τ12 6= τ21 ), and reported analytical results showing that the sta
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