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| Transitions from equilibrium to quasiperiodicity and from a two-cycle to a quasiperiodic regime are studied in a ring of unidirectionally-coupled nonidentical logistic maps. The former scenario is realized through a ?soft? (Neimark?Sacker) bifurcation, while the latter through a ?hard? (saddle-node) bifurcation. Special attention is paid on a noise-induced transition through ?hard? bifurcation, where a phenomenon of structural stabilization of the quasiperiodic system near the bifurcation point is observed and analyzed in detail.©2019 Elsevier B.V. All rights reserved.1. IntroductionIn complex systems composed of several interacting subsys-tems, new dynamical regimes, not observed in solitary units, can appear due to a cooperative effect. Such a situation occurs in many fields of science and engineering [1,3,2,4]. Even very sim-ple dynamical systems, e.g., low-dimensional maps being coupled exhibit behavioral transitions from equilibrium to quasiperiodic-ity (QP). In a standard scenario of such a transition, a QP regime slowly develops as a control parameter is changed. This occurs in a Neimark?Sacker ?soft? bifurcation [5,6]. However, another scenario is also possible, when a QP regime suddenly arises in a saddle-node ?hard? bifurcation [7].The transitions to QP were observed in many dynamical sys-tems, including mechanical [8], biological [9], economic [10], and climate [11]models, as well as in two symmetrically coupled iden-tical logistic maps [7] and in a ring of three unidirectionally cou-pled Duffing oscillators [12]. Recently, a constructive effect of noise on QP regimes was highlighted in a delayed logistic map near the Neimark?Sacker bifurcation [13]. It was shown that random distur-bances induce transitions between an equilibrium, a 4-cycle, and a closed invariant curve. The influence of noise on QP oscillations has also been studied in a map-based neuron model [14], where *Corresponding author.E-mail address:alexander.pisarchik@ctb.upm.es(A.N. Pisarchik).noise generates mixed-mode stochastic spiking oscillations. How-ever, a stochastic transition from a periodic attractor to QP has not yet been demonstrated, although a noise-induced backward stochastic bifurcation from a 7-cycle to a closed invariant curve has recently been found in the time-delayed logistic model [15]. Furthermore, the influence of noise on QP in a ring of coupled nonidentical maps was not yet studied at all, to the best of our knowledge.In the present paper, we investigate a constructive effect of noise in a system of three ring-coupled nonidentical logistic maps. We focus on the parameter range where this coupled system ex-hibits a stable equilibrium, a 2-cycle regime, and a QP attractor. First, in Sec.2, we consider a deterministic model and study the case when the isolated maps have a stable equilibrium and a 2-cycle. We are interested in how an increase in the coupling pa-rameter changes corporate dynamics and analyze transitions to QP-regimes from an equilibrium through the ?soft? (Neimark?Sacker) bifurcation or from a 2-cycle through ?hard? (saddle-node) bifur-cation. Then, in Sec.3 we focus on stochastic phenomena in the randomly forced coupled system. We consider a parametric zone near the ?hard? bifurcation and analyze noise-induced transitions from a 2-cycle to a QP regime. We explain these transitions by peculiarities of phase portraits of the unforced deterministic sys-tem and demonstrate the transformation of QP oscillations under increasing noise amplitude. Finally, main conclusions are given in Sec.4. | |
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Internacional
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JCR del ISI
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Si |
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Título de la revista
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Physics Letters a |
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ISSN
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0375-9601 |
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Factor de impacto JCR
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2,087 |
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Información de impacto
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Volumen
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383 |
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DOI
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Número de revista
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14 |
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