Nonconventional synchronization and energy localization in weakly coupled autogenerators

Margarita Kovaleva, Valery Pilipchuk, and Leonid Manevitch

Phys. Rev. E 94, 032223 – Published 30 September 2016

Abstract

The present work follows our previous study dealing with a new type of synchronization in a system of two weakly coupled generalized van der Pol–Duffing autogenerators. The essence of the effect revealed is that the synchronized oscillations are not stationary but accompanied by the most intensive energy exchange between the oscillators. The phase shift between the generators remains constant most of the time, except for vanishingly small transitional intervals. The current analysis deals with a generalized model in order to clarify the frequency detuning effect. We found that varying the frequency detuning, nonlinearity, and dissipation parameters can lead to structural changes in phase diagrams of the energy exchange dynamics, with important transitions from the intensive energy exchange to its localization on one of the two oscillators. The main conclusion is that stationary and nonstationary synchronizations associate with nonlinear normal and local modes, respectively. The analysis uses phase plane diagrams, including the concept of limiting phase trajectories, whose role in nonstationary synchronization appears to be similar to the role of nonlinear normal modes in conventional stationary states.

Received 25 June 2016

DOI:https://doi.org/10.1103/PhysRevE.94.032223

Physics Subject Headings (PhySH)

Coupled oscillators Synchronization Synchronization transition

Nonlinear DynamicsInterdisciplinary PhysicsGeneral Physics

Authors & Affiliations

Margarita Kovaleva^1,*, Valery Pilipchuk^2,†, and Leonid Manevitch^1,‡

¹N.N. Semenov Institute of Chemical Physics, Kosygin St., 4 Moscow 119991, Russia
²Wayne State University, 1200 Holden Street, Detroit, Michigan 48202, USA

^*makovaleva@chph.ras.ru
^†pilipchuk@wayne.edu
^‡manevitchleonid3@gmail.com

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Vol. 94, Iss. 3 — September 2016

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Images

Figure 1
Evolution of the phase plane of system (4) in the “weakly nonlinear” case, $k = 0.2$ , due to a gradual increase of the frequency detuning parameter, where the variables $θ$ and $Δ$ characterize the relation between the two amplitudes and the phase shift as $R_{1} = \sqrt{N} sin θ, R_{2} = \sqrt{N} cos θ$ , and $Δ = δ_{2} - δ_{1}$ , respectively: (a) nonlinearity parameter, the dissipative and detuning parameters are zero, $λ = 0$ and $g = 0$ – a conservative system of identical oscillators; (b) $λ = 0.1$ and $g = 0$ – weak dissipation with no detuning, the stationary points are unstable focuses, the limit cycle corresponding to LPT becomes attractor; (c) $λ = 0.1$ and $g = 0.1$ – the limit cycle is slightly deformed but still remains attractor; (d) $λ = 0.1$ and $g = 0.2$ – the limit cycle disappears by giving rise to the “running phase” trajectory, which becomes attractor; (e) $λ = 0.1$ and $g = 0.5$ – the NLM becomes the only attractor of the system with energy localization on the generator with lower frequency; (f) $λ = 0.1$ and $g = 1$ – further increase of the detuning does not affect the topology of the phase portrait; however, the energy localization effect becomes stronger.
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Figure 2
Evolution of the phase diagrams of system (4) due to the detuning increase under the greater nonlinearity level, $k = 0.4$ : (a) dissipative parameter $λ = 0$ , detuning $g = 0$ – conservative system of identical oscillators; (b) $λ = 0.1$ and $g = 0$ system with a weak dissipation but without detuning; in contrast to NNMs (Fig. 1), the NLMs are stable focuses, one of which attracts phase trajectories depending on the initial conditions; (c) $λ = 0.1$ and $g = 0.1$ – the attraction area of one of the two NLMs is shrinking; (d) $λ = 0.1$ and $g = 0.2$ , most of the energy localizes on the active oscillator with lower frequency.
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Figure 3
Evolution of the phase diagrams of system (4) under the weak nonlinearity condition, $k = 0.2$ , however, the greater dissipation term, $λ = 0.85$ : (a) no detuning case, $g = 0$ ; (b) $g = 0.02$ system with a weak detuning, red dots mark stationary points obtained analytically; (c) $g = 0.1$ ; and (d) $g = 0.5$ .
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Figure 4
Transitions to the nonstationary synchronization under different detuning magnitudes: (a, d, g, j) $g = 0$ ; (b, e, h, k) $g = 0.05$ ; (c, f, i, l) $g = 0.2$ ; numerical solutions of Eqs. (12) (black thick), and the same variables obtained from (6, 7, 8) by direct solution of system (1) (blue thin lines); in all the cases $ɛ = 0.01$ ; $α = 0.268$ ; $β = 1.0$ ; $b = 1.2$ ; $d = 0.8$ ; $λ = 0.1$ ; $E_{0} (0) = 0.5$ ; $Δ (0) = π - 0.001$ ; $P (0) = 0.128 844$ .
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Figure 5
Geometrical interpretation of the coherency index based on the direct numerical solution of system (1) on the interval $0 < t < 6000$ under the parameters corresponding to the zero detuning case in Fig. 4: (a) time history of the coherency index, where the letters identify sample areas for configuration plane diagrams; (b) breathing elliptic path, $5570 < t < 5670$ ; (c) the neighborhood of antiphase mode with the energy mostly on the second oscillator, $5700 < t < 5720$ ; and (d) – antiphase with the energy mostly on the first oscillator, $5860 < t < 5880$ .
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Figure 6
Numerical solutions of Eqs. (1) for different values of the detuning parameter: (a) $g = 0.0$ , (b) $g = 0, 2$ , and (c) $g = 1.0$ ; other parameters are the same in all the cases: $ɛ = 0.01$ ; $α = 0.268$ ; $β = 1.0$ ; $b = 1.2$ ; $d = 0.8$ ; $k = 0.2$ ; $λ = 0.1$ ; $E_{0} (0) = 0.5$ ; $u_{1} (0) = 1; v_{1} (0) = 0; u_{2} (0) = 0.000 01; v_{2} (0) = 0 .$
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