Recent zbMATH articles in MSC 34C15https://zbmath.org/atom/cc/34C152021-06-15T18:09:00+00:00WerkzeugSynchronization and locking in oscillators with flexible periods.https://zbmath.org/1460.920372021-06-15T18:09:00+00:00"Savinov, Mariya"https://zbmath.org/authors/?q=ai:savinov.mariya"Swigon, David"https://zbmath.org/authors/?q=ai:swigon.david"Ermentrout, Bard"https://zbmath.org/authors/?q=ai:ermentrout.bard-gSummary: Entrainment of a nonlinear oscillator by a periodic external force is a much studied problem in nonlinear dynamics and characterized by the well-known Arnold tongues. The circle map is the simplest such system allowing for stable \(N\):\(M\) entrainment where the oscillator produces \(N\) cycles for every \(M\) stimulus cycles. There are a number of experiments that suggest that entrainment to external stimuli can involve both a shift in the phase and an adjustment of the intrinsic period of the oscillator. Motivated by a recent model of \textit{J. D. Loehr}, \textit{E. W. Large} and \textit{C. Palmer} [``Temporal coordination and adaptation to rate change in music performance'', J. Exp. Psychol.: Hum. Percept. Perform. 37, No. 4, 1292--1309 (2011; \url{doi:10.1037/a0023102})], we explore a two-dimensional map in which the phase and the period are allowed to update as a function of the phase of the stimulus. We characterize the number and stability of fixed points for different \(N\):\(M\)-locking regions, specifically, 1:1, 1:2, 2:3, and their reciprocals, as a function of the sensitivities of the phase and period to the stimulus as well as the degree that the oscillator has a preferred period. We find that even in the limited number of locking regimes explored, there is a great deal of multi-stability of locking modes, and the basins of attraction can be complex and riddled. We also show that when the forcing period changes between a starting and final period, the rate of this change determines, in a complex way, the final locking pattern.
{\copyright 2021 American Institute of Physics}Solitary states in the mean-field limit.https://zbmath.org/1460.340432021-06-15T18:09:00+00:00"Kruk, N."https://zbmath.org/authors/?q=ai:kruk.n-p"Maistrenko, Y."https://zbmath.org/authors/?q=ai:maistrenko.yuri-l|maistrenko.yury"Koeppl, H."https://zbmath.org/authors/?q=ai:koeppl.heinzThe authors mostly consider the equations
\[\begin{aligned}
\dot{\phi}_i&=\omega_i,\\
\dot{\omega}_i&=-\xi\omega_i+(\sigma/N)\sum_{j=1}^N\sin{(\phi_j-\phi_i-\alpha)}\qquad \text{for }i=1,\dots N,
\end{aligned}\]
where each \(\phi_i\) is an angular variable, i.e. the Kuramoto model with inertia.
They show numerically that as \(\alpha\) and \(\sigma\) are varied
various ``solitary states'' can occur, in which oscillators form
clusters characterised by distinct phases and frequencies. The system may also be multistable.
The mechanism by which one or two solitary
oscillators appear is determined: via homoclinic bifurcations. Gaussian white noise is
added to the second equation above, the corresponding Fokker-Planck equation is written, and then studied numerically
and analytically. Phase transitions between disordered motion, partial synchrony, and
a solitary state with one frequency cluster are found numerically.
Reviewer: Carlo Laing (Auckland)Finite time blow-up in a mathematical model of sub-atomic particle oscillations.https://zbmath.org/1460.340442021-06-15T18:09:00+00:00"Ncube, Israel"https://zbmath.org/authors/?q=ai:ncube.israelThe author considers the nonlinear oscillator
\[
\ddot x+\frac{\alpha-\lambda\dot x^2}{1+\lambda x^2}x=0\tag{\(*\)}
\]
introduced by \textit{P. M. Mathews} and \textit{M. Lakshmanan} [Q. Appl. Math. 32, 215--218 (1974; Zbl 0284.34046)]. The influence of the real parameters \(\alpha\) and \(\lambda\) on the phase portrait of (\(*\)) is studied.
Reviewer: Klaus R. Schneider (Berlin)Spatiotemporal patterns in a 2D lattice with linear repulsive and nonlinear attractive coupling.https://zbmath.org/1460.940922021-06-15T18:09:00+00:00"Shepelev, I. A."https://zbmath.org/authors/?q=ai:shepelev.igor-aleksandrovich"Muni, S. S."https://zbmath.org/authors/?q=ai:muni.s-s"Vadivasova, T. E."https://zbmath.org/authors/?q=ai:vadivasova.tatiana-e|vadivasova.tatyana-evgenevnaSummary: We explore the emergence of a variety of different spatiotemporal patterns in a 2D lattice of self-sustained oscillators, which interact nonlocally through an active nonlinear element. A basic element is a van der Pol oscillator in a regime of relaxation oscillations. The active nonlinear coupling can be implemented by a radiophysical element with negative resistance in its current-voltage curve taking into account nonlinear characteristics (for example, a tunnel diode). We show that such coupling consists of two parts, namely, a repulsive linear term and an attractive nonlinear term. This interaction leads to the emergence of only standing waves with periodic dynamics in time and absence of any propagating wave processes. At the same time, many different spatiotemporal patterns occur when the coupling parameters are varied, namely, regular and complex cluster structures, such as chimera states. This effect is associated with the appearance of new periodic states of individual oscillators by the repulsive part of coupling, while the attractive term attenuates this effect. We also show influence of the coupling nonlinearity on the spatiotemporal dynamics.
{\copyright 2021 American Institute of Physics}Nonlinear energy harvester with coupled Duffing oscillators.https://zbmath.org/1460.700182021-06-15T18:09:00+00:00"Karličić, Danilo"https://zbmath.org/authors/?q=ai:karlicic.danilo-z"Cajić, Milan"https://zbmath.org/authors/?q=ai:cajic.milan"Paunović, Stepa"https://zbmath.org/authors/?q=ai:paunovic.stepa"Adhikari, Sondipon"https://zbmath.org/authors/?q=ai:adhikari.sondiponSummary: Structural vibrations are very common in aerospace and mechanical engineering systems, where dynamic analysis of modern aerospace structures and industrial machines has become an indispensable step in their design. Suppression of unwanted vibrations and their exploitation for energy harvesting at the same time would be the most desirable scenario. The dynamical system presented in this communication is based on a discrete model of energy harvesting device realized in such a manner as to achieve both vibration suppression and harvesting of vibration energy by introducing the nonlinear energy sink concept. The mechanical model is formed as a two-degree of freedom nonlinear oscillator with an oscillating magnet and harmonic base excitation. The corresponding mathematical model is based on the system of nonlinear nonhomogeneous Duffing type differential equations. To explore complex dynamical behaviour of the presented model, periodic solutions and their bifurcations are found by using the incremental harmonic balance (IHB) and continuation methods. For the detection of unstable periodic orbits, the Floquet theory is applied and an interesting harmonic response of the presented nonlinear dynamical model is detected. The main advantage of the presented approach is its ability to obtain approximated periodic responses in terms of Fourier series and estimate the voltage output of an energy harvester for a system with strong nonlinearity. The accuracy of the presented methodology is verified by comparing the results obtained in this work with those obtained by a standard numerical integration method and results from the literature. Numerical examples show the effects of different physical parameters on amplitude-frequency, response amplitude -- base amplitude and time response curves, where a qualitative change is explored and studied in detail. Presented theoretical results demonstrate that the proposed system has advanced performance in both system requirements -- vibration suppression, and energy harvesting.Asymptotic behavior of gradient flows on the unit sphere with various potentials.https://zbmath.org/1460.340632021-06-15T18:09:00+00:00"Huh, Hyungjin"https://zbmath.org/authors/?q=ai:huh.hyungjin"Kim, Dohyun"https://zbmath.org/authors/?q=ai:kim.dohyunIn this lengthy paper, a multi-agent system whose dynamics is governed by a gradient flow on the unit sphere associated with the interaction potential between positions of all agents measured by a weighted distance $|x_{i}-x_{j}|^{p+2}$ for any $p\neq 0$, is considered. Employing both attractive and repulsive couplings, the asymptotic behavior of the system accompanied by both $p>0$ (positive range) and $p<0$ (negative range) is studied.
Reviewer: Ioannis P. Stavroulakis (Ioannina)Two-community noisy Kuramoto model with general interaction strengths. II.https://zbmath.org/1460.340512021-06-15T18:09:00+00:00"Achterhof, S."https://zbmath.org/authors/?q=ai:achterhof.s"Meylahn, J. M."https://zbmath.org/authors/?q=ai:meylahn.j-mSummary: We generalize the study of the noisy Kuramoto model, considered on a network of two interacting communities, to the case where the interaction strengths within and across communities are taken to be different in general. Using a geometric interpretation of the self-consistency equations developed in Paper I of this series [the authors, ibid. 31, No. 3, 033115, 16 p. (2021; Zbl 1459.34114)] as well as perturbation arguments, we are able to identify all solution boundaries in the phase diagram. This allows us to completely classify the phase diagram in the four-dimensional parameter space and identify all possible bifurcation points. Furthermore, we analyze the asymptotic behavior of the solution boundaries. To illustrate these results and the rich behavior of the model, we present phase diagrams for selected regions of the parameter space.
{\copyright 2021 American Institute of Physics}KAM tori for completely resonant Hamiltonian derivative beam equations on \(\mathbb{T}^2\).https://zbmath.org/1460.370702021-06-15T18:09:00+00:00"Ge, Chuanfang"https://zbmath.org/authors/?q=ai:ge.chuanfang"Geng, Jiansheng"https://zbmath.org/authors/?q=ai:geng.jiansheng"Lou, Zhaowei"https://zbmath.org/authors/?q=ai:lou.zhaoweiSummary: In this paper, we study a class of completely resonant beam equations with derivative nonlinearities on \(\mathbb{T}^2\)
\[
u_{tt} + \Delta^2 u + u |\nabla u|^2 + u^2 \Delta u = 0.
\]
We will prove the existence of small-amplitude, quasi-periodic solutions for the above equations via KAM method.Effect of noise on the collective dynamics of a heterogeneous population of active rotators.https://zbmath.org/1460.340542021-06-15T18:09:00+00:00"Klinshov, V. V."https://zbmath.org/authors/?q=ai:klinshov.v-v"Zlobin, D. A."https://zbmath.org/authors/?q=ai:zlobin.d-a"Maryshev, B. S."https://zbmath.org/authors/?q=ai:maryshev.b-s"Goldobin, D. S."https://zbmath.org/authors/?q=ai:goldobin.denis-sSummary: We study the collective dynamics of a heterogeneous population of globally coupled active rotators subject to intrinsic noise. The theory is constructed on the basis of the circular cumulant approach, which yields a low-dimensional model reduction for the macroscopic collective dynamics in the thermodynamic limit of an infinitely large population. With numerical simulation, we confirm a decent accuracy of the model reduction for a moderate noise strength; in particular, it correctly predicts the location of the bistability domains in the parameter space.
{\copyright 2021 American Institute of Physics}Phase drift in networks of coupled colpitts oscillators.https://zbmath.org/1460.340412021-06-15T18:09:00+00:00"Coria, Lourdes"https://zbmath.org/authors/?q=ai:coria.lourdes"Lopez, Horacio"https://zbmath.org/authors/?q=ai:lopez.horacio"Palacios, Antonio"https://zbmath.org/authors/?q=ai:palacios.antonio"In, Visarath"https://zbmath.org/authors/?q=ai:in.visarath"Longhini, Patrick"https://zbmath.org/authors/?q=ai:longhini.patrickExplosive synchronization in interlayer phase-shifted Kuramoto oscillators on multiplex networks.https://zbmath.org/1460.340652021-06-15T18:09:00+00:00"Kumar, Anil"https://zbmath.org/authors/?q=ai:kumar.anil"Jalan, Sarika"https://zbmath.org/authors/?q=ai:jalan.sarikaSummary: Different methods have been proposed in the past few years to incite explosive synchronization (ES) in Kuramoto phase oscillators. In this work, we show that the introduction of a phase shift \(\alpha\) in interlayer coupling terms of a two-layer multiplex network of Kuramoto oscillators can also instigate ES in the layers. \(As \alpha \to \pi / 2\), ES emerges along with hysteresis. The width of hysteresis depends on the phase shift \(\alpha \), interlayer coupling strength, and natural frequency mismatch between mirror nodes. A mean-field analysis is performed to justify the numerical results. Similar to earlier works, the suppression of synchronization is accountable for the occurrence of ES. The robustness of ES against changes in network topology and natural frequency distribution is tested. Finally, taking a suggestion from the synchronized state of the multiplex networks, we extend the results to classical single networks where some specific links are assigned phase-shifted interactions.
{\copyright 2021 American Institute of Physics}Symmetry-independent stability analysis of synchronization patterns.https://zbmath.org/1460.340662021-06-15T18:09:00+00:00"Zhang, Yuanzhao"https://zbmath.org/authors/?q=ai:zhang.yuanzhao"Motter, Adilson E."https://zbmath.org/authors/?q=ai:motter.adilson-eThis paper concerns the determination of the stability of synchronised clusters in networks of coupled oscillators, for which the oscillators form two or more internally synchronised clusters that exhibit mutually distinct dynamics. Whether a given network supports such solutions depends on the symmetries of the network, and determining all such solutions based on the network symmetries is difficult, if not impossible, for moderate to large networks. The authors present a generalisation of the master stability function formalism, applicable even when the
oscillators and/or their interaction functions are nonidentical, which overcomes this obstacle. The new framework is based on finding the finest simultaneous block diagonalization of matrices in the variational equation, and leads to an algorithm hat is error-tolerant and orders of magnitude faster than existing symmetry-based algorithms. As an application, the stability of chimera states in networks with multiple types of interactions is rigorously characterised.
Reviewer: Carlo Laing (Auckland)The Kapitza equation for the inverted pendulum.https://zbmath.org/1460.340422021-06-15T18:09:00+00:00"Grundy, R. E."https://zbmath.org/authors/?q=ai:grundy.r-eSummary: It is well known that a simple pendulum can be made to perform finite amplitude oscillations about the `up' equilibrium position by subjecting the pivot to small amplitude high frequency oscillations. In this article, we show that the autonomous nonlinear ordinary differential equation, first derived by Kapitza in 1951 using the method of averaging and historically used to describe this phenomenon, is not uniformly valid in time and hence deviates appreciably from the exact solution. We explore why this is so and show that we can provide significantly improved temporal accuracy by way of a simple modification to the original Kapitza equation.Phase coalescence in a population of heterogeneous Kuramoto oscillators.https://zbmath.org/1460.340572021-06-15T18:09:00+00:00"Phogat, Richa"https://zbmath.org/authors/?q=ai:phogat.richa"Ray, Arnob"https://zbmath.org/authors/?q=ai:ray.arnob"Parmananda, P."https://zbmath.org/authors/?q=ai:parmananda.punit"Ghosh, Dibakar"https://zbmath.org/authors/?q=ai:ghosh.dibakarSummary: \textit{Phase coalescence} (PC) is an emerging phenomenon in an ensemble of oscillators that manifests itself as a spontaneous rise in the order parameter. This increment in the order parameter is due to the overlaying of oscillator phases to a pre-existing system state. In the current work, we present a comprehensive analysis of the phenomenon of phase coalescence observed in a population of Kuramoto phase oscillators. The given population is divided into responsive and non-responsive oscillators depending on the position of the phases of the oscillators. The responsive set of oscillators is then reset by a pulse perturbation. This resetting leads to a temporary rise in a macroscopic observable, namely, order parameter. The provoked rise thus induced in the order parameter is followed by unprovoked increments separated by a constant time \(\tau_{P C} \). These unprovoked increments in the order parameter are caused due to a temporary gathering of the oscillator phases in a configuration similar to the initial system state, i.e., the state of the network immediately following the perturbation. A theoretical framework corroborating this phenomenon as well as the corresponding simulation results are presented. Dependence of \(\tau_{P C}\) and the magnitude of spontaneous order parameter augmentation on various network parameters such as coupling strength, network size, degree of the network, and frequency distribution are then explored. The size of the phase resetting region would also affect the magnitude of the order parameter at \(\tau_{P C}\) since it directly affects the number of oscillators reset by the perturbation. Therefore, the dependence of order parameter on the size of the phase resetting region is also analyzed.
{\copyright 2021 American Institute of Physics}