Influence of coupling on the dynamics of three delayed oscillators

2021 ◽  
Vol 29 (6) ◽  
pp. 869-891
Author(s):  
Alexandra Kashchenko ◽  
Keyword(s):  
Coupled Oscillators ◽  
Main Part ◽  
Initial Conditions ◽  
Coupling Parameter ◽  
Original Model ◽  
Asymptotics Of Solutions ◽  
Finite Dimensional ◽  
Coupling Parameters ◽  
Equations With Delay ◽  
Dimensional Mapping

The purpose of this study is to construct the asymptotics of the relaxation regimes of a system of differential equations with delay, which simulates three diffusion-coupled oscillators with nonlinear compactly supported delayed feedback under the assumption that the factor in front of the feedback function is large enough. Also, the purpose is to study the influence of the coupling between the oscillators on the nonlocal dynamics of the model. Methods. We construct the asymptotics of solutions of the considered model with initial conditions from a special set. From the asymptotics of the solutions, we obtain an operator of the translation along the trajectories that transforms the set of initial functions into a set of the same type. The main part of this operator is described by a finite-dimensional mapping. The study of its dynamics makes it possible to refine the asymptotics of the solutions of the original model and draw conclusions about its dynamics. Results. It follows from the form of the constructed mapping that for positive coupling parameters of the original model, starting from a certain moment of time, all three generators have the same main part of the asymptotics — the generators are “synchronized”. At negative values of the coupling parameter, both inhomogeneous relaxation cycles and irregular regimes are possible. The connection of these modes with the modes of the constructed finite-dimensional mapping is described. Conclusion. From the results of the work it follows that the dynamics of the model under consideration is fundamentally influenced by the value of the coupling parameter between the generators.

scholarly journals Dependence of Dynamics of a System of Two Coupled Generators with Delayed Feedback on the Sign of Coupling

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1790
Author(s):  
Alexandra Kashchenko
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Coupling Parameter ◽  
Delayed Feedback ◽  
Structure Of Solutions ◽  
Asymptotics Of Solutions ◽  
Finite Dimensional ◽  
Method Of Steps ◽  
Delay Differential ◽  
Negative Coupling

In this paper, we study the nonlocal dynamics of a system of delay differential equations with large parameters. This system simulates coupled generators with delayed feedback. Using the method of steps, we construct asymptotics of solutions. By these asymptotics, we construct a special finite-dimensional map. This map helps us to determine the structure of solutions. We study the dependence of solutions on the coupling parameter and show that the dynamics of the system is significantly different in the case of positive coupling and in the case of negative coupling.

scholarly journals Finite-dimensional boundary uniform stabilization of the Boussinesq system in Besov spaces by critical use of Carleman estimate-based inverse theory

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Irena Lasiecka ◽  
Buddhika Priyasad ◽  
Roberto Triggiani
Keyword(s):  
Besov Space ◽  
Internal Control ◽  
Initial Conditions ◽  
Fluid Velocity ◽  
Critical Role ◽  
Boussinesq System ◽  
Heat Equations ◽  
Fluid Equation ◽  
Low Regularity ◽  
Finite Dimensional

Abstract We consider the 𝑑-dimensional Boussinesq system defined on a sufficiently smooth bounded domain and subject to a pair { v , u } \{v,\boldsymbol{u}\} of controls localized on { Γ ~ , ω } \{\widetilde{\Gamma},\omega\} . Here, 𝑣 is a scalar Dirichlet boundary control for the thermal equation, acting on an arbitrarily small connected portion Γ ~ \widetilde{\Gamma} of the boundary Γ = ∂ ⁡ Ω \Gamma=\partial\Omega . Instead, 𝒖 is a 𝑑-dimensional internal control for the fluid equation acting on an arbitrarily small collar 𝜔 supported by Γ ~ \widetilde{\Gamma} . The initial conditions for both fluid and heat equations are taken of low regularity. We then seek to uniformly stabilize such Boussinesq system in the vicinity of an unstable equilibrium pair, in the critical setting of correspondingly low regularity spaces, by means of an explicitly constructed, finite-dimensional feedback control pair { v , u } \{v,\boldsymbol{u}\} localized on { Γ ~ , ω } \{\widetilde{\Gamma},\omega\} . In addition, they will be minimal in number and of reduced dimension; more precisely, 𝒖 will be of dimension ( d - 1 ) (d-1) , to include necessarily its 𝑑-th component, and 𝑣 will be of dimension 1. The resulting space of well-posedness and stabilization is a suitable, tight Besov space for the fluid velocity component (close to L 3 ⁢ ( Ω ) \boldsymbol{L}^{3}(\Omega) for d = 3 d=3 ) and a corresponding Besov space for the thermal component, q > d q>d . Unique continuation inverse theorems for suitably over-determined adjoint static problems play a critical role in the constructive solution. Their proof rests on Carleman-type estimates, a topic pioneered by M. V. Klibanov since the early 80s.

Delay stabilization of periodic orbits in coupled oscillator systems

2010 ◽  
Vol 368 (1911) ◽  
pp. 319-341 ◽  
Author(s):  
B. Fiedler ◽  
V. Flunkert ◽  
P. Hövel ◽  
E. Schöll
Keyword(s):  
Normal Form ◽  
Numerical Simulations ◽  
Periodic Orbits ◽  
Coupled Oscillators ◽  
Coupled Oscillator ◽  
Non Invasive ◽  
Coupling Parameters ◽  
Necessary And Sufficient ◽  
Time Delayed Feedback ◽  
Coupled Oscillator Systems

We study diffusively coupled oscillators in Hopf normal form. By introducing a non-invasive delay coupling, we are able to stabilize the inherently unstable anti-phase orbits. For the super- and subcritical cases, we state a condition on the oscillator’s nonlinearity that is necessary and sufficient to find coupling parameters for successful stabilization. We prove these conditions and review previous results on the stabilization of odd-number orbits by time-delayed feedback. Finally, we illustrate the results with numerical simulations.

scholarly journals Asymptotic velocity for four celestial bodies

2018 ◽  
Vol 376 (2131) ◽  
pp. 20170426
Author(s):  
Andreas Knauf
Keyword(s):  
Celestial Mechanics ◽  
Integrable Systems ◽  
Energy Surface ◽  
Initial Conditions ◽  
Theme Issue ◽  
Asymptotic Velocity ◽  
Pair Potentials ◽  
Finite Dimensional ◽  
Celestial Bodies ◽  
Almost All

Asymptotic velocity is defined as the Cesàro limit of velocity. As such, its existence has been proved for bounded interaction potentials. This is known to be wrong in celestial mechanics with four or more bodies. Here, we show for a class of pair potentials including the homogeneous ones of degree − α for α ∈(0, 2), that asymptotic velocities exist for up to four bodies, dimension three or larger, for any energy and almost all initial conditions on the energy surface. This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’.

Solving linear systems from dynamical energy analysis - using and reusing preconditioners

2021 ◽  
Vol 263 (5) ◽  
pp. 1041-1052
Author(s):  
Martin Richter ◽  
Gregor Tanner ◽  
Bruno Carpentieri ◽  
David J. Chappell
Keyword(s):  
Wave Equations ◽  
Energy Analysis ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Iterative Solvers ◽  
Computational Method ◽  
Computational Time ◽  
Shell Elements ◽  
Finite Dimensional ◽  
Large Matrix

Dynamical energy analysis (DEA) is a computational method to address high-frequency vibro-acoustics in terms of ray densities. It has been used to describe wave equations governing structure-borne sound in two-dimensional shell elements as well as three-dimensional electrodynamics. To describe either of those problems, the wave equation is reformulated as a propagation of boundary densities. These densities are expressed by finite dimensional approximations. All use-cases have in common that they describe the resulting linear problem using a very large matrix which is block-sparse, often real-valued, but non-symmetric. In order to efficiently use DEA, it is therefore important to also address the performance of solving the corresponding linear system. We will cover three aspects in order to reduce the computational time: The use of preconditioners, properly chosen initial conditions, and choice of iterative solvers. Especially the aspect of potentially reusing preconditioners for different input parameters is investigated.

COEXISTENCE OF LOW- AND HIGH-DIMENSIONAL SPATIOTEMPORAL CHAOS IN A CHAIN OF DISSIPATIVELY COUPLED CHUA’S CIRCUITS

1994 ◽  
Vol 04 (03) ◽  
pp. 639-674 ◽  
Author(s):  
A.L. ZHELEZNYAK ◽  
L.O. CHUA
Keyword(s):  
Phase Space ◽  
Initial Conditions ◽  
Coupling Parameter ◽  
Reaction Diffusion ◽  
Cellular Neural Network ◽  
Spatiotemporal Chaos ◽  
Matrix Phase ◽  
High Dimensional ◽  
The Matrix ◽  
A Chain

Spatiotemporal dynamics of a one-dimensional cellular neural network (CNN) made of Chua’s circuits which mimics a reaction-diffusion medium is considered. An approach is presented to analyse the properties of this reaction-diffusion CNN through the characteristics of the attractors of an associated infinite-dimensional dynamical system with a matrix phase space. Using this approach, the spatiotemporal correlation dimension of the CNN’s spatiotemporal patterns is computed over various ranges of the diffusion coupling parameter, length of the chain, and initial conditions. It is shown that in a finite-dimensional projection of the matrix phase space of the CNN, both low- and high-dimensional attractors corresponding to different initial conditions coexist.

scholarly journals Coupled Thermo Mechanical Characterisation of Polymers Based on Inverse Analyses and IR Measurements

2011 ◽  
Vol 70 ◽  
pp. 393-398 ◽  
Author(s):  
Aurelien Maurel-Pantel ◽  
E. Baquet ◽  
Jerome Bikard ◽  
Noelle Billon
Keyword(s):  
Strain Rate ◽  
Heat Dissipation ◽  
Coupling Parameter ◽  
Superposition Principle ◽  
Rate Sensitivity ◽  
Mechanical Coupling ◽  
Mechanical Characterisation ◽  
Different Temperatures ◽  
Coupling Parameters ◽  
Time Temperature Superposition Principle

Heat dissipation during mechanical testing can disturb experimental characterisation of polymers. In this work it is demonstrated that these effects are not limited to extreme loading conditions such as impacts. A visco-hyperelastic, visco-plastic constitutive model is proposed that accounts for thermo mechanical coupling in a fully 3D thermodynamics approach. Strain-rate and temperature dependencies are coupled using a concept close to the well known time-temperature superposition principle. Constitutive and coupling parameters are identified at the same time using an inverse analysis protocol. An experimental data base is generated for mechanical measurements at different temperatures and strain rates but also for temperatures during tests measured using IR technology. Such a protocol allows investigation on the strain-rate sensitivity in a much more relevant manner than classical one and the value of the so-called Taylor-Quinney coupling parameter is discussed.

scholarly journals THE THEOREM OF AVERAGING IN THE CONDITION OF UNLIMETED SPEED FOR ALMOST-PERIODIC FUNCTIONS

2017 ◽  
Vol 19 (3) ◽  
pp. 53-57
Author(s):  
O.P. Filatov
Keyword(s):  
Differential Inclusion ◽  
Initial Conditions ◽  
Dimensional Space ◽  
Finite Dimensional Space ◽  
Almost Periodic ◽  
Independent Variables ◽  
Finite Dimensional ◽  
All Solutions ◽  
Independent Variable ◽  
The Right

It is proved that the limit of maximal mean is an independent variable of initial conditions if an axis exists from the convex hull of a set of permitted speeds out of a finite-dimensional space and the components of direction vector of the axis are the independent variables with respect to a spectrum of almost-periodic function. The set of permitted speeds is the right hand of differential inclusion. The limit of maximal mean is taken over all solutions of the Couchy problem for the differential inclusion.

scholarly journals Dynamics of Open Quantum Systems I, Oscillation and Decay

Quantum ◽  
2022 ◽  
Vol 6 ◽  
pp. 615 ◽  
Author(s):  
Marco Merkli
Keyword(s):  
Main Part ◽  
Open Quantum Systems ◽  
Companion Paper ◽  
Quantum Systems ◽  
Thermal Bath ◽  
Coupling Constant ◽  
Level System ◽  
Concrete Case ◽  
Finite Dimensional ◽  
Characteristic Features

We develop a framework to analyze the dynamics of a finite-dimensional quantum system S in contact with a reservoir R. The full, interacting SR dynamics is unitary. The reservoir has a stationary state but otherwise dissipative dynamics. We identify a main part of the full dynamics, which approximates it for small values of the SR coupling constant, uniformly for all times t≥0. The main part consists of explicit oscillating and decaying parts. We show that the reduced system evolution is Markovian for all times. The technical novelty is a detailed analysis of the link between the dynamics and the spectral properties of the generator of the SR dynamics, based on Mourre theory. We allow for SR interactions with little regularity, meaning that the decay of the reservoir correlation function only needs to be polynomial in time, improving on the previously required exponential decay.In this work we distill the structural and technical ingredients causing the characteristic features of oscillation and decay of the SR dynamics. In the companion paper [27] we apply the formalism to the concrete case of an N-level system linearly coupled to a spatially infinitely extended thermal bath of non-interacting Bosons.

Sign in / Sign up

Export Citation Format

Share Document