Attractors for a second order nonautonomous lattice dynamical system with nonlinear damping

We study, in the setting of a real Hilbert space H, the asymptotic behavior of trajectories of the second-order dissipative dynamical system with linear and gradient-driven nonlinear damping where λ > 0 and f, Φ: H → R are two convex differentiable functions. It is proved that if Φ is coercive and bounded from below, then the trajectory converges weakly towards a minimizer of Φ. In particular, we state that under suitable conditions, the trajectory strongly converges to the minimizer of Φ exponentially or polynomially.

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Integrable two-dimensional dynamical systems and the characteristic Lie algebras

Proceedings of the Mavlyutov Institute of Mechanics ◽

10.21662/uim2007.1.023 ◽

2007 ◽

Vol 5 ◽

pp. 195-200

Author(s):

A.V. Zhiber ◽

O.S. Kostrigina

Keyword(s):

Dynamical System ◽

Dynamical Systems ◽

Lie Algebra ◽

Lie Algebras ◽

Second Order ◽

System Of Equations ◽

Two Dimensional ◽

Finite Dimensional ◽

Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

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A new uniqueness theorem of wave profiles for a 2-D bistable lattice dynamical system

Applied Mathematics Letters ◽

10.1016/j.aml.2020.106958 ◽

2021 ◽

Vol 115 ◽

pp. 106958

Author(s):

Shuangting Lan ◽

Peixuan Weng ◽

Zhaoquan Xu

Keyword(s):

Dynamical System ◽

Uniqueness Theorem ◽

Lattice Dynamical System ◽

Wave Profiles

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Upper semicontinuity of the attractor for lattice dynamical systems of partly dissipative reaction diffusion systems

Journal of Applied Mathematics ◽

10.1155/jam.2005.273 ◽

2005 ◽

Vol 2005 (3) ◽

pp. 273-288 ◽

Cited By ~ 2

Author(s):

Ahmed Y. Abdallah

Keyword(s):

Dynamical System ◽

Hilbert Space ◽

Upper Semicontinuity ◽

Reaction Diffusion ◽

Dimensional Lattice ◽

Reaction Diffusion Systems ◽

Infinite Dimensional ◽

Diffusion Systems ◽

Lattice Dynamical System ◽

Lattice Dynamical Systems

We investigate the existence of a global attractor and its upper semicontinuity for the infinite-dimensional lattice dynamical system of a partly dissipative reaction diffusion system in the Hilbert spacel2×l2. Such a system is similar to the discretized FitzHugh-Nagumo system in neurobiology, which is an adequate justification for its study.

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Model Based Control of Laminar Wake Using Fluidic Actuation

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4002085 ◽

2010 ◽

Vol 5 (4) ◽

Cited By ~ 22

Author(s):

Imran Akhtar ◽

Ali H. Nayfeh

Keyword(s):

Dynamical System ◽

Stokes Equations ◽

Control Function ◽

Practical Importance ◽

Nonlinear Damping ◽

Open Loop ◽

Pole Placement ◽

Cylinder Surface ◽

Control Input ◽

Laminar Wake

Control of fluid-structure interaction is of practical importance from the perspective of wake modification and reduction of vortex-induced vibrations (VIVs). The aim of this study is to design a control to suppress vortex shedding. We perform a two-dimensional simulation of the flow past a circular cylinder using a parallel Computational Fluid Dynamics (CFD) solver. We record the velocity and pressure fields over a shedding cycle and compute the proper orthogonal decomposition (POD) modes of the divergence-free velocity and pressure, respectively. The Navier–Stokes equations are projected onto these POD modes to reduce the dynamical system to a set of ordinary-differential equations (ODEs). This dynamical system exhibits a limit cycle with negative linear damping and positive nonlinear damping. The reduced-order model is then modified by placing a pair of suction actuators and applying a control strategy using a control function method. We use the pressure POD mode distribution on the cylinder surface to optimally locate the actuators. We design a controller based on the linearized system and make it positively damped using pole-placement technique. The control-input settles to a constant value, suggesting constant suction through the actuators. We validate the results using CFD simulations in an open-loop setting and observe suppression of the hydrodynamic forces acting on the cylinder.

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