Quasi-periodic Forces Research Articles

Pullback and uniform exponential attractors for non-autonomous Oregonator systems

Abstract We consider the long-time global dynamics of non-autonomous Oregonator systems. This system is a coupled system of three reaction-diffusion equations, that arises from the Belousov-Zhabotinskii reaction. We first present some sufficient conditions for the existence of pullback and uniform exponential attractors for non-autonomous dynamical system. Then, we apply abstract results to prove the existence of a pullback exponential attractor for Oregonator systems affected by time-dependent forces and a uniform exponential attractor for Oregonator systems driven by quasi-periodic external forces.

Dynamical complexity of Anosov systems driven by a quasi-periodic force

Dynamical complexity of Anosov systems driven by a quasi-periodic force

Control of chaos and bifurcation by nonfeedback methods in an autocatalytic chemical system

AbstractNonfeedback methods of chaos and bifurcation control are suited for practical applications because of their speed, flexibility, no online monitoring, and processing requirements. In this paper, we analyze the control performance of various nonfeedback methods such as (i) adding a weak periodic force, (ii) adding a second periodic force, and (iii) adding a quasiperiodic force. We apply these methods to control chaos and bifurcation in an autocatalytic chemical system. By choosing the amplitudes of the external excitation as control parameters, we investigate what effect the amplitudes have on the dynamics of the chemical system with suitable system's parameters value. Controlling of chaotic and bifurcation behaviors has been investigated through the bifurcation diagram, phase portrait, and time series.

Random uniform exponential attractors for Schrödinger lattice systems with quasi-periodic forces and multiplicative white noise

<p style='text-indent:20px;'>We mainly consider the existence of random uniform exponential attractors for Schrödinger lattice systems with quasi-periodic forces and multiplicative white noise. We first give an existence criterion for a random uniform exponential attractor for a jointly continuous non-autonomous random dynamical system (NRDS) defined on the space of infinite sequences with complex-valued components. Then we transfer the considered stochastic Schrödinger lattice system into a random system with random parameters and quasi-periodic forces without white noise through Ornstein-Uhlenbeck process, whose solutions generate a jointly continuous NRDS. Thirdly, we prove the existence of a uniform absorbing random set for this NRDS. Fourthly, we construct a bounded closed random set and verify the Lipschitz continuity of the NRDS on this random set, and next we decompose the difference between two solutions into a sum of two parts including some random variables with bounded expectation. Finally, we obtain the existence of a random uniform exponential attractor for the considered system.</p>

Random uniform exponential attractors for non-autonomous stochastic discrete long wave-short wave resonance equations

In this paper, we study the dynamical behavior of the non-autonomous stochastic discrete long wave-short wave resonance system with multiplicative white noise and quasi-periodic forces. Firstly, we prove the uniqueness and existence for the solution of the system, which generates a jointly continuous random dynamical system $ \Phi $. Secondly, we prove the existence of a uniform absorbing random set for $ \Phi $ and estimate tail of the solution. Thirdly, we prove the Lipschitz continuity of the skew-product cocycle $ \pi $ defined on an extended space. Fourthly, we prove that the expectation of some random variables are bounded. Finally, we prove the existence of a random uniform exponential attractor for the considered system.

Random uniform exponential attractors for non-autonomous stochastic Schrödinger lattice systems in weighted space

<abstract><p>We mainly study the existence of random uniform exponential attractors for non-autonomous stochastic Schrödinger lattice system with multiplicative white noise and quasi-periodic forces in weighted spaces. Firstly, the stochastic Schrödinger system is transformed into a random system without white noise by the Ornstein-Uhlenbeck process, whose solution generates a jointly continuous non-autonomous random dynamical system $ \Phi $. Secondly, we prove the existence of a uniform absorbing random set for $ \Phi $ in weighted spaces. Finally, we obtain the existence of a random uniform exponential attractor for the considered system $ \Phi $ in weighted space.</p></abstract>

Quasi-periodic solutions for the incompressible Navier-Stokes equations with nonlocal diffusion

<abstract><p>This paper studied the incompressible Navier-Stokes (NS) equations with nonlocal diffusion on $ \mathbb{T}^d (d \ge 2) $. Driven by a time quasi-periodic force, the existence of time quasi-periodic solutions in the Sobolev space was established. The proof was based on the decomposition of the unknowns into the spatial average part and spatial oscillating one. The former were sought under the Diophantine non-resonance assumption, and the latter by the contraction mapping principle. Moreover, by constructing suitable time weighted function space and using the Banach fixed point theorem, the asymptotic stability of quasi-periodic solutions and the exponential decay of perturbation were proved.</p></abstract>

Strange Nonchaotic Attractors in Memristor-Based Shimizu–Morioka Oscillator

The existence of strange nonchaotic attractors (SNAs) is examined using a memristor-based Shimizu–Morioka oscillator. In particular, we analyze the birth of SNAs by varying one of the frequency of external quasi-periodic forces. The occurrence and its mechanism are further analyzed through rational approximation theory. Following that, we also perform different nonlinear dynamical characterizations, including Poincaré surface sections, singular continuous spectrum, separation of nearby trajectories, 0–1 test, as well as finite-time Lyapunov exponents for more precise validation. Finally, the experimental evidence for the occurrence of SNA is inspected through a real-time analog circuit in the laboratory.

Quasi-periodic forces and conditionally averaged characteristics of unsteady cavitation flow around a hydrofoil

Large-eddy simulations are performed to investigate the cavitating flow around two dimensional hydrofoil section with angle of attack of 9° and high Reynolds number of 1.3×106. We use the Schnerr-Sauer model for accurate phase transitions modelling. Instantaneous velocity fields are compared successfully with PIV data using the methodology of conditional averaging to take into account only the liquid phase characteristics as in PIV. The presence of two frequencies in a spectrum corresponding to the full and partial cavity detachments is analysed.

Random pullback and uniform exponential attractors for stochastic non-autonomous Navier-Stokes equations

We first prove the existence of a random pullback exponential attractor for the 2D Navier-Stokes equation with additive noise and non-autonomous forcing. Then, we introduce the definition and existence conditions of a random uniform exponential attractor for a jointly continuous non-autonomous random dynamical system, and prove the existence of a random uniform exponential attractor for the 2D Navier-Stokes equation with additive noise and quasi-periodic external forces.

The Navier\u2013Stokes Equation with Time Quasi-Periodic External Force: Existence and Stability of Quasi-Periodic Solutions

We prove the existence of small amplitude, time-quasi-periodic solutions (invariant tori) for the incompressible Navier–Stokes equation on the d-dimensional torus mathbb T^d, with a small, quasi-periodic in time external force. We also show that they are orbitally and asymptotically stable in H^s (for s large enough). More precisely, for any initial datum which is close to the invariant torus, there exists a unique global in time solution which stays close to the invariant torus for all times. Moreover, the solution converges asymptotically to the invariant torus for t rightarrow + infty , with an exponential rate of convergence O( e^{- alpha t }) for any arbitrary alpha in (0, 1).

Random uniform exponential attractors for non-autonomous stochastic lattice systems and FitzHugh–Nagumo lattice systems with quasi-periodic forces and multiplicative noise

We first give an existence criterion for a random uniform exponential attractor for a jointly continuous non-autonomous random dynamical system defined on the product space of [Formula: see text]-weighted spaces of infinite sequences. Then, based on this criterion, we prove the existence of random uniform exponential attractors for stochastic lattice systems and stochastic FitzHugh–Nagumo lattice systems that are both with quasi-periodic forces.

Random uniform exponential attractor for stochastic non-autonomous reaction-diffusion equation with multiplicative noise in ℝ3

Abstract We first introduce the concept of the random uniform exponential attractor for a jointly continuous non-autonomous random dynamical system (NRDS) and give a theorem on the existence of the random uniform exponential attractor for a jointly continuous NRDS. Then we study the existence of the random uniform exponential attractor for reaction-diffusion equation with quasi-periodic external force and multiplicative noise in ℝ3.

Strange nonchaotic attractors in a nonsmooth dynamical system

Strange nonchaotic attractors (SNAs) have fractal geometric structure, but are nonchaotic in the dynamical sense. Since Grebogi et al. discovered SNA in 1984, it has become one of the important topics in nonlinear dynamics. Now, the study of SNAs has been mainly confined to smooth dynamics with quasiperiodic excitation or random excitation. In this paper, we consider a class of single-degree-of-freedom gear dynamical system with quasiperiodic forcing. We show that the gear transmission system can be modeled as a three-dimensional piecewise linear system, which belongs to a typical class of nonsmooth system. We then show that SNAs do exist in such nonsmooth dynamical system with quasiperiodic force. The dynamical behavior of the nonsmooth system is analyzed as a parameter is varied. The dynamics is analyzed through phase diagrams and bifurcation diagrams, Lyapunov exponents, singular continuous power spectrum, phase sensitivity of time series and rational approximations.

Coherent structures and wind force generation of square-section building model

Typical flow topology of Karman type and arch type vortex shedding from two square-section building models with aspect ratios 2 and 6 is revealed by space-time analysis of fluctuating velocity fields and surface pressure signals obtained by synchronized time-resolved particle-image velocimetry (PIV) and multi-port pressure scanning. Proper orthogonal decomposition (POD) is applied to the PIV data with a pretreatment step to extract the two types of vortex patterns based on their inherent antisymmetry and symmetry features on the horizontal plane about the wake centerline. The antisymmetric POD modes have the general mode shapes of a row of single vortices of alternating rotation senses along the wake centerline, while the mode shapes of the symmetric POD modes show successive pairs of counter-rotating vortices along the wake centerline. When combined with the time-averaged base flow of the wake featuring two stationary recirculating bubbles, the two types of POD modes exhibit the respective vortex patterns of Karman and arch type. The roles of these coherent structures on the generation of fluctuating across-wind force, especially during occurrence of peak force magnitudes, are explored with reconstruction of peak flow events from the lower-order POD modes. The study provides a new insight into the evident connection between the quasi-cycles of antisymmetric POD modes and the generation of quasi-periodic across-wind force.

Simultaneous measurement of wind velocity field and wind forces on a square tall building

Vortex shedding from a tall building is known to be responsible for the quasi-periodic across-wind force exerted on the building. This article unveils the exact relationship between the vortex shedding pattern and the fluctuating across-wind force. Simultaneous particle-image velocimetry and pressure measurements are carried out on a square-plan tall building model in the wind tunnel toward an understanding of the velocity–pressure–force relation for across-wind force generation on the building. A collection of instantaneous wind flow patterns and synchronized wind pressure distributions suggests the existence of full periods of vortex shedding from the building. The results are further analyzed using the conditional sampling method by which the roles of development and shedding of large-scale vortices in the building wake on the generation of peak across-wind forces are evidently found. Furthermore, quasi-periodicity of across-wind excitation is clearly confirmed with Hilbert transform of the across-wind force signal. The phase averaging technique is applied to the particle-image velocimetry flow fields and distinct vortex shedding patterns from the building are observed for most of the measurement time, together with an evident phase relationship with the across-wind forces.

Observation of chaotic and strange nonchaotic attractors in a simple multi-scroll system

The dynamics of third-order chaotic circuit with threshold controller as a nonlinear element is reported. The circuit and the preliminary result reported by Chithra et al. (J Comput Electron 16:883---889, 2017), are considered as a prototype by adding quasiperiodic force in the original circuit. This present work focuses attention on exhibiting all possible nonlinear dynamical phenomena (i.e., periodic motion, chaotic, strange nonchaotic and multi-scroll attractor) for a different set of circuit parametric values. The approach helps in proof-of-experimental concept to implement all possible complex dynamical behaviors from a single circuit. Further, we also place emphasis on numerical simulation and physical circuit realization on the breadboard. The existence of strange nonchaotic attractor is quantitatively confirmed by calculating $${ 0}{-}{} { 1}$$0-1 test and singular-continuous spectrum analysis. We then find this circuit has potential to produce multi-scroll strange nonchaotic attractor by increasing threshold values. The experimental results are identical to numerical simulations.

Pullback and uniform exponential attractor for non-autonomous Boissonade system

In this paper, we first improve the existing sufficient conditions for the existence of pullback and uniform exponential attractors for non-autonomous dynamical systems. Then we prove the existence, continuity and boundedness of fractal dimension of a pullback exponential attractor for Boissonade system effected by time-dependent forces. Finally, we prove the existence and boundedness of fractal dimension of a uniform exponential attractor for Boissonade system driven by quasi-periodic external forces.

Response solutions for forced systems with large dissipation and arbitrary frequency vectors

We study the behaviour of one-dimensional strongly dissipative systems subject to a quasi-periodic force. In particular we are interested in the existence of response solutions, that is, quasi-periodic solutions having the same frequency vector as the forcing term. Earlier results available in the literature show that, when the dissipation is large enough and a suitable function involving the forcing has a simple zero, response solutions can be proved to exist and to be attractive provided some Diophantine condition is assumed on the frequency vector. In this paper we show that the results extend to the case of arbitrary frequency vectors.

Attractors for second order nonautonomous lattice system with dispersive term

In this paper, we prove the existence of pullback attractor, pullback exponential attractor and uniform attractor for second order non-autonomous lattice system with dispersive term and time-dependent forces. Then we prove the existence of uniform exponential attractor for the system driven by quasi-periodic external forces.