Numerical investigation of lagrangian coherent structures in steady rotation vortex shedding control

In this paper, vortex shedding and suppression are numerically investigated as autonomous and non-autonomous dynamical systems respectively. Lagrangian coherent structures (LCSs) are used as a numerical tool to analyze these systems. These structures are ridges of Finite time Lyapunov exponent (FTLE) which act as material surfaces that are transport barriers within the flow. Initially, the utility of LCSs is explored for revealing the coherent structures of these systems. Finally, an active flow control method, steady rotation is applied to the non-autonomous dynamical system with different speed ratios to mitigate vortex shedding magnitude. This will eventually turn the system into an autonomous system. Fixed saddle points, separation profiles essentially as unstable time variant manifolds attached to cylinder wall and evolution of other unstable manifolds with variant speed ratios are analyzed with reference to LCSs. It is revealed that speed ratio of 2.1 fully suppresses the von Karman vortex street at Reynolds number of 100 and system turns into an autonomous dynamical system with fixed saddle points and time-invariant manifolds.

Asymptotic behavior of nonlocal partial differential equations with long time memory

<p style='text-indent:20px;'>In this paper, it is first addressed the well-posedness of weak solutions to a nonlocal partial differential equation with long time memory, which is carried out by exploiting the nowadays well-known technique used by Dafermos in the early 70's. Thanks to this Dafermos transformation, the original problem with memory is transformed into a non-delay one for which the standard theory of autonomous dynamical system can be applied. Thus, some results about the existence of global attractors for the transformed problem are {proved}. Particularly, when the initial values have higher regularity, the solutions of both problems (the original and the transformed ones) are equivalent. Nevertheless, the equivalence of global attractors for both problems is still unsolved due to the lack of enough regularity of solutions in the transformed problem. It is therefore proved the existence of global attractors of the transformed problem. Eventually, it is highlighted how to proceed to obtain meaningful results about the original problem, without performing any transformation, but working directly with the original delay problem.</p>

Dynamical Stability and Geometrical Diagnostic of the Power Law K-Essence Dark Energy Model with Interaction

We investigate the cosmological evolution of the power law k-essence dark energy (DE) model with interaction in FRWL spacetime with the Lagrangian that contains a kinetic function F(X)=−X+X. Concretely, the cosmological evolution in this model are discussed by the autonomous dynamical system and its critical points, together with the corresponding cosmological quantities, such as Ωϕ, wϕ, cs2, and q, are calculated at each critical point. The evolutionary trajectories are drawn in order to show the dynamical process on the phases plan around the critical points. The result that we obtained indicates that there are four dynamical attractors, and all of them correspond to an accelerating expansion of universe for certain potential parameter and coupling parameter. Besides that, the geometrical diagnostic by the statefinder hierarchy S3(1) and S4(1) of this scalar field model are numerically obtained by the phase components, as an extended null diagnostic for the cosmological constant. This diagnostic shows that both the potential parameter λ and interaction parameter α play important roles in the evolution of the statefinder hierarchy.

Bianchi-I cosmology within f(T): Reconstruction method and dynamical study

This paper is fundamentally devoted to the cosmological reconstruction and dynamic studying in homogeneous BIANCHI-I space-time under the [Formula: see text] background. Its content is supported by the fact that in the General Relativity description of the standard cosmological paradigm, the evolution from an anisotropic universe into an Friedmann–Lemaitre–Robertson–Walker (FLRW) one can be achieved by a period of inflationary expansion. Nowadays, modified gravity theories like [Formula: see text] are widely accepted to provide a real description of some universe evolution phases like inflation era, matter-dominated era, etc. So, we aim to examine here what [Formula: see text] gravity model can accommodate with an anisotropic universe, an expanding universe and even the transition between both evolutions. To reach this goal, we use a reconstruction method based on dynamic equations in Bianchi-I space-time by assuming a particular form for the metric anisotropy and by specifying some time functions describing average scale factor. Most of the obtained models are consistent with certain known results in the literature but other add new results in this work. In the second part of this work, the dynamical behaviors of the Bianchi-I space-time are addressed through the reconstruction of an autonomous dynamical system. For an aleatory choice of anisotropic fluid, the numerical analysis of the system shows that the metric anisotropy decreases with expansion. Then, an attractor point is reached and becomes unstable by the end of inflation. Such interesting properties found in this work on Bianchi-I space-time are often interpreted as graceful exit from inflation which doesn’t occur in ordinary FLRW space-time.

Mixed-Mode Oscillations Based on Complex Canard Explosion in a Fractional-Order Fitzhugh-Nagumo Model.

This article highlights particular mixed-mode oscillations (MMO) based on canard explosion observed in a fractional-order Fitzhugh-Nagumo (FFHN) model. In order to rigorously analyze the dynamics of the FFHN model, a recently introduced mathematical notion, the Hopf-like bifurcation (HLB), which provides a precise definition for the change between a fixed point and an S−asymptotically T−periodic solution, is used. The existence of HLB in this FFHN model is proved and the appearance of MMO based on canard explosion in the neighborhoods of such HLB points are numerically investigated using a new algorithm: the global-local canard explosion search algorithm. This MMO is constituted of various patterns of solutions with an increasing number of small-amplitude oscillations when two key parameters of the FFHN model are varied simultaneously. On the basis of such numerical experiment, it is conjectured that chaos could occur in a two-dimensional fractional-order autonomous dynamical system, with the fractional-order close to one. Therefore, this very simple two-dimensional FFHN model, presents an incredible ability to mimic the complex dynamics of neurons.

A physical memristor based Muthuswamy–Chua–Ginoux system

In 1976, Leon Chua showed that a thermistor can be modeled as a memristive device. Starting from this statement we designed a circuit that has four circuit elements: a linear passive inductor, a linear passive capacitor, a nonlinear resistor and a thermistor, that is, a nonlinear “locally active” memristor. Thus, the purpose of this work was to use a physical memristor, the thermistor, in a Muthuswamy–Chua chaotic system (circuit) instead of memristor emulators. Such circuit has been modeled by a new three-dimensional autonomous dynamical system exhibiting very particular properties such as the transition from torus breakdown to chaos. Then, mathematical analysis and detailed numerical investigations have enabled to establish that such a transition corresponds to the so-called route to Shilnikov spiral chaos but gives rise to a “double spiral attractor”.

Using a first-order autonomous dynamical system to evaluate residence time of the Greenland freshwater anomaly in the Subpolar Gyre

&lt;p&gt;Increasing Greenland discharge has contributed more than 5000 km&lt;sup&gt;3&lt;/sup&gt; of surplus fresh water to the Subpolar North Atlantic since the early 1990s. The volume of this freshwater anomaly is projected to cause freshening in the North Atlantic leading to changes in the intensity of deep convection and thermohaline circulation in the subpolar North Atlantic. This is roughly half of the freshwater volume of the Great Salinity Anomaly of the 1970s that caused notable freshening in the Subpolar North Atlantic. In analogy with the Great Salinity Anomaly, it has been proposed that, over the years, this additional Greenland freshwater discharge might have a great impact on convection driving thermohaline circulation in the North Atlantic with consequent impact on climate. Previous numerical studies demonstrate that roughly half of this Greenland freshwater anomaly accumulates in the Subpolar Gyre. However, time scales over which the Greenland freshwater anomaly can accumulate in the subpolar basins is not known. This study estimates the residence time of the Greenland freshwater anomaly in the Subpolar Gyre by approximating the process of the anomaly accumulation in the study domain with a first order autonomous dynamical system forced by the Greenland freshwater anomaly discharge. General solutions are obtained for two types of the forcing function. First, the Greenland freshwater anomaly discharge is a constant function imposed as a step function. Second, the surplus discharge is a linearly increasing function. The solutions are deduced by utilizing results from the numerical experiments that tracked spreading of the Greenland fresh water with a passive tracer. The residence time of the freshwater anomaly is estimated to be about 10&amp;#8211;15 years. The main differences in the solutions is that under the linearly increasing discharge rate, the volume of the accumulated Greenland freshwater anomaly in the Subpolar Gyre does not reach a steady state. By contrast, solution for the constant discharge rate reaches a steady state quickly asymptoting the new steady state value for time exceeding the residence time. Estimated residence time is compared with the numerical experiments and observations.&lt;/p&gt;