Stability analysis of Navier–Stokes system

Analytical study on the motions around equilibrium points of restricted four-body problem

Abstract. Microbial-explicit soil organic carbon (SOC) cycling models are increasingly being recognized for their advantages over linear models in describing SOC dynamics. These models are known to exhibit oscillations, but it is not clear when they yield stable vs. unstable equilibrium points (EPs) – i.e., EPs that exist analytically but are not stable in relation to small perturbations and cannot be reached by transient simulations. The occurrence of such unstable EPs can lead to unexpected model behavior in transient simulations or unrealistic predictions of steady-state soil organic carbon (SOC) stocks. Here, we ask when and why unstable EPs can occur in an archetypal microbial-explicit model (representing SOC, dissolved OC (DOC), microbial biomass, and extracellular enzymes) and some simplified versions of it. Further, if a model formulation allows for physically meaningful but unstable EPs, can we find constraints in the model parameters (i.e., environmental conditions and microbial traits) that ensure stability of the EPs? We use analytical, numerical, and descriptive tools to answer these questions. We found that instability can occur when the resupply of a growth substrate (DOC) is (via a positive feedback loop) dependent on its abundance. We identified a conservative, sufficient condition in terms of model parameters to ensure the stability of EPs. Principally, three distinct strategies can avoid instability: (1) neglecting explicit DOC dynamics, (2) biomass-independent uptake rate, or (3) correlation between parameter values to obey the stability criterion. While the first two approaches simplify some mechanistic processes, the third approach points to the interactive effects of environmental conditions and parameters describing microbial physiology, highlighting the relevance of basic ecological principles for the avoidance of unrealistic (i.e., unstable) simulation outcomes. These insights can help to improve the applicability of microbial-explicit models, aid our understanding of the dynamics of these models, and highlight the relation between mathematical requirements and (in silico) microbial ecology.

Mono-unstable polyhedra with point masses have at least 8 vertices

A second-order model is developed to describe the essential behavior of the grid-following converter (GFLC) and the grid-forming converter (GFMC) for studying the synchronization characteristics of grid-connected converter systems. In this article, a general set of criteria in terms of the manifolds and saddle quantity is derived for studying the homoclinic bifurcation behavior of GFLCs and GFMCs. It is shown that three unstable periodic orbits and two homoclinic bifurcations may exist for the GFLC, while only one stable periodic orbit and one homoclinic bifurcation may exist for the GFMC. It is also found that after the onset of the first homoclinic bifurcation, the stable equilibrium point (SEP) is surrounded by the stable manifolds of the unstable equilibrium point (UEP), which forms the SEP's basin of attraction, for both GFLCs and GFMCs. In this case, the converters may lose their synchronization if the trajectory crosses over the UEP. Therefore, system parameters should be designed to avoid the onset of the aforementioned bifurcation so that the SEP's basin of attraction can be significantly enlarged to contain the UEP. As a result, the GFLC or GFMC can eventually resynchronize with the grid even when its trajectory passes the UEP. Finally, experimental results are provided to verify these theoretical findings.

AN INVESTIGATION OF STABILITY IN THE LARGE BEHAVIOUR OF A CONTROL SURFACE WITH STRUCTURAL NON-LINEARITIES IN SUPERSONIC FLOW

For a fourth-order autonomous nonlinear electric circuit, we present two evolution patterns to complexity associated with the three coexisting equilibrium points. In the first pattern, stable periodic movement with symmetric structure can be observed by Hopf bifurcation from the unstable equilibrium point, which may lead to chaos via cascading of period-doubling bifurcations. All the attractors, including the chaos, keep the symmetric property. While in the second evolution pattern, two limit cycles symmetric to each other may occur via Hopf bifurcations from the other two stable equilibrium points, which may also lead to two chaotic attractors, respectively. Comparing with the two evolution procedures associated with the two stable equilibrium points, not only the bifurcations keep the same pace, but also the attractors including the two final chaotic attractors are still symmetric to each other. With further variation of the parameters, the two chaotic attractors may interact with each other to form another enlarged chaotic attractor, which is qualitatively equivalent to the chaos in the first evolution pattern.

A closed form equation is derived that describes the stable manifold for each of the n-1 type-1 unstable equilibrium points (UEPs). These stable manifolds can be obtained without having to compute the postfault stable equilibrium points for each of the stable manifolds. The equation describing the stable manifold for each UEP can be written knowing only the postfault stable equilibrium point. The n-1 stable manifolds for the UEPs constrain the power across the cutset of branches that encircle each of the n-1 generators that are not assumed to be the swing machine. For a point to lie on the stable manifolds of a specific UEP, the sum of power flows across all branches in the cutset that surround the generator that corresponds to a specific UEP must equal the sum of the power flows across this cutset at the postfault equilibrium point. These results are intuitive since the equal-area criterion could be used to characterize the region of stability for a single-machine infinite-bus system in the exact same manner. >

This study presents a study of equilibrium points, periodic orbits, stabilities, and manifolds in a rotating plane-symmetric potential field. It has been found that the dynamical behaviour near equilibrium points is completely determined by the structure of the submanifolds and subspaces. The non-degenerate equilibrium points are classified into twelve cases. The necessary and sufficient conditions for linearly stable, non-resonant unstable and resonant equilibrium points are established. Furthermore, the results show that a resonant equilibrium point is a Hopf bifurcation point. In addition, if the rotating speed changes, two non-degenerate equilibria may collide and annihilate each other. The theory developed here is lastly applied to two particular cases, motions around a rotating, homogeneous cube and the asteroid 1620 Geographos. We found that the mutual annihilation of equilibrium points occurs as the rotating speed increases, and then the first surface shedding begins near the intersection point of the −x axis and the surface. The results can be applied to planetary science, including the birth and evolution of the minor bodies in the Solar system, the rotational breakup and surface mass shedding of asteroids, etc.

<p style='text-indent:20px;'>We study a predator-prey model with Holling type Ⅰ functional response, an alternative food source for the predator, and multiple Allee effects on the prey. We show that the model has at most two equilibrium points in the first quadrant, one is always a saddle point while the other can be a repeller or an attractor. Moreover, there is always a stable equilibrium point that corresponds to the persistence of the predator population and the extinction of the prey population. Additionally, we show that when the parameters are varied the model displays a wide range of different bifurcations, such as saddle-node bifurcations, Hopf bifurcations, Bogadonov-Takens bifurcations and homoclinic bifurcations. We use numerical simulations to illustrate the impact changing the predation rate, or the non-fertile prey population, and the proportion of alternative food source have on the basins of attraction of the stable equilibrium point in the first quadrant (when it exists). In particular, we also show that the basin of attraction of the stable positive equilibrium point in the first quadrant is bigger when we reduce the depensation in the model.

This paper introduces the swirling pendulum, a two-link, two degree-of-freedom mechanism which is under-actuated and has an unusual non-planar coupling with axis of rotation of the two links being perpendicular to each other. The swirling pendulum mechanism, while being simple to mathematically represent and easy to physically construct, exhibits several properties like loss of inertial coupling, loss of relative degree, multiple stable and unstable equilibrium points. These properties are unique as well as interesting from dynamics and controls point of view which make the swirling pendulum an excellent test-bed for testing various ideas in control and demonstrating several notions associated with systems and control theory. In this paper, we discuss the modeling of the swirling pendulum mechanism based on Lagrange’s equation along with an analysis related to equilibrium points and their stability. We also present simulation results for regulatory as well as tracking control tasks through simulations on a non-linear model using control methods like LQR, lead compensator and system inversion-based control to demonstrate the utility of the proposed mechanism in the area of systems, control and dynamics. Furthermore, we also discuss experimental results for controls applied on a real-time hardware setup.

In this paper, a class of discrete-time complex-valued neural networks with time-varying delays and impulses are considered.Based on M-matrix theory and analytic methods, several simple sufficient conditions checking the global exponential stability are obtained for the considered neural networks.The obtained results show that the stability still remains under certain impulsive perturbations for the neural network with stable equilibrium point, and the neural network with unstable equilibrium point can be stabilization by impose appropriate impulsive perturbations.

The method of Fourier transforms has been used in the investigation of the electrical forces between the platelets of the montmorillonite crystal. The potential-energy curve, as a function of the distance 2D between the platelets, is shown to have two turning points. The one with the smaller value of D is a minimum and is therefore a point of stable equilibrium, while the one with the larger value of D is a maximum and is a point of unstable equilibrium. For small separations of the platelets the potential energy is positive and increases with decreasing separation, while at large separations the potential energy is negative and decreases linearly with increasing separation. The foregoing is shown to be in complete accord with accumulated experimental findings over the years. Physically, the basis of the shape of the energy curve is that at small separations the repulsive ion—solvent and ion-image forces prevail whereas at intermediate distances the attractive surface-ion terms become important; finally the surface—surface terms dominate at larger distances. It is also shown that by suitable expansions the Fourier transform method can be used to generate the required sets of electrostatic images developed in an earlier work by the optical analog.

This article deals with the dynamics of prey and predator populations in a two patch environment, a protected area and an unprotected area for fishing. The prey disperses between the two patches and migrates easily. There are two predators, one is in the protected area and another is in the unprotected area. The predators cannot migrate. Both prey and predator in unprotected area are harvested with constant efforts. The dynamical behavior of the populations is stated as a system of differential equations. The existence of a positive equilibrium point and its stability are investigated. We discuss the local stability of the positive equilibrium point. The stable equilibrium point is then associated with optimal harvesting problems. Based on the analysis, we found that there exist a stable positive equilibrium point when there is no harvesting. For model with constant efforts for both prey and predator, we found that over fishing will maximize the profit but the predator in the unprotected area will be extinct. With the help of Pontryagin’s maximum principle in maximizing the present value of revenues, we found the extremal of the efforts that maximize the present value of revenues. This means that both prey and predator in the protected area as well as the prey and predator in the unprotected area are possibly coexist although the prey and the predator in the unprotected area are harvested with constant efforts. Some numerical simulations area given to confirm the result of analysis.

The classical theory of Kosambi–Cartan–Chern (KCC) developed in differential geometry provides a powerful method for analyzing the behaviors of dynamical systems. In the KCC theory, the properties of a dynamical system are described in terms of five geometrical invariants, of which the second corresponds to the so-called Jacobi stability of the system. Different from that of the Lyapunov stability that has been studied extensively in the literature, the analysis of the Jacobi stability has been investigated more recently using geometrical concepts and tools. It turns out that the existing work on the Jacobi stability analysis remains theoretical and the problem of algorithmic and symbolic treatment of Jacobi stability analysis has yet to be addressed. In this paper, we initiate our study on the problem for a class of ODE systems of arbitrary dimension and propose two algorithmic schemes using symbolic computation to check whether a nonlinear dynamical system may exhibit Jacobi stability. The first scheme, based on the construction of the complex root structure of a characteristic polynomial and on the method of quantifier elimination, is capable of detecting the existence of the Jacobi stability of the given dynamical system. The second algorithmic scheme exploits the method of semi-algebraic system solving and allows one to determine conditions on the parameters for a given dynamical system to have a prescribed number of Jacobi stable fixed points. Several examples are presented to demonstrate the effectiveness of the proposed algorithmic schemes.

In two recent papers, Brayton and Tong developed some significant results which are the basis of a constructive approach in the stability analysis of dynamical systems. Although their algorithm is powerful, it taxes the capabilities of most modern computers when the dimension of a system is high or even when the dimension is moderate in size. In this paper we remove these difficulties to a certain extent by generalizing the work of Brayton and Tong to interconnected dynamical systems. In our approach, the analysis of complex dynamical systems (which may be of high dimension) is accomplished in terms of the qualitative properties of the free subsystems and the interconnections of such systems.