The Swirling Pendulum: Conceptualization, Modeling, Equilibria and Control Synthesis

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.

Delays are a common physic effect that is present in a huge quantity of industrial systems with feedback control. Sometimes the impact of the delay presence in a system is neglected without any difference in the performance of the controller. But in some cases, the Delay quantity reaches levels that increase the overshoot significantly or destabilizes the system. For this system the CTCR method can be used to design a “delayed scheduled” controller able to reject the delays effect. To be able to apply this method it is necessary for both the system and the controller to be Linear or linearized. In this article the study case is an articulated inverted pendulum or also called “Pendubot”. This configuration of pendulum was selected because, in spite of being a simple pendulum, it has four equilibrium points, where two equilibrium points were selected to be controlled, the most unstable which has the maximum potential energy for each link and the most stable with the lowest potential energy. The purpose of work with those equilibrium points is compare the difference between the delay rejection in each point with similar setting times and overshoots from controllers. In order to get an accurate model, the state of the current is added along with the viscous friction terms for each joint in the pendulum. To tune the linear and non-linear model an experimental validation was carry on the physical prototype from the Universidad Industrial de Santander (VIE-5373 UIS). The control laws applied were a classical PID and LQR control. Due the controller sample frequency is extremely high in comparison to the states response, all the controllers and linear models were implemented in continuous space. In the first tuning process it was observed that the PID control gets a significantly lowest performance than the LQR control in the unstable equilibrium point, for that reason the comparison between points only was carried on with the LQR equation of feedback system, this can be done in different ways, but for the PID control the Mason theorems was applied and for LQR control only with matrix operation the equation can be obtained. After applying the “Delay scheduling” it was observed that the tuned LQR get a highest delay rejection that PID. An observer fact was that although the controllers for each point have a similar performance the delay pockets have completely different values due the final poles locations for each point. Also was observed that the system only gets one stability pocket, this could because only one delay in the actuator was induced.

Stability analysis of dynamical system is very useful and is able to classify the role of stable and unstable equilibrium points. In this work, Naiver–Stokes system has been studied by using KCC theory. The Jacobi stability and dynamics of the deviation vector near equilibrium points have been also studied. Further, the effect of bifurcation parameter on stability of Navier–Stokes system has been observed and found the limiting conditions for bifurcation. Numerical examples of particular interest have been taken to compare the results of Jacobi stability and linear stability. It is observed that Jacobi stability on the basis of KCC theory is more efficient than the linear stability.

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.

Qualitative analysis by modern methods of a stability problem in power-system analysis