Evolution from regular movement patterns to chaotic attractors in a nonlinear electrical circuit

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

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

<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.

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.

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

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.

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. >

We intend to study a modified version of the planar Circular Restricted Three-Body Problem (CRTBP) by incorporating several perturbing parameters. We consider the bigger primary as an oblate spheroid and emitting radiation while the small primary has an elongated body. We also consider the perturbation from a disk-like structure encompassing this three-body system. First, we develop a mathematical model of this modified CRTBP. We have found there exist five equilibrium points in this modified CRTBP model, where three of them are collinear and the other two are non-collinear. Second, we apply our modified CRTBP model to the Sun–Haumea system by considering several values of each perturbing parameter. Through our numerical investigation, we have discovered that the incorporation of perturbing parameters has resulted in a shift in the equilibrium point positions of the Sun–Haumea system compared to their positions in the classical CRTBP. The stability of equilibrium points is investigated. We have shown that the collinear equilibrium points are unstable and the stability of non-collinear equilibrium points depends on the mass parameter μ of the system. Unlike the classical case, non-collinear equilibrium points have both a maximum and minimum limit of μ for achieving stability. We remark that the stability range of μ in non-collinear equilibrium points depends on the perturbing parameters. In the context of the Sun–Haumea system, we have found that the non-collinear equilibrium points are stable.

&lt;abstract&gt;&lt;p&gt;Infectious diseases have a great impact on the economy and society. Dynamic models of infectious diseases are an effective tool for revealing the laws of disease transmission. Quarantine and nonlinear innate immunity are the crucial factors in the control of infectious diseases. Currently, there no mathematical models that comprehensively study the effect of both innate immunity and quarantine. In this paper, we propose and analyze an SEIQR epidemic model with nonlinear innate immunity. The boundedness and positivity of the solutions are discussed. Employing the next-generation matrix, we compute the expression of the basic reproduction number. Under certain conditions, the phenomenon of backward bifurcation may occur. That is to say, the stable disease-free equilibrium point and the stable endemic equilibrium point coexist when the basic reproduction ratio is less than one. And the basic reproduction number is no longer the threshold value to determine whether the disease breaks out. We investigate the globally asymptotical stability of the disease-free equilibrium point for the system by constructing Lyapunov function. Also, we research the global stability of the endemic equilibrium by using geometric approach. Numerical simulations are carried out to reveal the theoretical results and find some complex dynamics (for example, the existence of Hopf bifurcation) of the system. Both theoretical and numerical results indicate that the nonlinear innate immunity may cause backward bifurcation and Hopf bifurcation, which makes more difficult to eliminate the disease.&lt;/p&gt;&lt;/abstract&gt;

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.

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 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.

Although long-period population size cycles and chaotic fluctuations in abundance are common in ecological models, such dynamics are uncommon in simple population-genetic models where convergence to a fixed equilibrium is most typical. When genotype-frequency cycling does occur, it is most often due to frequency-dependent selection that results from individual or species interactions. In this paper, we demonstrate that fertility selection and genomic imprinting are sufficient to generate a Hopf bifurcation and complex genotype-frequency cycling in a single-locus population-genetic model. Previous studies have shown that on its own, fertility selection can yield stable two-cycles but not long-period cycling characteristic of a Hopf bifurcation. Genomic imprinting, a molecular mechanism by which the expression of an allele depends on the sex of the donating parent, allows fitness matrices to be nonsymmetric, and this additional flexibility is crucial to the complex dynamics we observe in this fertility selection model. Additionally, we find under certain conditions that stable oscillations and a stable equilibrium point can coexist. These dynamics are characteristic of a Chenciner (generalized Hopf) bifurcation. We believe this model to be the simplest population-genetic model with such dynamics.

The restricted four-body problem consists of an infinitesimal particle which is moving under the Newtonian gravitational attraction of three finite bodies, $m_{1}$, $m_{2}$ and $m_{3}$. The three bodies (called primaries) are moving in circular orbits around their common centre of mass fixed at the origin of the coordinate system. Moreover, according to the solution of Lagrange, these primaries are fixed at the vertices of an equilateral triangle. The fourth body does not affect the motion of the three bodies. In this paper, we deal with the photogravitational version of the problem with Stokes drag acting as a dissipative force. We consider the case where all the primaries are sources of radiation and that two of the bodies, $m_{2}$ and $m_{3}$, have equal masses ($m _{2} = m_{3} = \mu $) and equal radiation factors ($q_{2} = q_{3} = q$) while the dominant primary body $m_{1}$ is of mass $1 - 2\mu $. We investigate the dynamical behaviour of an infinitesimal mass in the gravitational field of radiating primaries coupled with the Stokes drag effect. It is found that under constant dissipative force, collinear equilibrium points do not exist (numerically and of course analytically) whereas the existence and positions of the non-collinear equilibrium points depend on the parameters values. The linear stability of the non-collinear equilibrium points ($L_{i},i = 1,2, \ldots ,8$) is also studied and it is found that they are all unstable except $L_{1}$, $L _{7}$ and $L_{8}$ which may be stable for a range of values of $\mu $ and various values of radiation factors. Finally, we justify the relevance of the model in astronomy by applying it to a stellar system (Ross 104-Ross775a-Ross775b), for which all the equilibrium points have been seen to be unstable.