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We propose to apply the Generalized Second Price (GSP) auction mechanism in the dynamic confirmation game on Bitcoin transactions. Our model is targeted to deal with the instability and low efficiency of the currently-adopted Generalized First Price (GFP) mechanism. Besides, we use the "rank-by-cost" rule to replace the "rank-by-fee" rule, where each transaction's cost is calculated by the fee and the waiting time. Aiming to probe users' equilibrium strategy, we first discuss the GSP game with complete information under synchronous submissions, and show that it has the Locally Envy-Free equilibrium. Then, we study the GSP game with incomplete information under asynchronous submissions, and define two types of strategies, i.e. the Farsighted Balanced (FB) strategy and the Instant Balanced (IB) strategy. The FB strategy is in line with users' needs of maximizing the long-term payoffs; however it cannot generate a stable equilibrium. Alternatively, the IB strategy focuses on the instant payoff maximization, and if all users follow the IB strategy, their equilibrium fees can finally converge to a stable profile. Finally, we design computational experiments to validate our theoretical analysis, and results indicate that this GSP mechanism is superior to the GFP mechanism, and it can converge under the IB strategy.

In this paper, stochastic two-strategy evolutionary games in which the players are interfered by multiplicative noise are studied. We propose a stochastic replicator dynamic model to investigate the almost sure exponential stability (ASES) of general two-strategy games. Compared with the deterministic games and stochastic games with additive noise, our most interesting results concern the impact of multiplicative noise on the cooperation behavior in large well-mixed populations. We establish sufficient conditions on stochastic noise which can ensure the stable equilibrium points of general two-strategy games are indeed ASES. The stochastic Lyapunov framework is used to prove the stochastic stability, where finding the suitable Lyapunov function is the key and challenging in our case. Three different types of two-strategy game models (snowdrift games, hunt stag games and prisoner’s dilemmas) are simulated to verify our theoretical results.

In this theoretical article, we develop a unified framework that encapsulates: i) — a system of heterogeneous cities, ii) — heterogeneous workers, iii) — the presence of agglomeration economies and congestion forces, and iv) — heterogeneous city composition preferences. We provide a full analytical characterization of the decentralized and centralized economies. Contrary to conventional wisdom, we show that social mixing can constitute a unique and stable equilibrium. We also disentangle the different economic factors that shape within- and across-city social/income dispersion. We also find that individuals’ preferences be overturned at an aggregate scale, which is consistent with Schelling paradoxical findings. We then offer a rationale for these counterintuitive results. Last, we demonstrate that the decentralized economy is not optimal but can be restored by local subsidies.

We investigate the impact of the extent of investors' short-termism on the market equilibrium in a dynamic rational expectations model. We show that multiple equilibria always arise irrespective of the degree of traders' myopia, and that short-termism alone may not be the cause of equilibrium multiplicity, but it does significantly affect the market quality of all equilibria. The liquidity, price efficiency and expected trading volume under the high trading intensity equilibrium (HTIE) are higher than those under the low trading intensity equilibrium (LTIE) no matter how myopic the informed traders are. As the extent of investors' short-termism increases, the difference between the HTIE and LTIE becomes larger. Interestingly, an increase in the degree of investors' myopia in the market on the one hand mitigates the instability of the unstable equilibrium --- the HTIE (upon market perturbations a higher degree of investors' myopia slows down the deviation from the original equilibrium under an HTIE), but on the other hand also dampens the stability of the stable equilibrium --- the LTIE (upon market perturbations a higher degree of investors’ myopia slows down the converging process to the original equilibrium under an LTIE).

The conditions on the mating matrix associated with a stable equilibrium are specified for an autosomal locus with four alleles. An example illustrates how Hardy–Weinberg proportions are maintained with nonrandom mating. The ABO blood group provides an illustration.

In this study, the isothermal saturation method was applied to obtain the data for solid–liquid equilibria in a system consisting of four solids (LiCl + NaCl + Li2SO4 + Na2SO4) and water at 288.15 K. The chemical composition of the liquid phase was experimentally determined, and the corresponding phase diagrams were constructed. The obtained diagram exhibited 6 crystallization areas, 10 univariant curves, and 5 invariant points. The 6 crystallization regions consisted of LiCl·2H2O, Li2SO4·H2O, NaCl, Na2SO4·10H2O, Li2SO4·3Na2SO4·12H2O and Li2SO4·Na2SO4. The solid-phase solubilities and phase equilibria were derived from the experimental data at 288.15 K using Pitzer’s equations. The calculated data were compared with the experiment, and the corresponding phase diagrams for both datasets were calculated. Experimental and theoretical diagrams were in good agreement.

We study how a rate of environmental change affects stability and persistence of an ecological community. Using an explicit rate of change in the environmental parameters of a spatially coupled ecological system, we study the rate-dependent responses associated with extinction of a metacommunity. Our results elucidate various scenarios of rate-induced extinction (R-tipping) such as tipping through increasing amplitude of oscillations, tipping through synchronized dynamics, and tipping through tracking unstable states. Further, we show rate-dependent delayed responses away from bifurcation points, but with tracking unstable states. Our results emphasize that the rate of environmental change can reveal the unexpected extinction dynamics which may not be predicted from stable equilibrium states and bifurcation points of the system. Thus, our results illustrate the importance of rate-dependent responses to understand stability and persistence of an ecological system.

Bistability is exhibited by an object when it can be resting in two stable equilibrium states. Certain composite laminates exhibit bistability by having two stable curvatures of opposite sign with the two axes of curvature perpendicular to each other. These laminates can be actuated from one state to the other. The actuation from the original post-cure shape to the second shape is called as ‘snap-through’ and the reverse actuation is called as ‘snap-back’. This phenomenon can be used in applications for morphing structures, energy harvesting, and other applications where there is a conflicting requirement of a structure that is load-carrying, light, and shape-adaptable. MW Hyer first reported this phenomenon in his paper in 1981. He found that thin unsymmetric laminates do not behave according to the predictions of the Classical Lamination Theory (CLT). The CLT is a linear theory and predicts the post-cure shape of thin unsymmetric laminates to be a saddle. MW Hyer developed a non-linear method called the “Extended Classical Lamination Theory” which accurately predicted the laminate to have two cylindrical shapes. Since then, a number of researchers have tried to identify the key parameters affecting the behavior of such laminates. Geometric parameters such as stacking sequence, fibre orientation, cure cycle, boundary conditions, and force of actuation, have all been studied. The objective of this research is to define a relation between the length, width and thickness of square and rectangular laminates required to achieve bistability. Using these relations, a 36 in × 36 in bistable laminate is fabricated with a thickness of 30 CFRP layers. Also, it is proved that a laminate does not lose bistability with an increase in aspect ratio, as long as both sides of the rectangular laminate are above a certain ‘critical length’. A bistable laminate with dimensions of 2 in × 50 in is fabricated. Further, for laminates that are bistable, it is necessary to be able to predict the curvature and force required for actuation. Therefore, a method is developed which allows us to predict the curvature of both stable shapes, as well as the force of actuation of laminates for which the thickness and dimensions are known. Finite Element Analysis is used to carry out the numerical calculations, which are validated by fabricating laminates. The curvature of these laminates is measured using a profilometer and the force of actuation is recorded using a universal test set-up.

Nonlinear systems can be studied by their equivalent linearized systems in the region of equilibrium points. This fact led the study of dynamical systems to be focused near these points. On the contrary Hidden attractors can be found in systems with no equilibrium points, with one stable equilibrium or in systems with lines of equilibrium points. This forces us to ``sail in uncharted waters'' and opens new perspectives in the study of dynamical systems.

The question of whether and how two competing consumers can coexist on a single limiting resource has a long tradition in ecological theory. We build on a recent seasonal (hybrid) model for one consumer and one resource, and we extend it by introducing a second consumer. Consumers reproduce only once per year, the resource reproduces throughout the"summer" season. When we use linear consumer reproduction between years, we find explicit expressions for the trivial and semi-trivial equilibria, and we prove that there is no positive equilibrium generically. When we use non-linear consumer reproduction, we determine conditions for which both semi-trivial equilibria are unstable. We prove that a unique positive equilibrium exists in this case, and we find an explicit analytical expression for it. By linear analysis and numerical simulation, we find bifurcations from the stable equilibrium to population cycles that may appear through period-doubling or Hopf bifurcations. We interpret our results in terms of climate change that changes the length of the"summer" season.

Use of a debris-tether-tug (DTT) system has been recognized as one of the most promising techniques for space debris removal. Attitude motion of the debris may give rise to severe disturbances to the debris removal system. In this case, a conventional DTT system with a single tether may no longer be capable of removing the debris in a stable manner. A hierarchical tether architecture is proposed to stabilize the debris with a main tether from the tug branched into multiple sub-tethers attached to the edges of the debris. The primary goal of this paper is to study the dynamics of a DTT system consisting of a tug, a rigid debris and the proposed tether architecture. First, a three-dimensional dynamics model is developed using Kane's method. A simplified model is studied analytically to determine the equilibrium configurations and their stability, and the frequencies of oscillation about the stable equilibrium points. Numerical simulations are then conducted to verify the analytical results with both rigid and elastic tethers. Additionally, the stabilization effect of the branched tether system is investigated considering various debris angular velocities.

In this paper, the multiple O ( t − α ) stability for fractional-order neural networks (FoNNs) with time-varying delays is formulated and explored. According to n pairs of bounded functions, which are obtained from FoNNs model as well as the maximum and minimum values of the activation functions, state space is partitioned into ∏ i = 1 n ( 2 M i + 1 ) parts, where Mi ≥ 0 is a nonnegative integer and is determined by bounded functions. Sufficient conditions are acquired to guarantee the existence of ∏ i = 1 n ( M i + 1 ) O ( t − α ) stable equilibrium points. The obtained criteria generalize and extend the existing results. Several illustrative examples are raised to confirm the effectiveness of the theoretical results.

We consider a family of maps used in the numerical simulation of a logistic equation with a delay. This equation is broadly used in problems of mathematical ecology. At the same time, the presented maps themselves can serve as models of the dynamics of populations; therefore, there is significant interest in studying them. The paper compares the properties of the trajectories of these mappings and the original equation with a delay. It is shown that the behavior of the solutions of maps can be quite complicated, while the logistic equation with a delay has only a stable equilibrium state or cycle.

This paper analyzes the probabilities of economies to lie into the middle-income trap by using a Logit Panel Data model. We define the trap with more encompassing criteria: average GDPs per capita relative to the US one and GDP per capita growth rates for ten-year averages in both cases. To determine those probabilities, we include two different sets of covariates. Firstly, a set of covariates that might be influenced by policymakers, e.g., inflation, exports, and education. Secondly, another set of covariates that are independent of policymakers’ influence; a trend, a dummy for the international financial crisis, growth rates of world leader economies (US and China), growth rates of continental leader economies, and dummies by continents. We find that domestic variables are essentialto determine the probability of lying into the middle-income trap. Also, the independent covariates are significant. The trend shows that the trap is a stable equilibrium where economies tend to stay for a long time. A global economic crisis can push the countries out of the trap, but the impulse makes them back to lower-income stages. Our model also finds that regional linkages with surrounding economies are essential for middle-income economies Keywords: Middle-Income Trap, International Economics, Logit Models, Panel Data

In this review, we categorize chaotic systems with special properties in their equilibrium into eight groups: Systems with no equilibrium (NE), systems with stable equilibrium (SE), systems with line equilibrium (LE), systems with curve equilibrium (CE), systems with plane equilibrium (PE), systems with surface equilibrium (ES), systems with unstable equilibrium (US), and systems which belong to more than one of the above categories (Chameleon systems). We introduce the pioneer works from each group. Also, we review some recent important papers related to those groups.

In this paper, we mainly investigate a delayed predator–prey model with stage structure for prey and Ivlev-type functional response. Four assumptions about this model are made as follows: (1) there are a single predator and a single prey population in the model; (2) the prey is divided by the age into two stage structures, immature and mature; (3) the trophic transfer from prey to predator is incomplete; (4) there are two time delays due to the time of maturation of prey and the time of gestation of predator. Some properties of equilibria for the model without delays are provided. Further, the stability of equilibria and existence of Hopf bifurcation are studied by discussing the different cases of time delays for the model with delays. We observe that delays can cause a stable equilibrium to become unstable one, even occur Hopf bifurcation when delays pass through their corresponding critical values. Meanwhile, we derive explicit formulae to determine the properties of Hopf bifurcation such as the direction of Hopf bifurcation and the stability of periodic solutions. Numerical simulations of all theoretical analyses are given for verifying our theoretical results. In the present work, we demonstrate that the validity and universality of oscillations induced by the time delays both theoretically and numerically. These results of this paper may be helpful for us to further understand the role of the critical values of time delays in stabilizing the predator–prey model.

Inspired by DNA data of the human cytomegalovirus we propose a model of a two-type parasite population distributed over its hosts. The parasite is capable to persist in its host till the host dies, and to reinfect other hosts. To maintain type diversity within a host, balancing selection is assumed. For a suitable parameter regime we show that in the limit of large host and parasite populations the host state frequencies follow a dynamical system with a globally stable equilibrium, guaranteeing that both types are maintained in the parasite population for a long time on the host time scale., Made a correction in the formulation of Theorem 3, added a specification concerning the topology in Theorem 1 and expanded the proofs of Theorem 1, Proposition 3.13 (which was Corollary 3.13 before) and Theorem 3 in order to improve the readability

We study the geometry of a stable drop of incompressible liquid constrained to lie in a spherical container. The energy functional is comprised of the surface tension, the wetting energy and the line tension. It is shown that the only stable equilibrium drops having the topology of a disc are flat discs and spherical caps. Sharp conditions for the stability of equilibrium spherical caps and flat discs are given.

Mathematical models and analyses can assist in designing the control strategies to prevent the spread of infectious disease. The present paper investigates the bifurcations and dynamics of a plant disease system under non-smooth control strategy. The generalized Lyapunov approach is employed to perform the analysis of the plant disease model with non-smooth control. It is found that the controlled disease system can have three types of equilibria. The globally asymptotically attractor for each of three types of equilibria is determined by constructing Lyapunov functions and using Green’s Theorem. It is shown that the disease system can exhibit rich dynamic behaviors including globally stable equilibrium, stable pseudo-equilibrium and sliding mode bifurcations. The solution of the disease system can converge to the disease-free equilibrium, endemic equilibrium or sliding equilibrium on discontinuous surfaces. Biological implications of the obtained results are discussed for implementing the control strategies to the infectious plant diseases.

This paper investigates the dynamics of an insurance market on which insurance companies may dishonestly deny eligible claims. Behaving dishonestly can increase the current profit but also entails the risk of losing profit in the future due to a worse reputation. Depending on the reputation cost imposed by policyholders, the analysis either predicts the emergence of reputation cycles or convergence to a stable equilibrium in which all eligible claims are accepted and the insurers’ reputations remain at a high level. I also show that policyholders may discipline insurers using a buying strategy based on an image-scoring rule. My results lead to important insights. For instance, reputation campaigns may have a pro-cyclic effect which leads to more severe reputation crises in the future.

The purpose of this article is to discuss two basic ideas of Henri Poincare in the theory of dynamical systems. The first one, the recurrence theorem, got at first a lot of attention but most scientists lost interest when finding out that long timescales were involved. We will show that recurrence can be a tool to find complex dynamics in resonance zones of Hamiltonian systems; this is related to the phenomenon of quasi-trapping. To demonstrate the use of recurrence phenomena we will explore the \begin{document}$ 2:2:3 $\end{document} Hamiltonian resonance near stable equilibrium. This will involve interaction of low and higher order resonance. A second useful idea is concerned with the characteristic exponents of periodic solutions of dynamical systems. If a periodic solution of a Hamiltonian system has more than two zero characteristic exponents, this points at the existence of an integral of motion besides the energy. We will apply this idea to examples of two and three degrees-of-freedom (dof), the Henon-Heiles (or Braun's) family and the \begin{document}$ 1:2:2 $\end{document} resonance.

The kinetics of ordering and concurrent ordering and phase separation are analyzed with an equation of motion initially developed to account for dissipative processes in quantum systems. A reduced-order energy eigenstructure, or pseudo-eigenstructure, is constructed from a static concentration wave method to describe the configuration-dependent energy in a binary alloy. This pseudo-eigenstructure is used in conjunction with an equation of motion that follows steepest entropy ascent to calculate the kinetic path that leads to ordering and phase separation in a series of hypothetical alloys. By adjusting the thermodynamic solution parameters, it is demonstrated that the model can predict: (a) the stable equilibrium state, (b) the unique thermodynamic path and kinetics of continuous or discontinuous ordering, and (c) the kinetics of concurrent processes involving simultaneous ordering and phase separation.

The paper investigates sufficient conditions on a differential inclusion which guarantee that the origin is a finite time stable equilibrium, namely a weak local one, a weak global one or a strong local one. The analysis relies on the existence of a Lyapunov function. A new Gronwall type results are used to estimate the settling time. An example of a neural network which is finite-time stable is given

A continuous attractor of recurrent neural network is a connected set of stable equilibrium points. It's widely used to interpret a lot of brain activities. A large number of literatures have studi...

This paper focuses on linguistic choice in book and journal publications on Sarawak. It is found that in general, English has lost its formerly quasi-monopoly in book publications on Sarawak in the period since the formation of Malaysia (1963-), in particular since the 1980s, although it is still of great importance. Malay has become an increasingly relevant language of book publications on Sarawak. There is also an interesting body of book publications in Chinese and Austronesian languages from Sarawak, including in Iban since the mid-2000s. Overall, there seems to be a rather stable equilibrium of languages since the early 2000s. In terms of academic journal articles on Sarawak as represented in the Malaysian Abstracting and Indexing database MyCite, a similarly stable equilibrium between English and Malay has been found. The specialised scholarly discourse on Sarawak in the examined contributions from the humanities and the social sciences is mostly conducted in Malay, not English, while a few publications in the social sciences and in all of the sciences, engineering, and medicine, are in English. This linguistic diversity in publications on Sarawak implies that scholars working on Sarawak should be fluent in more than one language. Keywords: Sarawak, linguistic diversity, English, Malay, Chinese, Iban, humanities, social sciences

Equilibrium explanations use an equilibrium to represent and explain a system’s dynamic behavior. They provide a system with the property of global stability: a system will converge towards and remain in equilibrium regardless of its initial conditions and dynamic process. Thus, equilibrium explanations are generally treated as non-causal explanations. There are two claims subsumed under that comprehensive thesis. The first claim is that equilibrium explanations do not identify any causes because a system with global stability resists manipulation. The second claim is that even if equilibrium explanations do identify causes by manipulation, those causes are embedded in a system’s deeper, underlying structural relationships, and those causes are irrelevant to explaining a system’s behavior. Only the system’s structural relationships are relevant. But equilibrium explanations are not monolithic. I compare dynamic systems with multiple, competing equilibria to systems with a globally stable equilibrium. Equilibrium explanations of the former use intervention on a system’s initial conditions and dynamic process to manipulate equilibria, which identifies causes. Furthermore, a system’s initial conditions and dynamic process are relevant to explaining why one equilibrium is selected instead of another. I then apply these lessons to systems with a globally stable equilibrium and discuss when their corresponding equilibrium explanations have a proper role in the parent sciences.

How to recover the overexploited species in the marine ecosystem is an enduring challenge. We have structured one prey-two predator’s model to study how disturbances made by different harvesting strategies affect the recovery dynamics. We explore how instantaneous and average return rates jointly study recovery dynamics under disturbances caused by different harvesting strategies. We show that disturbances made by prey harvesting induce reactivity of stable equilibrium. The stability of coexistence equilibrium changes at high reactive strength but not in a sustainable way. We prove that the stable equilibrium remains reactive following disturbance made by predators harvesting but sustainable. It is shown that the average return is higher than the instantaneous return rate under disturbance caused by predator harvesting, respectively. We evaluate the disturbance made by combined harvesting effort effect on reactivity and recovery dynamics.

Description, Test-trace-and-isolate and moderate contact restrictions offer sustainable exit strategies complementary to vaccination., The traditional long-term solutions for epidemic control involve eradication or population immunity. Here, we analytically derive the existence of a third viable solution: a stable equilibrium at low case numbers, where test-trace-and-isolate policies partially compensate for local spreading events and only moderate restrictions remain necessary. In this equilibrium, daily cases stabilize around ten or fewer new infections per million people. However, stability is endangered if restrictions are relaxed or case numbers grow too high. The latter destabilization marks a tipping point beyond which the spread self-accelerates. We show that a lockdown can reestablish control and that recurring lockdowns are not necessary given sustained, moderate contact reduction. We illustrate how this strategy profits from vaccination and helps mitigate variants of concern. This strategy reduces cumulative cases (and fatalities) four times more than strategies that only avoid hospital collapse. In the long term, immunization, large-scale testing, and international coordination will further facilitate control.

Cooperation is widely recognized to be challenging for the well-balanced development of human societies. The emergence of cooperation in populations has been largely studied in the context of the Prisoner's Dilemma game, where temptation to defect and fear to be betrayed by others often activate defective strategies. In this paper we analyze the decision making mechanisms fostering cooperation in the two-strategy Stag-Hunt and Chicken games, which include the mixed strategy Nash equilibrium, describing partially cooperative behavior. We find the conditions for which cooperation is asymptotically stable in both full and partial cases, and we show that the partially cooperative steady state is also globally stable in the simplex. Furthermore, we show that the last can be more rewarding than the first, thus making the mixed strategy effective, although people cooperate at a lower level with respect to the maximum allowed, as it is reasonably expected in real situations. Our findings highlight the importance of Stag-Hunt and Chicken games in understanding the emergence of cooperation in social networks.

In this paper, we consider two species competing for a limiting substrate such that each species impedes the growth of the other one (Mutual inhibition) in presence of a virus inhibiting one bacterial species. A system of ordinary differential equations is proposed as a mathematical model for this competition. A detailed local qualitative analysis of the system is carried out. We proved that for a general nonlinear growth rates, the Competitive Exclusion Principle still valid, that at least one species goes extinct. For some cases where we have two locally stable equilibrium points, initial species concentrations are important in determining which is the winning species. Obtained results were confirmed by some numerical simulations using Matlab software.

This paper presents Jacobi stability analysis of 23 simple chaotic systems with only one Lyapunov stable equilibrium by Kosambi-Cartan-Chern (KCC) theory, and analyzes the chaotic behavior of these systems from the geometric viewpoint. Different from Lyapunov stability, the unique equilibrium for each system is always Jacobi unstable. Moreover, the dynamical behaviors of deviation vector near equilibrium are discussed to reveal the onset of chaos for these 23 systems, and show furtherly the coexistence of unique Lyapunov stable equilibrium and chaotic attractor for each system geometrically. The obtaining results show that these chaotic systems are not robust to small perturbations of the equilibrium, indicating that the systems are extremely sensitive to internal environment. This reveals that the chaotic flows generated by these systems may be related to Jacobi instability of the equilibrium. It is hoped that the study of this paper can help reveal the true geometrical structure of hidden chaotic attractors.

The evolution of atherosclerotic plaque is in general a complex phenomenon, which is yet to be perceived completely. The present work deals with a simple reaction–diffusion model system to describe the early onset of atherosclerotic plaque formation. Both the non-spatial and spatial systems are studied analytically and numerically. The non-spatial system has been found to be globally stable, and hence, it can withstand considerable variation in parameter values leading to some assistance for various clinical investigations on atherosclerosis. The results based on model parameter values reveal several bifurcation diagrams with respect to significant model parameters with biological implications for the non-spatial system. Moreover, necessary condition for diffusive instability of a locally stable equilibrium is included in the present work to understand the dynamical behaviour of the system.

Steepest-entropy-ascent quantum thermodynamics, or SEAQT, is a unified approach of quantum mechanics and thermodynamics that avoids many of the inconsistencies that can arise between the two theories. Given a set of energy eigenlevels, i.e., energy eigenstructure, accessible to a given physical system, SEAQT predicts the unique kinetic path from any initial non-equilibrium state to stable equilibrium by solving a master equation that directs the system along the path of steepest entropy ascent. There are no intrinsic limitations on the length and time scales the method can treat so it is well-suited for calculations where the dynamics over multiple spacial scales need to be taken into account within a single framework. In this paper, the theoretical framework and its advantages are described, and several applications are presented to illustrate the use of the SEAQT equation of motion and the construction of a simplified, reduced-order, energy eigenstructure.

We address the problem of understanding the dynamics around typical singular points of 3D piecewise smooth vector fields. A model Z0 in 3D presenting a T-singularity is considered and a complete picture of its dynamics is obtained in the following way: (i) Z0 has an invariant plane $$\pi_0$$ filled up with periodic orbits (this means that the restriction $$Z_{0|\pi_0}$$ is a center around the singularity); (ii) All trajectories of Z0 converge to the surface $$\pi_0$$; (iii) given an arbitrary integer $$k \geq 0$$ then Z0 can be approximated by $$\pi_{0}$$-invariant piecewise smooth vector fields $$Z_{\varepsilon}$$ such that the restriction $$Z_{\varepsilon|\pi_0}$$ has exactly k-hyperbolic limit cycles; (iv) the origin can be chosen as an asymptotic stable equilibrium of $$Z_{\varepsilon}$$ when k = 0; and finally, (v) Z0 has infinite codimension in the set of all 3D piecewise smooth vector fields.

A family of mappings used in the numerical simulation of a delay logistic equation is considered. This equation finds wide applications in problems of mathematical ecology. The properties of solutions of these mappings and the original delay equation are compared. It is shown that the behavior of solutions of difference equations can be rather complex, while the delay logistic equation has only a stable equilibrium state or cycle. The constructed mappings can be used as models of population dynamics, so their study is of interest.

Utilizing ball-type auto-balancer in rotating systems is a popular method in industry in order to eliminate the rotational vibration. Recently, by using numerical methods, it is shown that a ball-type auto-balancer equipped with both the radial and peripheral springs, ball-spring automatic balancer (AB), is an improved model. In this paper, due to the advantages of analytical methods over the numerical ones, the complete system dynamics, i.e., the stability analysis and the time response, of the rotor equipped with ball-spring AB is analyzed by the multiple scales method as a unified technique. Therefore, with no need for implementing any other stability theories, the stability analysis is a definite advantage of this method due to the less computation cost. Moreover, the time responses show a good agreement with those obtained through the numerical method. Finally, for the first time, the influence of the peripheral springs on the time responses and the stable equilibrium points of the rotor with ball-spring AB is studied in details. The results show that the peripheral springs decrease the vibration amplitude at the transient and leave some residual imbalance at the steady state and accordingly, there is a compromise between the system working performance at the transient and the steady state.

To understand the thermodynamic characteristics of the brine located in Qaidam Basin salt lake, the stable equilibrium solubilities, densities, and refractive indices in the quaternary system Li2SO...

A nonlinear oscillating system with small perturbation is considered. It is assumed that the perturbation corresponds to external pumping which has a slow varying frequency. An asymptotics with respect to a small parameter is constructed for the solutions capturing in resonance in a neighborhood of stable equilibrium of the unperturbed system. The time interval of resonance is defined.

Different flow regimes existed for the magnetized fluidized bed with purely Geldart-B magnetizable particles (pure MFB) or their admixture with Geldart-B nonmagnetizable particles (admixture MFB). Several unresolved issues about the flow regime transition were addressed first for both the pure and admixture MFBs. Then the flow regime transition was thoroughly compared for the first time between the magnetization-FIRST and -LAST operation modes. The magnetic stabilization flow regime was observed to no longer exist under the magnetization-LAST operation mode. The comparison further indicated that the operating range covered in the literature and this work could be roughly divided into four zones. Particularly in operating zone II, the MFB state was found to depend not only on the magnetic field intensity and superficial gas velocity but also on the operation mode. Such a dependence feature in essence referred to the dependence of the MFB state on the path taken to achieve it. The reason why the bed state was path-dependent in operating zone II lay in that the MFB therein could have two different equilibrium states and different paths led to the two states, respectively. Of the two equilibrium states, the magnetic stabilization was demonstrated to be a metastable equilibrium state. Such unique characteristics of the magnetic stabilization state explained why it could not be obtained under the magnetization-LAST operation mode. Furthermore, the reason why the MFB could have an additional metastable equilibrium state aside from the stable equilibrium state in operating zone II lay in the difficulty in forming a new phase under the magnetization-FIRST operation mode.

An optimal power flow (OPF) problem can have one or multiple locally optimal solutions. It can also admit no solution. Few studies have addressed the existence of OPF solutions and derived a (tight) lower bound for the number of OPF solutions. The present paper is devoted to the analytical aspects of OPF solutions. Specifically, a necessary and sufficient condition for the existence of an OPF solution is presented and a tight lower bound for the number of OPF solutions is derived; in other words, an existence theorem is developed for the OPF solution and a computable lower bound for the number of solutions is derived by investigating a dynamical system called the quotient gradient system (QGS), which is built from the set of constraints. It is shown that an OPF solution exists if and only if the QGS has a regular stable equilibrium manifold (SEM) and the number of OPF solutions is not less than the number of regular SEMs for the QGS. Two test systems with 9 and 118 buses, respectively, are evaluated, and the numerical results are summarized to validate the presented analytical results and illustrate that multiple OPF solutions can lie in the same SEM.

This paper is an exploration of the genesis of Paul Samuelson’sFoundations of Economic Analysis(1947) from the perspective of his commitment to Edwin B. Wilson’s mathematics. The paper sheds new light on Samuelson’sFoundationsat two levels. First, Wilson’s foundational ideas, embodied in maxims that abound in Samuelson’s book, such as “Mathematics is a Language” or “operationally meaningful theorems,” unified the chapters ofFoundationsand gave a sense of unity to Samuelson’s economics. Second, Wilson influenced certain theoretical concerns of Samuelson’s economics. Particularly, Samuelson adopted Wilson’s definition of a stable equilibrium position of a system in terms of discrete inequalities. Following Wilson, Samuelson developed correspondences between the continuous and the discrete in order to translate the mathematics of the continuous of neoclassical economics into formulas of discrete magnitudes. InFoundations, the local and the discrete provided the best way ofoperationalizingmarginal and differential calculus. The discrete resonated intuitively with data; the continuous did not.

This paper experimentally compares the impact of the presence of strategic substitutes (GSS) and complements (GSC) on players’ ability to successfully play equilibrium strategies. By exploiting a simple property of the ordering on strategy spaces, our design allows us to isolate these effects by avoiding other confounding factors that are present in more complex settings, such as market games. We find that the presence of strategic complementarities significantly improves the rate of Nash play, but that this effect is driven mainly by early rounds of play. This suggests that GSS may be more difficult to learn initially, but that given sufficient time, the theoretically supported globally stable equilibrium offers a good prediction in both settings. We also show that increasing the degree of substitutability or complementarity does not significantly improve the rate of Nash play in either setting, which builds on the findings of previous studies.

In a three‐country customs union (CU) formation game, I introduce international trade in intermediate inputs and rules of origin (RoO) restrictions. In the case of symmetric countries, I show that as countries become more involved in global supply chains, global free trade is less likely to be a stable equilibrium outcome. RoO can help solve this problem. In the case of asymmetry, depending on the degree of the globalization, free riding (for high degree) or exclusion motive (for low degree) prevents global free trade. Correspondingly, I show that RoO can have helpful or detrimental effects on attaining global free trade.

This paper examines the location choice of cultural producers when cities differ in housing supply and income demographics. We develop a two-region spatial model containing three types of industries: a constant returns to scale traditional sector, a modern sector with external scale economies and a monopolistically competitive cultural sector. The model is initially analyzed when workers supply labor inelastically to their respective industry. Both integration and segregation are never a stable equilibrium, and the conditions for the stability of both concentrated and partially interior equilibrium are solved for. The model is extended to allow for cultural producers to divide their time between cultural production and moonlighting in the traditional sector, in order to smooth their income. Under this extension, there is an equilibrium where a share of cultural producers live isolated from a larger integrated market. We also identify an equilibrium where one region is able to sustain full-time cultural producers, while in the opposite region cultural producers must moonlight in the traditional sector. Under partially interior equilibria, the number of varieties of the cultural good is always larger in the region with the greater supply of housing.

The speed of China’s economic development is gradually accelerating, and the demand for energy is also constantly increasing, especially the demand for coal. In order to reveal whether the coal imports have an impact on China’s economic development, this paper constructs the VAR(6) model by selecting the quarterly data of coal imports (CIV) and gross domestic product (GDP) from 2002 to 2017, performing ADF (Augmented Dickey-Fuller) stationarity test and Johansen cointegration test. It shows that there is a long-term stable equilibrium relationship between coal imports and GDP. Then the impulse response function is used to obtain the relationship between coal imports and GDP. It is found that the impact of coal imports on GDP is greater than the impact of GDP on coal imports.

It is well known that a fractional-order system with a continuous right hand side does not have finite-time stable equilibria, but the discontinuous case has remained elusive in literature. Thus, based on novel mathematical tools, recently published in literature, it is demonstrated that attaining finite-time stable equilibria is not possible for a fractional-order system, not even in the case of an impulsive or discontinuous feedback. In consequence, it is demonstrated that, for a fractional-order system, a Lyapunov stable equilibrium cannot be at the same time finite-time stable.

Many population systems are subject to seasonally varying environments. As a result, many species exhibit seasonal changes in their life-history parameters. It is quite natural to try to understand how seasonal forcing affects population dynamics subject to stoichiometric constraints, such as nutrient/light availability and food quality. Here, we use a variation of a stoichiometric Lotka-Volterra type model, known as the LKE model, as a case study, focusing on seasonal variation in the producer's light-dependent carrying capacity. Positivity and boundedness of model solutions are studied, as well as numerical explorations and bifurcations analyses. In the absence of seasonal effects, the LKE model suggests that the dynamics are either stable equilibrium or limit cycles. However, through bifurcation analysis we observe that seasonal forcing can lead to complicated population dynamics, including periodic and quasi-periodic solutions.

The redistribution of impurity nanoparticles in a colloid mixture has been considered for two cases: (i) the mixture settles in a narrow channel first in the horizontal and then in the vertical position and (ii) the mixture is transferred from a vessel to a narrow channel. A flow that brings the system to stable equilibrium with a true vertical density gradient distorts the initially barometric distribution of impurity nanoparticles. It has been shown by numerical simulation that the initial inhomogeneity of the mixture to a great extent continues to persist in both cases.