Local stability of the Cournot solution with increasing heterogeneous competitors

In this paper we try to solve a paradox related to the results of Theocharis (1960). When the number of competitors increases the Cournot–Nash equilibrium loses stability. We relax the assumption about homogeneity in the decision mechanism and show that if we admit heterogeneity than by increasing the number of competitors the stability region on the parameters’ space may enlarge instead of shrinking.     

On the Tikhonov Well-Posedness of Concave Games and Cournot Oligopoly Games

The purpose of this paper is to investigate whether theorems known to guarantee the existence and uniqueness of Nash equilibria, provide also sufficient conditions for the Tikhonov well-posedness (T-wp). We consider several hypotheses that ensure the existence and uniqueness of a Nash equilibrium (NE), such as strong positivity of the Jacobian of the utility function derivatives (Ref. 1), pseudoconcavity, and strict diagonal dominance of the Jacobian of the best reply functions in implicit form (Ref. 2). The aforesaid assumptions imply the existence and uniqueness of NE. We show that the hypotheses in Ref. 2 guarantee also the T-wp property of the Nash equilibrium.As far as the hypotheses in Ref. 1 are concerned, the result is true for quadratic games and zero-sum games. A standard way to prove the T-wp property is to show that the sets of -equilibria are compact. This last approach is used to demonstrate directly the T-wp property for the Cournot oligopoly model given in Ref. 3. The compactness of -equilibria is related also to the condition that the best reply surfaces do not approach each other near infinity.     

Nonlinear Cournot oligopoly games with isoelastic demand function: The effects of different behavior rules

The analysis of asymptotical convergence for the oligopoly game has always been important to characterize the firms’ long-term behavior. In the nonlinear oligopoly competition possibly involving chaotic fluctuations, non-convergent trajectories are particularly undesirable since the resulting behavior will become unpredictable. In this paper, consistent with a traditional assumption that the firms update their outputs simultaneously, we at first construct an adjustment process and discuss the convergence to the equilibrium for a nonlinear Cournot duopoly game with the isoelastic demand function. We indicate that the tendency to instability does rise with the number of firms and the adjustment speeds. In particular, we alter this assumption from simultaneous decisions to sequential decisions so that the latter firms are able to observe the former ones at every time periods. We finally arrive at a conclusion that the unique equilibrium is convergent as long as the adjustment speeds are less than a fixed threshold, no matter what the number of the firms. Our findings show that the firms with sequential decisions can achieve the equilibrium more easily.     

不同发展阶段、双寡头竞争下的最优贸易政策

本运用非合作博弈思想，分析了本国在经济发展的三个阶段中，在本国和外国企业进行双寡头古诺竞争的情况下，本国最优贸易政策的确定问题。本研究发现：本国企业与外国企业的边际成本是影响最优关税税率的重要因素，并且在经济发展的不同阶段，它们对最优关税税率的作用也不同。     

Uniform global asymptotic stability of adaptive cascaded nonlinear systems with unknown high-frequency gains

We study adaptive tracking problems for nonlinear systems with unknown control gains. We construct controllers that yield uniform global asymptotic stability for the error dynamics, and hence tracking and parameter estimation for the original systems. Our result is based on a new explicit, global, strict Lyapunov function construction. We illustrate our work using a brushless DC motor turning a mechanical load. We quantify the effects of time-varying uncertainties on the motor electric parameters.     

An adaptive Euler-Maruyama scheme for SDEs: convergence and stability

J. C. Mattingly ^{The understanding of adaptive algorithms for stochastic differentialequations (SDEs) is an open area, where many issues relatedto both convergence and stability (long-time behaviour) of algorithmsare unresolved. This paper considers a very simple adaptivealgorithm, based on controlling only the drift component ofa time step. Both convergence and stability are studied. Theprimary issue in the convergence analysis is that the adaptivemethod does not necessarily drive the time steps to zero withthe user-input tolerance. This possibility must be quantifiedand shown to have low probability. The primary issue in thestability analysis is ergodicity. It is assumed that the noiseis nondegenerate, so that the diffusion process is elliptic,and the drift is assumed to satisfy a coercivity condition.The SDE is then geometrically ergodic (averages converge tostatistical equilibrium exponentially quickly). If the driftis not linearly bounded, then explicit fixed time step approximations,such as the Euler–Maruyama scheme, may fail to be ergodic.In this work, it is shown that the simple adaptive time-steppingstrategy cures this problem. In addition to proving ergodicity,an exponential moment bound is also proved, generalizing a resultknown to hold for the SDE itself.}     

Guaranteed asymptotic stability for a class of uncertain linear dynamical systems

We consider a class of linear dynamical systems with bounded, Lebesgue-measurable uncertainties in the system and input matrices as well as in the input itself. A state feedback control is derived, which guarantees global, uniform asymptotic stability of the zero state; this control is continuous, except at the zero state.This paper is based in part on research supported by the National Science Foundation.     

具有饱和项的互惠系统的分歧与稳定性

运用谱分析和分歧理论的方法,在齐次Dirichlet边界条件下,对具有饱和项的互惠系统的非负定态解的分歧及其稳定性进行研究.一方面,分别以生长率作为分歧参数,讨论了发自半平凡解的分歧;另一方面,以两物种的生长率作为分歧参数,利用Liapunov-Schmidt过程,研究了在二重特征值处的分歧;同时判定了这些分歧解的稳定性.     

Global asymptotic stability of a family of difference equations

We consider the family of difference equations of the form

     

Global asymptotic stability of a higher order rational difference equation

In this note, we consider the following rational difference equation:

     

Global asymptotic stability and a property of the SIS model on bipartite networks

In this paper, we study an SIS model on bipartite networks, in which the network structure and a connectivity-dependent infection scheme are considered. Applying the theory of the multigroup model, we prove the existence and the asymptotic stability of the endemic equilibrium. And then we examine the ratio between the densities of infected female and male individuals on the bipartite networks. In particular, we find that when the scale exponent (γ_F) of females is equal to and that of males (γ_M), the ratio is only determined by the scale exponents and the proportion between the infection rates of females and males (λ_F/λ_M). We also present a result for the ratio by numerical simulations when γ_F≠γ_M.     

Description of stability for linear time-invariant systems based on the first curvature

This paper focuses on using the first curvature κ(t) of trajectory to describe the stability of linear time-invariant system. We extend the results for two and three-dimensional systems (Wang, Sun, Song et al, arXiv:1808.00290) to n-dimensional systems. We prove that for a system $\text{◂=▸}\dot{r}(t)=\text{◂⋅▸}Ar(t)$, (a) if there exists a measurable set whose Lebesgue measure is greater than zero, such that $\text{◂≠▸}\underset{t\to +\infty }{\lim }\kappa (t)\ne 0$ or $\underset{t\to +\infty }{\lim }\kappa (t)$ does not exist for any initial value in this set, then the zero solution of the system is stable; (b) if the matrix A is invertible, and there exists a measurable set whose Lebesgue measure is greater than zero, such that $\text{◂=▸}\underset{t\to +\infty }{\lim }\kappa (t)=+\infty$ for any initial value in this set, then the zero solution of the system is asymptotically stable.     

Preservation of stability and asymptotic behaviour of perturbed integrodifferential equations in a Banach space

In this paper stability and asymptotic behaviour of solutions of the integrodifferential system in a Banach space is related to that of the integrodifferential system in a Banach space. The results obtained constitute a generalization of similar results for ordinary differential equations in a Banach space, which motivate the approach and proofs.     

Global asymptotic stability and oscillation of a family of difference equations

We consider the family of difference equations of the form

     

The stability of traveling waves with non-critical speeds for some competition systems

This paper is concerned with some quasilinear cross-diffusion systems which model competing species in mathematical ecology.By detailed spectral analysis,each traveling wave solution with non-critical speed is proved to be locally exponentially stable to perturbations in some exponentially weighted spaces.     

On the stability of nonautonomous difference equations

In this paper, we obtain new sufficient conditions for the asymptotic stability and instability of equilibrium points of the nonautonomous higher order difference equations by means of weak contractions and weak expansions.     

Mittag-Leffler stability and adaptive impulsive synchronization of fractional order neural networks in quaternion field

Fractional order quaternion-valued neural networks are a type of fractional order neural networks for which neuron state, synaptic connection strengths, and neuron activation functions are quaternion. This paper is dealing with the Mittag-Leffler stability and adaptive impulsive synchronization of fractional order neural networks in quaternion field. The fractional order quaternion-valued neural networks are separated into four real-valued systems forming an equivalent four real-valued fractional order neural networks, which decreases the computational complexity by avoiding the noncommutativity of quaternion multiplication. Via some fractional inequality techniques and suitable Lyapunov functional, a brand new criterion is proposed first to ensure the Mittag-Leffler stability for the addressed neural networks. Besides, the combination of quaternion-valued adaptive and impulsive control is intended to realize the asymptotically synchronization between two fractional order quaternion-valued neural networks. Ultimately, two numerical simulations are provided to check the accuracy and validity of our obtained theoretical results.     

The asymptotic stability of theoretical and numerical solutions for systems of neutral multidelay-differential equations

The asymptotic stability of theoretical and numerical solutions for neutral multidelay-differential equations (NMDEs) is dealt with. A sufficient condition on the asymptotic stability of theoretical solutions for NMDEs is obtained. On the basis of this condition, it is proved that A-stability of the multistep Runge-Kutta methods for ODEs is equivalent to NGP_k-stability of the induced methods for NMDEs. Project supported by the National Natural Science Foundation of China (Grant No. 19771034).