Stability of periodically switched discrete-time linear singular systems

In this paper we present some necessary and sufficient conditions for the stability of periodically switched discrete-time linear index-1 singular system, (PSSS). In particular, it is proved that, if at least one subsystem of a PSSS is asymptotically stable, then there is a switching rule, so that the whole system is also uniformly exponentially stable. Furthermore, for a periodically switched control system with no stable subsystems, there exist a switching rule and feedback matrices, such that the obtained PSSS is uniformly exponentially stable.     

Multistability in a system of two coupled oscillators with delayed feedback

In this paper, nonlocal dynamics of a system of two differential equations with a compactly supported nonlinearity and delay is studied. For some set of initial conditions asymptotics of solutions of considered system is constructed. By this asymptotics we build a special mapping. Dynamics of this mapping describes dynamics of initial system in general: it is proved that stable cycles of this mapping correspond to exponentially orbitally stable relaxation periodic solutions of initial system of delay differential equations. It is shown that amplitude, period of solutions of initial system, and number of coexisting stable solutions depend crucially on coupling parameter. Algorithm for constructing many coexisting stable solutions is described.     

On the dimension of the subspace of Liouvillian solutions of a Fuchsian system

The effect of small constantly acting random perturbations of white noise type on a dynamical system with locally stable fixed point is studied. The perturbed system is considered in the form of Itô stochastic differential equations, and it is assumed that the perturbation does not vanish at a fixed point. In this case, the trajectories of the stochastic system issuing from points near the stable fixed point exit from the neighborhood of equilibrium with probability 1. Classes of perturbations such that the equilibrium of a deterministic system is stable in probability on an asymptotically large time interval are described.     

Analysis and design of switched normal systems

In this paper, we study the stability property for a class of switched linear systems whose subsystems are normal. The subsystems can be continuous-time or discrete-time ones. We show that when all the continuous-time subsystems are Hurwitz stable and all the discrete-time subsystems are Schur stable, a common quadratic Lyapunov function exists for the subsystems and thus the switched system is exponentially stable under arbitrary switching. We show that when unstable subsystems are involved, for a desired decay rate of the system, if the activation time ratio between stable subsystems and unstable ones is less than a certain value (calculated using the decay rate), then the switched system is exponentially stable with the desired decay rate.     

Analysis and design of switched normal systems

In this paper, we study the stability property for a class of switched linear systems whose subsystems are normal. The subsystems can be continuous-time or discrete-time ones. We show that when all the continuous-time subsystems are Hurwitz stable and all the discrete-time subsystems are Schur stable, a common quadratic Lyapunov function exists for the subsystems and thus the switched system is exponentially stable under arbitrary switching. We show that when unstable subsystems are involved, for a desired decay rate of the system, if the activation time ratio between stable subsystems and unstable ones is less than a certain value (calculated using the decay rate), then the switched system is exponentially stable with the desired decay rate.     

The set of stable switching sequences for discrete-time linear switched systems

In this paper we study the characterization of the asymptotical stability for discrete-time switched linear systems. We first translate the system dynamics into a symbolic setting under the framework of symbolic topology. Then by using the ergodic measure theory, a lower bound estimate of Hausdorff dimension of the set of asymptotically stable sequences is obtained. We show that the Hausdorff dimension of the set of asymptotically stable switching sequences is positive if and only if the corresponding switched linear system has at least one asymptotically stable switching sequence. The obtained result reveals an underlying fundamental principle: a switched linear system either possesses uncountable numbers of asymptotically stable switching sequences or has none of them, provided that the switching is arbitrary. We also develop frequency and density indexes to identify those asymptotically stable switching sequences of the system.     

The stable set of the social conflict game with commitments: existence,uniqueness, and efficiency

We investigate the stable sets of social conflict games by employing the framework of the (abstract) system by Greenberg (Theory of social situations: an alternative game theoretic approach. Cambridge University Press, Cambridge, 1990). The social conflict game is a class of strategic games that includes the prisoners’ dilemma and the chicken game. We first show that the stable set generally fails to exist in a system that is directly derived from the social conflict game. In this system, the stable set exists if and only if the strong equilibrium exists in the underlying game. If the stable set exists, it coincides with the set of the strong equilibria that is equivalent to the core for the system. Then, we turn to a modified system where the players are allowed to make commitments. In the system with commitments, the stable set always exists, and it consists of efficient outcomes with a certain property. We also discuss the relationship between the core and the stable set for the system with commitments.     

New results on the parametric stability of nonlinear systems

In this paper, we derive some new results on the parametric stability of nonlinear systems. Explicitly, we derive a necessary and sufficient condition for a nonlinear system to be locally parametrically exponentially stable at an equilibrium point. We also derive a necessary condition for the nonlinear system to be locally parametrically asymptotically stable at an equilibrium point. Next, we derive some new results on the parametric stability of discrete-time nonlinear systems. As in the continuous case, we derive a necessary and sufficient condition for a discrete-time nonlinear system to be locally parametrically exponentially stable at an equilibrium point. We also derive a necessary condition for the discrete-time nonlinear system to be locally parametrically asymptotically stable at an equilibrium point. We illustrate our results with some classical examples from the bifurcation theory.     

适应镇定了的AR系统的稳定性

由鞅差序列驱动的 ARX系统经适应镇定后导致闭环系统在均方意义稳定 ,但此时系统的稳定性并不清楚 .在噪声满足一定的条件下 ,本文证明了此闭环系统就是稳定的     

Stability of superposed viscous-viscoelastic (Rivlin-Ericksen) fluids in the presence of suspended particles through a porous medium

The Rayleigh-Taylor instability of a Newtonian viscous fluid overlying a Rivlin-Ericksen viscoelastic fluid containing suspended particles in a porous medium is considered. As in both Newtonian viscous-viscoelastic fluids the system is stable in the potentially stable case and unstable in the potentially unstable case, this holds for the present problem also. The effects of a variable horizontal magnetic field and a uniform rotation are also considered. The presence of magnetic field stabilizes a certain wave-number band, whereas the system is unstable for all wave-numbers in the absence of the magnetic field for the potentially unstable configuration. However, the system is stable in the potentially stable case and unstable in the potentially unstable case for highly viscous fluids in the presence of a uniform rotation.     

Some minimal bimolecular mass-action systems with limit cycles

We discuss three examples of bimolecular mass-action systems with three species, due to Feinberg, Berner, Heinrich, and Wilhelm. Each system has a unique positive equilibrium which is unstable for certain rate constants and then exhibits stable limit cycles, but no chaotic behaviour. For some rate constants in the Feinberg–Berner system, a stable equilibrium, an unstabe limit cycle, and a stable limit cycle coexist. All three networks are minimal in some sense.By way of homogenising these three examples, we construct bimolecular mass-conserving mass-action systems with four species that admit a stable limit cycle. The homogenised Feinberg–Berner system and the homogenised Wilhelm–Heinrich system admit the coexistence of a stable equilibrium, an unstable limit cycle, and a stable limit cycle.     

Stability performance for primary frequency regulation of hydro-turbine governing system with surge tank

This paper aims to study the stability for primary frequency regulation of hydro-turbine governing system with surge tank. Firstly, a novel nonlinear mathematical model of hydro-turbine governing system considering the nonlinear characteristic of penstock head loss is introduced. The nonlinear state equations under opening control mode and power control mode are derived. Then, the nonlinear dynamic performance of nonlinear hydro-turbine governing system is investigated based on the stable domain for primary frequency regulation. New feature of the nonlinear hydro-turbine governing system caused by the nonlinear characteristic of penstock head loss is described by comparing with a linear model, and the effect mechanism of nonlinear characteristic of penstock head loss is revealed. Finally, the concept of critical stable sectional area of surge tank for primary frequency regulation is proposed and the analytical solution is derived. The combined tuning and optimization method of governor parameters and sectional area of surge tank is proposed. The results indicate that for the primary frequency regulation under opening control mode and power control mode, the nonlinear hydro-turbine governing system is absolutely stable and conditionally stable, respectively. The stability of the nonlinear hydro-turbine governing system and linear hydro-turbine governing system is the same under opening control model and different under power control model. The nonlinear characteristic of penstock head loss mainly affects the initial stage of dynamic response process of power output, and then changes the stability of the nonlinear system. The critical stable sectional area of surge tank makes the system reach critical stable state. The governor parameters and critical stable sectional area of surge tank jointly determine the distributions of stability states.     

参激白噪声作用下蔡电路的脉冲控制

考察了参激白噪声和脉冲信号联合作用下蔡电路的渐近P阶矩稳定性问题,得到该随机脉冲系统的比较系统,从而可由该确定性比较系统的稳定性得到原随机脉冲系统的渐近P阶矩稳定性.并从理论上得到能使该随机脉冲系统渐近P阶矩稳定的参数取值范围,即在稳定区域内取值的参数组合能够用脉冲方法对该随机蔡电路实现混沌控制.最后用数值仿真验证了理论结果的正确性.     

一类具分布型滞量的中立型微分系统的稳定性

研究了一类具分布型滞量的线性中立型系统,得得到了系统零解的一致稳定性及渐近稳定性的充分条件.     

The evolution of invariant manifolds in Hamiltonian-Hopf bifurcations

We consider the evolution of the stable and unstable manifolds of an equilibrium point of a Hamiltonian system of two degrees of freedom which depends on a parameter, ν. The eigenvalues of the linearized system are complex for ν<0 and pure imaginary for ν>0. Thus, for ν<0 the equilibrium has a two-dimensional stable manifold and a two-dimensional unstable manifold, but for ν>0 these stable and unstable manifolds are gone. If the sign of a certain term in the normal form is positive then for small negative ν the stable and unstable manifolds of the system are either identical or must have transverse intersection. Thus, either the system is totally degenerate or the system admits a suspended Smale horseshoe as an invariant set.     

Stabilization of Linear Systems by a Multiplicative Random Noise

We consider a linear system of differential equations multiplicatively perturbed by a random noise and obtain formulas for the expectation and the correlation function of the solution. We show that, under a multiplicative perturbation, a Lyapunov unstable linear system can become asymptotically stable in mean, and a stable linear system can become unstable in mean. The corresponding examples of such systems are given.     

Dynamics of the cournot oligopoly with multi-product firms

The stability of the Cournot equilibrium for a linear oligopoly with multiproduct firms is analyzed. Under certain conditions, the equilibrium is proven to be stable independently of the values of the adjustment coefficients in the case of a continuous time system. However, it is stable only in duopoly for a discrete system of adjustment.     

Existence of traveling waves in compressible Euler equations with viscosity and capillarity

Given any Lax shock of the compressible Euler dynamics equations, we show that there exists the corresponding traveling wave of the system when viscosity and capillarity are suitably added. For a traveling wave corresponding to a given Lax shock, the governing viscous–capillary system is reduced to a system of two differential equations of first-order, which admits an asymptotically stable equilibrium point and a saddle point. We then develop the method of estimating attraction domain of the asymptotically stable equilibrium point for the compressible Euler equations and show that the saddle point in fact lies on the boundary of this set. Then, we establish a saddle-to-stable connection by pointing out that there is a stable trajectory leaving the saddle point and entering the attraction domain of the asymptotically stable equilibrium point. This gives us a traveling wave of the viscous–capillary compressible Euler equations.     

Stable Solution of Nonlinear Age-structured Forest Evolution System

This paper studies the dynamical behavior of a class of total area dependent nonlinear age-structured forest evolution model. We give the problem of equal value for the forest system, and discuss the stable solution of system. We obtained the necessary and sufficient conditions for there exists the stable solution.