Qingwen Hu

Fixed point theorems in CAT(0)(

In this paper, noncompact CAT(0) versions of the Fan-Browder fixed point theorem are established. As applications, we obtain new minimax inequalities, a saddle point theorem, a fixed point theorem for single-valued mappings, best approximation theorems, and existence theorems of φ-equilibrium points for multiobjective noncooperative games in the setting of noncompact CAT(0) spaces. These results generalize many well-known theorems in the literature.MSC:91A10, 47H04, 47H10, 54H25.

Stabilization in a State-Dependent Model of Turning Processes

We consider a two-degree-of-freedom model for turning processes which involves a system of differential equations with state-dependent delay. Depending on process parameters (e.g., spindle speed, depth of cut) the cutting tool can exhibit unwanted vibrations, resulting in a nonsmooth surface of the workpiece. In this paper we propose a feedback law to stabilize the turning process for a large range of system parameters. The feedback law introduces a generic nonhyperbolic stationary point into the model, which generates the main technical challenge of this work. We establish the stability equivalence between the differential equations with state-dependent delay and a corresponding nonlinear system with the delay fixed at its stationary value. Then we show the stability of that nonlinear system with constant delay by computing its normal form. Finally, we obtain conditions on system parameters which guarantee the stability of the state-dependent delay model at the nonhyperbolic stationary point.

Fixed point theorems in CAT(0) spaces with applications

In this paper, noncompact CAT(0) versions of the Fan-Browder fixed point theorem are established. As applications, we obtain new minimax inequalities, a saddle point theorem, a fixed point theorem for single-valued mappings, best approximation theorems, and existence theorems of φ-equilibrium points for multiobjective noncooperative games in the setting of noncompact CAT(0) spaces. These results generalize many well-known theorems in the literature.MSC:91A10, 47H04, 47H10, 54H25.

Global stability lobes of turning processes with state-dependent delay

We obtain global stability lobes of two models of turning processes with inherit nonsmoothness due to the presence of state-dependent delays. In the process, we transform the models with state-dependent delays into systems of differential equations with both discrete and distributed delays and develop a procedure to determine analytically the global stability regions with respect to parameters. We find that the spindle speed control strategy that we investigated in [SIAM J. Appl. Math., 72 (2012), pp. 1--24] can provide essential improvement on the stability of turning processes with state-dependent delay, and furthermore we show the existence of a proper subset of the stability region which is independent of system damping. Numerical simulations are presented to illustrate the general results.

A model of regulatory dynamics with threshold-type state-dependent delay

We model intracellular regulatory dynamics with threshold-type state-dependent delay and investigate the effect of the state-dependent diffusion time. A general model which is an extension of the classical differential equation models with constant or zero time delays is developed to study the stability of steady state, the occurrence and stability of periodic oscillations in regulatory dynamics. Using the method of multiple time scales, we compute the normal form of the general model and show that the state-dependent diffusion time may lead to both supercritical and subcritical Hopf bifurcations. Numerical simulations of the prototype model of Hes1 regulatory dynamics are given to illustrate the general results.

A Model of Cold Metal Rolling Processes with State-Dependent Delay

We model multistand cold metal rolling processes and investigate chatter vibrations using a system of differential equations which involves state-dependent interstand transportation time delay. We show that the model with state-dependent delay can be transformed into a system of equations with both retarded and advanced delays and show that the equilibria are unstable if the strip velocity is not controlled. We further show that a delayed feedback control of the strip velocity can stabilize the equilibria under certain conditions. Local Hopf bifurcation shows that the variation of interstand tension can lead to chatter. Numerical simulations are presented to illustrate the results of Hopf bifurcation.

Global Hopf Bifurcation for Differential-Algebraic Equations with State-Dependent Delay

We develop a global Hopf bifurcation theory for differential equations with a state-dependent delay governed by an algebraic equation, using the

Fixed point theorems in [InlineEquation not available: see fulltext.] spaces with applications

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