In this paper, the Hopf bifurcation and control of the magnetic bearing system under an uncertain parameter are investigated. Firstly, the two-degree-of-freedom magnetic bearing system model with uncertain parameter is established. The method of orthogonal polynomial approximation is used to obtain the equivalent magnetic bearing model which is deterministic. Secondly, combining mathematical analysis tools and numerical simulations, the Hopf bifurcation of the equivalent model is analyzed. Finally, a hybrid feedback control method (linear feedback control method combined with nonlinear stochastic feedback control method) is introduced to control the Hopf bifurcation behavior of the magnetic bearing system.

PurposeWith the growing demand for infrastructure and public services in recent years, PPP-UP have attracted a great deal of attention. However, while the user focuses on the payment for use and the private sector is concerned with its return on investment, the public sector pays more attention to the efficient utilization of public funds. In order to analyze the willingness of each stakeholder to join PPP-UP, an evolutionary game model involving the three parties is constructed.Design/methodology/approachAn evolutionary game model is established that considers the users and the public and private sectors in user-pay public-private-partnership projects (PPP-UP). Eight scenarios of equilibriums and the game's evolutionary stable strategies are analyzed, and the corresponding stability conditions are then obtained. A situation where all three players are willing to cooperate in theory is also examined. The key influencing parameters that affect cooperation behaviors are further discussed.FindingsFirst, the results illustrate that by properly adjusting the influencing factors, the cooperation status among the three parties can be changed along with certain evolutionary trends. Second, it is hard to modify unsatisfactory evolutionary stability by small changes in both the price compensation of and the construction and operation compensation. Third, it is necessary to involve the users in the decision-making process in PPP-UP and take their demands regarding benefits and payments into account.Originality/valueIn this paper, we focus on PPP-UP to research interactions among the public and private sectors and the users. Based on the analysis of the evolutionary game, to facilitate the successful implementation and development of a project, several conditions are needed to ensure tripartite cooperation. Several recommendations are then proposed for decision-makers in PPP-UP.

Evolution of provincial governments' discretion: The parallel system of fiscal decentralization and political concentrationThe administrative system in China is made up of multiple levels of central and local governments.As these governments' administrative behaviors are usually embodied in the form of financial expenditures, administrative restructuring and fiscal reform tend to go hand in hand (Zhang, 2003). As illustrated by Zhu and Peyrache (2017), three typical stages can be used to explain the evolutionary process of intergovernmental fiscal and administrative relations.In the first stage of the socialist period before 1979, the central and provincial governments were under a consolidated fiscal system. The revenues were primarily derived from state-owned enterprises' profits and taxes, and the spending priorities were established by the central government; local governments were just state-owned enterprises and had little discretion. In this case, the administrative system was driven through top-down hierarchical control, and the functions of territorial administrative jurisdictions were marginal (Zhu and Peyrache, 2017). In the second stage, from 1980 to 1993, China underwent a transition period of central-local government fiscal relationships with a market-oriented transformation. Specifically, a fiscal revenue sharing system replaced the highly centralized system for collecting revenue in 1980. Beyond that, a reform called the "fiscal contracting system" was completed during the period from 1988-1993, under which only a part of the provincial fiscal revenues was handed over to the central government. Local governments still kept the surplus revenues and had full discretion in spending. Both of these reforms were attempts at decentralization between central and local governments and an incentive from the central government for local governments to develop (Song, 2013). In the third stage of 1994, the tax-sharing reform changed the authority boundary between central and local governments, as financial power was transferred upward and administrative power was delegated (Zhang, 2006). Specifically, the tax revenues were separated into central revenue, local revenue, and shared revenue. The central government obtained the majority of tax revenues through the tax-sharing system, redistributing them to local governments through transfer payments, with the local governments still maintaining high discretionary power over expenditures. Additionally, 2 Information Classification: General local governments were equipped with more rights regarding administrative affairs, including formulating their economic growth planning, promoting regional development, and providing public services. It should be noted that the devolvement of administrative power from central to local governments is a process of delegating autonomy rather than hollowing out of the state. As a one-party state, China currently maintains vertical control through cadre management and requires local governments to be responsible to upper-level governm...

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