^{1}

^{2}

^{1}

^{2}

^{1}

^{2}

This paper establishes a model on the upstream and downstream relationship among private enterprises, provincial and local officials, and the central government in the real estate market using the population ecology theory of mutual relations among individual species from the perspective of business ecosystem. A dynamic model is introduced and the complex dynamical behaviors of such a predator-prey model are investigated by means of numerical simulation. The local stability conditions and complex dynamics are investigated, and the existence of chaos is discussed in the sense of Marotto theorem; bifurcation diagrams, Lyapunov exponents, sensitivity analysis for initial values, and time history figure of the system are mapped out and discussed. This shows that there are two routes to complicated dynamics, one of which is the cascade of flip bifurcations resulting in periodic cycles (and chaos), and the other one is Neimark-Sacker bifurcation which produces attractive invariant closed curves. We arrive at conclusions that the phenomenon of chaos is harmful to private enterprises, and unstable behavior is often unfavorable. Thus, linear feedback control is applied to drive the model to a stable state when the system exhibits chaotic behaviors, achieving the goal of eliminating the negative effects to a large extent.

In 2017, the real estate policy adheres to the keynote that “the house is used for living, not for speculation” and that the local area is dominated by urban agglomerations. The monthly average transaction area of 50 representative urban commercial residential markets was 29.43 million square meters, down 24.2% from the same period last year, with an absolute value lower than the same period in 2015, from January to November 2017 according to preliminary statistics. According to different levels of cities, the number of first-tier representative cities has fallen most obviously, and the absolute level is the same as that of 2011. Sales in second-tier cities fell to the level of 2015. Turnover of third-tier cities is lower than that of 2016 but has relatively high absolute scale. In the Beijing-Tianjin-Hebei region, the Yangtze river delta, the pearl river delta, the middle reaches of the Yangtze river, and the five urban Chengdu-Chongqing agglomerations, large-sized private enterprises occupy more than 60% of the total area and they focus on the major urban agglomeration market from the perspective of distribution.

In 2017, the concentration of the real estate industry will continue to improve, and new changes will take place in the development pattern of enterprises. As for the private enterprises, they will continue to grasp the incremental market space, expand the key city circle and urban agglomeration deeply, focusing on different urban development processes, and improving the competitiveness.

Chaos has become a hot topic in the competition of economics. Puu [

Agiza et al. [

Cotter and Roll [

The topic of this study is within the field of complexity, and relevant papers have been reviewed, such as [

The relationships among private enterprises, provincial and local officials, and center government are just similar to the relationship in the food chain. Thus, we propose the 3D continuous predator-prey model to analyze the evolution in the real estate market. Three innovations can be obtained in this paper:

This paper is designed as follows: in Section

There are relationships similar to the ecological food chain relationships between species in the real estate market. Simulate ecology features, and introduce ecological equations.

In this model, all the parameters are considered to be positive constants.

We use the following nondimensionalized variables and parameters to simplify the model:

Thus, we obtain

According to the regional characteristics, in particular, there are many differences in the development process between the first-tier cities and backward area. The first-tier cities have rapid economic development and easily accept new knowledge. In order to study the dynamic system, we can analyze it from discrete and continuous systems, respectively. In this paper, we analyze it from discrete system. Parameters

By the dynamics equation

The Jacobian matrix can be written as

According to Routh-Hurwitz criteria, we can obtain the conditions for the local stability of equilibrium:

The stability of the system is guaranteed under condition (

Denote

It follows that

The equilibrium point becomes unstable in certain conditions and varies with adjustment parameters in real estate market. In particular, a flip bifurcation occurs and Neimark-Sacker bifurcation takes place and varies with parameters in the dynamic system in real estate market. It will become more and more unstable and difficult to control with the increasing of adjustment parameters, which indicates that the faster the development is for the private enterprises, the more fluctuating the evolution becomes.

Stability means that the state is fixed in the real estate market and that every behavior can get fixed benefit in every time period; thus, stability is beneficial for private enterprisers in the real estate market to make long-time strategies. For provincial and local officials and the central government, it will be easy to make some principles or regulations in the real estate market. Chaos means that the real estate market is irregular and vibratile; it is difficult for behaviors to make long-term strategies. Therefore, from the perspective of strategy making and adjustment, stability is much better and chaos is much worse. The appearance of chaos in the economic system is harmful to private enterprises, provincial and local officials, and central government. Thus, in order to avert the risk, it is expedient for the behaviors to maintain at a stable state.

Numerical efforts are devoted to the analysis of the abundant complex dynamics. When

where

From

Figure

Lyapunov index of dynamic system in real estate market when

The first Lyapunov index when

The second Lyapunov index when

The third Lyapunov index when

Lyapunov index when

We continue to consider the influence of the adjustment parameters

Flip bifurcation diagram for private enterprises, provincial and local officials, and central government, respectively, when

Figure

Flip bifurcation diagram of provincial and local officials in the dynamic system when

Figure

Neimark-Sacker bifurcations in the dynamic system in real estate market.

Neimark-Sacker bifurcation of private enterprises with variations of parameter

Neimark-Sacker bifurcation of provincial and local officials with variations of parameter

Phase portraits according to the flip bifurcation diagram with different parameters in the dynamic system in the real estate market when

Projection onto the

Projection onto the

Projection onto the

Projection onto the

Projection onto the

Projection onto the

Projection onto the

3D view of attractor in the

Attractors of dynamic system in the real estate market when

Projection onto the

Projection onto the

Projection onto the

Fractal dimension is

As can be seen from Figure

Sensitivity analysis for private enterprises in the real estate market when

Figure

Sensitivity analysis for provincial and local officials when

Sensitivity analysis for central government when

Sensitivity analysis for private enterprises in the real estate market when

Sensitivity analysis for provincial and local officials when

Sensitivity analysis for central government when

Figure

Time history figure of the dynamic system in the real estate market.

There is a steady state in the complex process of competition and cooperation in the real estate market. It is similar to the ecological evolution process. There are variable factors (such as government policy and the advent of the financial crisis) making the temporary stable condition broken; thus, the system rushes into a chaotic state. All of the private enterprises in the real estate market will be unpredictable. In reality, there are all kinds of competition in the process of evolution. Thus, the private enterprises should take measures to resist risks, such as cooperating with the other private enterprises, building a competitive team, or taking the merger and assimilation of other private enterprises to enhance their strength. Chaotic behavior will be detrimental to the entire real estate market and puts a huge negative impact on the whole process of economic operation in our country. Therefore, taking corresponding measures to control the chaos will be needed.

Thus, we put controller

Figure

Lyapunov index with parameter

Here,

Future policy in real estate market will continue the clear line of “the house is used for living, not for speculation,” maintaining the continuity and stability of policy, stabilizing the real estate market, defusing the risk of bubbles, and guiding market expectations.

In this paper, a 3D continuous dynamic system is proposed and analyzed. Bifurcations and other chaotic phenomena are generated by communication via computing Lyapunov exponents, with Lyapunov dimension showing the chaotic attractors, sensitive dependence on initial conditions, and time history figure of the system. Once chaos occurs, the stable state will be broken and the market will become unpredictable, irregular, and abnormal. Finally, a linear feedback control is applied to drive or restore the chaos to be delayed or even eliminated completely, achieving the goal of a steady state.

We can draw some conclusions.

The implementation of the long-term mechanism will be further accelerated in 2018. At the same time, short-term regulations and long-term mechanisms are linked closely, maintaining the stability of the real estate market and establishing a more stable foundation.

The authors declare that they have no conflicts of interest.

_{n}