Haken model

The Haken model, proposed by the founder of synergetic theory, Hermann Haken, is a valuable model for measuring the orderliness of a system. It evaluates the evolutionary stage of a system by identifying order parameters. Order parameters are parameters that dominate the macroscopic orderliness or patterns of the system. Based on the principles of order parameters, the complex self-organizing and cooperative evolution process of a system can be effectively studied. The main analysis process of the Haken model is as follows⁴⁹:

Assuming that the behavior of a system at a certain time t, denoted as q(t), depends solely on the external force F(t) at that time and decays over time, F(t) = ae^−δt, where a is a constant and δ is the damping coefficient, the solution to q˙(t) = γq + F(t) is:

Due to the instantaneous nature of the system’s response to external forces, there is no time for energy exchange to occur during this process. Therefore, this response process is referred to as an “adiabatic” process. Assuming that the rate at which the system’s behavior decays over time is much faster than the rate at which the external force decays over time, then:

The assumption γ ≫ δ is the prerequisite for using the adiabatic elimination method to eliminate fast variables. This principle is known as the adiabatic approximation principle.

Haken performed mathematical treatment on system parameters. Assuming q₁ represents the internal force of a subsystem and q₂ is controlled by this internal force, the system satisfies the following motion equation:

where q₁ and q₂ represent state variables, while a, b, γ₁, and γ₂ are control variables. The parameters a and b reflect the strength of the interaction between q₁ and q₂. γ₁ and γ₂ represent the damping coefficients of the two subsystems, with |γ₂| ≥|γ₁| (γ₂ > 0) referred to as the “adiabatic approximation assumption” of the motion system. If the adiabatic approximation assumption holds true, removing q₂ suddenly results in q₁ not having enough time to change. By setting q˙2 = 0, we can obtain:

q₁, which represents the order parameter, can be substituted into Eq. (8) to obtain the evolution equation of the order parameter:

The above equation indicates that q₁ determines q₂, and q₂ changes correspondingly with the variations in q₁. Therefore, q₁ is the order parameter of the system, governing and dominating the process of cooperative evolution in the system.

Based on the system’s motion equation and the order parameter, the system’s potential function is determined to assess its state. Integrating the negative of q˙1 yields the potential function of the system, which effectively determines the overall state of the system:

Since the physical equations are formulated for continuous random variables, they need to be discretized when applied to urbanization and ecological environment analysis. This involves converting them into discrete form:

Based on the potential function, we can determine the stable point A of the system. The distance between any point B on the potential function and stable point A determines the system’s state, i.e., the level of synergy. We can calculate the distance between two points using a distance formula:

The larger the value of d is, the less synergy there is in the system. Conversely, a higher value of d indicates a higher level of synergy in the system. To facilitate further analysis, we perform a reverse transformation on the d value by using the following formula:

where S represents the degree of synergy.