Lyapunov function is a real function defined on the time index configuration system. At each time step, the Lyapunov function assigns a value to the configuration. If the configuration changes, that is, if the model is not in equilibrium, the value of Lyapunov function will decrease by a fixed amount. Lyapunov function also has a minimum value, which means that at some point, its value will eventually stop decreasing. When this happens, the model reaches equilibrium.

Bottom-by-bottom competition game: there are n game participants, and each game participant should propose a support level for each period, with the range of {0, 1, …, 100}. It is proposed that the game participants who are closest to 2/3 of the average support level can get rewards during this period. Lyapunov function describes the strategic environment of players in this game, which makes each player more willing to provide a slightly lower than average level.

Second, the local majority model.

The Lyapunov function in the local majority model is defined as the total inconsistency in the population, that is, the sum of all the cells adjacent to the cells in the opposite state. Even though some elements may bring more inconsistency, the total inconsistency satisfies the conditions of Lyapunov function. Therefore, the local majority model will inevitably converge to equilibrium, not only sometimes or most of the time, but all the time.

Third, the self-organizing activity mode.

This model consists of a group of people and a set of activities that everyone can choose to do. The key assumption of this model is that everyone likes less crowded activities. For example, fewer people take part in the same activity, which means there is no need to wait in the gym, and there is no need to queue in the bakery and coffee shop.

In order to prove that this model is convergent, we should prove that the total crowding degree, that is, the sum of the crowding degree of the whole crowd satisfies the condition of Lyapunov function. When a person lowers his congestion level, his contribution to the total congestion level will decrease, and the congestion level of everyone he no longer meets will decrease by 1, while the congestion level of everyone he newly meets will increase by 1.

Generally speaking, we can't guarantee that the system can find an effective equilibrium, but this self-organizing activity model almost always converges to the configuration with the minimum total congestion.

Fourth, pure exchange economy.

Pure exchange economy consists of a group of consumers, each of whom has his own commodity endowment and preference. Every transaction requires both parties to pay a certain amount of time and energy. In order to make both parties have the motivation to complete the transaction, each party must benefit and get a certain amount that exceeds the transaction cost.

Whether Lyapunov function exists in trading environment depends on the size of negative externalities. The existence of externalities means that we can't directly say whether the system will reach equilibrium. When the transaction in the market contains negative externalities, the transaction may not necessarily improve the overall happiness.

Verb (abbreviation of verb) has no model of Lyapunov function.

Sometimes, we try to construct Lyapunov function for the model, but we can't succeed. Nevertheless, we can still accumulate knowledge. Usually, we can at least understand why the model does not produce equilibrium. In the game of life, some configurations will produce equilibrium, while others will not. When a configuration does produce equilibrium, the unique Lyapunov function of this configuration can be written.

The failure to construct Lyapunov function does not mean that the model or system cannot reach equilibrium. Some systems can reach equilibrium under all known conditions, but no one can construct Lyapunov function.

Lyapunov function can not only help us prove whether a system or model can reach equilibrium, but also tell us how quickly it can reach equilibrium. Even if the attempt to construct Lyapunov function fails, this attempt is meaningful. They can provide some clues to the causes of complexity. Using Lyapunov function, we can infer the reason why the system tends to be balanced, and we can also explain the speed at which the system converges to balance, design an information system (such as the time-sharing tour reservation system adopted by Disney World), take action (such as the trading desk is not desirable), how the communication system reaches balance, predict the time when the system reaches balance and explore.