Abstract

This paper studies Lyapunov stability at a point and its application in the Cournot duopoly game. We first explore the relationship between Lyapunov stability at a point, nonsensitivity and non-Devaney chaos and find that a dynamical system is nonsensitive and non-Devaney chaotic if there is a point in this system such that it is Lyapunov stable at that point. We next prove a group of equivalent characterizations of Lyapunov stability at a point to deduce the composite theorems and product theorems of Lyapunov stability at a point, and then we prove three equivalent characterizations of the Cournot duopoly system to demonstrate that this system is Lyapunov stable at its unique nonzero fixed point (Cournot equilibrium point) when the unit costs of the Cournot double oligarchies satisfy certain conditions. Therefore, we conclude that the Cournot duopoly system is safe relative to both sensitivity and Devaney chaos. The robustness of our results are also verified by conducting numerical simulations in the Cournot duopoly game.

1. Introduction

A general concept of Lyapunov stability at a point was introduced by Ethan and Sergii in 2003 [1] through studies of Li-York sensitivity that link the Li-York versions of chaos with the notion of sensitivity to initial conditions. The Lyapunov stability (hereafter, L-stability) at a fixed point was formally defined by Clark Robinson in 2012 [2] for a state description of differential equation solutions of being close to an equilibrium point. In the following years, this concept emerged in various studies, and these studies mainly focused on different types of systems, for example, topologic dynamical systems, differential systems, neural network systems, and epidemiology systems (see References [3–6]). However, there is limited evidence of the nature and characterization of L-stability at a point in the literature.

To the best of our knowledge, this paper is the first to study the nature and characterization of L-stability at a point and its applications in the Cournot duopoly game. In Section 2, we explore the relationship between L-stability at a point, nonsensitive dependence on initial condition (hereafter, nonsensitivity), and nonchaos in the sense of Devaney (hereafter, non-Devaney chaos) by introducing the concept of sensitive dependence on initial condition (hereafter, sensitivity) at a point. First, we find that a system is nonsensitive at a point if and only if it is L-stable at this point (see Proposition 1). Next, the following two conclusions are deduced in Proposition 3: (1) a system is nonsensitive (i.e., nonglobally sensitive) if it is nonsensitive at a given point in the system and (2) a system is non-Devaney chaotic if the system is nonsensitive. Hence, we infer that a system is nonsensitive and non-Devaney chaotic as long as there is a point that is L-stable in this system. That is, we declare a system’s nonsensitivity and non-Devaney chaos as long as we can find a point in the system that is L-stable. In Section 3, we first prove a group of equivalent characterizations of L-stability at a point (see Theorem 1). By using this theorem, we obtain the composite theorems (Theorem 2 and Corollary 1) and product theorems (Theorem 3 and Corollary 2) of L-stability at a point, and then three equivalent characterizations of L-stability at a point of the Cournot duopoly game system are obtained (Theorem 4). In Section 4, based on the previous theorems and the knowledge of differential calculus, we show that the system of the Cournot duopoly game is L-stable at its Cournot equilibrium point if the unit costs of two oligopolies are within a certain range (see Theorem 6). That is, the system of the Cournot duopoly game is safe relative to sensitivity and Devaney chaos if the condition of unit costs in Theorem 6 is satisfied. In Section 5, we conduct numerical simulations to show that our results are robust, as the sequences of the Cournot duopoly game generated by the iterations are either convergent or periodic if the condition of Theorem 6 is satisfied. It is well known that system security is very important for economics and other sciences. Therefore, our study might provide research methods for future research on the relative security of a system.

2. Relationship between L-Stability at a Point, Nonsensitivity, and Non-Devaney Chaos

Conventionally, a dynamical system is a pair where is a nonvoid compact metric space with metric and is a surjective and continuous map. denotes the set of all natural numbers (i.e., , both and denote positive real numbers, where is usually a sufficiently small positive number and is any positive number. The symbol denotes the neighborhood centered on with radial in a metric space , where is the distance between and in , and the symbol denotes the smallest upper bound of a set of some real numbers. Let and be two propositions, then, symbol denotes that inevitably holds if holds.

To elucidate the relationship between L-stability at a point, nonsensitivity, and non-Devaney chaos, we define the following precise concepts:

Definition 1 (see [1]). Assume that is a dynamical system, , the system is L-stable at , if for any , there is such that for each , where .

Definition 2 (see [7]). The system is Devaney chaotic if the following properties hold: (D1) is topological transitive, i.e., to any pair of open sets there is such that ; (D2) The periodic points are dense in ; (D3) has sensitivity, i.e., there is such that for any and , there is and such that .We deduce the following equivalent characterizations from Definition 1:

Proposition 1. Assuming that is a dynamical system, , the system is L-stable at if only and if for any , there is such that , for each and for each .

Therefore, the following three equivalent characterizations naturally hold:

Proposition 2. Assume that is a dynamical system, , the following three propositions are equivalent:(1)System is not L-stable at;(2)There is such that for any , there is and such that ;(3)There is such that for any , there is and such that .

From Definition 2, we can see that sensitivity is one of three conditions of Devaney chaos. In other words, a system is non-Devaney chaotic if it is nonsensitive. Sensitivity is a global concept relative to a system . Thus, for any point , we can discuss its localization at a point as follows:

Definition 3. A system is sensitive at a point if there is such that for any , there is and such that .
To distinguish the two sensitivities, we define (D3) of Definition 2 as global sensitivity and Definition 3 as local sensitivity at a point. In addition, surprisingly, we find that Definition 3 is identical to condition (3) of Proposition 2. Therefore, the following relationships are obviously true:

Proposition 3. Assuming that is a system, , then is L-stable at the system is nonsensitive at is nonglobally sensitive the system is non-Devaney chaos.

Proposition 3 shows that L-stability at a point is a sufficient condition that a system is nonsensitive and further non-Devaney chaotic. From Definition 2, it is clear that sensitivity at (i.e., £nonstability at ) can be explained intuitively when , as follows:

If there is a positive number , for an arbitrarily sufficient small neighborhood centered on with radial , there will be at least one point and one moment such that the distance between both and at moment becomes equal or greater to the positive number . We can see that this is a precise mathematical description of an explosion phenomenon when is considered as a sufficiently large real number. In other words, L-stability at point implies that any sufficiently small neighborhood at is safe relative to sensitivity and Devaney chaos because it is nonsensitive at , i.e., an explosion will never happen at . Therefore, system is nonglobally sensitive, as well as non-Devaney chaotic.

3. L-Stability at a Point

In this section, we first prove the composite theorems and the product theorems of L-stable points in dynamic systems. Then, using these theorems, we further show three equivalent characterizations of L-stable at a point of Cournot duopoly mapping.

To achieve this, we first prove the following equivalent characterizations:

Theorem 1. If is a dynamical system, , the following assertions are equivalent:(1)The system is L-stable at ;(2)For any , there is such that for each and for each , ;(3)For any , there is such that for each .

Proof. It is apparent that . Therefore, we only need to prove .
In fact, for any , there is such that for any , we haveThen, for each and for any , the following inequation holds:Eventually, we have that .
According to Theorem 1, we naturally obtain the following composite theorem:

Theorem 2. Assume that is a compact dynamic system, , then is L-stable at if and only if is L-stable at .

Proof. For an arbitrary , since is L-stable at , there is such that for any and for each , we have , thus . In addition, because , then the inequationholds. Therefore, is L-stable at . For an arbitrary , since is uniformly continuous, there is , such that when for each pair , we haveSince point is an L-stable point in , for , there is , such that for each and each , the following inequations are satisfied:Then, for each Thus, we haveThat is, is L-stable at the point .
The following corollary naturally follows from Theorem 2:

Corollary 1. Assume that is a compact dynamic system, , then is L-stable at if and only if the system is L-stable at for any .

For a nature number , let be compact dynamic systems, Cartesian set , and product mapping . Then, for , is a product system with the metric , where is a metric on . Therefore, we have Theorem 3.

Theorem 3. Product system is L-stable at point , if and only if for , each is L-stable at point .

Proof. Arbitrarily taking a natural number and a real number , since is L-stable at , there is such that for any and any natural number , the following inequation holds: For an arbitrary , we take when , and have ; then,Therefore, we haveThat is, is L-stable at the point . For any , since each is L-stable at , there is such that for any and for any , the following inequation holds:Let ; then for and any , we have the following equation:where and .
Subsequently, for any , we haveTherefore, system is L-stable at point .
When in this theorem, the following corollary is obviously true:

Corollary 2. Product system is L-stable at point , if and only if each is L-stable at point for .

Assuming that and are two continuous mappings, where and are two compact metric spaces, the product mapping is said to be a Cournot duopoly mapping, if for any pair , . The system is defined as the Cournot duopoly game system, and is defined as Cournot duopoly mapping where the metric on satisfies the following: for , and are the metrics on and , respectively. We have the subsequent theorem:

Theorem 4. Let to be the Cournot duopoly mapping, ; then, the following items are equivalent:(1) is L-stable at the point ;(2) is L-stable at the point ;(3) and are L-stable at both and , respectively.

Proof. Theorem 2 implies the equivalence of (1) and (2). Thus, we only need to prove the equivalence of (2) and (3):
In fact, for any pair , we have the following quation:That is, . Then, according to Corollary 2, the equivalence of (2) and (3) is shown to be true.

4. L-Stability at Equilibrium in the Cournot Duopoly Game System

In this section, we discuss the L-stability at equilibrium in the Cournot duopoly game system. Assume that duopoly firms produce homogenous goods that are perfect substitutes and offer goods at discrete time periods. Therefore, duopoly firms encounter the same market demands. They both choose the adaptive expectation rule to decide the amount of goods in the next period as their response strategy. Assume that the demanded quantity is reciprocal to price . This represents an “iso-elastic” demand function reflecting a case where consumers always spend a constant sum on the commodity, regardless of price. Inverting the demand function, we obtainwhere the total quantity in the denominator is the sum of the supplies, and , of two firms.

The revenues of the two firms equal price times quantity, that is, and . We assume that the firms operate under constant unit costs, denoted as and . Their total costs are accordingly and and their profits becomerespectively. To maximize the profits of the two firms, we have to set the partial derivative of the previous with respect to and to zero, that is, let and . Then, we obtain the two reaction functions as follows:

Let , where for any and for any . We have

It is a continuous self-mapping on the compact metric space . .

According to Theorem 4, we first state the following lemmas to prove that point is an L-stable point of .

Lemma 1. Let , for any and . Then,(1), if and only if , where is the derivative of to ;(2) if and only if .

Proof. The following two inferences are true:(1),and(2).

Lemma 2. Assume that for , for , and , then,(1) if and only if , and(2), where .

Proof. Using a similar method as Lemma 1 (1), it is obvious that , if and only if . Then, , if and only if , and a series of equivalence relations follows:Therefore, (1) is true.
In addition, the following equivalence relations hold:Therefore, (2) is also true.
Let UCF =  which is the unit cost field of , UCF is an unbounded triangle open area on the real plane as shown in Figure 1.
For each UCF, and . According to Lemma 1 (2) and Lemma 2 (2), we define , the subsequent two inequalities hold:(i) and(ii).LetIt is easy to see that from Lemma 1 (2) and Lemma 2 (2). The following conclusion is true when :