Advances in Systems Science and Applications (2013) Vol.13 No.1 21-36 Systems Defined on Dynamic Sets and Analysis of Their Characteristics Xiaojun Duan1 and Yi Lin2 1Department of Mathematics and Systems Science, National University of Defense Technology,Changsha 410073, PR China 2Department of Mathematics,Slippery Rock University Slippery Rock, PA 16057 USA Abstract (Lin, 1999) describes the static structures of general systems and investigates systems dynamics using the concept of time systems without clearing pointing out where systems attributes would come into play. To resolve this problem, in this paper, we introduce the concept of dynamic sets so that the evolutions and changing structures of systems can be adequately described by using object sets, relation sets, attribute sets, and environments of systems. On the basis of this background, we revisit some of the fundamental properties of systems, including systems emergence, stability, etc. Keywords Dynamic set, system description frame, attribute set, system property 1 Introduction Systems research represents such a science that it investigates the structure, evo- lution, and control of systems. Its goal is to study the interactions of the parts or relevant matters or events that constitute a whole and the consequent emerging holistic properties and behaviors of the whole. This whole that is made up of the parts, relevant matters or events is known as a system. In order to investigate different behaviors of systems, as of this writing, there has been a good amount of published literature, constituting a well developed theory, see (Lin, 1999)[1] and listed references there[2-4]. Zadeh (1962)[5] believes that the main goal of systems science is to study the organization and structure of systems, that is, how parts of a system are organized together to form the whole. From the basis of set theory, (Lin, 1999)[1] develops a multi-relation systems theory of rigor by establishing and analyzing the elementary concepts and fundamental character- istics of systems. That is, Lin establishes the basic concepts and properties of systems science on the solid foundation and rigor of mathematics. The general systems theory maintains (Lin, 1999)[1] that each system stand- s for such a whole that it consists of a set of objects and a set of interactions between the objects. Symbolically, a system can be written as S = (M,R) (1.1) where the set M = {mi : i ∈ I} contains all the elements that make up of the system S, known as the object set, I is an index set, R the set of all relations 22 Xiaojun Duan: Systems Defined on Dynamic Sets and Analysis of Their Characteristics between the objects in M that describe the system S. For any relation r ∈ R, there is an ordinal number n(r), which is a function of r, such that r is a sub- set of the Cartesian product of n(r) copies of M : r ⊆ Mn(r). Both the object and relation sets M and R together completely describe the given system S, its component parts, and the interactions between the parts. On this basis, leveled systems, time systems, etc., are effectively described and studied in general sys- tems theory. Because the objects of systems are highly abstract, the only difference between sets of objects will be their index sets. That is why such general theory can be powerfully employed to investigate systems with parts of identical properties. That is, the general system, as given in equ.(1.1), grasps the commonality of objects by ignoring their individual specifics. However, in terms of a realistic system, the objects’ specifics might play important roles in the operations of the system. They influence not only the composite of the system, but also the struc- ture and functionality of the system, producing the relevant dynamic behaviors of evolution of the system. For example, a system that is made up of pure oxy- gen gas or of pure hydrogen gas is fundamentally different of that consisting of both oxygen and hydrogen gases. In particular, the interactions of the objects of the former system are mainly the repulsive and attractive forces between the gas molecules, while in the latter system, other than the similar repulsive and at- tractive forces, there are also following chemical reactions when the environment provides the needed condition: 2H2 +O2 → 2H2O (1.2) The resultant system after the chemical reactions is different of the system that existed before the reactions in terms of the object set and also the relation set. What is more important is that the property of further chemical reactions in the system resulted from the first round of chemical reactions is essentially d- ifferent from that of the system that existed before the first round of chemical reactions. If the system that existed before the chemical reactions is written as S0 = (M0, R0) and the system that existed after as S1 = (M1, R1), then it is obvious that M0 ̸= M1 and R0 ̸= R1. Systems S0 and S1 are very different due to that the objects in S0 have undergone substantial change with 2H2O molecules added. Additionally, the interactions between the objects of S1 are totally differ- ent of those in S0. Other than repulsive and attractive reactions between the gas molecules, the interactions of the objects of S1 also include those between liquid molecules and gas molecules, and those between liquid molecules. From this example, it follows that for certain circumstance, it is just not e- nough to employ only object and relation sets to describe the systems and their behaviors, especially if changes and evolutionary behaviors of the systems are concerned with. It is because in the current general systems theory, both H2 and Advances in Systems Science and Applications (2013) Vol.13 No.1 23 O2 are treated as abstract objects in S0, while ignoring their differences in other aspects. When such differences do not affect the evolution of the systems much, the description of systems in equ.(1.1) will be most likely sufficient. However, the undeniable fact is that there are many such systems that the interactions between the objects of specific attributes great affect the structures, functionalities, and evolutions of the systems, just as the differences described by the two gases in equ.(1.2). Therefore, we need to develop such a systems theory to deal with this situ- ation. We have to further deliberate the description of the object and relation sets of systems so that the consequent evolutions of systems can be adequately investigated. 2 Attributes of Systems Because the usage of rigorous mathematical language in the discussion of prop- erties of general systems is very advantageous, we will continue to employ this approach. In this paper, we will not consider the case when a set is empty; and all sets are assumed to be well defined. That is, we do not consider such paradoxical situations as the set containing all sets as its elements. To this end, those readers who are interested in set theory and the rigorous treatment of general systems theory are advised to consult with (Lin, 1999)[1]. As discussed earlier, our main focus here is the difference between various parts of a system, that is, the differences between the system’s objects and between the relations of the objects. We will employ the concept of attributes to describe such differences. By attribute, it means a particular property of the objects and their interactions and the overall behavior and evolutionary characteristic of the system related to the property. If speaking in the abstract language of mathematics, the attributes of a system are a series of propositions regarding the system’s objects, relations between the objects, and the system itself. Due to the differences widely existing between systems’ objects, between systems’ relations, and between systems themselves, these propositions can vary greatly. For example, a proposition might hold true for some objects, and become untrue for others. In this case, we say that these objects have the particular attribute, while the others do not. As another exam- ple, a proposition can be written as a mapping from the object set to the set of all real numbers so that different objects are mapped onto different real numbers. In this case, we say that the system’s objects contain differences. Additionally, the discussion above also indicates that each system evolves with time so that its objects, relations, and attributes should all change with time. So, when we consider the general description of systems, we must include the time factor. By doing so, the structure of a system at each fixed time moment is 24 Xiaojun Duan: Systems Defined on Dynamic Sets and Analysis of Their Characteristics embodied in the object set, relation set, and attribute set, while the evolution of the system is shown in the changes of these sets with time. Summarizing what is analyzed above, we can define the system of our concern as follows: Definition 2.1. Assume that T is a connected subset of the interval [0,+∞), on which a system S exists. Then, for t ∈ T , the system S is defined as the following order triplet: St = (Mt, Rt, Qt) (2.1) where Mt stands for set of all objects of the system S at the time moment t, Rt the set of the relations between the objects in Mt, and Qt the set of all attributes of the system S. For the sake of convenience of communication, T is referred to as the life span and St the momentary system of the system S. What needs to be emphasized is that we study not only the evolution of systems with time, but also the evolution of the system along with continuous changes of some conditions, such as temperature, density, etc. In such cases, T will be understood as one of those external conditions. The object set of the system of our concern is made up of the system’s funda- mental units. So, for each t ∈ T , the objects in Mt are fixed. Let us write M̂ = {mt,a : a ∈ It, t ∈ T} = ∪ t∈T Mt (2.2) where Mt = {mt,a : a ∈ It} is the object set of the momentary system St, and It the index set of Mt as a function of time t. If for any t ∈ T , Mt = M , for some set M , then we say that the object set of the system S is fixed. One of the most elementary relations between objects is binary, relating each pair of objects. Such a relation can be written by using the 2-dimensional Carte- sian product of the system’s object set: R0 t,2 = {(mt,a,mt,b) ∈ Mt ×Mt : Φ 0 t,2(mt,a,mt,b)} ⊆ M2 t (2.3) for each t ∈ T , where Φ0 t,2(, ) is a proposition that defines R0 t,2. Corresponding to different properties, the system St might contain different binary relations as subsets of the 2-dimensional Cartesian product M2 t of the object set. The set of all the binary relations in St is written as follows: Rt,2 = {Rk t,2 : k ∈ K2} (2.4) where K2 is the index set of all binary relations of St. Similarly, the system St might contain trinary relations: R0 t,3 = { (mt,a,mt,b,mt,c) ∈ M3 t : Φ0 t,3 (mt,a,mt,b,mt,c) } ⊆ M3 t (2.5) Advances in Systems Science and Applications (2013) Vol.13 No.1 25 for each t ∈ T , where Φ0 t,3(, , ) is a proposition that defines R0 t,3 . The set of all trinary relations of St is written as follows: Rt,3 = {Rk t,3 : k ∈ K3} (2.6) where K3 is the index set of all trinary relations of St. Higher order relations of St can be introduced similarly. For convenience, let us define unitary relations of St as follows: R0 t,1 = {(mt,a) ∈ Mt : Φ 0 t,1(mt,a)} ⊆ Mt (2.7) for each t ∈ T , where Φ0 t,1() is a proposition that defines R0 t,1. The set of all unitary relations of St is written as follows: Rt,1 = {Rk t,1 : k ∈ K1} (2.8) whereK1 is the index set of all unitary relations of St. Now, the set Rt of relations of St can be written as follows: Rt = ∪ α∈Ord Ri,α (2.9) where Ord stands for the set of all ordinal numbers. When Ord is taken to be N = the set of all natural numbers, equ. (2.9) stands for the set of all finite relations the momentary system St contains. From the discussion above, it can be seen that all kinds of algebras and spaces studied in mathematics are special cases of equ. (2.9) with Ord replaced by N . Without loss of generality, let us assume that Mt ∈ Rt,1. With this convention, it can be seen that in the following when we talk about the attributes of a system St, we only need to mean the attributes of the relations of St, because the object set is also considered a relation, a unitary relation and the attributes of objects are now also those of relations. The attribute set Qt of the momentary system St are a series of propositions about the relations. Symbolically, we can write: Qt = {qt(r1, r2, ..., rα, ...) : qt(...) is a proposition of rα ∈ Rt, α = 1, 2, 3...} (2.10) Without any doubt, with this notation in place, these propositions in Qt embody all aspects of the momentary system St. For example, the concept of mass of object in physics is a mapping: µt : Rt → R+ (2.11) which assigns each element in an unitary relation the mass of the element, and each element x⃗ = (x1, x2, ..., xα, ...) in an n-nary relation the sum of the masses 26 Xiaojun Duan: Systems Defined on Dynamic Sets and Analysis of Their Characteristics of the objects contained in the element, if the sum exists. That is, for any r ∈ Rt, and any x⃗ = (x1, x2, ..., xα, ...) ∈ r, µt(x⃗) = µt(x1, x2, ..., xα, ...) = ∑ α µt(xα) (2.12) assuming that the sum on the right hand side converges. Let us look at the network model of systems as an example, where only unitary and binary relations are considered. In this model, all objects of the system of concern are treated as nodes; each binary relation is modeled as the set of edges of the network. By doing so, each binary relation of the system corresponds to a network or a graph; and different binary relations correspond to different sets of edges. Such correspondence can be seen as an attribute of the edges. Accordingly, different unitary relations of the system correspond to different attributes of nodes in the network, such as size of the nodes, flow intensities of the nodes, etc. If different types of edges are treated as identical, then the network can be expressed as G = (V,E,Q), where V stands for the set of all nodes, E the set of edges, and Q some attributes of either the nodes or the edges or both, such as weights of the edges. The ordered pair (V,E) completely describes the topological structure of the network, which is sufficient for some applications. However, if the system we investigate is quite specific, for example, it is a network of railroads, a network of human relationships, etc., we may very well need to model multiple relations. In this case, the attribute set Q can be employed to describe the scales of the stations in the railroad network, the traffic conditions or transportation capabilities between stations, etc. If the system is a network of human relationships, then Q can be utilized to represent the social status of each individual person, the intensity of interaction between two chosen persons, etc. As a matter of fact, the concept of systems, as defined in equ.(2.1), is a generalization of that as defined in equ.(1.1) (Lin, 1999)[1]. In other words, we can rewrite equ.(1.1) in the format of equ.(2.1) as follows. Let all object sets be static. So, for any t ∈ T , we have Mt = M and Rt = R. And what is interesting is how an attribute is introduced. To this end, we can introduce a proposition q0 on the Cartesian product M̂ = ∑ α∈OrdM α of the object set M so that for any r ∈ M̂ , q0(r) = { 1, if r ∈ R 0, otherwise (2.13) Then, we take Q = {q0}. That is, the attribute set is a singleton. Now, each system written in the format of equ. (1.1) is rewritten in the format of equ. (2.1). Here, the attribute q0 describes if an arbitrarily chosen relation in the Cartesian product M̂ belongs to the system’s relation set R or not. In essence, it restates Advances in Systems Science and Applications (2013) Vol.13 No.1 27 the membership relation to the relation set R from the angle of attributes. In particular, because the relation set R of the system S is a subset of M̂ ; now the membership in the relation set R is determined by a proposition q0, while such a description is an attribute of the system S. That is, the general systems theory developed on set theory (Lin, 1999)[1] has already implicitly introduced the con- cept of attributes. What we do here is to make this fact explicit. And because the systems we are interested in can have multiple attributes, our contribution to the general systems theory is to make the concept of attributes more general as a set Q of attributes, including more than just the particular attribute q0. 3 Subsystems Just like each set has its own subsets, every system has subsystems, which can be constructed from the object set, relation set, and the attribute set of the system. In short, a system s is a subsystem of the system S, provided that the object set, relation set, and attribute set of s are corresponding subsets of those of S so that the restrictions of the attributes of S on s agree with the attributes of s. Symbolically, we have Definition 3.1. Let st = (mt, rt, qt) and St = (Mt, Rt, Qt) are two systems, for any t ∈ T . If the following hold true: mt ⊆ Mt, rt ⊆ Rt|st , and qt ⊆ Qt|st ,∀t ∈ T (3.1) then st is known as a subsystem of St, denoted st < St, ∀t ∈ T, or s < S (3.2) where Rt|st and Qt|st represent respectively the restrictions of the relations and attributes in Rt and Qt on the system St and are defined as follows: Rt|st = {r|mt : r ∈ Rt} and Qt|st = {q|mt∪rt : q ∈ Qt}. In this definition, the notation of less than of mathematics is employed for the relationship of subsystems, because the relation of subsystems can be seen as a partial ordering on the collection of all systems. Similarly, when equ. (3.1) does not hold true, we say that s is not a subsystem of S, denoted st ≮ St, ∀t ∈ T , or s ≮ S. Let the set of all subsystems of S be S , then we have S ≡ {s : s < S} = {st : st < St, ∀t ∈ T} (3.3) For any given system S = (M,R,Q), where M = {Mt : t ∈ T}, R = {Rt : t ∈ T}, and Q = {Qt : t ∈ T}, as defined by equ. (2.1), take a subset set of its object set 28 Xiaojun Duan: Systems Defined on Dynamic Sets and Analysis of Their Characteristics A = {At ⊆ Mt : t ∈ T}. Then by restricting the relation set R and the attribute set Q on A, we obtain the following subsystem of S induced by A: S|A = {st = (At, Rt|At , Qt|(At,Rt|At ) ) : t ∈ T} (3.4) where the restrictions Rt|At and Qt|At are assumed respectively to be Rt|At = {r|At : each element in r|Athas the same length, r ∈ Rt} (3.4a) and Qt|(At,Rt|At ) = {q|(At,Rt|At ) : qt|(At,Rt|At ) is a well defined proposition on st, q ∈ Qt} (3.4b) Proposition 3.1. The induced subsystem S|A is the maximum subsystem in- duced by A. Proof. Let s = {(At, rt,A, qt,A) : t ∈ T} < S be an arbitrary subsystem with the entire A as its object set. According equ. (3.1) we have rt,A ⊆ Rt|s, and qt,A = Qt|s = Qt|(At,rt,A), ∀t ∈ T So, from equ. (3.4a), it follows that s ≤ S|A only when rt,A = Rt|s, the equal sign holds true. QED. Proposition 3.2. Assume that A and B are subsets of M satisfying that B ⊆ A ⊆ M , then S|B ≤ S|A. Proof. According equ. (3.4), we have S|B = {st = (Bt, Rt|Bt , Qt|(Bt,Rt|Bt ) ) : t ∈ T}.B ⊆ A implies that Bt ⊆ At, ∀t ∈ T . So, it follows that Rt|Bt ⊆ Rt|At and consequently Rt|Bt ⊆ (Rt|At)|Bt and Qt|(Bt,Rt|Bt ) = (Qt|(At,Rt|At ) )|(Bt,Rt|Bt ) . Therefore, S|B ≤ S|A. QED. Proposition 3.3. Let S be a system. Then the collection of all subsystems of S forms a partially ordered set by the subsystem relation “<”. Proof. This result is a straightforward consequence of Proposition 3.2. QED. Assume that S1, S2, and S are systems with the same time span such that S1 < S and S2 < S. Let Si, i = 1, 2, denote the set of all subsystems of Si. Then each element in the set S1 − S2 = {s : s < S1, s ≮ S2} is a subsystem of S1 but not a subsystem of S2. Similarly, each element in the set S2 − S1 = {s : s < S2, s ≮ S1} is a subsystem of S2 but not a subsystem of S1. Let S1∆S2 = {s : s < S1, s ≮ S2} ∪ {s : s < S2, s ≮ S1} be the union of the previous two sets of subsystems of either S1 or S2; and S1 ∩ S2 = {s : s < S1, s < S2} the set of all subsystems of both S1 and S2. Proposition 3.4. Assume that S1 < S and S2 < S. Then S1 ∪ S2 = {s : s < S1} ∪ {s : s < S2} is a subset of S|M1∪M2 . Proof. S|M1∪M2 is a maximal element in the partially ordered set (S|(M1∪M2),⊆) , satisfying ∀s ∈ S|M1∪M2 , s < S|M1∪M2 . On the contrary,∀s < S|M1∪M2 , we Advances in Systems Science and Applications (2013) Vol.13 No.1 29 have s ∈ S|M1∪M2 . So, ∀s ∈ S1 ∪ S2, we have s ∈ S|M1∪M2 . QED. What this result indicates is that the union S1 ∪ S2 of the sets of subsystems of two subsystems S1 and S2 is a subset of the S|M1∪M2 of the subsystems of the induced system on the union M1 ∪M2. Proposition 3.5. Given two arbitrary systems S1 and S2, there is always a system S12 such that S1 < S12 and S2 < S12. Proof. Without loss of generality, assume that S1 = {(M1 t , R 1 t , Q 1 t ) : t ∈ T 1} and S2 = {(M2 t , R 2 t , Q 2 t ) : t ∈ T 2}. To construct the system S12 = {(M12 t , R12 t , Q12 t ) : t ∈ T 12}, T 12 = T 1 ∪ T 2, we first assume that the object sets M1 t and M2 t are disjoint, that is, M1 t ∩ M2 t = ∅, for any t ∈ T 1 ∩ T 2. Then, the desired system S12 is defined as follows: for t ∈ T 1 ∩ T 2,  M12 t = M1 t ∪M2 t R12 t = R1 t ∪R2 t Q12 t = Q1 t ∪Q2 t (3.5) for t ∈ T 1−T 2, define M12 t = M1 t , R 12 t = R1 t , and Q12 t = Q1 t , and for t ∈ T 2−T 1, define M12 t = M2 t , R 12 t = R2 t , and Q12 t = Q2 t , and is denoted as S12 = S1 ⊕ S2. Now, if M1 t ∩ M2 t ̸= ∅, for some t ∈ T 1 ∩ T 2, we simply take two systems ∗S1 = {(∗M1 t , ∗R1 t , ∗Q1 t ) : t ∈ T 1} and ∗S2 = {(∗M2 t , ∗R2 t , ∗Q2 t ) : t ∈ T 2} with M1 t ∩ M2 t = ∅, for any t ∈ T 1 ∩ T 2, such that ∗Si is similar to Si, i = 1, 2, where similar systems are defined in the same fashion as in [1]. Then, we define S12 =∗ S1⊕∗S2. Up to a similarity, the system S12 is uniquely defined. Therefore, it can be seen as well constructed such that S1 < S12 and S2 < S12, where the time spans T 1 and T 2 are seen as the same as T 12 such that when t ∈ T 1 − T 2 (respectively, t ∈ T 2 −T 1), we treat S2 t (respectively,S1 t ) as a system with empty object set. QED. From this proposition, it follows that when the interactions of some given systems are considered, these systems can always be seen as subsystems of a larger system. On the other hand, this proposition also shows that there is always some kind of interaction between two given systems, which is embodied in the fact that they are subsystems of a certain system. 4 Interactions between Systems When there is an interaction between two objects of a system S = (M,R,Q) (of time span T ), where M = {Mt : t ∈ T}, R = {Rt : t ∈ T}, and Q = {Qt : t ∈ T}, it can be described by using a binary relation of the system. We say that objects m1 and m2 ∈ M , which means either mi = (mt)t∈T such that mt ∈ Mt, for each t ∈ T , or mi = mt ∈ Mt, for a particular t ∈ T, i = 1, 2, interacts with respect to an attribute q ∈ Q, it means that the proposition q holds true for a binary relation 30 Xiaojun Duan: Systems Defined on Dynamic Sets and Analysis of Their Characteristics r ∈ R that contains either (m1,m2) or (m2,m1) or both. The idea of interactions between systems is a natural generalization of that between two objects. Based on the discussion of the previous section, we will discuss interactions of systems in the framework of subsystems. Assume that Si < S, i = 1, 2. Now, let us look at how these subsystems could interact with each other. Definition 4.1. The system S1 is said to have a weak effect on the system S2 with respect to an attribute q ∈ Q, provided that for any r2 ∈ R2 there is r1 ∈ R1 such that the proposition q holds true for the ordered pair (r1, r2). When no confusion is caused, we simply say that the system S1 affects the system S2 weakly without mentioning q. If the system S2 also exerts a weak affect of S1, then we say that these systems interact with each other weakly. Definition 4.2. The system S1 is said to have a strong effect on the system S2 with respect to an attribute q ∈ Q, provided that for any r2 ∈ R2 and any r1 ∈ R1, the proposition q holds true for the ordered pair (r1, r2). When no confusion is caused, we simply say that the system S1 affects the system S2 strongly without mentioning q. If the system S2 also exerts a strong affect of S1 , then we say that these systems interact with each other strongly. From these definitions, it follows that strong interaction requires interactions between every ordered pair of objects, which is a more rigorous requirement than that of weak interactions. Also, to maintain the intuition behind the concepts of interactions, in Definitions 4.1 and 3.2, we only look at two relations ri ∈ Ri, i = 1, 2. In order to capture the general spirit, these individual relations should be replaced by subsets {ri ∈ Ri : Φi(ri)} , where Φi() stands for the proposition that defines the set, for i = 1, 2. By doing so, what are discussed in Definitions 4.1 and 4.2 become special cases. Definition 4.3. Given a subsystem s = (m, r, q) of a system S = (M,R,Q), the totality of all objects in M −m, each of which interacts weakly with at least one object in m, is known as the environment of the subsystem s in S, denoted ES . 5 Systems Properties Based on Dynamic Set Theory 5.1 Basic Properties For two given systems S1 = {S1 t = (M1 t , R 1 t , Q 1 t ) : t ∈ T 1} and S2 = {S2 t = (M2 t , R 2 t , Q 2 t ) : t ∈ T 2} , let us consider Definition 5.1. These systems S1 and S2 are equal, provided that M1 t = M2 t , R 1 t = R2 t , Q 1 t = Q2 t , and T 1 = T 2, (5.1) for each t ∈ T 1 = T 2. The systems S1 and S2 are said to be identical on the time period T ⊆ T 1∩T 2, it means that M1 t = M2 t , R 1 t = R2 t , Q 1 t = Q2 t , for each t ∈ T. (5.2) Advances in Systems Science and Applications (2013) Vol.13 No.1 31 Definition 5.2. The system S1 is said to be homomorphically embeddable into the system S2, provided that there is a non-decreasing mapping f : T 1 → T 2 such that for any t1 ∈ T 1, if t2 = f(t1) ∈ T 2, then S1 t1 = S2 t2 , or equivalently, M1 t1 = M2 t2 , R 1 t1 = R2 t2 , and Q1 t1 = Q2 t2 (5.3) The mapping f is referred to as an embedding mapping from S1 into S2. If the system S1 can be homomorphically embeddable into S2 and S2 into S1, then the systems S1 and S2 are said to be homophorhically equivalent. Evidently, equal systems are homomorphically equivalent with the identity mapping on the time set as the canonical embedding mapping. Proposition 5.1. If the embedding mapping f : T 1 → T 2 from the system S1 into S2 is bijective, then the systems are homomorphically equivalent. Proof. It suffers to show that the inverse mapping f−1 : T 2 → T 1 is an embedding mapping from S2 into S1. Because f is bijective, it is strictly increasing from T 1 into T 2; and its inverse f−1 is also a strictly increasing mapping from T 2 into T 1 satisfying for any t2 ∈ T 2, if t1 = f−1(t2) ∈ T 1, then M2 t2 = M1 t1 , R 2 t2 = R1 t1 , and Q2 t2 = Q1 t1 . Therefore, f−1 : T 2 → T 1 is an embedding mapping from S2 into S1. QED. Definition 5.3. A system S = {St = (Mt, Rt, Qt) : t ∈ T} is said to be cyclic or periodic, provided that there is time tc > 0 such that St+tc = St, for any t ∈ T . Evidently, in this case, for any natural number n ∈ N , ntc is also a period of the system S. The minimum period is named the period of S, denoted Tc. 5.2 Systemic Emergence By systemic emergence, it means the properties of the whole system that parts of the system do not have. Such a property might suddenly appear at a particular time moment. In terms of the system, the holistic emergence is mainly created and excited by the system’s specific organization of its parts and how these parts interact, supplement, and constrain on each other. It is a kind of effect of rele- vance, the organizational effect, and the structural effect. To be specific, let P represent such a property. It is defined on the entire system S. Because of the system’s dynamic characteristics, the interactions between the system’s objects, relations, attributes, and the environment change constantly. So, the value of P also varies accordingly. Define P (St) = { 1, if S has this property 0, otherwise , ∀t ∈ T (5.4) It satisfies the following properties: P (st) = 1 → P (St) = 1, ∀s < S, ∀t ∈ T (5.5) 32 Xiaojun Duan: Systems Defined on Dynamic Sets and Analysis of Their Characteristics That is, as long as a subsystem s has this property, the overall system S also has the property. That is another way to say that the whole is greater than the sum of parts. Of course, there are also such properties that some subsystems have, while the overall system does not have. For instance, let P be a property the system S does not have. Define P−1(1) = {s < S : s ̸= S} be the collection of all proper subsystems of S, where the superscript (−1) stands for the inverse operation. Then, for any s ∈ P−1(1), s < S and P (s) = 1. However, P (S) = 0 does not satisfy equ. (5.5). When a property P is said to be system S’s holistic emergence, provided that for any subsystem s < S, P (st) = 0, ∀t ∈ T , and P (St0) = 1 for at least one t0 ∈ T . In the traditional static description of systems, there is another definition of systemic emergence. In particular, consider a monotonically increasing sequence of subsystems ∅ ̸= s1 < s2 < ... < sk < ... < S, there is a k0 such that P (sk) = 0, ∀k < k0, P (sk0) = 1 (5.6) In this case, the system S is said to have the emergent property P , while its parts sk, k < k0, do not share this property until they reach the whole sk0 of certain scale. From a detailed analysis, it follows readily that what is just presented is a special case of the systemic emergence of dynamic systems, where we can surely treat the collection of all the superscript k as a subset of the time index T so that the sequence {sk : k = 1, 2, 3, ...} of subsystems a subsystem of a dynamic system by letting St = st, for t = 1, 2, 3, ... From equ. (5.6), it follows that P (st) = 0, for t = 1, 2, 3, ... < k0, and P (sk0) = 1 That is, the property P emerges at the time moment k0. 5.3 Stability of Systems By stability, it means the maintenance or continuity of a certain measure of the dynamic system on a certain time scale. That is, under small disturbances, the measure does not undergo noticeable changes. Let us look at the trajectory sys- tem of a single point, where the focus is how the point moves under the influence of an external force. If a disturbance is given to a portion of the trajectory that is not at a threshold point, then the disturbed trajectory will not differ from the original trajectory much. However, if the same disturbance is given to the trajectory at an extremely unstable extreme point, then a minor change in the disturbed value could cause major deviations in the following portion of the tra- jectory. Thus, only the trajectory of motion at stable critical extrema is stable, while the trajectory systems with instable critical points, such as saddle point, are instable. Advances in Systems Science and Applications (2013) Vol.13 No.1 33 In general, assume that an attribute q ∈ Q of the system S satisfies that q is a real-valued function defined for each momentary system St, for any t ∈ T . Let t0 ∈ T . If for any ε > 0, there is a δt0,ε > 0 such that |q(st0)− q(st)| < ε, ∀t ∈ (t0 − δt0,ε, t0 + δt0,ε) (5.7) then the attribute q of the system S is regionally stable over time at t0 ∈ T . If for any ε > 0, there is δ = δ(ε) = δε > 0 such that |q(st1)− q(st2)| < ε, ∀t1, t2 ∈ T such that |t1 − t2| < δε (5.8) then the attribute q is said to be holistically stable over time or uniformly stable over time. When no confusion can be caused, the previous concepts of stability of the attribute q are respectively referred to as that the system S is regionally stable at t0 ∈ T , or uniformly stable over T . In addition, the structural stability of systems can also be defined. In partic- ular, Let d ∈ Q be such that d : St → R+ is a positively real-valued function defined for each subsystem of the momentary system St, t ∈ T , satisfying (1)d(∅) = 0,where ∅ stands for the the subsystems with the empty set as their object set; (2)∀S1, S2 < S, if S1 < S2, then d(S1) ≤ d(S2); (3)∀S1, S2 < S, if S1△S2 = ∅, then d(S1 ∪ S2) = d(S1) + d(S2). It can be readily seen that such an attribute d can be employed to measure the difference between subsystems of S; and for any chosen S1 < S, and for any S2 ∈ S, the greater the attribute value d(S1△S2) is, the more different the sys- tems S1 and S2 are. By making use of such a d ∈ Q, which satisfies the previous properties, for any chosen s ∈ S and any real number δ > 0, we can define the δ -neighborhood Nbrdd(s, δ) of s as follows: Nbrdd(s, δ) = {s′ ∈ S : d(s′△s) < δ} (5.9) The attribute q ∈ Q of the system S is said to be structurally stable in the neighborhood of a subsystem s < S with respect to attribute d ∈ Q, provided that for any ε > 0, there is δ = δε > 0 such that |q(s)− q(s′)| < ε, ∀s′ ∈ Nbrdd(s, δε) (5.10) Similarly, the concepts of regionally structural stability and uniformly struc- tural stability of a system S at all of its subsystems can be defined. If a system S is both uniformly stable over time and uniformly structural stable, then the system is referred to as a uniformly stable system. 34 Xiaojun Duan: Systems Defined on Dynamic Sets and Analysis of Their Characteristics 5.4 Evolution of Systems When systems’ evolution is concerned with, the focus is how the system devel- ops over time. Speaking rigorously, each system can be described by using the dynamic format in equ.(2.1). As for those systems, which do not seem to change with time, when seen from the angle of dynamics, they are either evolving with time extremely slowly or considered within a very short period of time. That is how they project a mistakenly incorrect sense of being static. The evolution of systems takes two main forms: One is the transition of the system from one structure or form to another structure or form, and the other is that a system appears from its earlier state of non-existence and grows from a immature state to a more mature state. Both of these two forms of evolution can be described and illustrated by using the language of the dynamic systems in equ.(2.1). Assume that the system of our concern is indeed described in the form of equ.(2.1). Then for the first form of evolution, the relation set Rt and the attribute set Qt change with time, while for the second form of evolution, the elements in the object set Mt develop over time together of course with changes of the relation and attributes sets Rt and Qt. They represent two specific cases of the evolution of dynamic systems. 5.5 Boundary of Systems By boundary, it stands for the separation between a system and its environmen- t. The boundary is a part of the system and interacts with the environment more closely than the interior of the system. By reviewing the definition of en- vironments in Definition 4.3, we can define the boundary of a system as follows. Assume that s < S is a subsystem of the system S over the time span T . Let us denote the subsystem s and the system S respectively as st = (ms t , r s t , q s t ) and S = (M,R,Q), for t ∈ T . For the sake of convenience of communication, we write s = (ms, rs, qs) for a fixed time moment. Then, the (external) environment Es of s within the system S is defined to be: Es = {(M −ms, r|mE , q|mE s ,rE ) : r ∈ R(r|ms /∈ rs), q ∈ Q(q|ms /∈ qs)} = (mE , rE , qE) where mE = M − m, rE = r|mE , and qE = q|mE ,rE , for r ∈ R(r|mE /∈ rs) and q ∈ Q(q|mE /∈ qs). Assume that there is an attribute µ ∈ Qt of the system S such that it assigns each relation to a positive real number µ(r) : Rt → R+. Intuitively, this attribute µ is an index that measures the intensity of each relation of the objects of S. Now, let us fix a threshold value µ0 for the relational intensity, then by combining with the concept of weak interactions between systems (Definition 4.1), we can obtain the set of all relations in S of intensity at least µ0 that relate the subsystem s Advances in Systems Science and Applications (2013) Vol.13 No.1 35 and its environment Es as follows: rb = {r ∈ R : r|ms ∈ rs, Supp(r) ∩ms ̸= ∅ ̸= Supp(r) ∩mE , µ(r) ≥ µ0} (5.11) where Supp(r) stands for the support of the relation r, which is the set of all objects that appear in the relation r. The set of all objects of s that interact with the environment Es of intensity of at least µ0 is given by the following: mb = ∪ r∈rb Supp(r)−mE (5.12) Then the system ∂s = (mb, rb|mb , qb|mb,rb|mb ) (5.13) satisfies that ∂s < s, and is referred to as the boundary (system) of s within the system S. The intensity of its interaction with the environment is no less than µ0, while any other subsystem of s interacts with the environment Es with strictly less intensity than µ0. If in the evolution of the system, ∂s has good stability, then we say that the boundary of the system is clear. Otherwise, we say that the boundary of the system is fuzzy. Symbolically, let q ∈ Q be an attribute and t0 ∈ T chosen. If for any ε > 0, there is δ = δt0,ε > 0 such that |q(∂st0)− q(∂st)| < ε, ∀t ∈ (t0 − δt0,ε, t0 + δt0,ε) (5.14) then we say that the boundary of the system s at time t0 ∈ T is definite. Other- wise, the boundary is said to be fuzzy at t0 ∈ T . It is not hard to see that the definiteness of a system’s boundary is defined by using the stability of the boundary system ∂s. Therefore, the stability of the boundary system at one time moment corresponds to the definiteness of the sys- tem’s boundary. Similarly, we can study the concepts of regional definiteness of boundaries over time and structural definiteness of boundaries. 6 Some Final Words By using the traditional set theory, the static structures and relations of systems are described. Then, the dynamic changes of systems are studied by using the concept of time systems. In this paper, on the basis of the concept of systems, we describe the dynamic structure and change of a general system from a more delicate angle by determining the objects, relations, and attributes. By consid- ering the affects of environment, we establish the basic concepts of systems using the idea of dynamic sets that are related to the variable of time. Then on top of these developed concepts, we revisit the description and analysis of some of the most fundamental properties of systems, while making the necessary comparisons 36 Xiaojun Duan: Systems Defined on Dynamic Sets and Analysis of Their Characteristics with the studies of systems developed on the classical set theory. As for the analysis of systems properties under the new setting, additional rig- orous deduction of various results of systems is badly needed in order to enrich the relevant systems theory. Acknowledgements This work is supported by the National Science Foundation of China (60974124), the Program for New Century Excellent in University, and the Project-sponsored by SRF for ROCS, SEM in China. References [1] Lin. Y. (1999), General Systems Theory: A Mathematical Approach, New York: Kluwer Academic and Plenum Publishers. [2] Mickens. R. E. (1990), Mathematics and Science, Singapore: World Scien- tific. [3] Quastler. H. (1965), General Principles of Systems Analysis, In: T. H. Wa- terman and H. [4] J. Morrowits (eds.), Theoretical and Mathematical Biology, New York: Blais- dell Publishing. [5] Zadeh. L. (1962), “From circuit theory to systems theory”, Proc. IRE, Vol.50, pp.856-865. Corresponding Author Xiaojun Duan can be contacted at: xjduan@nudt.edu.cn.