Java Based Distributed Learning Platform Adv Syst Sci Appl 2019; 01; 116-140 Published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/591 Models of Continuous Dynamics on the 2-Simplex and Applications in Economics Denis Stijepic*1 1) University of Hagen, Department of Macroeconomics, Hagen, Germany E-mail: denis@stijepic.com Received May 16, 2018; Revised January 21, 2019; Published April 15, 2019 Abstract: In this paper, we discuss the models of continuous dynamics on the 2-simplex that arise when different qualitative restrictions are imposed on the (continuous) functions that generate the dynamics on the 2-simplex. We consider three types of qualitative restrictions: inequality (or set-theoretical) conditions, monotonicity/curvature (or differential-geometrical) conditions, and topological conditions (referring to (transversal) non-(self-)intersection of trajectories). We discuss the implications of these restrictions for transitional and limit dynamics on the 2-simplex and the wide range of potential and existing applications of the resulting system-theoretical models in economics and, in particular, in economic growth and development theory. Keywords: dynamics, trajectory, 2-simplex, continuous, monotonous, intersection, self-intersection, Poincaré-Bendixson theory, economics 1. INTRODUCTION In this paper, we discuss the models of continuous dynamics on the 2-simplex that arise when different qualitative restrictions are imposed on the (continuous) vector function x(t) ≡ (x1(t), x2(t), x3(t)) that generates the dynamics on the 2-simplex (where t represents time). In particular, there are three major types of qualitative conditions that can be imposed on this function: (1.) inequality conditions of the type ∀t ∈ A ∀i ∈ B xi(t) ≶ ai = const., which can be treated by using set-theoretical concepts (referring to the points or segments of the corresponding trajectory and the partitions of the 2-simplex); (2.) (strict) monotonicity conditions referring to all or some of the functions xi(t), which can be treated by (differential) geometrical concepts of tangential vector angles and curvature; and * Corresponding author: denis@stijepic.com mailto:denis@stijepic.com MODELS OF CONTINUOUS DYNAMICS ON THE 2-SIMPLEX AND APPLICATIONS IN ECONOMICS 117 Copyright ©2019 ASSA. Adv. in Systems Science and Appl. (2019) (3.) conditions regarding (transversal) trajectory non-(self-)intersections, which can be treated by using topological concepts (e.g., homeomorphisms). We discuss the implications of these restrictions for the transitional and limit dynamics on the 2-simplex (among others, fixed points, waves, or (limit) cycles may arise). The models that result from this discussion are relatively simple from the mathematical point of view, yet they seem widely applicable in economic growth and development theory and, thus, may be regarded as powerful system-theoretical constructs. As discussed in Section 6, many core topics of long-run economic theory (e.g., savings rate dynamics, functional income distribution dynamics, and sector dynamics) can be modeled by continuous trajectories on simplexes, to which our system-theoretical results apply. In general, the system-theoretical analysis of economic dynamics seems a highly valuable complement to the standard approaches of economic modeling, which rely on quantitative theoretical micro-foundations: System-theoretical models can be relatively crude and do not necessarily require detailed/quantitative assumptions about the nature of economic phenomena. This can be an advantage, since, often, detailed/quantitative assumptions about economic agents and economic environments cannot be supported by empirical evidence and, thus, may open the door for speculation, ideology, and prediction errors. For these reasons, system-theoretical results are not only transferable across many economic topics, but can also be relatively robust in comparison to the results of quantitative micro-founded economic models, and this fact is particularly useful in prediction of long-run economic dynamics (see [20] and Section 6 for a detailed discussion). Moreover, the dynamics on the 2-simplex merit a detailed consideration from the (application oriented) mathematical point of view: Although the 2-simplex can be regarded as a bounded subset of a plane (in ℝ3), the description of the dynamics on the 2-simplex requires a greater variety of analytical concepts in comparison to the description of the dynamics in ℝ2 (see, e.g., the discussion of the monotonicity concepts in Sections 2.3 and 3). The rest of the paper is organized as follows. In Section 2, we discuss the characterization of trajectory families on the 2-simplex via geometrical and topological concepts. Sections 3-5 discuss the implications of these concepts for transitional and limit dynamics on the 2-simplex. This discussion yields system-theoretical models. The potential and existing applications of these models in economics and, in particular, in growth and development theory are discussed in Section 6. Concluding remarks are provided in Section 7. 2. CHARACTERIZATION OF THE TRAJECTORIES ON THE 2-SIMPLEX In Section 2, we summarize the concepts that can be used to characterize continuous dynamics on the 2-simplex as applied by [13-15] in structural change modeling. While there are different mathematical notational conventions, we choose the following notation for reasons of simplicity: small letters (e.g., x), bold small letters (e.g., x), capital letters (e.g., X), and Greek letters (e.g., α) denote scalars, vectors/points, sets, and vector angles, respectively. cl(A) denotes the closure of the set A. If I denotes an open interval (e.g., (a, b)), then [I], [I), and (I] denote the corresponding closed (e.g., [a, b]), left-closed (e.g., [a, b)), and right-closed (e.g., (a, b]) interval, respectively. 118 D.STIJEPIC Copyright ©2019 ASSA. Adv. in Systems Science and Appl. (2019) 2.1 Trajectories on the 2-Simplex The (standard) 2-simplex (S), which is defined by (1), is a triangle in ℝ3, as depicted by Fig. 1. The Cartesian coordinates of the simplex vertices v1, v2, and v3 are stated by (2). S := {(x1, x2, x3) ∈ ℝ3: x1 + x2 + x3 = 1 ∧ ∀i ∈ {1, 2, 3} 0 ≤ xi ≤ 1} (1) v1 := (1, 0, 0) (2a) v2 := (0, 1, 0) (2b) v3 := (0, 0, 1) (2c) We define the vector function x(t, j) as follows: x(t, j) ≡ (x1(t, j), x2(t, j), x3(t, j)): T × J → S (3a) 0 ∈ T ⊆ ℝ (3b) J ⊆ S (3c) The trajectory X(T, j) and the trajectory segment X(T.+, j) are defined by (4). ∀j ∈ J X(T, j) := {x(t, j) ∈ S: t ∈ T} (4a) ∀j ∈ J X(T +, j) := {t ∈ T: t ≥ 0} (4b) x3 v3 v2 v1 x2x1 Fig. 1. The standard simplex S in ℝ3. In fact, (4a) defines a trajectory family indexed by the set J, where each trajectory X(T, j) describes a path on S that is traversed over the period T. X(T +, j) is the segment of this path that is traversed over t ≥ 0. 2.2 Set-Theoretical Trajectory Classification Equations (5) introduce a partitioning of S, which can be used for describing the location of relevant trajectory points or segments (e.g., initial segment/state, empirically observed segment, or segment representing the future dynamics), as we will see later. ∀i ∈ {1, 2, 3} Svi := {(x1, x2, x3) ∈ S: xi > 1/2} (5a) Sv0 := S \ (Sv1 ∪ Sv2 ∪ Sv3) (5b) MODELS OF CONTINUOUS DYNAMICS ON THE 2-SIMPLEX AND APPLICATIONS IN ECONOMICS 119 Copyright ©2019 ASSA. Adv. in Systems Science and Appl. (2019) (5a) and (1) imply that the partition Svi contains all the points of S that are dominated by xi; i.e., if a point (x1, x2, x3) is located in partition Svi, then ∀j ∈ {1, 2, 3}\i xi > xj. The geometrical interpretation of the partitioning (5) is depicted in Fig. 2. As we can see, for i ∈ {1, 2, 3}, the partition Svi contains all the points of S that are closer to the vertex vi than to the other vertices (vj, j ≠ i). The following (set-theoretical) definitions allow us to assess the prediction range of monotonous models, as we will see later. Let a(K) denote the area function, which assigns to a set K ⊆ S the (real number indicating the) area of K. B(T, F) := ⋃j∈F X(T, j) is the image of the family F of trajectories X(T, j), j ∈ F ⊆ J (cf. (4)). Among all the path-connected and closed subsets of S that cover B(T, F), let M(T, F) denote one of the sets that cover the smallest area of S. a*(T, F) := a(M(T, F)) is the family image size of the family F. 23113 12332 31221 vwvv vwvv vwvv    Sv1 Sv2 Sv3 Sv0 v3 v2v1 w31 w23 w12 Fig. 2. The partitioning of S. 2.3 Differential-Geometrical Trajectory Classification While the previous discussion can be used for a set-theoretical characterization of trajectories, we focus now on a differential-geometrical characterization of trajectories referring to the angles of the tangential vectors and expressing the monotonicity characteristics and the curvature of a trajectory. We say that the trajectory X(T, j) is continuous if for the given j, x(t, j) is continuous in t on the time interval T (cf. (4a)). Moreover, a trajectory family is continuous if all the trajectories belonging to this family are continuous. Let (a) d(t, j) be the directional (or tangential) vector associated with the point x(t, j), (b) ℓ be a line through the point x(t, j) that is parallel to the simplex edge v1-v2, and (c) δ(t, j) := ∡(d(t, j), ℓ) ∈ [0°, 360°] be the angle between the directional vector d(t, j) and the line ℓ (cf. Fig. 3). 120 D.STIJEPIC Copyright ©2019 ASSA. Adv. in Systems Science and Appl. (2019) X(T, j) ℓ|| x(t, j) d(t, j) δ(t, j) v2v1 v3 Fig. 3. The vector angle δ(t, j). Moreover, we define the angles αi and the angle intervals Ii by relying on the ‘saw tooth’ function ϕ: ℝ → ℝ as follows (cf. Fig. 4): ϕ(z) = [(z – 1)/6 – floor((z – 1)/6)]360° (6a) ∀i ∈ ℕ Ii ≡ (αi, αi+1) := (ϕ(i), ϕ(i + 1)) (6b) ∀i ∈ ℕ ∀j ∈ {n ∈ ℕ: i < n} [Ii~j] := ⋃ j k = i [Ik] ∧ [Ii~j) := ⋃ j k = i [Ik] \ αj+1 ∧ (Ii~j]:= ⋃ j k = i [Ik] \ αi ∧ Ii~j := ⋃ j k = i [Ik] \ αi \ αj+1 (6c) By using our definition of the vector angle δ(t, j) and the vector angles and intervals (6), we can formulate the Properties 1-3 reflecting the relation between the tangential vector angles and the dynamics of x(t, j) in the case that ẋ(t, j) ≠ 0 (cf. Fig. 1, 3, and 4). Property 1: If ẋ(t, j) ≠ 0, then (a) δ(t, j) ∈ I3~5 ⟺ ẋ1(t, j) > 0, (b) δ(t, j) ∈ I0~2 ⟺ ẋ1(t, j) < 0, and (c) δ(t, j) ∈ {α3, α6} ⟺ ẋ1(t, j) = 0. Property 2: If ẋ(t, j) ≠ 0, then (a) δ(t, j) ∈ I5~7 ⟺ ẋ2(t, j) > 0, (b) δ(t, j) ∈ I2~4 ⟺ ẋ2(t, j) < 0, and (c) δ(t, j) ∈ {α2, α5} ⟺ ẋ2(t, j) = 0. Property 3: If ẋ(t, j) ≠ 0, then (a) δ(t, j) ∈ I1~3 ⟺ ẋ3(t, j) > 0 (b) δ(t, j) ∈ I4~6 ⟺ ẋ3(t, j) < 0, and (c) δ(t, j) ∈ {α1, α4} ⟺ ẋ3(t, j) = 0. We rely on the following definitions of monotonicity. First, xi(t, j) is monotonous (in t) if (∀t ∈ T ẋi(t, j) ≥ 0) or (∀t ∈ T ẋi(t, j) ≤ 0). Second, xi(t, j) is strictly monotonous (in t) if either (∀t ∈ T ẋi(t, j) > 0) or (∀t ∈ T ẋi(t, j) < 0) but not both. Third, the trajectory X(T, j) (associated with the function x(t, j)) is (strictly) monotonous in one dimension if (a) there exists an i ∈ {1, 2, 3} such that xi(t, j) is (strictly) monotonous and (b) for all k ∈ {1, 2, 3}\i xk(t, j) is not (strictly) monotonous. Fourth, the trajectory X(T, j) is (strictly) monotonous in two MODELS OF CONTINUOUS DYNAMICS ON THE 2-SIMPLEX AND APPLICATIONS IN ECONOMICS 121 Copyright ©2019 ASSA. Adv. in Systems Science and Appl. (2019) dimensions if (a) there exist an i ∈ {1, 2, 3} and a k ∈ {1, 2, 3}\i such that xi(t, j) and xk(t, j) are (strictly) monotonous and (b) xl(t, j) is not (strictly) monotonous with l ∈ {1, 2, 3}\{i, k}. Fifth, the trajectory X(T, j) is (strictly) monotonous (in three dimensions) if ∀i ∈ {1, 2, 3} xi(t, j) is (strictly) monotonous. ϕ(z)/100 z α1 = 0 α2 = 60 = α8 α3 = 120 α4 = 180 α5 = 240 α6 = 300 = α0 I6 = (300 , 360 ) = I0 I1 = (0 , 60 ) = I7 I2 = (60 , 120 ) I3 = (120 , 180 ) I4 = (180 , 240 ) I5 = (240 , 360 ) . S5(x0) S6(x0) = S0(x0) S4(x0) S3(x0) S2(x0) S1(x0) = S7(x0) v1 v2 v3 x0 ≡ (x01, x02, x03) fe d c b L1(x0) = L2(x0) = L3(x0) = L4(x0) = L5(x0) = L6(x0) = a Fig. 4. The function ϕ(z), the angle intervals Ii, the line segments Li(x0), and the sets Si(x0). 122 D.STIJEPIC Copyright ©2019 ASSA. Adv. in Systems Science and Appl. (2019) Instead of using the curvature definition that is widespread in differential geometry (and which is difficult to apply in the proofs of our theorems), we use the following definition of curvature relying on vector angles: Let β(q, r, j) denote the angle between the two tangential vectors d(q, j) and d(r, j) associated with the (monotonous) trajectory X(T, j) on S (cf. (4)), where q, r ∈ T. Among all the tangential vector pairs (d(t, j), d(s, j)) associated with the (monotonous) trajectory X(T, j), where t, s ∈ T, let d(t*, j) and d(s*, j) be among the ones that are characterized by the largest angle β, i.e., β(t*, s*, j) =: κ(T, j) is the maximal tangential vector angle β associated with the trajectory X(T, j). The greater κ(T, j) is, the greater the curvature of the trajectory X(T, j) is. Obviously, a linear trajectory has a curvature of 0. We say that a strictly monotonous trajectory X(T, j) (or a trajectory segment) that describes a clockwise (counterclockwise) movement on S has a positive (negative) signed curvature and write κ(T, j) > 0 (κ(T, j) < 0). Let F be a family of trajectories X(T, j), j ∈ F ⊆ J (cf. (4)). Then, κ*(T, F) := max(cl({κ(T, j): j ∈ F})) is the maximum curvature of the family F on the time interval T. 2.4 Topological Trajectory Classification Here, the topological characterization refers to the question whether a trajectory family is non-(self-)intersecting, which is a characteristic that can be expressed by homeomorphisms. Moreover, it is deciding for the applicability of the Poincaré-Bendixson theory (cf. Section 5) that the 2-simplex is homeomorphic to a bounded (and closed) subset of a plane. Two trajectories X(T, j) and X(U, k) are non-intersecting if X(T, j) ∩ X(U, k) = ∅, where U ⊆ ℝ. Otherwise they are intersecting. A trajectory X(T, j) is self-intersecting if ∃(r, s, t) ∈ T 3 r < s < t ∧ x(r, j) = x(t, j) ≠ x(s, j). Otherwise, the trajectory is non-self-intersecting. A trajectory X(T, j) is transversally self-intersecting if ∃(t, s) ∈ T 2 t ≠ s ∧ x(t, j) = x(s, j) ∧ δ(t, j) ≠ δ(s, j). Otherwise, the trajectory is not transversally self-intersecting. According to these definitions, a closed trajectory corresponding to a Jordan curve is self-intersecting but not transversally self-intersecting. 3. IMPLICATIONS OF MONOTONICITY In contrast to monotonous and bounded trajectories in ℝ2, monotonous trajectories on the 2-simplex can have (a) a wide range of different shapes and (b) omega limit sets consisting of more than only one (fixed) point. In this section, we discuss the geometrical aspects of the transitional and limit dynamics associated with continuous trajectories that are monotonous in one, two, or three dimensions. As we will see, these geometrical properties have interesting applications in economic dynamics modeling. 3.1 General Properties of Monotonous Trajectories on the 2-Simplex In this section, we show that continuous trajectories that are monotonous in three dimensions (two dimensions) are characterized by relatively low curvatures, allow for relatively weak waves, and are placed in relatively small subsets of the 2-simplex in comparison to the ‘related’ trajectories that are monotonous in two dimensions (one dimension). Propositions 1-3 and Corollary 1 summarize these results formally. The MODELS OF CONTINUOUS DYNAMICS ON THE 2-SIMPLEX AND APPLICATIONS IN ECONOMICS 123 Copyright ©2019 ASSA. Adv. in Systems Science and Appl. (2019) readers who are less interested in this formal discussion can also go directly to the discussion of Figure 4 (see the paragraphs below Proposition 3), which elaborates on the intuitive/graphical interpretation of these geometrical properties. Given a point x0 ≡ (x01, x02, x03) ∈ S, (7) defines different subsets of S. As we will see (in Proposition 3), each of the subsets S1-S6 defined by (7) corresponds to the closure of one of the vector angle intervals I1-I6 defined by (6), and each of the line segments L1-L6 defined by (7) corresponds to one of the angles α1-α6 defined by (6) (cf. Fig. 4). L1(x0) := {(x1, x2, x3) ∈ S: x1 ≤ x01 ∧ x3 = x03} (7a) S1(x0) := {(x1, x2, x3) ∈ S: x2 ≥ x02 ∧ x3 ≥ x03} (7b) L2(x0) := {(x1, x2, x3) ∈ S: x1 ≤ x01 ∧ x2 = x02} (7c) S2(x0) := {(x1, x2, x3) ∈ S: x1 ≤ x01 ∧ x2 ≤ x02} (7d) L3(x0) := {(x1, x2, x3) ∈ S: x1 = x01 ∧ x2 ≤ x02} (7e) S3(x0) := {(x1, x2, x3) ∈ S: x1 ≥ x01 ∧ x3 ≥ x03} (7f) L4(x0) := {(x1, x2, x3) ∈ S: x1 ≥ x01 ∧ x3 = x03} (7g) S4(x0) := {(x1, x2, x3) ∈ S: x2 ≤ x02 ∧ x3 ≤ x03} (7h) L5(x0) := {(x1, x2, x3) ∈ S: x1 ≥ x01 ∧ x2 = x02} (7i) S5(x0) := {(x1, x2, x3) ∈ S: x1 ≥ x01 ∧ x2 ≥ x02} (7j) L6(x0) := {(x1, x2, x3) ∈ S: x1 = x01 ∧ x2 ≥ x02} (7k) S6(x0) := {(x1, x2, x3) ∈ S: x1 ≤ x01 ∧ x3 ≤ x03} (7l) S0(x0) := S6(x0) S7(x0) := S1(x0) (7m) ∀i ∈ ℕ ∀j ∈ {n ∈ ℕ: i < n} Si~j(x0) := ⋃j k = i Sk(x0) (7n) In Propositions 1-3, we derive the characteristics of monotonous trajectories and, in particular, the ranges of tangential vector angles δ(t, j) and the sets in which the trajectory segments X(T +, j) are located. Later, we will discuss the geometrical/graphical interpretation of these characteristics and their applications in economic dynamics analysis. Proposition 1 focuses on the trajectories that are continuous and monotonous in one dimension on S and shows that there are only nine trajectory types belonging to this monotonicity class, where each type is identified by one of the Condition-Sets P11-P19. Proposition 1: Assume that (a) the trajectory X(T, j) defined by (4a) is continuous and monotonous in one dimension on S, (b) x(0, j) = x0 ≡ (x01, x02, x03) ∈ S, and (c) ∀t ∈ T ẋ(t, j) ≠ 0. Then, X(T, j) and X(T +, j) (cf. (4b)) satisfy one and only one of the Condition Sets P11-P19, which are defined as follows (cf. (6) and (7)): a) for i ∈{1, 2, …, 6}, Condition Set P1i is: (∀t ∈ T δ(t, j) ∈ [I(i–1)~(i+1)]) ∧ (∃t ∈ T δ(t, j) ∈ I(i–1)~(i+1)) ∧ (∃(r, s) ∈ T 2 δ(r, j) ∈ [Ii–1) ∧ δ(s, j) ∈ (Ii+1]) ∧ X(T +, j) ⊂ S(i–1)~(i+1)(x0); b) for i ∈ {7, 8, 9}, Condition Set P1i is: (∀t ∈ T δ(t, j) ∈ {αi–6, αi–3}) ∧ (∃(p, q) ∈ T 2 δ(p, j) ∈ {αi–6} ∧ δ(q, j) ∈ {αi–6}) ∧ X(T +, j) ⊆ Li–6(x0) ∪ Li–3(x0). Proof. As defined in Section 2, X(T, j) is monotonous in one dimension if for one and only one i ∈ {1, 2, 3}, the function xi(t, j) is monotonous while for all other i, xi(t, j) is non-monotonous. Thus, for proving Proposition 1, we have to consider only three alternative scenarios of monotonicity in one dimension: (A) x1(t, j) is monotonous, (B) x2(t, j) is 124 D.STIJEPIC Copyright ©2019 ASSA. Adv. in Systems Science and Appl. (2019) monotonous, and (C) x3(t, j) is monotonous. Moreover, since a monotonous function can be monotonously increasing or monotonously decreasing (or both), we have three alternative sub-scenarios for each of the three scenarios (A)-(C): (a) monotonously increasing, (b) monotonously decreasing, and (c) both, monotonously increasing and monotonously decreasing (which means constant). Thus, overall, we have nine sub-scenarios: (Aa)-(Ac), (Ba)-(Bc), and (Ca)-(Cc). According to Properties 1-3, each of the Condition Sets P11-P19, to which Proposition 1 refers, represents one of the nine sub-scenarios (Aa)-(Cc). For example, Condition Sets P12, P15, and P17 represent the sub-scenarios (Ca), (Cb), and (Cc), respectively. Consider first the sub-scenario (Ca), i.e., assume that x3(t, j) increases monotonously. Property 3 and (6b) imply that (8) is valid in sub-scenario (Ca). (∀t ∈ T δ(t, j) ∈ [I1~3]) ∧ (∃t ∈ T δ(t, j) ∈ I1~3) (8) Moreover, according to the definition of ‘monotonicity in one dimension’, to which Proposition 1 refers, (9) is valid in sub-scenario (Ca). x1(t, j) and x2(t, j) are non-monotonous. (9) The interval [I1~3], to which (8) refers, can be partitioned into three subintervals [I1), [I2], and (I3]. Properties 1 and 2 and (6b) imply (10). ∀t ∈ T δ(t, j) ∈ (I3] ⇒ x1(t, j) is monotonous. (10a) ∀t ∈ T δ(t, j) ∈ [I1) ⇒ x2(t, j) is monotonous. (10b) ∀t ∈ T δ(t, j) ∈ [I2] ⇒ x1(t, j) and x2(t, j) are monotonous. (10c) The statements (10) imply statement (11). x1(t, j) or x2(t, j) is monotonous if for all t ∈ T, δ(t, j) is within one and only one of the sub-intervals [I1), [I2], and (I3]. (11) (9) and (11) imply that over the period T, the tangential vectors δ(t, j) cannot stay within one and the same subinterval, i.e., at least one subinterval switch must occur over the period T. Given the three subintervals [I1), [I2], and (I3], the set of all possible subinterval switches is: (i) switch from [I1) to [I2], (ii) switch from [I1) to (I3], (iii) switch from [I2] to [I1), (iv) switch from [I2] to (I3], (v) switch from (I3] to [I1), and (vi) switch from (I3] to [I2]. We analyze now these interval switches. In case (i), i.e., if (1.) initially, the tangential vector angles are within the interval [I1) and (2.) at some later time point, the tangential vector angles switch to the interval [I2], x1(t, j) is monotonous (cf. Property 1). This contradicts (9). Analogously, it can be shown that cases (iii), (iv), and (vi) contradict (9), since: in case (iii), x1(t, j) is monotonous; in case (iv), x2(t, j) is monotonous; in case (vi), x2(t, j) is monotonous. Only, in cases (ii) and (v), both, x2(t, j) and x1(t, j), are non-monotonous, which is consistent with (9). In each of the cases (ii) and (v), (12) is true. ∃(r, s) ∈ T 2 δ(r, j) ∈ [I1) ∧ δ(s, j) ∈ (I3] (12) The fact that x3(t, j) increases monotonously in sub-scenario (Ca) implies that ∀t ≥ 0 x3(t, j) ≥ x3(0, j), where x3(0, j) = x03 according to the assumptions made in Proposition 1. In other words, in sub-scenario (Ca), X(T +, j) ⊂ {(x1, x2, x3) ∈ S: x3 ≥ x03} =: SCa(x0) (cf. Proposition 1). If X(T +, j) ⊂ SCa(x0) ⇒ X(T +, j) ⊂ S1~3(x0), then (13) is valid in sub-scenario (Ca). X(T +, j) ⊂ S1~3(x0) (13) MODELS OF CONTINUOUS DYNAMICS ON THE 2-SIMPLEX AND APPLICATIONS IN ECONOMICS 125 Copyright ©2019 ASSA. Adv. in Systems Science and Appl. (2019) We prove now that X(T +, j) ⊂ SCa(x0) ⇒ X(T +, j) ⊂ S1~3(x0). Given the point x0 ≡ (x01, x02, x03) ∈ S (cf. Proposition 1), the definition of SCa(x0) (and (1)) implies that (14)-(16) are true if x(t, j) ∈ SCa(x0). Either x3(t, j) > x03 or x3(t, j) = x03 but not both. (14) x2(t, j) < x02 or x2(t, j) > x02 (or x2(t, j) = x02 ). (15) x1(t, j) < x01 or x1(t, j) > x01 (or x1(t, j) = x01). (16) The statement (16) can be divided into the two (disjunctive) cases (17a) and (17b). Either x1(t, j) > x01 or x1(t, j) = x01 but not both. (17a) x1(t, j) < x01 (17b) If (14), (15), and (17a) are true and x(t, j) ∈ S, then x(t, j) ∈ {(x1, x2, x3) ∈ S: x1 ≥ x01 ∧ x3 ≥ x03} and, thus, x(t, j) ∈ S3(x0) (cf. (7f)). We consider now the cases in which (14), (15), and (17b) are true. These cases are: x1(t, j) < x01 ∧ x3(t, j) > x03 ∧ x2(t, j) > x02 (18a) x1(t, j) < x01 ∧ x3(t, j) > x03 ∧ x2(t, j) < x02 (18b) x1(t, j) < x01 ∧ x3(t, j) > x03 ∧ x2(t, j) = x02 (18c) x1(t, j) < x01 ∧ x3(t, j) = x03 ∧ x2(t, j) > x02 (18d) x1(t, j) < x01 ∧ x3(t, j) = x03 ∧ x2(t, j) < x02 (18e) x1(t, j) < x01 ∧ x3(t, j) = x03 ∧ x2(t, j) = x02 (18f) Obviously, the cases (18e) and (18f) violate (1). Thus, if (18e) or (18f) is true, then x(t, j) ∉ S. If (18b) or (18c) is true and x(t, j) ∈ S, then x(t, j) ∈ {(x1, x2, x3) ∈ S: x1 < x01 ∧ x2 ≤ x02 ∧ x3 > x03} =: Sbc(x0). If x(t, j) ∈ S2(x0), then x3(t, j) ≥ x03, since, otherwise, (1) is violated (cf. (7d)). In other words, S2(x0) = {(x1, x2, x3) ∈ S: x1 ≤ x01 ∧ x2 ≤ x02 ∧ x3 ≥ x03}. Obviously, Sbc(x0) ⊂ S2(x0). Thus, if (18b) or (18c) is true and x(t, j) ∈ S, then x(t, j) ∈ S2(x0). Analogously, if (18a), (18c), or (18d) is true and x(t, j) ∈ S, then x(t, j) ∈ {(x1, x2, x3) ∈ S: x1 < x01 ∧ x2 ≥ x02 ∧ x3 ≥ x03} =: Sacd(x0). Moreover, if x(t, j) ∈ S1(x0), then x1(t, j) ≤ x01, since, otherwise, (1) is violated (cf. (7b)). In other words, S1(x0) = {(x1, x2, x3) ∈ S: x1 ≤ x01 ∧ x2 ≥ x02 ∧ x3 ≥ x03}. Obviously, Sacd(x0) ⊂ S1(x0). Thus, if (18a), (18c), or (18d) is true and x(t, j) ∈ S, then x(t, j) ∈ S1(x0). Overall, we have shown that if x(t, j) ∈ SCa(x0) ⊆ S, then the statements (14)-(17) are valid, which imply several feasible cases. In each of these cases, x(t, j) is in one of the sets S1(x0), S2(x0), and S3(x0), i.e., x(t, j) ∈ SCa(x0) ⇒ x(t, j) ∈ S1(x0) ∪ S2(x0) ∪ S3(x0). This implies that X(T +, j) ⊂ SCa(x0) ⇒ X(T +, j) ⊂ S1(x0) ∪ S2(x0) ∪ S3(x0), since X(T +, j) is the union of the points x(t, j) ∈ S for which the statements (14)-(17) (and (1)) hold (cf. Proposition 1). According to (7n), S1~3(x0) = S1(x0) ∪ S2(x0) ∪ S3(x0). This completes the proof that X(T +, j) ⊂ SCa(x0) ⇒ X(T +, j) ⊂ S1~3(x0). By now, we have shown that in the sub-scenario (Ca), the statements (8), (12), and (13) must be true. These three statements reduce to Condition Set P12. It can be shown in the same way that (1.) the sub-scenarios (Cb) and (Cc) correspond to Condition Sets P15 and P17, respectively, and (2.) each of the sub-scenarios (Ba)-(Cc) corresponds to one and only one of the Condition Sets P11, P13, P14, P16, P18, and P19. 126 D.STIJEPIC Copyright ©2019 ASSA. Adv. in Systems Science and Appl. (2019) This completes the proof that each of the alternative (sub-)scenarios of monotonicity in one dimension (i.e., each of the sub-scenarios (Aa)-(Cc)) corresponds to one and only one of the Condition Sets P11-P19. □ Now, we turn to Proposition 2, which is similar to Proposition 1 except for the fact that it focuses on the trajectories that are continuous and monotonous in two dimensions on S. In particular, Proposition 2 derives the corresponding ranges of tangential vector angles δ(t, j) and the sets in which the trajectory segments X(T +, j) are located and shows that there are only six types of trajectories that are continuous and monotonous in two dimensions on S, where each type is identified by one of the Condition-Sets P21-P26. Proposition 2: Assume that (a) the trajectory X(T, j) defined by (4a) is continuous and monotonous in two dimensions on S, (b) x(0, j) = x0 ≡ (x01, x02, x03) ∈ S, and (c) ∀t ∈ T ẋ(t, j) ≠ 0. Then, X(T, j) and X(T +, j) (cf. (4b)) satisfy one and only one of the Condition Sets P21-P26, where for i ∈ {1, 2, …, 6}, Condition Set P2i is: (∀t ∈ T δ(t, j) ∈ [Ii~(i+1)]) ∧ (∃(r, s) ∈ T 2 δ(r, j) ∈ [Ii) ∧ δ(s, j) ∈ (I(i+1)]) ∧ X(T +, j) ⊂ Si~(i+1)(x0) (cf. (6)/(7)). Proof. According to the definition of monotonicity in two dimensions, two of the functions x1(t, j), x2(t, j), and x3(t, j) must be monotonous, while the remaining one must be non-monotonous. Thus, we have to consider only three cases: (A) x1(t, j) and x2(t, j) are monotonous (while x3(t, j) is non-monotonous), (B) x1(t, j) and x3(t, j) are monotonous (while x2(t, j) is non-monotonous), and (C) x2(t, j) and x3(t, j) are monotonous (while x1(t, j) is non-monotonous). For each of these cases, we must distinguish between four subcases. For example, in case (A), we can distinguish between the following subcases: (a) x1(t, j) and x2(t, j) are monotonously increasing, (b) x1(t, j) is monotonously increasing, while x2(t, j) is monotonously decreasing, (c) x1(t, j) and x2(t, j) are monotonously decreasing, and (d) x1(t, j) is monotonously decreasing, while x2(t, j) is monotonously increasing. Subcases (a) and (c) are infeasible, since they violate (1): for example, if x1(t, j) and x2(t, j) are monotonously increasing, then x3(t, j) must be monotonously decreasing (instead of being non-monotonous), since x1(t, j) + x2(t, j) + x3(t, j) must be equal to 1 for all t. Thus, we must consider only the subcases (b) and (d) of case (A). Properties 1 and 2 imply that in subcase (b) of case (A), the statement (19) is valid (cf. (6)). ∀t ∈ T δ(t, j) ∈ [I3~4] (19) Moreover, since case (A) requires that x3(t, j) is non-monotonous, Property 3 implies that (20) is valid in case (A). ∃(r, s) ∈ T 2 ẋ3(r, j) > 0 ∧ ẋ3(s, j) < 0 (20) According to (6), the interval [I3~4], to which (19) refers, can be partitioned into the following partitions: [I3), α4, and (I4]. Property 3 and (6) imply (21). ∀t ∈ T δ(t, j) ∈ [I3) ∨ δ(t, j) ∈ [I3) ∪ α4 ⇒ ∀t ∈ T ẋ3(t, j) ≥ 0 (21a) ∀t ∈ T δ(t, j) ∈ (I4] ∨ δ(t, j) ∈ α4 ∪ (I4] ⇒ ∀t ∈ T ẋ3(t, j) ≤ 0 (21b) ∀t ∈ T δ(t, j) ∈ α4 ⇒ ∀t ∈ T ẋ3(t, j) = 0 (21c) MODELS OF CONTINUOUS DYNAMICS ON THE 2-SIMPLEX AND APPLICATIONS IN ECONOMICS 127 Copyright ©2019 ASSA. Adv. in Systems Science and Appl. (2019) The statements (21) imply that if (19) and (20) are true, the tangential vectors δ(t, j) cannot stay within one and only one of the subintervals [I3), [I3) ∪ α4, α4, α4 ∪ (I4], and (I4] for all t ∈ T, and there must occur a switch from subinterval [I3) to subinterval (I4] or from subinterval (I4] to subinterval [I3) over the period T. Thus, (22) is valid. ∃(r, s) ∈ T 2 δ(r, j) ∈ [I3) ∧ δ(s, j) ∈ (I4] (22) Since in subcase (b) of case (A), x1(t, j) increases monotonously and x2(t, j) decreases monotonously, the assumptions made in Proposition 2 and (4b) imply that X(T +, j) ⊂ {(x1, x2, x3) ∈ S: x1 ≥ x01 ∧ x2 ≤ x02} =: SAb(x0). SAb(x0) can be partitioned as follows: SAb(x0) = SAb1(x0) ∪ SAb2(x0), where SAb1(x0) ∩ SAb2(x0) = ∅ and SAb1(x0) := {(x1, x2, x3) ∈ S: x1 ≥ x01 ∧ x2 ≤ x02 ∧ x3 ≥ x03} and SAb2(x0) := {(x1, x2, x3) ∈ S: x1 ≥ x01 ∧ x2 ≤ x02 ∧ x3 < x03}. We can see immediately that SAb1(x0) ⊂ S3(x0) (cf. (7f)) and SAb2(x0) ⊂ S4(x0) (cf. (7h)). Thus, SAb(x0) ⊂ S3(x0) ∪ S4(x0). This result, (7n), and the previously shown fact that X(T +, j) ⊂ SAb(x0) imply (23). X(T +, j) ⊂ S3~4(x0) (23) Overall, we have shown that in the subcase (b) of case (A), Condition Set P23 must be true (cf. (19), (22), and (23)). Analogously, it can be shown that in all the feasible subcases of cases (A)-(C), one and only one of the statements P21, P22, P24, P25, and P26 is true, which proves Proposition 2. □ Finally, we focus on the trajectories that are continuous and monotonous in three dimensions on S. In particular, Proposition 3 derives the corresponding ranges of tangential vector angles δ(t, j) and the sets in which the trajectory segments X(T +, j) are located and shows that there are twelve types of trajectories that are continuous and monotonous in three dimensions on S, where each type is identified by one of the Condition-Sets P31-P312. Proposition 3: Assume that (a) the trajectory X(T, j) defined by (4a) is continuous and monotonous (in three dimensions) on S, (b) x(0, j) = x0 ≡ (x01, x02, x03) ∈ S, and (c) ∀t ∈ T ẋ(t, j) ≠ 0. Then, X(T, j) and X(T +, j) (cf. (4b)) satisfy one and only one of the Condition Sets P31-P312, where (cf. (6) and (7)): a) for i ∈ {1, 2, …, 6}, Condition Set P3i is: (∀t ∈ T δ(t, j) ∈ [Ii]) ∧ (∃s ∈ T δ(s, j) ∈ Ii) ∧ X(T +, j) ⊂ Si(x0); b) for i ∈ {7, 8, …, 12}, Condition Set P3i is: ∀t ∈ T δ(t, j) ∈ {αi–6} ∧ X(T +, j) ⊆ Li–6(x0). Proof. According to our definition of monotonicity (in three dimensions), x1(t, j), x2(t, j), and x3(t, j) must be monotonous if X(T, j) is monotonous (in three dimensions) on S. Since a monotonous function can be (a) monotonously increasing, (b) monotonously decreasing, or (c) both (monotonously increasing and monotonously decreasing and, thus, constant), we have per function xi(t, j) three cases ((a)-(c)). Moreover, we have three functions x1(t, j), x2(t, j), and x3(t, j). Thus, overall, there are 33 possible combinations. This set of 27 combinations contains the combination (A) ∀i ẋi(t, j) ≤ 0, the combination (B) ∀i ẋi(t, j) ≥ 0, three times the combination (C) ẋi(t, j) ≥ 0 ∧ ẋk(t, j) ≥ 0 ∧ ẋl(t, j) = 0 ∧ i ≠ k ≠ l, three times the combination (D) ẋi(t, j) ≤ 0 ∧ ẋk(t, j) ≤ 0 ∧ ẋl(t, j) = 0 ∧ i ≠ k ≠ l, six times the combination (E) ẋi(t, j) = ẋk(t, 128 D.STIJEPIC Copyright ©2019 ASSA. Adv. in Systems Science and Appl. (2019) j) = 0 ∧ ẋl(t, j) ≠ 0 ∧ i ≠ k ≠ l, and the combination (F) ∀i ẋi(t, j) = 0. The combinations (A)-(E) are infeasible, since they violate (1) unless they reduce to combination (F). The combination (F) represents a fixed point (ẋ(t, j) = 0) and is excluded by the assumptions made in Proposition 3. In the rest of the proof, we have to consider the remaining 12 combinations.1 Each of these 12 combinations is covered by one of the Conditions Sets P31-P312. We leave it to the reader to prove the validity of Proposition 3 in all these 12 cases; we prove the validity in only two representative cases. Consider the case ∀t ∈ T ẋ1(t, j) ≤ 0 ∧ ẋ2(t, j) ≤ 0 ∧ ẋ3(t, j) ≥ 0, where ∃(r, s, p) ∈ T 3 ẋ1(r, j) < 0 ∧ ẋ2(s, j) < 0 ∧ ẋ3(p, j) > 0. Then, (a) Properties 1-3 imply almost directly that the tangential vector angles δ(t, j) satisfy the Condition Set P32, and (b) the assumptions made in Proposition 3 and (4b) imply that X(T +, j) ⊂ {(x1, x2, x3) ∈ S: x1 ≤ x01 ∧ x2 ≤ x02 ∧ x3 ≥ x03} =: Sz(x0), and, thus, (7d) implies that Sz(x0) ⊂ S2(x0); thus, X(T +, j) ⊂ S2(x0) as stated by the Condition Set P32. Alternatively, consider the case ∀t ∈ T ẋ1(t, j) ≤ 0 ∧ ẋ2(t, j) ≥ 0 ∧ ẋ3(t, j) = 0, where ∃r ∈ T ẋ1(r, j) < 0 ∧ ẋ2(r, j) > 0.2 Properties 1-3 imply almost directly that in this case, the tangential vector angles δ(t, j) satisfy the Condition Set P37. Moreover, the assumptions made in Proposition 3, (4b), and (7a) imply that X(T +, j) ⊆ {(x1, x2, x3) ∈ S: x1 ≤ x01 ∧ x2 ≥ x02 ∧ x3 = x03} ⊆ L1(x0). Thus, X(T +, j) ⊆ L1(x0) as stated by Condition Set P37. □ We discuss now the geometrical interpretation of Propositions 1, 2, and, 3, as depicted by Fig. 4. To construct Fig. 4, we choose an arbitrary point (x0) in the interior of S. Then, we draw three line segments going through x0 and each being parallel to one of the simplex edges v1-v2, v2-v3, and v3-v1. The intersections of the line segments with the simplex edges are denoted by the points a-f. We can see that the line segments that connect x0 with one of the points a-f are the line segments L1(x0)-L6(x0), which are defined by (7) and which localize the six (closed) subsets S1(x0)-S6(x0) defined by (7). The angles between the line segments L1(x0)-L6(x0) and the simplex edge v1-v2 (according to the definition of tangential vector angles and intervals (6)) are depicted in the middle panel of Fig. 4. (7n) and Fig. 4 imply almost directly that (1.) each of the six sets Si~(i+1)(x0), to which Proposition 2 refers, is simply the union of two neighboring sets Sj(x0) and Sk(x0), (2.) each of the six sets S(i–1)~(i+1)(x0), to which Proposition 1 refers, is simply the union of three neighboring sets Sj(x0), Sk(x0), and Sm(x0). In particular, Propositions 1-3 can be interpreted easily by using Fig. 4: 1.) Proposition 3 implies three geometrical properties of a trajectory segment X(T +, j) that is monotonous in three dimensions. First, X(T +, j) is located in one of the line segments L1(x0)-L6(x0) or in one of the sets S1(x0)-S6(x0). Second, if X(T +, j) is in Li(x0), then for all t ≥ 1 These feasible combinations are: (1.) ẋ1 ≤ 0 ∧ ẋ2 ≤ 0 ∧ ẋ3 ≥ 0, (2.) ẋ1 ≤ 0 ∧ ẋ2 ≥ 0 ∧ ẋ3 ≤ 0, (3.) ẋ1 ≤ 0 ∧ ẋ2 ≥ 0 ∧ ẋ3 ≥ 0, (4.) ẋ1 ≤ 0 ∧ ẋ2 ≥ 0 ∧ ẋ3 = 0, (5.) ẋ1 ≤ 0 ∧ ẋ2 = 0 ∧ ẋ3 ≥ 0, (6.) ẋ1 ≥ 0 ∧ ẋ2 ≤ 0 ∧ ẋ3 ≤ 0, (7.) ẋ1 ≥ 0 ∧ ẋ2 ≤ 0 ∧ ẋ3 ≥ 0, (8.) ẋ1 ≥ 0 ∧ ẋ2 ≤ 0 ∧ ẋ3 = 0, (9.) ẋ1 ≥ 0 ∧ ẋ2 ≥ 0 ∧ ẋ3 ≤ 0, (10.) ẋ1 ≥ 0 ∧ ẋ2 = 0 ∧ ẋ3 ≤ 0, (11.) ẋ1 = 0 ∧ ẋ2 ≤ 0 ∧ ẋ3 ≥ 0, and (12.) ẋ1 = 0 ∧ ẋ2 ≥ 0 ∧ ẋ3 ≤ 0, where ∃t ∈ T ẋi(t) < 0 if it is stated that ẋi ≤ 0, and, analogously, ∃t ∈ T ẋi(t) > 0 if it is stated that ẋi ≥ 0. 2 Note that the cases ẋ1(r, j) < 0 ∧ ẋ2(r, j) = ẋ3(r, j) = 0 and ẋ2(r, j) > 0 ∧ ẋ1(r, j) = ẋ3(r, j) = 0 are infeasible (see the discussion of combinations (A)-(E)). MODELS OF CONTINUOUS DYNAMICS ON THE 2-SIMPLEX AND APPLICATIONS IN ECONOMICS 129 Copyright ©2019 ASSA. Adv. in Systems Science and Appl. (2019) 0, the tangential vector angles δ(t, j) associated with X(T +, j) are equal to the angle that is associated to the line segment Li(x0) in Fig. 4 +/–180°. For example, if X(T +, j) is in L3(x0), then δ(t, j) ∈ {120°, 300°} for t ≥ 0. Third, if X(T +, j) is located in one of the sets Si(x0), then for t ≥ 0, the tangential vector angles δ(t, j) associated with X(T +, j) are within the angle range indicated by the angles associated to the line segments Li(x0) and Li+1(x0) that bound the set Si(x0) in Fig. 4. For example, if the trajectory segment X(T +, j) that is monotonous in three dimensions is in S3(x0), then δ(t, j) is within the angle range [120°, 180°] for t ≥ 0 (cf. Proposition 3 and Condition Set P33). 2.) The geometrical interpretation of Proposition 2 is analogous. In particular, the trajectory segment X(T +, j) that is monotonous in two dimensions is located in two neighboring sets Sj(x0) and Sk(x0), and for all t ≥ 0, the tangential vector angles δ(t, j) of X(T+, j) are within the angle range indicated by the angles associated to the two line segments Lj(x0) and Lk+1(x0) that bound the union of the sets Sj(x0) and Sk(x0) in Fig. 4. For example, if the trajectory segment X(T +, j) that is monotonous in two dimensions is in S3~4(x0) = S3(x0) ∪ S4(x0), then δ(t, j) is within the angle range [120°, 240°] for t ≥ 0 (cf. Proposition 2 and Condition Set P23). 3.) Analogously, Proposition 1 implies that the trajectory segment X(T +, j) that is monotonous in one dimension is located in three neighboring sets Sj(x0), Sk(x0), and Sm(x0). Moreover, for all t ≥ 0, the tangential vector angles δ(t, j) associated with this trajectory segment are within the angle range indicated by the angles associated to the two line segments Lj(x0) and Lm+1(x0) that bound the union of the sets Sj(x0), Sk(x0), and Sm(x0) in Fig. 4. For example, if the trajectory segment X(T +, j) that is monotonous in one dimension is in S3~5(x0) = S3(x0) ∪ S4(x0) ∪ S4(x0), then δ(t, j) is within the angle range [120°, 300°] for t ≥ 0 (cf. Proposition 1 and Condition Set P14). This graphical interpretation highlights important implications of Propositions 1-3: First, a trajectory that is monotonous (in three dimensions) is captured in a smaller subset of S than a related trajectory that is monotonous in two dimensions. Second, a trajectory that is monotonous in two dimensions is captured in a smaller subset of S than a related trajectory that is monotonous in one dimension. Moreover, the maximum curvature κ* of trajectories that are monotonous in three dimensions (two dimensions) is greater than the maximum curvature of related trajectories that are monotonous in two dimensions (one dimension). This intuitive discussion does not explicitly define the meaning of the term ‘related’. Thus, we define the meaning of this term and then formulate Corollary 1 (which is implied by Propositions 1-3) on the basis of this definition such that the discussion becomes more precise. Let F(x0) ⊆ J be a family of continuous trajectory segments X(T +, j) ⊂ S, j ∈ F(x0), satisfying ∀j ∈ F(x0) x(0, j) = x0 ∈ S and ∀t ∈ T + ∀j ∈ F(x0) ẋ(t, j) ≠ 0 (cf. (4)). Moreover, let P11(x0), P12(x0), …, P16(x0), P21(x0), P22(x0), …, P26(x0), P31(x0), P32(x0), …, and P36(x0) denote the subfamilies of F(x0) satisfying the Conditions Sets P11, P12, …, P16, P21, P22, …, P26, P31, P32, …, and P36, respectively. That is, j ∈ Pcd(x0) ⊂ F(x0) implies that X(T +, j) satisfies the Condition Set Pcd, where c ∈ {1, 2, 3} and d ∈ {1, 2, …, 6}. For (h, k) ∈ {1, 2, …, 6}2, we say that the families P1h(x0) and P2k(x0) are related if ∃i ∈ {1, 2, 3} ∀j ∈ P1h(x0) ∪ P2k(x0) (∀t ∈ T + ẋi(t, j) ≥ 0) ∨ (∀t ∈ T + ẋi(t, j) ≤ 0) ∧ (∃tj ∈ T + ẋi(tj, j) ≠ 0). That is, a family defined by Proposition 1 is related to a family defined by Proposition 2 if there exists an i for 130 D.STIJEPIC Copyright ©2019 ASSA. Adv. in Systems Science and Appl. (2019) which the monotonicity characteristics of xi(t, j) are identical in both families. For example, the families P12(x0) and P21(x0) are characterized by a monotonously increasing x3(t, j), i.e., ∀j ∈ P12(x0) ∪ P21(x0) (∀t ∈ T + ẋ3(t, j) ≥ 0) ∧ (∃tj ∈ T + ẋ3(tj, j) > 0); thus, P12(x0) and P21(x0) are related. We define the relations between the families defined by Propositions 2 and 3 analogously: For (p, q) ∈ {1, 2, …, 6}2, we say that the families P2p(x0) and P3q(x0) are related if ∃(v, w) ∈ {1, 2, 3}2 ∀j ∈ P2p(x0) ∪ P3q(x0) (∀t ∈ T + ẋv(t, j) ≥ 0) ∨ (∀t ∈ T + ẋv(t, j) ≤ 0) ∧ (∀t ∈ T + ẋw(t, j) ≥ 0) ∨ (∀t ∈ T + ẋw(t, j) ≤ 0) ∧ (∃tj ∈ T + ẋv(tj, j) ≠ 0) ∧ (∃sj ∈ T + ẋi(sj, j) ≠ 0) ∧ v ≠ w. That is, a family defined by Proposition 2 is related to a family defined by Proposition 3 if (a) there exists a v for which the monotonicity characteristics of xv(t, j) are identical in both families and (b) there exists a w ≠ v for which the monotonicity characteristics of xw(t, j) are identical in both families. For example, as implied by (6), Properties 1 and 3, and Propositions 2 and 3, the families P21(x0) and P31(x0) are characterized by (a) a monotonously decreasing x1(t, j), i.e., ∀j ∈ P21(x0) ∪ P31(x0) (∀t ∈ T + ẋ1(t, j) ≤ 0) ∧ (∃tj ∈ T + ẋ1(tj, j) < 0), and (b) a monotonously increasing x3(t, j), i.e., ∀j ∈ P21(x0) ∪ P31(x0) (∀t ∈ T + ẋ3(t, j) ≥ 0) ∧ (∃sj ∈ T + ẋ3(sj, j) > 0). Thus, P21(x0) and P31(x0) are related. Corollary 1: a) Consider the trajectory family P1h(x0), where h ∈ {1, 2, …, 6} and x0 ≡ (x01, x02, x03) ∈ int(S). There exist two trajectory families P2k(x0) and P2m(x0), (k, m) ∈ {1, 2, …, 6}2, k ≠ m, that are related to P1h(x0) and satisfy the following condition: ∀n ∈ {k, m} a*(T +, P1h(x0)) > a*(T +, P2n(x0)) ∧ κ*(T +, P1h(x0)) > κ*(T +, P2n(x0)) (cf. Sections 2.2. and 2.3). b) Consider the trajectory family P2p(x0), where p ∈ {1, 2, …, 6} and x0 ≡ (x01, x02, x03) ∈ int(S). There exist two trajectory families P3q(x0) and P3r(x0), (q, r) ∈ {1, 2, …, 6}2, q ≠ r, that are related to P2p(x0) and satisfy the following condition: ∀u ∈ {q, r} a*(T +, P2p(x0)) > a*(T +, P3u(x0)) ∧ κ*(T +, P2p(x0)) > κ*(T +, P3u(x0)). Proof. We only sketch here the proof. Starting with Corollary 1a, assume that h = 2, i.e., consider the family P12(x0). According to our definition of relatedness, P12(x0) is related to P21(x0) and P22(x0), since (6), Property 3, and Propositions 1 and 2 imply that P12(x0), P21(x0), and P22(x0) are characterized by a monotonously increasing x3(t, j). According to Proposition 1, (7b), (7d), (7f), (7n), and the definitions of a and M given in Section 2.2, the following statements are true: M(T +, P12(x0)) ⊆ S1~3(x0) = S1(x0) ∪ S2(x0) ∪ S3(x0) = {(x1, x2, x3) ∈ S: x3 ≥ x03} (24) M(T +, P21(x0)) ⊆ S1~2(x0) = S1(x0) ∪ S2(x0) ⊂ S1~3(x0) (25) M(T +, P22(x0)) ⊆ S2~3(x0) = S2(x0) ∪ S3(x0) ⊂ S1~3(x0) (26) As implied by (24), all the trajectories belonging to the family P12(x0) are located in S1~3(x0), where the latter is a triangle obtained by constructing a line on S going through x0 and being parallel to the simplex edge v1-v2 (cf. Property 3a, (6), and Fig. 3 and 5). According to Proposition 1 and (6), all the trajectories belonging to the family P12(x0) satisfy the following vector angle condition: ∀j ∈ P12(x0) (∀t ∈ T + δ(t, j) ∈ [0°, 180°]) ∧ (∃(r, s) ∈ T + × T + δ(r, j) ∈ [0°, 60°) ∧ δ(s, j) ∈ (120°, 180°]) (27) MODELS OF CONTINUOUS DYNAMICS ON THE 2-SIMPLEX AND APPLICATIONS IN ECONOMICS 131 Copyright ©2019 ASSA. Adv. in Systems Science and Appl. (2019) If we allow for non-smooth trajectories and, in particular, trajectories that are unions of line segments, it is easy to show geometrically by referring to Fig. 5 that such trajectories can be constructed to any point on S1~3(x0) while satisfying the condition (27).3 Thus, M(T +, P12(x0)) = S1~3(x0) and, thus, a*(T +, P12(x0)) = a(S1~3(x0)) (cf. Section 2.2). Moreover, (25) and (26) imply that a*(T +, P21(x0)) ≤ a(S1~2(x0)) < a(S1~3(x0)) and a*(T +, P22(x0))) ≤ a(S2~3(x0)) < a(S1~3(x0)). Thus, (28) is true. a*(T +, P12(x0)) > a*(T +, P21(x0)) ∧ a*(T +, P12(x0)) > a*(T +, P22(x0)) (28) If we require that the trajectories belonging to the family P12(x0) are smooth (i.e., ∀j ∈ P12(x0) ∀t ∈ T + x(t, j) is differentiable with respect to t), then it is not possible to construct a trajectory that obeys (27) and goes through the points/vertices (x01, 0, x03) ∈ S1~3(x0) and (0, x02, x03) ∈ S1~3(x0), which can be easily proven by referring to Fig. 4 and 5. That is, the smooth trajectories belonging to the family P12(x0) cannot cover two infinitesimally small areas of S1~3(x0). However, even in this case, it is still ensured that a*(T +, P12(x0)) > a*(T +, P21(x0)), since (a) S1~3(x0) = S1~2(x0) ∪ S3(x0) (cf. (24) and (25)), (b) S3(x0) (cf. (24)) is not infinitesimally small (in generic cases), and (c) a*(T +, P21(x0)) ≤ a(S1~2(x0)). The definition of κ* (cf. Section 2.3) and (27) imply that κ*(T +, P12(x0)) = 180°. Analogously, Proposition 2, the definition of κ*, and (6) imply that κ*(T +, P21(x0)) = 120°. Thus, κ*(T +, P12(x0)) > κ*(T +, P21(x0)). Overall, by now, we have (heuristically) proven Corollary 1a for h = 2. The proof is analogous for h ∈ {1, 3, 4, 5, 6}. The proof of Corollary 1b is very similar to the proof of Corollary 1a. Thus, we omit it here. □ Overall, among related trajectories and trajectory families the following is true: the higher the dimension of monotonicity, the smaller is (a) the family image (and, thus, the set of predicted states) and (b) the maximal curvature (and, thus, the potential strength of waves). If two families are unrelated, then a higher degree of monotonicity does not necessarily imply a smaller family image and a smaller maximum curvature. We can see that Corollary 1 does not categorize all the Condition Sets postulated by Propositions 1 and 3 and, in particular, not the Condition Sets P17-P9 and P37-P312. These Condition Sets imply that one of the xi(t, j) is constant for all t ∈ T + and, thus, the trajectory segments X(T +, j) are located on line segments. Obviously, the constancy requirement is much stronger than a monotonicity requirement; thus, in the cases represented by the Condition Sets P17-P9 and P37-P312, the size of the family image M(.) is relatively small and the curvature is zero. 3 Exactly speaking, (a) each of the line segments constituting such a trajectory is characterized by an angle to the v1-v2-edge of S in the range of [0°, 180°], (b) each trajectory contains a line segment that has an angle in the range of [0°, 60°), and (c) each trajectory contains a line segment that has an angle in the range (120°, 180°] (cf. (27)). 132 D.STIJEPIC Copyright ©2019 ASSA. Adv. in Systems Science and Appl. (2019) . S1~3(x0) v1 v2 v3 x0 ≡ (x01, x02, x03). .(x01, 0, x03) (0, x02, x03) Fig. 5. The set S1~3(x0). 3.2 Implications for Prediction of Transitional Dynamics If we regard t = 0 as now and t > 0 as the future (and, thus, T + as the predicted trajectory segment), Corollary 1 implies that the trajectories that are monotonous in one dimension (two dimensions) are harder to predict than the trajectories that are monotonous in two dimensions (three dimensions), ceteris paribus, since (a) the set of all possible future states is relatively great and (b) relatively stronger curvatures/waves may arise in the former case (in comparison to the latter case). However, besides the monotonicity characteristics, the location of the initial state x0 is decisive for the predictability of the future dynamics. In particular, the family image M and its size a* depend on x0 (cf. Corollary 1 and its proof). In general, monotonicity implies that the system moves from x0 along the trajectory segment T + towards a vertex or an edge of the 2-simplex. Thus, if the initial state x0 is relatively close to this vertex/edge, T + is captured in a relatively small set, i.e., the set of potential future states of the system is relatively small. This is almost a direct implication of the boundedness of the 2-simplex. As implied by Propositions 1-3, (6), and Section 2.3 (cf. Proof of Corollary 1), the maximum curvature κ* of trajectories that are monotonous in three dimensions, two dimensions, and one dimension is 60°, 120°, and 180°, respectively. Thus, the cyclical behaviors corresponding to a transversally self-intersecting trajectory or a closed trajectory (Jordan curve) are prohibited in all cases of monotonicity, since these types of cyclical behavior require a curvature greater than 180°. Yet, monotonous trajectories allow for a cyclical behavior corresponding to waves on the simplex (see Fig. 6). The angle range of 180° associated with monotonicity in one dimension allows for waves of high amplitude and short wavelength on the 2-simplex. In contrast, monotonicity in three dimensions allows only for relatively low-amplitude/long-wavelength waves (cf. Fig. 6). MODELS OF CONTINUOUS DYNAMICS ON THE 2-SIMPLEX AND APPLICATIONS IN ECONOMICS 133 Copyright ©2019 ASSA. Adv. in Systems Science and Appl. (2019) . S3(x0) S2(x0) S1(x0) v1 v2 v3 x0 d c b L1(x0) = || L2(x0) = || L3(x0) = || L4(x0) = || a X(T +, j) monotonous in one dimension (Condition Set P12) . S2(x0) S1(x0) v1 v2 v3 x0 c b a X(T +, j) monotonous in two dimensions (Condition Set P21) . S1(x0) v1 v2 v3 x0 b monotonous in three dimensions (Condition Set P31) a X(T +, j) Fig. 6. Examples of monotonous waves. Obviously, all three types of monotonicity allow for curved trajectories on the 2-simplex. However, only monotonicity in three dimensions allows for unidirectional linear trajectories, while monotonicity in one dimension allows for linear (non-smooth) trajectories yet requires at least one direction change. 134 D.STIJEPIC Copyright ©2019 ASSA. Adv. in Systems Science and Appl. (2019) 3.3 Limit Dynamics In all three cases of monotonicity (monotonicity in one, two, and three dimensions), the following two facts are true for continuous dynamics. First, the cyclical limit dynamics where the omega limit set is a Jordan curve are excluded, since such cycles require that x1(t), x2(t), and x3(t) are non-monotonous in the limit. Second, the system may converge to a fixed point or reach the fixed point in finite time (and stay there). The proof of these facts is obvious. In the case of monotonicity in three dimensions, only the fixed point outcome is possible, as implied by the monotone convergence theorem: since each of the functions x1(t), x2(t), and x3(t) is monotonous and restricted by an upper/lower limit of 0 and 1, each of the x1(t), x2(t), and x3(t) converges to its fixed point (x1 *, x2 *, and x3 *, respectively) or reaches it in finite time (and stays there). Thus, x(t) converges to a fixed point x* ≡ (x1 *, x2 *, x3 *) ∈ S or reaches it in finite time. Additionally, in the case of monotonicity in one dimension, the system may converge to a line segment if the trajectory is a wave. In this case, the wavelength decreases and the vector-angle range converges to the range of 180° as the system converges to the line segment (see the first part of Fig. 6). Note that in the case of monotonicity in two dimensions, the convergence to a line segment is not possible, as explained in the following. If the trajectory converges to a line segment, the tangential vector angle range must increase to a range of 180° which is prohibited by the definition of monotonicity in two dimensions, which allows only for a vector angle range of 120°. The former fact follows from the definition of the omega limit set, where for each of the points on the line segment (constituting the omega limit set), a sequence of points on the wave must be found that converges to it. 4. IMPLICATIONS OF NON-SELF-INTERSECTION FOR TRANSITIONAL DYNAMICS While non-intersecting trajectories and limit dynamics are treated in Section 5, we focus, now, on the implications of (transversal) non-self-intersection for transitional dynamics. The class of transversally non-self-intersecting continuous trajectories on the 2-simplex is a subclass of the class of continuous trajectories on the 2-simplex. Moreover, the class of non-self-intersecting continuous trajectories on the 2-simplex is a subclass of the class of transversally non-self-intersecting trajectories on the 2-simplex, since the former does not allow for Jordan-curves in contrast to the latter. Thus, by imposing the condition of (transversal) non-self-intersection, we can reduce the set of feasible trajectories on the 2-simplex, which can be exploited in prediction of dynamics, as explained in the following. Obviously, the non-self-intersection is an important constraint in systems of continuous trajectories on two-dimensional domains. In the case of discontinuous trajectories, non-self-intersection still may reduce the class of feasible trajectories significantly depending on the type of discontinuity and the physical/social system being analyzed. However, in extreme cases and, in particular, in the case of point sequences on the MODELS OF CONTINUOUS DYNAMICS ON THE 2-SIMPLEX AND APPLICATIONS IN ECONOMICS 135 Copyright ©2019 ASSA. Adv. in Systems Science and Appl. (2019) 2-simplex (e.g., discrete-time paths), non-self-intersection becomes obsolete as a restraint (in natural and social sciences where the exact position of a system on the simplex is not measurable). For the same reasons, non-self-intersection is an obsolete restraint in three- or higher-dimensional dynamical system domains (see, e.g., [13]). 4.1 Qualitative Simulation The (transversal) non-self-intersection constraint on continuous trajectories on the 2-simplex can be understood as a dynamic constraint: at each point of time t ∈ T +, we have a restriction on system dynamics x(t, j) preventing certain type of dynamics (namely the dynamics that correspond to a self-intersection of the trajectory). The constraint is dynamic in the sense that it changes over time. In particular, it depends on the current position of the system on the 2-simplex and the form of the trajectory segment X(T –, j) representing the dynamics over the past time period T – (e.g., the longer the latter segment, the stronger is the constraint on the current dynamics), i.e., the constraint is updated continuously. This fact can be used in qualitative simulation, as discussed in detail by [7-8]. 4.2 Non-Self-Intersection in Combination with a Determined Trajectory-Segment The concept of (transversal) non-self-intersection can be very useful even if we do not assume the dynamic constraint view discussed in Section 4.1. In particular, assume that the trajectory segment (X(T –, j)) representing past dynamics is given by empirical data on past dynamics or by empirical laws. Then, in general, X(T –, j) can be used as a basis for a partitioning of the 2-simplex. For example, since the lines that are parallel to the 2-simplex edges have a clear intuitive interpretation, X(T –, j) and such lines can constitute an intuitively meaningful partitioning of the simplex (see [13]). Then, paths on the 2-simplex can be understood as sequences of partition switches, and the non-self-intersection constraint as an exclusion of certain switches, as demonstrated in Fig. 7, where (immediate) switches between the partitions A and C are prohibited by the non-self-intersection constraint. If we interpret t = 0 as present, T – as past, and T + as future, this prohibition corresponds to infeasible future scenarios, i.e., non-self-intersection can be used in prediction of future dynamics (cf. [13]). Obviously, depending on the (natural/social sciences) topic analyzed by these concepts, a certain length and positioning of X(T –, j) may be necessary to derive significant predictions. In particular, if X(T –, j) is relatively short or located in a relatively small or peripheral subset of the 2-simplex, it may not be possible to establish a relevant partitioning inducing a prohibition of paths that is of significant relevance for the topic/theory being analyzed (cf. [13]). These requirements are well known in statistics, where the length of the past time series and avoidance of outliers is important for the (statistical) significance of the predictions based on empirical (time-series) data. 136 D.STIJEPIC Copyright ©2019 ASSA. Adv. in Systems Science and Appl. (2019) X(T –, j) x(0, j) X(T +, j) v2v1 v3 D C A B Fig. 7. Paths as partition switches. 5. POINCARÉ-BENDIXSON THEORY The Poincaré-Bendixson theory, which is one of the fundaments of the dynamical systems theory, can be used to predict the qualitative properties of the limit dynamics of a smooth dynamical system in the plane. It applies to continuous systems, yet requires additional restrictions on the system (to ensure a sufficient degree of smoothness). We discuss here these requirements from a rather topological point of view applying the concepts discussed in Section 2.4. For a general, discussion of the requirements and predictions/statements of the Poincaré-Bendixson theory, see, e.g., [1, pp.362f.], [2], [3, p.45], [4, p.55], and [21, Chapter 7.3]. Assume that the dynamics on the 2-simplex are representable by a (relatively) smooth autonomous differential equation system in terms of the coordinates (y1, y2) of a two-dimensional coordinate system that is parallel to the 2-simplex (see Fig. 8). Then, the Poincaré-Bendixson theory states that the limit dynamics of this system are either cyclical or transitory. In particular, the omega limit set of a trajectory generated by such a system consists of a fixed point, a Jordan curve, or a homo-/heteroclinic union (of curves and fixed points). The geometrical interpretation of the requirement of the representability by a smooth differential equation system in y1-y2-coordinates is that the trajectories of the dynamical system on 2-simplex constitute a simple covering (of a connected subset) of the 2-simplex. In particular, such a simple covering consists of non-intersecting and transversally non-self-intersecting trajectories, where the union of these trajectories is a connected subset of the 2-simplex (see [16,20] for a detailed discussion and literature references). 6. APPLICATIONS IN ECONOMICS The mathematical theories of continuous dynamics on the standard 2-simplex developed in the previous sections have almost direct applications in the analysis of economic dynamics. In particular, they can be used in prediction of economic structural change, discussion of MODELS OF CONTINUOUS DYNAMICS ON THE 2-SIMPLEX AND APPLICATIONS IN ECONOMICS 137 Copyright ©2019 ASSA. Adv. in Systems Science and Appl. (2019) sectoral production functions, assessment of structural change costs, and design of cost-minimal structural change policies, as discussed in the following sections. v3 v2v1 y2 y1 ẏ(t) = φ(y(t)) y(t) ≡ (y1(t), y2(t)) φ: U → U' U, U' ⊂ ℝ2 Fig. 8. Representation of a dynamical system on S by a two-dimensional differential equation system. 6.1 Economic Topics Covered by the Models of Continuous Dynamics on the 2-Simplex A major pillar of economics is the study of long-run economic dynamics, where short-run fluctuations are neglected and the dynamic patterns that persist over long periods of time (e.g., 100 years) are studied. In this context, the concept of structural change is essential, where not only aggregate economic indices (e.g., gross domestic product, trade volume, and economy-wide employment) are studied but also their structure. In particular, the aggregate indices are subdivided into components and the significance of these components for the aggregate index is indicated by the components’ shares in the aggregate index. Many of these ‘shares’, such as savings rate, investment rate, and sectoral employment shares, are well known even in public debates. In general, structural change refers to the dynamics of these ‘shares’, where the shares satisfy the conditions stated by (1). In other words, economic structural change, i.e., the long-run dynamics of the ‘shares’ can be depicted by trajectories on standard simplexes. Moreover, the assumption of continuous-time frameworks and continuous functional forms is a general convention in long-run economic dynamics modeling (although there are exceptions from this convention), which, in general, yields continuous dynamics of the shares on standard simplexes. For an overview of the topics that are covered by the system-theoretical models of continuous trajectories on standard simplexes and for corresponding references from the economics literature, see [15]. To provide some details and references on the economic applications of the system-theoretical models derived in the previous sections, we focus on a specific sort of economic structural change, namely long-run labor allocation dynamics in the three-sector framework, in Section 6.2. 138 D.STIJEPIC Copyright ©2019 ASSA. Adv. in Systems Science and Appl. (2019) 6.2 Example: Long-run Labor Allocation Dynamics The three-sector framework is one of the major concepts for studying economic structural change (for an overview of the literature, see, e.g., [5-6,9-12]. It is based on the assumption that economic activities can be divided into three categories or ‘sectors’: agriculture, manufacturing, and services. One of the major indices for studying the importance of these sectors are the shares of each of the sectors in aggregate employment (abbr. ‘employment shares’). These shares and their dynamics can be depicted by continuous trajectories on the 2-simplex (cf. [13,14]). Thus, we can use the concepts discussed in Sections 2-5 in the analysis of labor allocation dynamics as discussed in the following. While it is difficult to derive any consensus statements on the quantitative characteristics of labor allocation dynamics, the literature implies that there seem to be some empirically observable qualitative laws of labor allocation dynamics, which can be expressed by using the geometrical concepts discussed in our paper. In particular, [19] shows that the typical long-run labor allocation dynamics of a nowadays highly developed country over the last two centuries can be described by a trajectory that has the following characteristics: (1.) it is monotonous in two dimensions (cf. Sections 2.3 and 3) and (thus) non-self-intersecting (cf. Sections 2.4 and 4); (2.) it has a negative signed curvature κ (cf. Section 2.3); (3.) its initial segment is located in the simplex partition Sv1 (cf. (5) and Fig. 2); and (4.) its final segment is located in partition Sv3 (cf. (5) and Fig. 2). As discussed by [14], these empirical observations can be interpreted as “natural” laws of structural change (since, among others, they are supported by the theoretical literature consensus) and, thus, can be exploited for predictions of structural change. For example, Proposition 2 and Corollary 1 (cf. [14]) and the approach discussed in Section 4.2 (cf. [13]) can be used to predict the future (transitional) dynamics of labor allocation in developing and developed economies. Moreover, [16,20] uses the empirical findings of [19] and the topological approach discussed in Section 5 for a discussion of the applicability of the Poincaré-Bendixson theory in the prediction of limit dynamics of labor allocation. Beside these applications, which focus on prediction of structural change, the models of continuous dynamics discussed in our paper have further applications in in structural change modeling: [17] shows that labor allocation trajectories that are monotonous in three dimensions minimize the structural change costs (e.g., unemployment, geographical relocation costs, and environmental pollution) and uses this result to elaborate a development policy minimizing the labor reallocation costs in a developing economy. [18] uses the model of monotonous and continuous trajectories and the concept of curvature (cf. Section 2.3) to discuss a widespread assumption in theoretical structural change modeling (namely the assumption of Cobb-Douglas production functions) by applying an axiomatic-geometrical approach. Finally, [15] discusses how the topological concepts discussed in Section 2.4 can be interpreted and applied in the context of labor allocation dynamics modeling. MODELS OF CONTINUOUS DYNAMICS ON THE 2-SIMPLEX AND APPLICATIONS IN ECONOMICS 139 Copyright ©2019 ASSA. Adv. in Systems Science and Appl. (2019) 7. CONCLUSIONS In this paper, we have studied models of continuous dynamics on the 2-simplex that arise when set-theoretical, differential-geometrical, or topological restrictions are imposed on the trajectories of the model. We focused on the qualitative properties of transitional and limit dynamics of these models and discussed their applications in long-run economic dynamics modeling. Many of our results (in general, the results of Section 3 and their applications in economics) can be extended to discrete or discontinuous systems or higher-dimensional simplexes. Yet, the rather topological concepts discussed in Sections 4 and 5 (e.g., the Poincaré-Bendixson theory) are, in general, not applicable or not useful in discrete or higher-dimensional systems and their applications (cf. [13,15-16,20]). In the latter systems, the concept of chaos as well as existence theorems on fixed points are of interest. Thus, further research could focus on them and, in particular, their system-theoretical significance for long-run economic dynamics. Empirical evidence implies that there are fluctuations of the labor allocation shares that correspond to the waves on the 2-simplex discussed in Sections 3.2 and 3.3. 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