Dynamics of the Shapovalov mid-size firm model

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY-NC-ND 4.0 https://creativecommons.org/licenses/by-nc-nd/4.0/ Dynamics of the Shapovalov mid-size firm model © 2020 Elsevier Ltd. All rights reserved. Accepted version (Final draft) Alexeeva, Tatyana A.; Barnett, William A.; Kuznetsov, Nikolay V.; Mokaev, Timur N. Alexeeva, T. A., Barnett, W. A., Kuznetsov, N. V., & Mokaev, T. N. (2020). Dynamics of the Shapovalov mid-size firm model. Chaos, Solitons and Fractals, 140, Article 110239. https://doi.org/10.1016/j.chaos.2020.110239 2020 Dynamics of the Shapovalov mid-size firm model Tatyana A. Alexeeva a, William A. Barnettb,c, Nikolay V. Kuznetsov d,e,f,, Timur N. Mokaev d aSt. Petersburg School of Mathematics, Physics and Computer Science, National Research University Higher School of Economics, 194100 St. Petersburg, Kantemirovskaya ul., 3, Russia bDepartment of Economics, University of Kansas, Lawrence, KS 66045, USA cCenter for Financial Stability, New York, NY 10036, USA dFaculty of Mathematics and Mechanics, St. Petersburg State University, 198504 Peterhof, St. Petersburg, Russia eDepartment of Mathematical Information Technology, University of Jyv¨askyl¨a, 40014 Jyv¨askyl¨a, Finland fInstitute for Problems in Mechanical Engineering RAS, 199178 St. Petersburg, V.O., Bolshoj pr., 61, Russia Abstract One of the main tasks in the study of financial and economic processes is forecasting and analysis of the dynamics of these processes. Within this task lie important research questions including how to determine the qualitative properties of the dynamics (stable, unstable, deterministic chaotic, and stochastic process) and how best to estimate quantitative indicators: dimension, entropy, and correlation characteristics. These questions can be studied both empirically and theoretically. In the empirical approach, one considers the real data represented by time series, identifies patterns of their dynamics, and then forecasts short- and long-term behavior of the process. The second approach is based on postulating the laws of dynamics for the process, deriving mathematical dynamic models based on these laws, and conducting subsequent analytical investigation of the dynamics generated by the models. To implement these approaches, both numerical and analytical methods can be used. It should be noted that while numerical methods make it possible to study complex models, the possibility of obtaining reliable results using them is significantly limited due to calculations being performed only over finite-time intervals, numerical integration errors, and the unbounded space of possible initial data sets. In turn, analytical methods allow researchers to overcome these problems and to obtain exact qualitative and quantitative characteristics of the process dynamics. However, their effective applications are often limited to low-dimensional models (in the modern scientific literature on this subject, two-dimensional dynamic systems are the most often studied). In this paper, we develop analytical methods for the study of deterministic dynamic systems based on the Lyapunov stability theory and on chaos theory. These methods make it possible not only to obtain analytical stability criteria and to estimate limiting behavior (localization of self-excited and hidden attractors, study of multistability), but also to overcome the difficulties related to implementing reliable numerical analysis of quantitative indicators (such as Lyapunov exponents and Lyapunov dimension). We demonstrate the effectiveness of the proposed methods using the “mid-size firm” model suggested recently by V.I. Shapovalov as an example. Keywords: mid-size firm model, forecasting, global stability, chaos, absorbing set, Lyapunov exponents, multistability Email address: Corresponding author: [email protected] (Nikolay V. Kuznetsov ) This work was done under the auspices of the Institute for Nonlinear Dynamical Inference at the International Center for Emerging Markets Research (http://icemr.ru/institute-for-nonlinear-dynamical-inference/). 1. Introduction Understanding and predicting the behavior of complex systems is one important task of current research in various fields. The events of the last decade have demonstrated the dangers of unpredictable developments in economic and financial systems, which can lead to systemic failures and even to collapses in the global financial-economic system. As a result, researchers have posed a number of conceptual questions, including search for approaches to forecasting critical transitions and determining stability indicators in complex systems [8, 52]. Usually these characteristics are initially determined analytically, with subsequent experimental verification using real systems. At the same time, investigating and forecasting the dynamics of current financial and economic systems allows one to effectively control the systems, discovering and tracking stable, unstable, deterministic chaotic and stochastic processes. Studying current systems also allows quantitative estimation of the processes, using characteristics of their dimension, entropy and correlation. To answer these conceptual questions, it is necessary to develop and apply reliable forecasting procedures, which will significantly reduce the costs of unpredictable behavior of financial and economic systems, including in times of crises, and allow researchers to offer recommendations for stabilizing the dynamics of these systems. In the second half of the last century, after the discovery of chaotic processes in dynamic systems by Ueda and Lorenz [48, 59], chaos theory began to be actively developed, and helped to explain the complexity and unpredictability of dynamic systems behavior. The complexity of dynamic systems, which are associated with the non-linearity and limited predictability of their behavior, as well as a number of open problems, have spurred significant interest in chaos theory among economists. Research aimed to study and reveal non-trivial economic effects and to stabilize irregular processes (see, e.g. [9, 10, 12, 13, 15, 16, 17, 18, 23, 24, 25, 57]). Developments in this field from the 1970s to the present can be traced through the studies of many famous economists [5, 6, 7, 10, 14, 16, 24, 27, 49] who have explored numerous examples of deterministic economic models that could generate nonperiodic fluctuations. For example, chaotic dynamics has been studied from the angles of economic growth and development, market structure and game theory [3, 12, 17, 23], rational expectations models [9, 14], open economy New Keynesian models [6, 7] based on the Gali and Monachelli model [26], and non-linear heterogeneous agent models (HAMs) with periodic and chaotic asset price fluctuations [14, 31]. Researchers have also focused on the issue of monetary chaos by applying tools based on both the metric (correlation dimension and Lyapunov exponents) and topological (recurrence plots) approaches to chaos [3]. There have been attempts to identify business cycles by looking at financial time series [15], to analyze the stock returns of the markets of G7 countries [58] and to forecast high-frequency trading transactions in foreign exchange markets [30]. More recently, research has developed in two main directions. An empirical direction is interested in whether some actual economic time series are characterized by chaotic dynamics and in developing statistical tests for chaos, applying them to macroeconomic and financial time series. A more theoretical approach has focused on demonstrating the possibility of cyclical and chaotic dynamic behaviors that can occur in a wide range of theoretical models. These studies attempt to understand whether mathematical non-linear deterministic models can demonstrate the types of fluctuations commonly found in economic data. On the one hand, the theoretical direction has included analyses of low-dimensional dynamic models reconstructed from time series of real data [49]. On the other hand, the theoretical approach is based on the construction of mathematical models that incorporated a-priori assumptions about the dynamics of the process under study. The main goal of both approaches, empirical and theoretical, has been to build reliable forecasts of the behavior of dynamic systems based on concepts and methods of chaos theory. The central complexity of the empirical approach lies in the difficulty of distinguishing between chaotic behavior of a deterministic dynamic system and random fluctuations caused by 2 measurement or sampling errors [18, 57]. The theoretical approach suffers from the fact that only low-dimensional systems allow derivations of analytical results (usually, two-dimensional systems are considered [3, 15, 49]). The history of the development of chaos theory and its applications shows that the complexity of describing the dynamics of economic phenomena is largely associated with the difficulty of constructing the adequate mathematical models of the phenomena. Moreover, attempts to fully describe such models and to observe real data in order to reconstruct models from it typically lead to high-dimensional dynamic models, including stochastic ones. As a rule, only quantitative analysis using numerical procedures is possible for these models. It is well known that in chaos theory, confirming the reliability of results obtained by numerical methods requires separate clarification, including aspects connected with the use of shadowing theory and analysis of computational errors (caused by a finite precision arithmetic and numerical integration of differential equations) [50]. Computational procedures have a number of significant limitations. First, calculations are performed over finite-time intervals, which makes it difficult to distinguish between a process of transition and established (limiting) chaotic behavior. Second, the use of numerical integration algorithms is associated with computational instability, which inevitably leads to approximation errors. Third, the theoretically unbounded space of possible initial conditions does not allow efficient forecasting of the limiting behavior of a dynamic system in the phase space. At the same time, for low-dimensional systems, it is possible to apply rigorous nonlinear methods from the dynamic systems theory, which allows one to obtain exact analytical results. This enables a derivation of the effective criteria for stability and absence of chaotic behavior in such systems. Thus, both qualitative exploration and estimation of the quantitative characteristics of the dynamics of the process can be performed numerically, as well as analytically. In particular, it is possible to calculate the Lyapunov dimension of an attractor using numerical procedures and to analytically localize the attractor via determination of the absorbing set. In this paper, we use an analytical approach to analyze local and global stability of the dynamics of a mid-size firm model [53, 54]. We perform analytical localization of the attractor of the system and investigate the global stability of its dynamics. This allows us to obtain parameter domains in which the system demonstrates various types of behavior: stable, unstable, or deterministic chaotic dynamics. In addition, we solve the problem of forecasting established (limiting) behavior of this dynamic system, obtaining the condition of the global conversion to the stationary set and bouded localization of non-trivial attractor. Thus, we overcome the challenge of unboundedness of the set of initial data and implement reliable numerical analysis of the model, including the study of its chaotic dynamics. We use the adaptive algorithm of the finite-time Lyapunov dimension and Lyapunov exponents computation for the values of the model’s parameters at which a chaotic attractor can be confirmed. Thus, we obtain a number of quantitative estimates, which allow us to calculate characteristics including the Lyapunov dimension and entropy. 2. Problem statement Consider the model of V.I. Shapovalov proposed in [53] which describes behavior of a mid-size firm      ˙x=−σx +δy, ˙y=µx +µy −βxz, ˙z=−γz +αxy. (1) Here α,β,σ,δ,µ,γare positive parameters, and the variables x,y,zdenote the growth of three main factors of production: the loan amount x, fixed capital yand the number of employees 3 z(as an increase in human capital). An increase in the loan amount is proportional to the amount of capital and the size of the loan taken out. The coefficient with the variable yis positive on the premise that, with an increase in capital, the company is more likely to grant loans on the lending market; the coefficient for the variable xis negative and indicates the losses that the company incurs when taking out a new loan, which is associated with the requirement to pay interest, as well as the fact that the company is less willing to give credit when it has many loan obligations. The capital gain is proportional to the income from the investment of available capital and the loan taken, as well as expenses for labor remuneration and loan repayment. The coefficients for the sum of the variables xand yare positive, since they show a positive effect of investing in the development of production; the coefficient for the product of the variables xand zis negative, since it indicates the costs of the company. The increase in the number of employees is proportional to the capital, the loan taken and the current number of employees. The coefficient for the product of the variables xand yis positive, based on the assumption that the company may spend part of the amount of capital and the loan taken on attracting additional employees. A negative coefficient for the variable zindicates that the outflow from the current number of employees due to dismissal or on their own initiative should be taken into account. Coefficients at variables are control parameters: αreflects a combination of factors that contribute to creating a company image that will be attractive to new employees; βsummarizes factors that influence cost allocation; µdescribes the effectiveness of capital investments (the effects of various taxes should be taken into account); γsummarizes factors related to difficulties obtaining a loan; for example, a high interest rate, etc. As part of the study of system (1), [29, 53, 54] formulated the task of nonlinear analysis of the system and its limit dynamics in order to predict the stability of the Shapovalov model (1) and determine the conditions under which the system has some predictable dynamics (the Shapovalov problem of a mid-size firm dynamics forecasting). The non-triviality of this problem lies in the fact that the system has an unstable state of equilibrium and may exhibit of chaotic dynamics. 3. System transformation A significant number of papers has studied the behavior of three dimensional nonlinear dynamic systems. It is important to verify (see, e.g. [42]) which known systems can be reduced to system (1) using linear coordinate transformation. System (1) can be reduced to a Lorenz-like system        ˙x=−cx +cy, ˙y=rx +y−xz, where c=σ µ, r =δ σ, b =γ µ, ˙z=−bz +xy, (2) using the following coordinate transformation (x, y, z)→µ √αβ x, µσ δ√αβ y, µσ δβ z, t →t µ.(3) System (2) differs from the classical Lorenz system [48] in the sign of the coefficient at yin the second equation, which is 1 here, while in the Lorenz system this coefficient is -1. Accordingly, the inverse transformation (x, y, z)→√αβ µx, r√αβ µy, rβ µz, t →µt (4) 4 reduces system (2) to system (1) with coefficients σ=cµ, δ =rcµ, γ =bµ 2. In addition, system (1) with parameters satisfying the relations σ2/(σ−δ) = µand δ < σ < µ can be reduced to the well-known Chen system [20]        ˙x=−ax +ay, ˙y= (c−a)x+cy −xz, with a=σ, c =σ2 σ−δ=µ, b =γ, a < c, ˙z=−bz +xy, (5) using coordinate substitutions (x, y, z)→1 √αβ x, σ δ√αβ y, σ δβ z.(6) The possibility of reducing system (1) to the Chen system (5) under the above conditions shows the complexity of studying the mid-size firm model. The Chen system demonstrates much more complex behavior in terms of constructing its absorbing set [2] than the Lorenz system, and the problem of analytical calculation of the dimension of its attractor [42] is still opened. 4. Sustainability analysis Further, we analyze the system (2) and apply the inverse transformation (4) to obtain conditions on the parameters of system (1). To solve the Shapovalov problem, using the standard stability analysis of dynamic systems, we calculate the equilibria of system (1). System (1) always has three equilibria O(1) 1= (0,0,0), O(1) 2,3= ±sγµ(σ+δ) αβσ ,±sγµσ(σ+δ) αβδ2,µ(σ+δ) βδ !.(7) Accordingly, system (2) also has three equilibria O(2) 1= (0,0,0), O(2) 2,3=±pb(r+ 1),±pb(r+ 1), r + 1.(8) For the Jacobian matrix of system (1) J= −σ δ 0 µ−βz µ −βx αy αx −γ (9) the characteristic polynomial det(J−Is)has form χ(s, x, y, z) = s3+p(1) 1(x, y, z)s2+p(1) 2(x, y, z)s+p(1) 3(x, y, z),(10) where p(1) 1(x, y, z) = σ+γ−µ, p(1) 2(x, y, z) = σ(γ−µ)−µ(γ+δ) + α2x2+αδz, p(1) 3(x, y, z) = −γµ(σ+δ) + α2σx2+α2δxy +αδγz. (11) 2Transformations (3) and (4) do not change the direction of time, which is essential for the analysis of the Lyapunov exponents and dimension [42]. 5 Lemma 1. The equilibrium state O(1) 1= (0,0,0) of system (1) is unstable for all parameter values. Proof. Consider system (2) obtained from (1) by changing variables (3), with the Jacobian matrix J= −c c 0 r−z1−x y x −b .(12) At the point O(2) 1= (0,0,0), the coefficients of the characteristic polynomial of the Jacobian matrix (12) have the following form p(2) 1(0,0,0) = c+b−1, p(2) 2(0,0,0) = c(b−r)−(c+b), p(2) 3(0,0,0) = −cb(1 + r). (13) Since inequality p(2) 3(0,0,0) <0holds for any admissible values of the parameters of system (2), the Routh-Hurwitz conditions are not satisfied and the equilibrium O(2) 1is always unstable. Using (4) we obtain the statement of Lemma for system (1). Lemma 2. If one of the relations     r > c(3 −(c+b)) b−(c+ 1) , b > c + 1, r < c(3 −(c+b)) b−(c+ 1) ,3−c<b<c+ 1 (14) holds for system (2) then the equilibria O(1) 2,3of system (1) are stable. If both relations (14) are not satisfied, then the equilibria O(1) 2,3of system (1) are unstable. Proof. Similar to Lemma 1 we consider system (2) obtained from (1) by changing variables (3). At points O(2) 2,3=±pb(r+ 1),±pb(r+ 1), r + 1the coefficients of the characteristic polynomial of the Jacobian matrix of system (2) are the following p(2) 1(O(2) 2,3) = c+b−1, p(2) 2(O(2) 2,3) = b(c+r), p(2) 3(O(2) 2,3)=2cb(r+ 1). (15) If c+b > 1is true then p(2) 1(O(2) 2,3)>0,p(2) 2(O(2) 2,3)>0, and p(2) 3(O(2) 2,3)>0hold for any positive values of c, b, r. Using the second relation from (14) the condition p(2) 1(O(2) 2,3)p(2) 2(O(2) 2,3)−p(2) 3(O(2) 2,3) = c(c+b−3) + r(b−(c+ 1)) >0(16) holds. Hence, we obtain the stability conditions (14) for the equilibria O(2) 2,3. Using (4) we obtain the statement of Lemma for system (1). 6 5. Analytical localization of the global attractor It is important to show that system (1) does not have trajectories tending to infinity either for a finite or for an infinite period of time for a correct mathematical description of economic processes in a model and the possibility of studying its limit dynamics. Next, we distinguish the domain of the parameters of system (1) for which all trajectories are bounded and, moreover, which over time fall into a limited closed region called an absorbing set [2]. For the corresponding set of parameters system (1) has a global attractor. Using the ideas presented in [11, 40, 55], we can prove the following Lemma 3. If γ < 2σ, then for any solution of system (1) we have the following estimate lim inf t→+∞z(t)−α 2δx2(t)≥0.(17) Proof. For system (2) and the Lyapunov function V(x, z) = z−x2 2c we have ˙ V(x(t), z(t)) = −b V (x(t), z(t)) + 1−b 2cx2(t). If b < 2cthen V(x(t), z(t)) ≥exp(−bt)V(x(0), z(0)), that yields estimate (17). Thus, the global attractor is located in the positive invariant set representing a parabolic cylinder (Fig. 1) Ω1=(x, y, z)∈R3|z≥x2 2c.(18) Using (4) we obtain the statement of Lemma for system (1). Theorem 1. If γ > 2µand γ < 2σ, then all solutions of system (1) eventually fall into a bounded closed set. Proof. For system (2) we consider the Lyapunov function V(x, y, z) = 1 2hAx2−2B x y +y2+z−r+ (A+B)c−B2i,(19) where Aand Bare arbitrary positive parameters. Case 1. If A > B2then V(x, y, z) = 1 2hA(x−B Ay)2+ (1 −B2 A)y2+z−r+ (A+B)c−B2i→ ∞ (20) as |(x, y, z)|→∞. For an arbitrary solution u(t) = (x(t), y(t), z(t)) of system (2) by Lemma 3 we have ˙ V(x, y, z) = −(Ac +Br)x2−(Bc −1)y2−(b−2Bc)z2+ (r+ (A+B)c−B)bz −2c B zz−x2 2c ≤−(Ac +Br)x2−(Bc −1)y2−(b−2Bc)z2+ (r+ (A+B)c−B)bz. 7 In order to have Bc −1>0,b−2Bc > 0, we choose B∈1 c,b 2cunder the assumptions b > 2, b < 2c. Suppose that ε∈0, b −2Bcand λ= min Ac +Br, Bc −1,(b−2Bc)−ε>0. Then ˙ V(x, y, z) = −(Ac +Br)x2−(Bc −1)y2−(b−2Bc −ε)z2−εz2+ (r+ (A+B)c−B)bz =−(Ac +Br)x2−(Bc −1)y2−(b−2Bc −ε)z2−√εz −(r+ (A+B)c−B)b 2√ε2 +(r+ (A+B)c−B)2b2 4ε≤ −λ(x2+y2+z2) + (r+ (A+B)c−B)2b2 4ε. Suppose that x2+y2+z2≥R2. Then a positive κexists such that ˙ V(x, y, z)≤ −λR2+(r+ (A+B)c−B)2b2 4ε<−κfor R2>1 λ (r+ (A+B)c−B)2b2 4ε. We choose η > 0such that {(x, y, z)|V(x, y, z)≤η} ⊃ (x, y, z)|x2+y2+z2≤R2. Thus, the relation x2+y2+z2≤R2implies that A(x−B Ay)2+ (1 −B2 A)y2+z−r+ (A+B)c−B2= A(x−B Ay)2+ (1 −B2 A)y2+z2−2r+ (A+B)c−Bz+r+ (A+B)c−B2≤2η. According to the Cauchy-Bunyakovsky-Schwarz inequality, we have A(x−B Ay)2≤2A(x2+B2 A2y2)≤2A(1 + B2 A2)R2 and, since −2r+ (A+B)c−Bz≤2r+ (A+B)c−B|z| ≤ 2r+ (A+B)c−BR, it is sufficient to choose η≥1 2h(2A+2+B2 A)R2+ 2r+ (A+B)c−BR+r+ (A+B)c−B2i. If Rand ηare chosen as shown above, then system (2) has the following compact ellipsoidal absorbing set: B0=(x, y, z)V(x, y, z) = 1 2hAx2−2B x y +y2+z−r+ (A+B)c−B2i≤η. Case 2. If A=B2then, using technics from Case 1, we attain the attractor of dynamic system (1) is located in the positive invariant set representing an elliptic cylinder: Ω2=(x, y, z)V(x, y, z) = 1 2hB2x−1 By2+z−r+B2c+B(c−1)2i≤η. Condition (20) holds for all (x, y, z)except for line x=1 Bywhich is symmetry axis of the elliptic cylinder Ω2. Hence, we obtain a number of the following positive invariant sets: the absorbing set B= Ω1TB0and the elliptic cylinder Ω2(Fig. 1). Using (4) we obtain the statement of Theorem for system (1). Thus, system (1) generates a dynamic system, the solutions of system (1) exist for t∈[0,+∞)and system (1) possesses a global attractor, which contains all equilibria and nontrivial (local) attractors (see, e.g. [21, 43]). 8 existence of multistability with hidden attractors in the model under consideration, as well as for the classical Lorenz system, is an open problem [19, 36, 42, 43, 56]. Conclusion The complexity of analyses of the dynamics of financial and economic systems is often due to the presence of multistability, when, for different initial data, the system trajectories can converge to distinct attractors. The coexistence of local attractors complicates forecasting the behavior of a dynamic system and estimating its quantitative characteristics, for instance, the Lyapunov dimension of the attractor. Recent results obtained in the field of nonlinear methods of dynamic systems theory allow one to successfully analyze global and local dynamics using analytical procedures. These studies have extensive applications including the analysis of global dynamics in a number of economic models [4, 47]. One efficient method for analysis of these models is discovering global attractors by constructing their absorbing sets, and subsequently implementing effective analytical and numerical procedures for investigation bounded sets of initial conditions. In this paper, we perform a global analysis of the stability of the mid-size firm model. We then derive the absorbing set, which makes it possible to localize the global attractor of system (1), and the domains of the model parameters for which stability and instability are observed. To characterize the chaotic dynamics of system (1), we calculate the Lyapunov dimension of attractor for specific values of parameters. Our work relies on an analytical approach to analyze the behavior of the model considered, which at the same time allows us to perform reliable numerical calculations of the Lyapunov dimension of the attractor of the mid-size firm model. We believe that ongoing efforts in the field of forecasting dynamics of real processes should be focused on the development of efficient analytical and numerical procedures that allow researchers to obtain the most complete and reliable information about the dynamics of the processes. This approach can help to expand the applicability of analytical procedures and overcome the disadvantages of numerical analysis. Hence, it opens a space for designing new control strategies for both stable regimes (including multistability, one of the most exciting phenomena in dynamic systems) and crisis processes. Acknowledgments We dedicate this paper to the memory of Gennady A. Leonov (1947-2018), with whom we began this work in 2016. We acknowledge support from the Russian Science Foundation (project 19-41-02002). References [1] Alexeeva, T., Barnett, W., Kuznetsov, N., and Mokaev, T. (2020). Time-delay control for stabilization of the Shapovalov mid-size firm model. arXiv preprint arXiv:2003.08484. (https://arxiv.org/pdf/2003.08484.pdf, accepted to IFAC WC 2020). [2] Barboza, R. and Chen, G. (2011). On the global boundedness of the Chen system. International Journal of Bifurcation and Chaos, 21(11):3373–3385. [3] Barkoulas, J. T. (2008). Testing for deterministic monetary chaos: Metric and topological diagnostics. 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