Abstract
Inspired in dynamic systems theory and Brewer’s contributions to apply it to economics, this paper establishes a bond graph model. Two main variables, a set of inter-connectivities based on nodes and links (bonds) and a fractional order dynamical perspective, prove to be a good macro-economic representation of countries’ potential performance in nowadays globalization. The estimations based on time series for 50 countries throughout the last 50 decades confirm the accuracy of the model and the importance of scale for economic performance.
Similar content being viewed by others
References
Acemoglu, D., Johnson, S., Robinson, J.A.: The colonial origins of comparative development: an empirical investigation. Am. Econ. Rev. 91(5), 1369–1401 (2001)
Acemoglu, D., Robinson, J.: Why Nations Fail: The Origins of Power, Prosperity, and Poverty. Profile Books, New York (2012)
Allen, R.C.: International competition in iron and steel, 1850–1913. J. Econ. Hist. 39(4), 911–937 (1979)
Allen, R.C.: American exceptionalism as a problem in global history. Oxford Discussion Paper Series, Number 689. University of Oxford, Department of Economics (2013). http://www.economics.ox.ac.uk/materials/papers/13161/paper689.pdf
Bacharach, M., Dempster, A., Enos, J. (eds.): Mathematical Models in Economics. Oxford University Press, Oxford (1994)
Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J.: Fractional Calculus: Models and Numerical Methods. Series on Complexity, Nonlinearity and Chaos. World Scientific Publishing Company, Singapore (2012)
Borutzky, W.: Bond Graph Methodology: Development and Analysis of Multidisciplinary Dynamic System Models. Springer, London (2010)
Breedveld, P.C.: Thermodynamic bond graphs and the problem of thermal lnertance. J. Frankl. Inst. 314(1), 15–40 (1982)
Brewer, J.W.: Bond graphs of microeconomic systems. In: 75-WA/Aut-8. American Society of Mechanical Engineering, Houston (1975)
Brewer, J.W.: Structure and cause and effect relations in social systems simulations. IEEE Trans. Syst. Man Cybern. 7(6), 468–474 (1977)
Brewer, J.W.: Progress in the bond graph representations of economics and population dynamics. J. Frankl. Inst. 328(5/6), 675–696 (1991)
Brewer, J.W., Craig, P.P.: Bilinear, dynamic single-ports and bond graphs of economic systems. J. Frankl. Inst. 313(4), 185–196 (1982)
Broadberry, S.: The Productivity Race: British Manufacturing in International Perspective. Cambridge University Press, Cambridge (1997)
Bruton, L.T.: Network transfer functions using the concept of frequency dependent negative resistance. IEEE Trans. Circuit Theory 16, 406–408 (1969)
Chen, Y., Moore, K.: Discretization schemes for fractional-order differentiators and integrators. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 49(3), 363–367 (2002)
Chua, L.O.: Memristor—the missing circuit element. IEEE Trans. Circuit Theory 18(2), 507–519 (1971)
Chua, L.O.: Nonlinear circuit foundations for nanodevices, Part I: the four-element torus. Proc. IEEE 91(11), 1830–1859 (2003)
Chua, L.O., Kang, S.M.: Memristive devices and systems. Proc. IEEE 64(2), 209–223 (1976)
Connolly, T.J., Contréras, J.A.: Bond graph primitives for modeling systems with fractional differential equations. Fract. Calc. Appl. Anal. 12(4), 391–408 (2009)
Crafts, N.: Steam as a general purpose technology: a growth accounting perspective. Econ. J. 114(495), 338–351 (2004)
Deskur, J.: Models of magnetic circuits and their equivalent electrical diagrams. Int. J. Comput. Math. Electr. Electron. Eng. 18(4), 600–610 (1999)
Fogel, R.W.: Capitalism and democracy in 2040—forecasts and speculations. NBER Working Paper Series. The National Bureau of Economic Research, Cambridge (2007). http://www.nber.org/papers/w13184
Gawthrop, P.J., Bevan, G.P.: Bond-graph modeling: a tutorial introduction for control engineers. IEEE Control Syst. Mag. 27(2), 24–45 (2007)
Gawthrop, P.J., Smith, L.S.: Metamodelling: Bond Graphs and Dynamic Systems. Prentice Hall, Englewood Cliffs (1996)
Goldberg, D.E.: Genetic Algorithms in Search Optimization, and Machine Learning. Addison-Wesley, Reading (1989)
Guijarro, N., Dauphin-Tanguy, G.: Approximation methods to embed the non-integer order models in bond graphs. Sig. Process. 83(11), 2335–2344 (2003)
Hayek, F.: The use of knowledge in society. Am. Econ. Rev. 35(4), 519–530 (1945)
Holland, J.H.: Adaptation in Natural and Artificial Systems. The University of Michigan Press, Ann Arbor (1975)
Ionescu, C.: The Human Respiratory System: An Analysis of the Interplay between Anatomy, Structure, Breathing and Fractal Dynamics. Series in BioEngineerin. Springer, London (2013)
Jeltsema, D., Dòria-Cerezo, A.: Memristive port-Hamiltonian systems. Math. Comput. Model. Dyn. Syst. 16(2), 75–93 (2010)
Jeltsema, D., Dòria-Cerezo, A.: Port-Hamiltonian formulation of systems with memory. Proc. IEEE 100(6), 1928–1937 (2012)
Jesus, I.S., Machado, J.A.T.: Development of fractional order capacitors based on electrolyte processes. Nonlinear Dyn. 56(1–2), 45–55 (2009)
Karnopp, D., Rosenberg, R.C.: System Dynamics: A Unified Approach. Wiley, New York (1975)
Kilbas, A., Srivastava, H., Trujillo, J.: Theory and Applications of Fractional Differential Equations, vol. 204. North-Holland Mathematics Studies, Elsevier (2006)
Long, J., Ferrie, J.: Intergenerational occupational mobility in Britain and the United States since 1850. Am. Econ. Rev. 103(4), 1109–1137 (2013)
Ma, C., Hori, Y.: Application of bond graph models to the representation of buildings and their use. In: Proceedings of American Control Conference, Boston, pp. 2901–2906 (2004)
Machado, J.A.T.: Calculation of fractional derivatives of noisy data with genetic algorithms. Nonlinear Dyn. 57(1–2), 253–260 (2009)
Machado, J.A.T.: Fractional derivatives: probability interpretation and frequency response of rational approximations. Commun. Nonlinear Sci. Numer. Simul. 14(9–10), 3492–3497 (2009)
Machado, J.A.T.: Fractional generalization of memristor and higher order elements. Commun. Nonlinear Sci. Numer. Simul. 18(12), 264–275 (2013)
Machado, J.A.T., Galhano, A.M.: Fractional order inductive phenomena based on the skin effect. Nonlinear Dyn. 68(1–2), 107–115 (2012)
Machado, J.T.: Analysis and design of fractional-order digital control systems. Syst. Anal. Model. Simul. 27(2–3), 107–122 (1997)
Machado, J.T., Galhano, A.M., Trujillo, J.J.: On development of fractional calculus during the last fifty years. Scientometrics 98, 577–582 (2013)
Machado, J.T., Kiryakova, V., Mainardi, F.: Recent history of fractional calculus. Commun. Nonlinear Sci. Numer. Simul. 16(3), 1140–1153 (2011)
Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. Imperial College Press, London (2010)
Méhauté, A.L., Nigmatullin, R.R., Nivanen, L.: Flèches du temps et géométrie fractale. Hermès, Paris (1998)
Miller, K., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)
Moreau, X., Khemane, F., Malti, R., Serrier, P.: Approximation of a fractance by a network of four identical RC cells arranged in gamma and a purely capacitive cell. In: Baleanu, D., Machado, J.A.T., Güvenç, Z.B. (eds.) New Trends in Nanotechnology and Fractional Calculus Applications, pp. 107–120. Springer, Dordrecht (2001)
Mukherjee, A., Karmakar, R., Samantaray, A.K.: Bond Graph in Modeling, Simulation and Fault Identification. CRC Press, New Delhi (2006)
Oldham, K., Spanier, J.: The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order. Academic Press, New York (1974)
O’Rourke, K.: Tariffs and growth in the late 19th century. Econ. J. 110(463), 456–483 (2000)
O’Rourke, K., Williamson, J.G.: Globalization and History: The Evolution of a Nineteenth-Century Atlantic Economy. MIT Press, Cambridge (2001)
Oustaloup, A.: Systèmes asservis linéaires d’ordre fractionnaire: théorie et pratique. Masson, Paris (1983)
Paynter, H.: An Epistemic Prehistory of Bond Graphs. North-Holland, Amsterdam (1992)
Pease, D.E.: The New American Exceptionalism. Critical American Studies. University of Minnesota Press, Minneapolis (2009)
Podlubny, I.: Fractional differential equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution, Mathematics in Science and Engineering, vol. 198. Academic Press, San Diego (1998)
Podlubny, I.: Fractional-order systems and PI\(^{\lambda }\)D\(^{\mu }\)-controllers. IEEE Trans. Autom. Control 44(1), 208–213 (1999)
Samko, S., Kilbas, A., Marichev, O.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers, Amsterdam (1993)
Say, J.B.: Traité dÉconomie Politique (translated as A Treatise on Political Economy). Batoche Books, Kither, Paris (1803, 17, 19, 26, 41 (2001))
Senani, R.: On the realization of floating active elements. IEEE Trans. Circuits Syst. 33(3), 323–324 (1986)
Seymour, M. (ed.): Dynamic Factors in Industrial Productivity. Basil Blackwell, Oxford (1956)
Shakespeare, W.: As You Like It (Act 2, Scene 7). Cambridge University Press, New York ([1599] 2000)
Soliman, A.M., Saad, R.A.: Two new families of floating fdnr circuits. J. Electr. Comput. Eng. 2010(Article ID 563761), 7 p (2010)
Stiglitz, J.E.: Information and the change in the paradigm in economics. Am. Econ. Rev. 92(3), 460–501 (2002)
Strukov, D.B., Snider, G.S., Stewart, D.R., Williams, R.S.: The missing memristor found. Nature 97, 80–83 (2008)
Tarasov, V.: Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer, New York (2010)
Temin, P.: General equilibrium models in economic history. J. Econ. Hist. 31(1), 58–75 (1971)
Thoma, J., Bouamama, B.O.: Modelling and Simulation in Thermal and Chemical Engineering: A Bond Graph Approach. Springer, Berlin (2010)
Thoma, J.U.: Introduction to Bond Graphs and Their Applications. Pergamon Press, Oxford (1975)
Tsai, J.J.H., Gero, J.S.: Unified Energy-Based Qualitative Representation for Building Analysis. VDM Verlag, Saarbrucken (2009)
Tsai, J.J.H., Gero, J.S.: A qualitative energy-based unified representation for buildings. Autom. Constr. 19(1), 20–42 (2010)
Valério, D., da Costa, J.S.: An Introduction to Fractional Control. IET, Stevenage (2012)
Ventra, M.D., Pershin, Y.V., Chua, L.O.: Circuits elements with memory: memristors, memcapacitors and meminductors. Proc. IEEE 97(10), 1717–1724 (2009)
Walras, L.: Elements dÉconomie Politique Pure (translated to Elements of Pure Economics). Homewwod, Irwin (1874, 1877 (1954))
Wellstead, P.E.: Introduction to Physical System Modelling. Academic Press, London (1979)
Williams, R.S.: How we found the missing memristor. IEEE Spectr. 45(12), 28–35 (2008)
Wong, Y.K.: Application of bond graph models to economics. Int. J. Model. Simul. 21(3), 181–190 (2001)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Machado, J.A.T., Mata, M.E. A fractional perspective to the bond graph modelling of world economies. Nonlinear Dyn 80, 1839–1852 (2015). https://doi.org/10.1007/s11071-014-1334-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-014-1334-0