{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,18]],"date-time":"2026-04-18T08:26:26Z","timestamp":1776500786657,"version":"3.51.2"},"reference-count":65,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2015,7,29]],"date-time":"2015-07-29T00:00:00Z","timestamp":1438128000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>This paper examines modern economic growth according to the multidimensional scaling (MDS) method and state space portrait (SSP) analysis. Electing GDP per capita as the main indicator for economic growth and prosperity, the long-run perspective from 1870 to 2010 identifies the main similarities among 34 world partners\u2019 modern economic growth and exemplifies the historical waving mechanics of the largest world economy, the USA. MDS reveals two main clusters among the European countries and their old offshore territories, and SSP identifies the Great Depression as a mild challenge to the American global performance, when compared to the Second World War and the 2008 crisis.<\/jats:p>","DOI":"10.3390\/e17085402","type":"journal-article","created":{"date-parts":[[2015,7,29]],"date-time":"2015-07-29T10:25:31Z","timestamp":1438165531000},"page":"5402-5421","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":76,"title":["Fractional State Space Analysis of Economic Systems"],"prefix":"10.3390","volume":"17","author":[{"given":"J.","family":"Machado","sequence":"first","affiliation":[{"name":"Institute of Engineering, Polytechnic of Porto, Rua Dr. Ant\u00f3nio Bernardino de Almeida, 431, 4200-072 Porto, Portugal"}]},{"given":"Maria","family":"Mata","sequence":"additional","affiliation":[{"name":"Nova SBE, Universidade Nova de Lisboa, Faculdade de Economia, Campus de Campolide, Lisbon, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7359-4370","authenticated-orcid":false,"given":"Ant\u00f3nio","family":"Lopes","sequence":"additional","affiliation":[{"name":"UISPA-LAETA\/INEGI, Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2015,7,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Acemoglu, D. (2012). The World Our Grandchildren Will Inherit: The Rights Revolution and Beyond, National Bureau of Economic Research. Technical Report.","DOI":"10.3386\/w17994"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.strueco.2015.01.003","article-title":"Nonlinear multiplier\u2013accelerator model with investment and consumption delays","volume":"33","author":"Matsumoto","year":"2015","journal-title":"Struct. Chang. Econ. Dyn."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Dassios, I.K., Zimbidis, A.A., and Kontzalis, C.P. (2014). The Delay Effect in a Stochastic Multiplier\u2013Accelerator Model. J. Econ. Struct., 3.","DOI":"10.1186\/s40008-014-0007-y"},{"key":"ref_4","unstructured":"Holcombe, R.G. (1989). Economic Models and Methodology, Greenwood Press."},{"key":"ref_5","unstructured":"Reiter, M. (2012). The Dynamics of Business Cycles: Stylized Facts, Economic Theory, Econometric Methodology and Applications, Springer."},{"key":"ref_6","first-page":"73","article-title":"A probabilistic interpretation of the fractional-order differentiation","volume":"6","author":"Machado","year":"2003","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"3492","DOI":"10.1016\/j.cnsns.2009.02.004","article-title":"Fractional derivatives: Probability interpretation and frequency response of rational approximations","volume":"14","author":"Machado","year":"2009","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_8","first-page":"225","article-title":"A Revised Estimate of Italian Economic Growth 1861\u20131989","volume":"44","author":"Maddison","year":"1991","journal-title":"PSL Q. Rev."},{"key":"ref_9","unstructured":"Maddison, A. (1995). Monitoring the World Economy, 1820\u20131992, Development Centre of the Organisation for Economic Co-operation and Development."},{"key":"ref_10","unstructured":"Maddison, A. (2010). Statistics on world population, GDP and per capita GDP, Technical Report; University of Groningen."},{"key":"ref_11","unstructured":"Maddison, A. (1982). Phases of Capitalist Development, Oxford University Press."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Barro, R.J., and Urs\u00faa, J.F. (2008). Macroeconomic Crises since 1870, National Bureau of Economic Research. NBER working-paper no. 13940.","DOI":"10.3386\/w13940"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1017\/S096856501200008X","article-title":"The international propagation of the financial crisis of 2008 and a comparison with 1931","volume":"19","author":"Allen","year":"2012","journal-title":"Financ. Hist. Rev."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"711","DOI":"10.1016\/j.asieco.2007.07.002","article-title":"Economic integration and synchronization of business cycles in East Asia","volume":"18","author":"Rana","year":"2007","journal-title":"J. Asian Econ."},{"key":"ref_15","first-page":"33","article-title":"Business Cycle Synchronization in the Euro Area and the Impact of the Financial Crisis","volume":"2","author":"Riedl","year":"2012","journal-title":"Monet. Policy Econ."},{"key":"ref_16","unstructured":"Maddison, A. (1991). Dynamic Forces in Capitalist Development: A Long-Run Comparative View, Oxford University Press."},{"key":"ref_17","unstructured":"Prados de la Escosura, L. Mismeasuring Long Run Growth: The Bias from Spliced National Accounts. Available online: http:\/\/papers.ssrn.com\/sol3\/papers.cfm?abstract_id=2501620."},{"key":"ref_18","unstructured":"Lenin, V.I. (1999). Imperialism: The Highest Stage of Capitalism, Resistance Books."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Acemoglu, D., Robinson, J.A., and Woren, D. (2012). Why Nations Fail: The Origins of Power, Prosperity and Poverty, SciELO Chile.","DOI":"10.1355\/ae29-2j"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1231","DOI":"10.1162\/003355302320935025","article-title":"Reversal of Fortune: Geography and Institutions in the Making of the Modern World Income Distribution","volume":"117","author":"Acemoglu","year":"2002","journal-title":"Q. J. Econ."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1016\/j.gfj.2011.10.012","article-title":"A historical overview of financial crises in the United States","volume":"22","author":"Shachmurove","year":"2011","journal-title":"Glob. Financ. J."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Tallman, E.W. (2012). The Panic of 1907, Federal Reserve Bank of Cleveland. Technical report.","DOI":"10.26509\/frbc-wp-201228"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"627","DOI":"10.1111\/1468-0289.12032","article-title":"The Maddison Project: Collaborative research on historical national accounts","volume":"67","author":"Bolt","year":"2014","journal-title":"Econ. Hist. Rev."},{"key":"ref_24","unstructured":"The Maddison Project. Available online: http:\/\/www.ggdc.net\/maddison\/maddison-project\/home.htm."},{"key":"ref_25","unstructured":"Torgerson, W. (1958). Theory and Methods of Scaling, Wiley."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1007\/BF02289621","article-title":"The analysis of proximities: Multidimensional scaling with an unknown distance function","volume":"27","author":"Shepard","year":"1962","journal-title":"Psychometrika"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF02289565","article-title":"Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis","volume":"29","author":"Kruskal","year":"1964","journal-title":"Psychometrika"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Kruskal, J.B., and Wish, M. (1978). Multidimensional Scaling, Sage Publications.","DOI":"10.4135\/9781412985130"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Cox, T.F., and Cox, M.A.A. (2001). Multidimensional Scaling, Chapman & Hall\/CRC.","DOI":"10.1201\/9781420036121"},{"key":"ref_30","unstructured":"Borg, I., and Groenen, P.J. (2005). Modern Multidimensional Scaling-Theory and Applications, Springer."},{"key":"ref_31","unstructured":"Martinez, W.L., and Martinez, A.R. (2005). Exploratory Data Analysis with MATLAB, Chapman & Hall\/CRC."},{"key":"ref_32","unstructured":"Machado, J., and Mata, M. (2013, January 2\u20135). A multidimensional scaling perspective of Rostow\u2019s forecasts with the track-record (1960s\u20132011) of pioneers and latecomers. Proceedings of the 12th International Conference on Dynamical Systems Theory and Applications, \u0141\u00f3d\u017a, Poland."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Machado, J.A.T., and Mata, M.E. (2013). Multidimensional Scaling Analysis of the Dynamics of a Country Economy. Sci. World J., 2013.","DOI":"10.1155\/2013\/594587"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1145\/584091.584093","article-title":"A mathematical theory of communication","volume":"5","author":"Shannon","year":"2001","journal-title":"ACM SIGMOBILE Mob. Comput. Commun. Rev."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"3892","DOI":"10.3390\/e15093892","article-title":"Analysis and Visualization of Seismic Data Using Mutual Information","volume":"15","author":"Machado","year":"2013","journal-title":"Entropy"},{"key":"ref_36","first-page":"583","article-title":"Cluster ensembles\u2014A knowledge reuse framework for combining multiple partitions","volume":"3","author":"Strehl","year":"2003","journal-title":"J. Mach. Learn. Res."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Polderman, J.W., and Willems, J.C. (1998). Introduction to Mathematical Systems Theory: A Behavioral Approach, Springer.","DOI":"10.1007\/978-1-4757-2953-5"},{"key":"ref_38","unstructured":"Holoborodko, P. Smooth Noise Robust Differentiators. Available online: http:\/\/www.holoborodko.com\/pavel\/numerical-methods\/numerical-derivative\/smooth-low-noise-differentiators\/."},{"key":"ref_39","unstructured":"North, D.C. (1974). Growth and Welfare in the American Past. A New Economic History, Princeton-Hall."},{"key":"ref_40","unstructured":"Aldcroft, D.H. (1987). From Versailles to Wall Street, 1919\u20131929, Penguin Books."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Brunner, K. (1981). The Great Depression Revisited, Klwer Nijhoff Publishers.","DOI":"10.1007\/978-94-009-8135-5"},{"key":"ref_42","unstructured":"Kindleberger, C.P. (1987). The World in Depression, 1929\u20131939, Penguin Books."},{"key":"ref_43","unstructured":"Eichengreen, B.J. (1998). Globalizing Capital: A History of the International Monetary System, Princeton University Press."},{"key":"ref_44","unstructured":"Van der Wee, H. (1986). Prosperity and Upheaval: The World Economy, 1945\u20131980, University of California Press."},{"key":"ref_45","unstructured":"Baleanu, D., Diethelm, K., Scalas, E., and Trujillo, J.J. (2012). Models and Numerical Methods, World Scientific."},{"key":"ref_46","unstructured":"Kenneth, M., and Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley New York."},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Mainardi, F. (2010). Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, World Scientific.","DOI":"10.1142\/9781848163300"},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Luo, Y., and Chen, Y. (2012). Fractional Order Motion Controls, Wiley.","DOI":"10.1002\/9781118387726"},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Sheng, H., Chen, Y., and Qiu, T. (2011). Fractional Processes and Fractional-Order Signal Processing: Techniques and Applications, Springer.","DOI":"10.1007\/978-1-4471-2233-3"},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Ionescu, C.M. (2013). The Human Respiratory System: An Analysis of the Interplay between Anatomy, Structure, Breathing and Fractal Dynamics, Springer.","DOI":"10.1007\/978-1-4471-5388-7"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"998","DOI":"10.1177\/1077546312473323","article-title":"Fractional order models of leaves","volume":"20","author":"Lopes","year":"2014","journal-title":"J. Vib. Control"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1007\/s11071-004-3770-8","article-title":"Fractional order control of a hexapod robot","volume":"38","author":"Silva","year":"2004","journal-title":"Nonlinear Dyn."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"647","DOI":"10.1016\/j.camwa.2013.02.003","article-title":"Fractional dynamics and MDS visualization of earthquake phenomena","volume":"66","author":"Lopes","year":"2013","journal-title":"Comput. Math. Appl."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"979","DOI":"10.1016\/j.cnsns.2009.05.027","article-title":"On fractional filtering versus conventional filtering in economics","volume":"15","author":"Nigmatullin","year":"2010","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_55","unstructured":"Omay, T., and Baleanu, D. (2009). Innovation Policies, Business Creation and Economic Development, Springer."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"1399","DOI":"10.1007\/s11071-014-1749-7","article-title":"Finite-time stabilizing a fractional-order chaotic financial system with market confidence","volume":"79","author":"Xin","year":"2014","journal-title":"Nonlinear Dyn."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"396","DOI":"10.1016\/j.cnsns.2014.08.032","article-title":"Pseudo Phase Plane and Fractional Calculus modeling of western global economic downturn","volume":"22","author":"Machado","year":"2015","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"2445","DOI":"10.1007\/s11071-012-0632-7","article-title":"Chaos suppression in a fractional order financial system using intelligent regrouping PSO based fractional fuzzy control policy in the presence of fractional Gaussian noise","volume":"70","author":"Pan","year":"2012","journal-title":"Nonlinear Dyn."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"238459","DOI":"10.1155\/2014\/238459","article-title":"A review of definitions for fractional derivatives and integrals","volume":"2014","author":"Machado","year":"2014","journal-title":"Math. Probl. Eng"},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"1140","DOI":"10.1016\/j.cnsns.2010.05.027","article-title":"Recent history of fractional calculus","volume":"16","author":"Machado","year":"2011","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"1827","DOI":"10.1140\/epjst\/e2013-01967-y","article-title":"Fractional calculus: A survey of useful formulas","volume":"222","author":"Trujillo","year":"2013","journal-title":"Eur. Phys. J. Special Top."},{"key":"ref_62","doi-asserted-by":"crossref","unstructured":"Petras, I. (2011). Fractional-order Nonlinear Systems: Modeling, Analysis and Simulation, Springer.","DOI":"10.1007\/978-3-642-18101-6"},{"key":"ref_63","unstructured":"Loverro, A. (2004). Fractional Calculus: History, Definitions and Applications for the Engineer, Univeristy of Notre Dame. Rapport technique."},{"key":"ref_64","unstructured":"Dor\u010d\u00e1k, V. (2002). Numerical models for the simulation of the fractional-order control systems, arXiv: math\/0204108."},{"key":"ref_65","unstructured":"Podlubny, I. (1999). Fractional Differential Equations, Academic Press."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/17\/8\/5402\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T20:49:53Z","timestamp":1760215793000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/17\/8\/5402"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,7,29]]},"references-count":65,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2015,8]]}},"alternative-id":["e17085402"],"URL":"https:\/\/doi.org\/10.3390\/e17085402","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,7,29]]}}}