Datta, Soumya (2014): Robustness and Stability of Limit Cycles in a Class of Planar Dynamical Systems.
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Abstract
Using the AndronovHopf bifurcation theorem and the PoincaréBendixson Theorem, this paper explores robust cyclical possibilities in a generalized KolmogorovLotkaVolterra class of models with positive intraspecific cooperation in the prey population. This additional feedback effect introduces nonlinearities which modify the cyclical outcomes of the model. Using an economic example, the paper proposes an algorithm to symbolically construct the topological normal form of AndronovHopf bifurcation. In case the limit cycle turns out to be unstable, the possibilities of the dynamics converging to another limit cycle is explored.
Item Type:  MPRA Paper 

Original Title:  Robustness and Stability of Limit Cycles in a Class of Planar Dynamical Systems 
Language:  English 
Keywords:  KolmogorovLotkaVolterra model, predatorprey, AndronovHopf bifurcation, Limit cycles 
Subjects:  C  Mathematical and Quantitative Methods > C6  Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C62  Existence and Stability Conditions of Equilibrium C  Mathematical and Quantitative Methods > C6  Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C69  Other 
Item ID:  56970 
Depositing User:  Dr. Soumya Datta 
Date Deposited:  29 Jun 2014 05:42 
Last Modified:  12 Aug 2024 20:17 
References:  Asada, T., Chen, P., Chiarella, C. & Flaschel, P. (2006), ‘Keynesian dynamics and the wageprice spiral: A baseline disequilibrium model’, Journal of Macroeconomics 28(1), 90–130. Asada, T. & Yoshida, H. (2003), ‘Coefficient Criterion for Fourdimensional Hopf Bifurcations: A Complete Mathematical Characterization and Applications to Economic Dynamics’, Chaos, Solitons and Fractals 18, 525–536. Barnett, W. A. & He, Y. (1998), Bifurcations in ContinuousTime Macroeconomic Systems, Macroeconomics 9805018, EconWPA. *http://ideas.repec.org/p/wpa/wuwpma/9805018.html Barnett, W. & He, Y. (2006), Existence of Bifurcation in Macroeconomic Dynamics: Grandmont was Right, Working papers series in theoretical and applied economics 200610, University of Kansas, Department of Economics. *http://ideas.repec.org/p/kan/wpaper/200610.html Benhabib, J. & Miyao, T. (1981), ‘Some New Results on the Dynamics of the Generalized Tobin Model’, International Economic Review 22(3), 589–96. Benhabib, J. & Nishimura, K. (1979), ‘The Hopf Bifurcation and the Existence and Stability of Closed Orbits in Multisector Models of Optimal Economic Growth’, Journal of Economic Theory 21, 421–444. Chiarella, C. & Flaschel, P. (2000), The Dynamics of Keynesian Monetary Growth: Macro Foundations, Cambridge University Press. Chiarella, C., Flaschel, P. & Franke, R. (2005), Foundations for a Disequilibrium Theory of the Business Cycle: Qualitative Analysis and Quantitative Assessment, Cambridge University Press, Cambridge, U.K. Datta, S. (2012), Cycles and crises in a model of debtfinanced investmentled growth, MPRA Paper 50200, University Library of Munich, Germany. *http://mpra.ub.unimuenchen.de/id/eprint/50200 Edneral, V. F. (2007), An Algorithm for Construction of Normal Forms, in V. G. Ganzha, E. W. Mayr & E. V. Vorozhtsov, eds, ‘CASC’, Vol. 4770 of Lecture Notes in Computer Science, Springer, pp. 134–142. Franke, R. (1992), ‘Stable, Unstable and Persistent Cyclical Behavior in a KeynesWicksell Monetary Growth Model’, Oxford Economic Papers 44, 242–256. Hofbauer, J. & So, J. W. H. (1990), ‘Multiple Limit Cycles for Predatorprey Models’, Mathematical Biosciences 99(1), 71–75. Hsu, S.B. & Hwang, T.W. (1999), ‘Hopf Bifurcation for a Predatorprey System of Holling and Leslie Type’, Taiwanese Journal of Mathematics 3(1), 35–53. Kind, C. (1999), ‘Remarks on the Economic Interpretation of Hopf Bifurcations’, Economics Letters 62, 147–154. Kuznetsov, Y. A. (1997), Elements of Applied Bifurcation Theory, Vol. 112 of Applied Mathematical Sciences, second edn, SpringerVerlag, New York. Kuznetsov, Y. A. (2006), ‘AndronovHopf bifurcation’, Scholarpedia 1(10), 1858. *http://www.scholarpedia.org/article/AndronovHopf_bifurcation Leijonhufvud, A. (1973), ‘Effective Demand Failures’, Swedish Journal of Economics 75, 27–48. Minagawa, J. (2007), A determinantal criterion of Hopf bifurcations and its application to economic dynamics, in T. Asada & T. Ishikawa, eds, ‘Time and Space in Economics’, Springer, pp. 160–172. Munkres, J. R. (2000), Topology, second edn, Pearson Education, Inc. Velupillai, K. (2006), ‘A Disequilibrium Macrodynamic Model of Fluctuations’, Journal of Macroeconomics 28(4), 752–767. Wiggins, S. (1990), Introduction to Applied Nonlinear Dynamical Systems and Chaos, SpringerVerlag, New York, Inc. Yuquan, W., Zhujun, J. & Chan, K. (1999), ‘Multiple Limit Cycles and Global Stability in PredatorPrey Model’, Acta Mathematicae Applicatae Sinica 15(2), 206–219. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/56970 
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Robustness and Stability of Limit Cycles in a Class of Planar Dynamical Systems. (deposited 20 Oct 2013 13:20)
 Robustness and Stability of Limit Cycles in a Class of Planar Dynamical Systems. (deposited 29 Jun 2014 05:42) [Currently Displayed]