In this work we introduce a topological method for the search of fixed points and periodic points for continuous maps defined on generalized rectangles in finite dimensional Euclidean spaces. We name our technique "Stretching Along the Paths" method, since we deal with maps that expand the arcs along one direction. Such theory was developed in the planar case by Papini and Zanolin in [11,12] and it has been extended to the Ndimensional framework by the author and Zanolin in [16]. In the bidimensional setting, elementary theorems from plane topology suffice, while in the higher dimension some results from degree theory are needed, leading to the study of the socalled "Cutting Surfaces" [16]. Our method is also significant from a dynamical point of view, as it allows to detect complex dynamics. As it is wellknown, a prototypical example of chaotic system is represented by the Smale horseshoe. However, in order to prove conjugacy with the shift map, it requires the verification of hyperbolicity conditions, which are difficult or impossible to prove in practical cases. For such reason more general and less stringent definitions of horseshoe have been suggested so as to reproduce some geometrical features typical of the Smale horseshoe while discarding the hyperbolicity hypotheses. This led to the study of the socalled "topological (or geometrical) horseshoes" [2,5]. In particular, different characterizations have been proposed by various authors in order to establish the presence of complex dynamics for continuous maps defined on subsets of the Ndimensional Euclidean space (see, for instance, [10,21,23] and the references therein). The tools employed in these and related works range from the Conley index [10] to the Lefschetz fixed point theory [20]. On the other hand, our approach, although mathematically rigorous, avoids the use of more advanced topological theories and it is relatively easy to apply to specific models arising in applications. For example we have employed such method to study discrete and continuoustime models arising from economics and biology [9,18]. In more details, the topics considered along the thesis can be summarized as follows. The description of the Stretching Along the Paths method and suitable variants of it can be found in Chapter 1. In Chapter 2 we discuss which are the chaotic features that can be obtained for a given map when our technique applies. In particular, we are able to prove semiconjugacy to the Bernoulli shift and thus positivity of the topological entropy, the presence of topological transitivity and sensitivity with respect to initial conditions, density of periodic points. Moreover we show the mutual relationships among various classical notions of chaos (such as those by Devaney, LiYorke, etc.). We also introduce an alternative geometrical framework related to the socalled "Linked Twist Maps" [3,4,22], where it is possible to employ our method in order to detect complex dynamics. The theoretical results obtained so far find an application to discrete and continuoustime systems in Chapters 3 and 4. As regards the former, in Chapter 3 we deal with some onedimensional and planar discrete economic models, both of the Overlapping Generation and of the Duopoly Game classes. The bidimensional models are taken from [8,19] and [1], respectively. On the other hand, in Chapter 4, with respect to continuoustime models, we study some nonlinear ODEs with periodic coefficients through a combination of a careful but elementary phaseplane analysis with the results on chaotic dynamics for Linked Twist Maps from Chapter 2. In more details, we consider a modified version of the Volterra predatorprey model, in which a periodic harvesting is included, as well as a simplification of the LazerMcKenna suspension bridges model [6,7] from [13,14]. When dealing with ODEs with periodic coefficients, our method is applied to the associated Poincaré map. The contents of the present thesis are based on the papers [9,13,16,17,18] and partially on [14], where maps expansive along several directions were considered. [1] H.N. Agiza and A.A. Elsadany, Chaotic dynamics in nonlinear duopoly game with heterogeneous players, Appl. Math. Comput. 149 (2004), 843860. [2] K. Burns and H. Weiss, A geometric criterion for positive topological entropy, Comm. Math. Phys. 172 (1995), 95118. [3] R. Burton and R.W. Easton, Ergodicity of linked twist maps, In: Global theory of dynamical systems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979), pp. 3549, Lecture Notes in Math., 819, Springer, Berlin, 1980. [4] R.L. Devaney, Subshifts of finite type in linked twist mappings, Proc. Amer. Math. Soc. 71 (1978), 334338. [5] J. Kennedy and J.A. Yorke, Topological horseshoes, Trans. Amer. Math. Soc. 353 (2001), 25132530. [6] A.C. Lazer and P.J. McKenna, Large scale oscillatory behaviour in loaded asymmetric systems, Ann. Inst. Henry Poincar e, Analyse non lineaire 4 (1987), 244274. [7] A.C. Lazer and P.J. McKenna, Largeamplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis, SIAM Review 32 (1990), 537578. [8] A. Medio, Chaotic dynamics. Theory and applications to economics, Cambridge University Press, Cambridge, 1992. [9] A. Medio, M. Pireddu and F. Zanolin, Chaotic dynamics for maps in one and two dimensions. A geometrical method and applications to economics, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 19 (2009), 32833309. [10] K. Mischaikow and M. Mrozek, Isolating neighborhoods and chaos, Japan J. Indust. Appl. Math. 12 (1995), 205236. [11] D. Papini and F. Zanolin, On the periodic boundary value problem and chaoticlike dynamics for nonlinear Hill's equations, Adv. Nonlinear Stud. 4 (2004), 7191. [12] D. Papini and F. Zanolin, Fixed points, periodic points, and cointossing sequences for mappings defined on twodimensional cells, Fixed Point Theory Appl. 2004 (2004), 113134. [13] A. Pascoletti, M. Pireddu and F. Zanolin, Multiple periodic solutions and complex dynamics for second order ODEs via linked twist maps, Electron. J. Qual. Theory Differ. Equ., Proc. 8'th Coll. Qualitative Theory of Diff. Equ. 14 (2008), 132. [14] A. Pascoletti and F. Zanolin, Example of a suspension bridge ODE model exhibiting chaotic dynamics: a topological approach, J. Math. Anal. Appl. 339 (2008), 11791198. [15] M. Pireddu and F. Zanolin, Fixed points for dissipativerepulsive systems and topological dynamics of mappings defined on Ndimensional cells, Adv. Nonlinear Stud. 5 (2005), 411440. [16] M. Pireddu and F. Zanolin, Cutting surfaces and applications to periodic points and chaoticlike dynamics, Topol. Methods Nonlinear Anal. 30 (2007), 279319. [17] M. Pireddu and F. Zanolin, Some remarks on fixed points for maps which are expansive along one direction, Rend. Istit. Mat. Univ. Trieste 39 (2007), 245274. [18] M. Pireddu and F. Zanolin, Chaotic dynamics in the Volterra predatorprey model via linked twist maps, Opuscula Math. 28/4 (2008), 567592. [19] P. Reichlin, Equilibrium cycles in an overlapping generations economy with production, J. Econom. Theory 40 (1986), 89102. [20] R. Srzednicki, A generalization of the Lefschetz fixed point theorem and detection of chaos, Proc. Amer. Math. Soc. 128 (2000), 12311239. [21] R. Srzednicki and K. Wojcik, A geometric method for detecting chaotic dynamics, J. Differential Equations 135 (1997), 6682. [22] S. Wiggins, Chaos in the dynamics generated by sequence of maps, with application to chaotic advection in flows with aperiodic time dependence, Z. angew. Math. Phys. 50 (1999), 585616. [23] P. Zgliczy nski and M. Gidea, Covering relations for multidimensional dynamical systems, J. Differential Equations 202 (2004), 3258.
Fixed points and chaotic dynamics for expansivecontractive maps in Euclidean spaces, with some applications
PIREDDU, MARINA
2009
Abstract
In this work we introduce a topological method for the search of fixed points and periodic points for continuous maps defined on generalized rectangles in finite dimensional Euclidean spaces. We name our technique "Stretching Along the Paths" method, since we deal with maps that expand the arcs along one direction. Such theory was developed in the planar case by Papini and Zanolin in [11,12] and it has been extended to the Ndimensional framework by the author and Zanolin in [16]. In the bidimensional setting, elementary theorems from plane topology suffice, while in the higher dimension some results from degree theory are needed, leading to the study of the socalled "Cutting Surfaces" [16]. Our method is also significant from a dynamical point of view, as it allows to detect complex dynamics. As it is wellknown, a prototypical example of chaotic system is represented by the Smale horseshoe. However, in order to prove conjugacy with the shift map, it requires the verification of hyperbolicity conditions, which are difficult or impossible to prove in practical cases. For such reason more general and less stringent definitions of horseshoe have been suggested so as to reproduce some geometrical features typical of the Smale horseshoe while discarding the hyperbolicity hypotheses. This led to the study of the socalled "topological (or geometrical) horseshoes" [2,5]. In particular, different characterizations have been proposed by various authors in order to establish the presence of complex dynamics for continuous maps defined on subsets of the Ndimensional Euclidean space (see, for instance, [10,21,23] and the references therein). The tools employed in these and related works range from the Conley index [10] to the Lefschetz fixed point theory [20]. On the other hand, our approach, although mathematically rigorous, avoids the use of more advanced topological theories and it is relatively easy to apply to specific models arising in applications. For example we have employed such method to study discrete and continuoustime models arising from economics and biology [9,18]. In more details, the topics considered along the thesis can be summarized as follows. The description of the Stretching Along the Paths method and suitable variants of it can be found in Chapter 1. In Chapter 2 we discuss which are the chaotic features that can be obtained for a given map when our technique applies. In particular, we are able to prove semiconjugacy to the Bernoulli shift and thus positivity of the topological entropy, the presence of topological transitivity and sensitivity with respect to initial conditions, density of periodic points. Moreover we show the mutual relationships among various classical notions of chaos (such as those by Devaney, LiYorke, etc.). We also introduce an alternative geometrical framework related to the socalled "Linked Twist Maps" [3,4,22], where it is possible to employ our method in order to detect complex dynamics. The theoretical results obtained so far find an application to discrete and continuoustime systems in Chapters 3 and 4. As regards the former, in Chapter 3 we deal with some onedimensional and planar discrete economic models, both of the Overlapping Generation and of the Duopoly Game classes. The bidimensional models are taken from [8,19] and [1], respectively. On the other hand, in Chapter 4, with respect to continuoustime models, we study some nonlinear ODEs with periodic coefficients through a combination of a careful but elementary phaseplane analysis with the results on chaotic dynamics for Linked Twist Maps from Chapter 2. In more details, we consider a modified version of the Volterra predatorprey model, in which a periodic harvesting is included, as well as a simplification of the LazerMcKenna suspension bridges model [6,7] from [13,14]. When dealing with ODEs with periodic coefficients, our method is applied to the associated Poincaré map. The contents of the present thesis are based on the papers [9,13,16,17,18] and partially on [14], where maps expansive along several directions were considered. [1] H.N. Agiza and A.A. Elsadany, Chaotic dynamics in nonlinear duopoly game with heterogeneous players, Appl. Math. Comput. 149 (2004), 843860. [2] K. Burns and H. Weiss, A geometric criterion for positive topological entropy, Comm. Math. Phys. 172 (1995), 95118. [3] R. Burton and R.W. Easton, Ergodicity of linked twist maps, In: Global theory of dynamical systems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979), pp. 3549, Lecture Notes in Math., 819, Springer, Berlin, 1980. [4] R.L. Devaney, Subshifts of finite type in linked twist mappings, Proc. Amer. Math. Soc. 71 (1978), 334338. [5] J. Kennedy and J.A. Yorke, Topological horseshoes, Trans. Amer. Math. Soc. 353 (2001), 25132530. [6] A.C. Lazer and P.J. McKenna, Large scale oscillatory behaviour in loaded asymmetric systems, Ann. Inst. Henry Poincar e, Analyse non lineaire 4 (1987), 244274. [7] A.C. Lazer and P.J. McKenna, Largeamplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis, SIAM Review 32 (1990), 537578. [8] A. Medio, Chaotic dynamics. Theory and applications to economics, Cambridge University Press, Cambridge, 1992. [9] A. Medio, M. Pireddu and F. Zanolin, Chaotic dynamics for maps in one and two dimensions. A geometrical method and applications to economics, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 19 (2009), 32833309. [10] K. Mischaikow and M. Mrozek, Isolating neighborhoods and chaos, Japan J. Indust. Appl. Math. 12 (1995), 205236. [11] D. Papini and F. Zanolin, On the periodic boundary value problem and chaoticlike dynamics for nonlinear Hill's equations, Adv. Nonlinear Stud. 4 (2004), 7191. [12] D. Papini and F. Zanolin, Fixed points, periodic points, and cointossing sequences for mappings defined on twodimensional cells, Fixed Point Theory Appl. 2004 (2004), 113134. [13] A. Pascoletti, M. Pireddu and F. Zanolin, Multiple periodic solutions and complex dynamics for second order ODEs via linked twist maps, Electron. J. Qual. Theory Differ. Equ., Proc. 8'th Coll. Qualitative Theory of Diff. Equ. 14 (2008), 132. [14] A. Pascoletti and F. Zanolin, Example of a suspension bridge ODE model exhibiting chaotic dynamics: a topological approach, J. Math. Anal. Appl. 339 (2008), 11791198. [15] M. Pireddu and F. Zanolin, Fixed points for dissipativerepulsive systems and topological dynamics of mappings defined on Ndimensional cells, Adv. Nonlinear Stud. 5 (2005), 411440. [16] M. Pireddu and F. Zanolin, Cutting surfaces and applications to periodic points and chaoticlike dynamics, Topol. Methods Nonlinear Anal. 30 (2007), 279319. [17] M. Pireddu and F. Zanolin, Some remarks on fixed points for maps which are expansive along one direction, Rend. Istit. Mat. Univ. Trieste 39 (2007), 245274. [18] M. Pireddu and F. Zanolin, Chaotic dynamics in the Volterra predatorprey model via linked twist maps, Opuscula Math. 28/4 (2008), 567592. [19] P. Reichlin, Equilibrium cycles in an overlapping generations economy with production, J. Econom. Theory 40 (1986), 89102. [20] R. Srzednicki, A generalization of the Lefschetz fixed point theorem and detection of chaos, Proc. Amer. Math. Soc. 128 (2000), 12311239. [21] R. Srzednicki and K. Wojcik, A geometric method for detecting chaotic dynamics, J. Differential Equations 135 (1997), 6682. [22] S. Wiggins, Chaos in the dynamics generated by sequence of maps, with application to chaotic advection in flows with aperiodic time dependence, Z. angew. Math. Phys. 50 (1999), 585616. [23] P. Zgliczy nski and M. Gidea, Covering relations for multidimensional dynamical systems, J. Differential Equations 202 (2004), 3258.File  Dimensione  Formato  

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https://hdl.handle.net/20.500.14242/106794
URN:NBN:IT:UNIMIB106794