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February  2020, 25(2): 651-670. doi: 10.3934/dcdsb.2019259

Periodic orbits of discrete and continuous dynamical systems via Poincaré-Miranda theorem

Armengol Gasull 1,, and  Víctor Mañosa 2,

1. 

Departament de Matemàtiques, Universitat Autònoma de Barcelona, Facultat de Ciències, 08193 Bellaterra (Barcelona), Spain
2. 

Departament de Matemàtiques, Universitat Politècnica de Catalunya, Colom 1, 08222 Terrassa, Spain

* Corresponding author: Armengol Gasull

Received  September 2018 Revised  December 2018 Published  November 2019

We present a systematic methodology to determine and locate analytically isolated periodic points of discrete and continuous dynamical systems with algebraic nature. We apply this method to a wide range of examples, including a one-parameter family of counterexamples to the discrete Markus-Yamabe conjecture (La Salle conjecture); the study of the low periods of a Lotka-Volterra-type map; the existence of three limit cycles for a piecewise linear planar vector field; a new counterexample of Kouchnirenko conjecture; and an alternative proof of the existence of a class of symmetric central configuration of the $ (1+4) $-body problem.

Keywords: Poincaré-Miranda theorem, periodic orbits, Lotka-Volterra maps, Thue-Morse maps, discrete Markus-Yamabe conjecture, Kouchnirenko conjecture, Limit cycles, planar piecewise linear systems, central configurations.
Mathematics Subject Classification: Primary: 37C25, 39A23; Secondary: 13P15, 34D23, 70F15, 70K05.
Citation: Armengol Gasull, Víctor Mañosa. Periodic orbits of discrete and continuous dynamical systems via Poincaré-Miranda theorem. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 651-670. doi: 10.3934/dcdsb.2019259
References:
[1]

Y. Avishai and D. Berend, Transmission through a Thue-Morse chain, Phys. Rev. B., 45 (1992), 2717-2724.  doi: 10.1103/PhysRevB.45.2717.  Google Scholar

[2]

F. Balibrea, J. L. García Guirao, M. Lampart and J. Llibre, Dynamics of a Lotka-Volterra map, Fundamenta Mathematicae, 191 (2006), 265-279.  doi: 10.4064/fm191-3-5.  Google Scholar

[3]

J. Bernat and J. Llibre, Counterexample to Kalman and Markus-Yamabe conjectures in dimension larger than 3, Dynam. Contin. Discrete Impuls. Systems, 2 (1996), 337-379.   Google Scholar

[4]

J. Casasayas, J. Llibre and A. Nunes, Central configurations of the planar 1+n-body problem, Celestial Mech. Dynam. Astronom., 60 (1994), 273-288.  doi: 10.1007/BF00693325.  Google Scholar

[5]

A. Cima, A. Gasull and F. Mañosas, The discrete Markus-Yamabe problem, Nonlinear Anal. TMA, 35 (1999), 343-354.  doi: 10.1016/S0362-546X(97)00715-3.  Google Scholar

[6]

A. Cima, A. Gasull and F. Mañosas, On the global asymptotic stability of difference equations satisfying a Markus-Yamabe condition, Publ. Mat., 58 (2014), 167-178.  doi: 10.5565/PUBLMAT_Extra14_09.  Google Scholar

[7]

A. Cima, A. van den Essen, A. Gasull, E. Hubbers and F. Mañosas, A polynomial counterexample to the Markus-Yamabe conjecture, Adv. Math., 131 (1997), 453-457.  doi: 10.1006/aima.1997.1673.  Google Scholar

[8]

J. M. Cors, J. Llibre and M. Ollé, Central configurations of the planar coorbital satellite problem, Celestial Mech. Dynam. Astronom., 89 (2004), 319-342.  doi: 10.1023/B:CELE.0000043569.25307.ab.  Google Scholar

[9]

A. Dickenstein, J. M. Rojas, K. Rusek and J. Shih, Extremal Real Algebraic Geometry and $\mathcal{A}$-Discriminants, Moscow Math. Journal, 7 (2007), 425-452,574.  doi: 10.17323/1609-4514-2007-7-3-425-452.  Google Scholar

[10]

G. H. Erjaee and F. M. Dannan, Stability analysis of periodic solutions to the nonstandard discretized model of the Lotka-Volterra predator-prey system, Int. J. Bifurcation and Chaos, 14 (2004), 4301-4308.  doi: 10.1142/S0218127404011946.  Google Scholar

[11]

E. Freire, E. Ponce and F. Torres, The discontinuous matching of two planar linear foci can have three nested crossing limit cycles, Publ. Mat., 58 (2014), 221-253.  doi: 10.5565/PUBLMAT_Extra14_13.  Google Scholar

[12]

J. García-Saldaña, A. Gasull and H. Giacomini, Bifurcation diagram and stability for a one-parameter family of planar vector fields, J. Math. Anal. Appl., 413 (2014), 321-342.  doi: 10.1016/j.jmaa.2013.11.047.  Google Scholar

[13]

J. D. García-Saldaña, A. Gasull and H. Giacomini, Bifurcation values for a familiy of planar vector fields of degree five, Discrete Contin. Dyn. Syst. A, 35 (2015), 669-701.  doi: 10.3934/dcds.2015.35.669.  Google Scholar

[14]

A. Gasull, M. Llorens and V. Mañosa, Periodic points of a Landen transformation, Commun. Nonlinear Sci. Numer. Simulat., 64 (2018), 232-245.  doi: 10.1016/j.cnsns.2018.04.020.  Google Scholar

[15]

A. Granas and J. Dugundji, Fixed Point Theory, Springer Monographs in Mathematics, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21593-8.  Google Scholar

[16]

C. Gutiérrez, A solution to the bidimensional global asymptotic stability conjecture, Ann. Inst. H. Poincaré Anal. Non Linéaire, 12 (1995), 627-671.  doi: 10.1016/S0294-1449(16)30147-0.  Google Scholar

[17]

B. Haas, A simple counterexample to Kouchnirenko's conjecture, Beiträge zur Algebra und Geometrie, 43 (2002), 1-8.   Google Scholar

[18]

S.-M. Huan and X.-S. Yang, On the number of limit cycles in general planar piecewise linear systems, Discrete Contin. Dyn. Syst. A, 32 (2012), 2147-2164.  doi: 10.3934/dcds.2012.32.2147.  Google Scholar

[19]

E. Isaacson and H. B. Keller, Analysis of Numerical Methods, Dover Publications, Inc., New York, 1994.  Google Scholar

[20]

A. G. Khovanskiǐ, On a class of systems of transcendental equations, Doklady Akad. Nauk. SSSR, 255 (1980), 804-807.   Google Scholar

[21]

W. Kulpa, The Poincaré-Miranda theorem, Amer. Math. Month., 104 (1997), 545-550.  doi: 10.2307/2975081.  Google Scholar

[22]

J. P. La Salle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1976.  Google Scholar

[23]

T.-Y. Li, J. M. Rojas and X. S. Wang, Counting real connected components of trinomial curve intersections and m-nomial hypersurfaces, Discrete Comput. Geom., 30 (2003), 379-414.  doi: 10.1007/s00454-003-2834-8.  Google Scholar

[24]

J. Llibre, On the central configurations of the n-body problem, Appl. Math. Nonlinear Sci., 2 (2017), 509-518.  doi: 10.21042/AMNS.2017.2.00042.  Google Scholar

[25]

J. Llibre and E. Ponce, Three nested limit cycles in discontinuous piecewise linear differential systems with two zones, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 19 (2012), 325-335.   Google Scholar

[26]

P. Maličký, Interior periodic points of a Lotka-Volterra map, J. Difference Eq. Appl., 18 (2012), 553-567.  doi: 10.1080/10236198.2011.583241.  Google Scholar

[27]

L. Markus and H. Yamabe, Global stability criteria for differential systems, Osaka Math. Journal, 12 (1960), 305-317.   Google Scholar

[28]

C. Miranda, Un'osservazione su un teorema di Brouwer, Boll. Unione Mat. Ital., 3 (1940), 5-7.   Google Scholar

[29]

H. Poincaré, Sur certaines solutions particulieres du probléme des trois corps, Bull. Astronomique, 1 (1884), 63-74.   Google Scholar

[30]

H. Poincaré, Sur les courbes définies par une équation différentielle Ⅳ, J. Math. Pures Appl., 85 (1886), 151-217.   Google Scholar

[31]

A. N. Sharkovskiǐ, Low dimensional dynamics, Tagungsbericht 20/1993, Proceedings of Mathematisches Forschungsinstitut Oberwolfach, (1993), 17. Google Scholar

[32]

J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, Third edition, Texts in Applied Mathematics, 12. Springer-Verlag, New York, 2002. doi: 10.1007/978-0-387-21738-3.  Google Scholar

[33]

M. N. Vrahatis, A short proof and a Generalization of Miranda's existence Theorem, Proc. Amer. Math. Soc., 107 (1989), 701-703.  doi: 10.2307/2048168.  Google Scholar

show all references

References:
[1]

Y. Avishai and D. Berend, Transmission through a Thue-Morse chain, Phys. Rev. B., 45 (1992), 2717-2724.  doi: 10.1103/PhysRevB.45.2717.  Google Scholar

[2]

F. Balibrea, J. L. García Guirao, M. Lampart and J. Llibre, Dynamics of a Lotka-Volterra map, Fundamenta Mathematicae, 191 (2006), 265-279.  doi: 10.4064/fm191-3-5.  Google Scholar

[3]

J. Bernat and J. Llibre, Counterexample to Kalman and Markus-Yamabe conjectures in dimension larger than 3, Dynam. Contin. Discrete Impuls. Systems, 2 (1996), 337-379.   Google Scholar

[4]

J. Casasayas, J. Llibre and A. Nunes, Central configurations of the planar 1+n-body problem, Celestial Mech. Dynam. Astronom., 60 (1994), 273-288.  doi: 10.1007/BF00693325.  Google Scholar

[5]

A. Cima, A. Gasull and F. Mañosas, The discrete Markus-Yamabe problem, Nonlinear Anal. TMA, 35 (1999), 343-354.  doi: 10.1016/S0362-546X(97)00715-3.  Google Scholar

[6]

A. Cima, A. Gasull and F. Mañosas, On the global asymptotic stability of difference equations satisfying a Markus-Yamabe condition, Publ. Mat., 58 (2014), 167-178.  doi: 10.5565/PUBLMAT_Extra14_09.  Google Scholar

[7]

A. Cima, A. van den Essen, A. Gasull, E. Hubbers and F. Mañosas, A polynomial counterexample to the Markus-Yamabe conjecture, Adv. Math., 131 (1997), 453-457.  doi: 10.1006/aima.1997.1673.  Google Scholar

[8]

J. M. Cors, J. Llibre and M. Ollé, Central configurations of the planar coorbital satellite problem, Celestial Mech. Dynam. Astronom., 89 (2004), 319-342.  doi: 10.1023/B:CELE.0000043569.25307.ab.  Google Scholar

[9]

A. Dickenstein, J. M. Rojas, K. Rusek and J. Shih, Extremal Real Algebraic Geometry and $\mathcal{A}$-Discriminants, Moscow Math. Journal, 7 (2007), 425-452,574.  doi: 10.17323/1609-4514-2007-7-3-425-452.  Google Scholar

[10]

G. H. Erjaee and F. M. Dannan, Stability analysis of periodic solutions to the nonstandard discretized model of the Lotka-Volterra predator-prey system, Int. J. Bifurcation and Chaos, 14 (2004), 4301-4308.  doi: 10.1142/S0218127404011946.  Google Scholar

[11]

E. Freire, E. Ponce and F. Torres, The discontinuous matching of two planar linear foci can have three nested crossing limit cycles, Publ. Mat., 58 (2014), 221-253.  doi: 10.5565/PUBLMAT_Extra14_13.  Google Scholar

[12]

J. García-Saldaña, A. Gasull and H. Giacomini, Bifurcation diagram and stability for a one-parameter family of planar vector fields, J. Math. Anal. Appl., 413 (2014), 321-342.  doi: 10.1016/j.jmaa.2013.11.047.  Google Scholar

[13]

J. D. García-Saldaña, A. Gasull and H. Giacomini, Bifurcation values for a familiy of planar vector fields of degree five, Discrete Contin. Dyn. Syst. A, 35 (2015), 669-701.  doi: 10.3934/dcds.2015.35.669.  Google Scholar

[14]

A. Gasull, M. Llorens and V. Mañosa, Periodic points of a Landen transformation, Commun. Nonlinear Sci. Numer. Simulat., 64 (2018), 232-245.  doi: 10.1016/j.cnsns.2018.04.020.  Google Scholar

[15]

A. Granas and J. Dugundji, Fixed Point Theory, Springer Monographs in Mathematics, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21593-8.  Google Scholar

[16]

C. Gutiérrez, A solution to the bidimensional global asymptotic stability conjecture, Ann. Inst. H. Poincaré Anal. Non Linéaire, 12 (1995), 627-671.  doi: 10.1016/S0294-1449(16)30147-0.  Google Scholar

[17]

B. Haas, A simple counterexample to Kouchnirenko's conjecture, Beiträge zur Algebra und Geometrie, 43 (2002), 1-8.   Google Scholar

[18]

S.-M. Huan and X.-S. Yang, On the number of limit cycles in general planar piecewise linear systems, Discrete Contin. Dyn. Syst. A, 32 (2012), 2147-2164.  doi: 10.3934/dcds.2012.32.2147.  Google Scholar

[19]

E. Isaacson and H. B. Keller, Analysis of Numerical Methods, Dover Publications, Inc., New York, 1994.  Google Scholar

[20]

A. G. Khovanskiǐ, On a class of systems of transcendental equations, Doklady Akad. Nauk. SSSR, 255 (1980), 804-807.   Google Scholar

[21]

W. Kulpa, The Poincaré-Miranda theorem, Amer. Math. Month., 104 (1997), 545-550.  doi: 10.2307/2975081.  Google Scholar

[22]

J. P. La Salle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1976.  Google Scholar

[23]

T.-Y. Li, J. M. Rojas and X. S. Wang, Counting real connected components of trinomial curve intersections and m-nomial hypersurfaces, Discrete Comput. Geom., 30 (2003), 379-414.  doi: 10.1007/s00454-003-2834-8.  Google Scholar

[24]

J. Llibre, On the central configurations of the n-body problem, Appl. Math. Nonlinear Sci., 2 (2017), 509-518.  doi: 10.21042/AMNS.2017.2.00042.  Google Scholar

[25]

J. Llibre and E. Ponce, Three nested limit cycles in discontinuous piecewise linear differential systems with two zones, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 19 (2012), 325-335.   Google Scholar

[26]

P. Maličký, Interior periodic points of a Lotka-Volterra map, J. Difference Eq. Appl., 18 (2012), 553-567.  doi: 10.1080/10236198.2011.583241.  Google Scholar

[27]

L. Markus and H. Yamabe, Global stability criteria for differential systems, Osaka Math. Journal, 12 (1960), 305-317.   Google Scholar

[28]

C. Miranda, Un'osservazione su un teorema di Brouwer, Boll. Unione Mat. Ital., 3 (1940), 5-7.   Google Scholar

[29]

H. Poincaré, Sur certaines solutions particulieres du probléme des trois corps, Bull. Astronomique, 1 (1884), 63-74.   Google Scholar

[30]

H. Poincaré, Sur les courbes définies par une équation différentielle Ⅳ, J. Math. Pures Appl., 85 (1886), 151-217.   Google Scholar

[31]

A. N. Sharkovskiǐ, Low dimensional dynamics, Tagungsbericht 20/1993, Proceedings of Mathematisches Forschungsinstitut Oberwolfach, (1993), 17. Google Scholar

[32]

J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, Third edition, Texts in Applied Mathematics, 12. Springer-Verlag, New York, 2002. doi: 10.1007/978-0-387-21738-3.  Google Scholar

[33]

M. N. Vrahatis, A short proof and a Generalization of Miranda's existence Theorem, Proc. Amer. Math. Soc., 107 (1989), 701-703.  doi: 10.2307/2048168.  Google Scholar

Figure 1.  Intersection of the curves $ P(x,y) = 0 $ (in blue) and $ Q(x,y) = 0 $ (in magenta). The PM boxes $ I_1\times I_5 $, $ I_2\times I_4 $ (left and right respectively, in red)
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Figure 2.  Intersection the curves $ g_1(x,y;0.9) = 0 $ (in blue) and $ g_2(x,y;0.9) = 0 $ (in magenta). It can be seen that there are seven intersections corresponding to one fixed point and two different $ 3 $-periodic orbits. In red, a PM box of one of the solutions of system (4)
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Figure 3.  The PM box of a solution of system (6) used in the proof of Theorem 4.1 (in red). It corresponds to the intersection of the curves defined by the curves $ T_{5,1}(x,y) = 0 $ (in blue) and $ T_{5,2}(x,y) = 0 $ (in magenta)
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Figure 4.  Two $ 5 $-periodic orbits of the map (5) in $ \mathrm{Int}(\triangle) $ (Orbit 1 in red and Orbit 2 in green). They correspond to the intersection of the curves defined by the curves $ T_{5,1}(x,y) = 0 $ (in blue) and $ T_{5,2}(x,y) = 0 $ (in magenta). The fixed point $ (1,2) $ (in brown). The PM box containing the point $ P_{9,10} $ used in the proof of Theorem 4.1 (in red)
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Figure 5.  Three $ 6 $-periodic orbits of the map (5) in $ \mathrm{Int}(\triangle) $ (Orbit 1, 2 and 3 in brown, red and green, respectively). They correspond to the intersection of the curves defined by the curves $ T_{6,1}(x,y) = 0 $ (in blue) and $ T_{6,2}(x,y) = 0 $ (in magenta). The fixed point $ (1,2) $ (in black)
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Figure 6.  Left part: Intersection points between $ g_{1}(u,v) = 0 $ (in blue) and $ g_{2}(u,v) = 0 $ (in magenta) and some PM boxes containing them. Right part: the 3 limit cycles of system (10)
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Figure 7.  Left figure: graph of $ f(\theta) $ in $ (0,\pi] $. Right figure: intersection of the curves $ g_1(u,v) = 0 $ (in blue) and $ g_2(u,v) = 0 $ (in magenta). In red, a PM box
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