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

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

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.

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. BalibreaJ. L. García GuiraoM. 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. CasasayasJ. 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. CimaA. 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. CimaA. 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. CimaA. van den EssenA. GasullE. 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. CorsJ. 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. DickensteinJ. M. RojasK. 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. FreireE. 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-SaldañaA. 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-SaldañaA. 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. GasullM. 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. LiJ. 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. BalibreaJ. L. García GuiraoM. 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. CasasayasJ. 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. CimaA. 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. CimaA. 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. CimaA. van den EssenA. GasullE. 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. CorsJ. 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. DickensteinJ. M. RojasK. 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. FreireE. 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-SaldañaA. 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-SaldañaA. 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. GasullM. 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. LiJ. 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)
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)
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)
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)
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)
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)
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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