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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Stability for one-dimensional discrete dynamical systems revisited
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 |
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 and 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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [17] |
B. Haas,
A simple counterexample to Kouchnirenko's conjecture, Beiträge zur Algebra und Geometrie, 43 (2002), 1-8.
|
| [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. |
| [19] |
E. Isaacson and H. B. Keller, Analysis of Numerical Methods, Dover Publications, Inc., New York, 1994. |
| [20] |
A. G. Khovanskiǐ,
On a class of systems of transcendental equations, Doklady Akad. Nauk. SSSR, 255 (1980), 804-807.
|
| [21] |
W. Kulpa,
The Poincaré-Miranda theorem, Amer. Math. Month., 104 (1997), 545-550.
doi: 10.2307/2975081. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [27] |
L. Markus and H. Yamabe,
Global stability criteria for differential systems, Osaka Math. Journal, 12 (1960), 305-317.
|
| [28] |
C. Miranda,
Un'osservazione su un teorema di Brouwer, Boll. Unione Mat. Ital., 3 (1940), 5-7.
|
| [29] |
H. Poincaré,
Sur certaines solutions particulieres du probléme des trois corps, Bull. Astronomique, 1 (1884), 63-74.
|
| [30] |
H. Poincaré,
Sur les courbes définies par une équation différentielle Ⅳ, J. Math. Pures Appl., 85 (1886), 151-217.
|
| [31] |
A. N. Sharkovskiǐ, Low dimensional dynamics, Tagungsbericht 20/1993, Proceedings of Mathematisches Forschungsinstitut Oberwolfach, (1993), 17. |
| [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. |
| [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. |
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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [17] |
B. Haas,
A simple counterexample to Kouchnirenko's conjecture, Beiträge zur Algebra und Geometrie, 43 (2002), 1-8.
|
| [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. |
| [19] |
E. Isaacson and H. B. Keller, Analysis of Numerical Methods, Dover Publications, Inc., New York, 1994. |
| [20] |
A. G. Khovanskiǐ,
On a class of systems of transcendental equations, Doklady Akad. Nauk. SSSR, 255 (1980), 804-807.
|
| [21] |
W. Kulpa,
The Poincaré-Miranda theorem, Amer. Math. Month., 104 (1997), 545-550.
doi: 10.2307/2975081. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [27] |
L. Markus and H. Yamabe,
Global stability criteria for differential systems, Osaka Math. Journal, 12 (1960), 305-317.
|
| [28] |
C. Miranda,
Un'osservazione su un teorema di Brouwer, Boll. Unione Mat. Ital., 3 (1940), 5-7.
|
| [29] |
H. Poincaré,
Sur certaines solutions particulieres du probléme des trois corps, Bull. Astronomique, 1 (1884), 63-74.
|
| [30] |
H. Poincaré,
Sur les courbes définies par une équation différentielle Ⅳ, J. Math. Pures Appl., 85 (1886), 151-217.
|
| [31] |
A. N. Sharkovskiǐ, Low dimensional dynamics, Tagungsbericht 20/1993, Proceedings of Mathematisches Forschungsinstitut Oberwolfach, (1993), 17. |
| [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. |
| [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. |
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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