American Institute of Mathematical Sciences

Advanced
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)
Figure Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Mats Gyllenberg, Ping Yan. On the number of limit cycles for three dimensional Lotka-Volterra systems. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 347-352. doi: 10.3934/dcdsb.2009.11.347
[2]

Victoriano Carmona, Soledad Fernández-García, Antonio E. Teruel. Saddle-node of limit cycles in planar piecewise linear systems and applications. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5275-5299. doi: 10.3934/dcds.2019215
[3]

Song-Mei Huan, Xiao-Song Yang. On the number of limit cycles in general planar piecewise linear systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2147-2164. doi: 10.3934/dcds.2012.32.2147
[4]

Shimin Li, Jaume Llibre. On the limit cycles of planar discontinuous piecewise linear differential systems with a unique equilibrium. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5885-5901. doi: 10.3934/dcdsb.2019111
[5]

Tongren Ding, Hai Huang, Fabio Zanolin. A priori bounds and periodic solutions for a class of planar systems with applications to Lotka-Volterra equations. Discrete & Continuous Dynamical Systems - A, 1995, 1 (1) : 103-117. doi: 10.3934/dcds.1995.1.103
[6]

Peter Ashwin, Xin-Chu Fu. Symbolic analysis for some planar piecewise linear maps. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1533-1548. doi: 10.3934/dcds.2003.9.1533
[7]

Michael Y. Li, Xihui Lin, Hao Wang. Global Hopf branches and multiple limit cycles in a delayed Lotka-Volterra predator-prey model. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 747-760. doi: 10.3934/dcdsb.2014.19.747
[8]

Jaume Llibre, Lucyjane de A. S. Menezes. On the limit cycles of a class of discontinuous piecewise linear differential systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1835-1858. doi: 10.3934/dcdsb.2020005
[9]

Rui Xu, M.A.J. Chaplain, F.A. Davidson. Periodic solutions of a discrete nonautonomous Lotka-Volterra predator-prey model with time delays. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 823-831. doi: 10.3934/dcdsb.2004.4.823
[10]

Qihuai Liu, Dingbian Qian, Zhiguo Wang. Quasi-periodic solutions of the Lotka-Volterra competition systems with quasi-periodic perturbations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1537-1550. doi: 10.3934/dcdsb.2012.17.1537
[11]

Chiun-Chuan Chen, Yin-Liang Huang, Li-Chang Hung, Chang-Hong Wu. Semi-exact solutions and pulsating fronts for Lotka-Volterra systems of two competing species in spatially periodic habitats. Communications on Pure & Applied Analysis, 2020, 19 (1) : 1-18. doi: 10.3934/cpaa.2020001
[12]

Norimichi Hirano, Wieslaw Krawcewicz, Haibo Ruan. Existence of nonstationary periodic solutions for $\Gamma$-symmetric Lotka-Volterra type systems. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 709-735. doi: 10.3934/dcds.2011.30.709
[13]

Victoriano Carmona, Emilio Freire, Soledad Fernández-García. Periodic orbits and invariant cones in three-dimensional piecewise linear systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 59-72. doi: 10.3934/dcds.2015.35.59
[14]

Lih-Ing W. Roeger, Razvan Gelca. Dynamically consistent discrete-time Lotka-Volterra competition models. Conference Publications, 2009, 2009 (Special) : 650-658. doi: 10.3934/proc.2009.2009.650
[15]

Xiaoli Liu, Dongmei Xiao. Bifurcations in a discrete time Lotka-Volterra predator-prey system. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 559-572. doi: 10.3934/dcdsb.2006.6.559
[16]

M. R. S. Kulenović, J. Marcotte, O. Merino. Properties of basins of attraction for planar discrete cooperative maps. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020202
[17]

Allyson Oliveira, Hildeberto Cabral. On stacked central configurations of the planar coorbital satellites problem. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3715-3732. doi: 10.3934/dcds.2012.32.3715
[18]

Shaohua Chen, Runzhang Xu, Hongtao Yang. Global and blowup solutions for general Lotka-Volterra systems. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1757-1768. doi: 10.3934/cpaa.2016012
[19]

Bang-Sheng Han, Zhi-Cheng Wang, Zengji Du. Traveling waves for nonlocal Lotka-Volterra competition systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1959-1983. doi: 10.3934/dcdsb.2020011
[20]

Suqing Lin, Zhengyi Lu. Permanence for two-species Lotka-Volterra systems with delays. Mathematical Biosciences & Engineering, 2006, 3 (1) : 137-144. doi: 10.3934/mbe.2006.3.137

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (140)
  • HTML views (94)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences