May  2015, 20(3): 915-944. doi: 10.3934/dcdsb.2015.20.915

Continuous separation for monotone skew-product semiflows: From theoretical to numerical results

1. 

Departamento de Matemática Aplicada, E. Ingenierías Industriales, Universidad de Valladolid, 47011 Valladolid, Spain

2. 

Departamento de Matemática Aplicada, Escuela de Ingenierías Industriales and member of IMUVA, Instituto de Matemáticas, Universidad de Valladolid, 47011 Valladolid

3. 

Departamento de Didáctica de las Ciencias Experimentales, Sociales y de la Matemática, Facultad de Educación and member of IMUVA, Instituto de Matemáticas, Universidad de Valladolid, 34004 Palencia, Spain

Received  September 2013 Revised  May 2014 Published  January 2015

This paper investigates relevant dynamical properties of nonautonomous linear cooperative families of ODEs and FDEs based on the existence of a continuous separation. It provides numerical algorithms for the computation of the dominant one-dimensional subbundle of the continuous separation and the upper Lyapunov exponent of the semiflow. The extension of the theory to general linear cooperative families of ODEs and FDEs without strong monotonicity is also given. Finally these methods and results are applied in the study of nonlinear families of neural networks of Hopfield type with sigmoidal activation function.
Citation: Juan A. Calzada, Rafael Obaya, Ana M. Sanz. Continuous separation for monotone skew-product semiflows: From theoretical to numerical results. Discrete and Continuous Dynamical Systems - B, 2015, 20 (3) : 915-944. doi: 10.3934/dcdsb.2015.20.915
References:
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H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976), 620-709. doi: 10.1137/1018114.

[2]

O. Arratia, R. Obaya and M. E. Sansaturio, Applications of the exponential ordering in the study of almost-periodic delayed Hopfield neural networks, Physica D, 241 (2012), 1551-1566. doi: 10.1016/j.physd.2012.06.007.

[3]

G. Benettin, L. Galgani, A. Giorgilli and J. M. Strelcyn, Lyapunov exponents for smooth dynamical systems and for hamiltonian systems; a method for computing all of them. Part 1: Theory, Meccanica, 15 (1980), 9-20. doi: 10.1007/BF02128236.

[4]

G. Benettin, L. Galgani, A. Giorgilli and J. M. Strelcyn, Lyapunov exponents for smooth dynamical systems and for hamiltonian systems; a method for computing all of them. Part 1: numerical applications, Meccanica, 15 (1980), 21-30.

[5]

D. Breda, S. Maset and R. Vermiglio, Approximation of eigenvalues of evolution operators for lineal retarded functional differential equations, SIAM J. Numer. Anal., 50 (2012), 1456-1483. doi: 10.1137/100815505.

[6]

D. Breda and E. Van Vleck, Approximating Lyapunov exponents and Sacker-Sell spectrum for retarded functional differential equations, Numer. Math, 126 (2014), 225-257. doi: 10.1007/s00211-013-0565-1.

[7]

D. Breda, S. Maset and R. Vermiglio, Pseudospectral approximation of eigenvalues of derivative operators with non-local boundary conditions, Appl. Numer. Math., 56 (2006), 318-331. doi: 10.1016/j.apnum.2005.04.011.

[8]

L.O. Chua and L. Yang, Cellular neural network: Theory, IEEE Trans. Circuits. Syst., 35 (1988), 1257-1272. doi: 10.1109/31.7600.

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L. O. Chua and L. Yang, Cellular neural network: Applications, IEEE Trans. Circuits. Syst., 35 (1988), 1273-1290. doi: 10.1109/31.7601.

[10]

I. Chueshov, Monotone Random Systems: Theory and Applications, Springer-Verlag, Berlin, 2002. doi: 10.1007/b83277.

[11]

L. Dieci, R. D. Russell and E. S. Van Vleck, On the computation of Lyapunov exponents for continuous dynamical systems, SIAM J. Numer. Anal. 34 (1997), 402-423. doi: 10.1137/S0036142993247311.

[12]

L. Dieci and E. S. Van Vleck, Perturbation theory for approximation of Lyapunov exponent by QR methods, J. Dyn. Diff. Equat., 18 (2006), 815-840. doi: 10.1007/s10884-006-9024-3.

[13]

C. Elia and R. Fabbri, Rotation number and exponential dichotomy for linear hamiltonian systems: From theoretical to numerical results, J. Dynam. Differential Equations, 25 (2013), 95-120. doi: 10.1007/s10884-013-9290-9.

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R. Ellis, Lectures on Topological Dynamics, Benjamin, New York, 1969.

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K. Engelborghs and D. Roose, On stability of LMS methods and characteristic roots of delay differential equations, SIAM J. Numer. Anal., 40 (2002), 629-650. doi: 10.1137/S003614290037472X.

[16]

W. H. Enright, A new error-control for initial value solvers, Appl. Math. Comput., 31 (1989), 288-301. doi: 10.1016/0096-3003(89)90123-9.

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J. D. Farmer, Chaotic attractors of an infinite-dimensional system, Physica D, 4 (1981/82), 366-393. doi: 10.1016/0167-2789(82)90042-2.

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M. Fiedler, Special Matrices and Their Applications in Numerical Mathematics, Dover Publications, Inc., Mineola, New York, 2008.

[19]

K. Geist, U. Parlitz and W. Lauterborn, Comparison of different methods for computing Lyapunov exponents, Prog. Theor. Phys., 83 (1990), 875-893. doi: 10.1143/PTP.83.875.

[20]

K. Gopalsamy and X. Z. He, Stability in asymmetric Hopfield nets with transmission delays, Physica D, 76 (1994), 344-358. doi: 10.1016/0167-2789(94)90043-4.

[21]

S. Grossberg, Nonlinear networks: Principles, mechanisms and architectures, Neural Netw., 1 (1988), 17-61. doi: 10.1016/0893-6080(88)90021-4.

[22]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, 99, Springer-Verlag, Berlin, Heidelberg, New York, 1993. doi: 10.1007/978-1-4612-4342-7.

[23]

J. J. Hopfield, Neural networks and physical systems with emergent collective computational abilities, Proc. Nat. Acad. Sci. USA, 79 (1982), 2554-2558. doi: 10.1073/pnas.79.8.2554.

[24]

J. J. Hopfield, Neurons with graded response have collective computational properties like those of two-state neurons, Proc. Nat. Acad. Sci. USA, 81 (1984), 2554-2558. doi: 10.1073/pnas.79.8.2554.

[25]

T. Insperger and G. Stépán, Semi-Discretization for Time-Delay Systems. Stability and Engineering Applications, Applied Mathematical Sciences, 178, Springer, New York, 2011. doi: 10.1007/978-1-4614-0335-7.

[26]

R. Johnson and F. Mantellini, Non-autonomous differential equations, in Dynamical Systems, Lecture Notes in Math., 1822, Springer, Berlin, 2003, 173-229. doi: 10.1007/978-3-540-45204-1_3.

[27]

R. Johnson, K. J. Palmer and G. R. Sell, Ergodic properties of linear dynamical systems, SIAM J. Math. Anal., 18 (1987), 1-33. doi: 10.1137/0518001.

[28]

W. Liu and E. S. Van Vleck, Exponential dichotomy for asymptotically hyperbolic two-dimensional linear systems, J. Dyn. Diff. Equat., 22 (2010), 697-722. doi: 10.1007/s10884-010-9170-5.

[29]

T. Luzyanina, K. Engelborghs and D. Roose, Computing stability of differential equations with bounded distributed delays, Numer. Algorithms, 34 (2003), 41-66. doi: 10.1023/A:1026194503720.

[30]

C. M. Marcus and R. M. Westervelt, Stability of analog neural networks with delay, Phys. Rev. A, 39 (2009), 347-359. doi: 10.1103/PhysRevA.39.347.

[31]

J. Mierczyński and W. Shen, Lyapunov exponents and asymptotic dynamics in random Kolmogorov models, J. Evol. Equ., 4 (2004), 371-390. doi: 10.1007/s00028-004-0160-0.

[32]

V. Nemytskii and V. Stepanoff, Qualitative Theory of Differential Equations, Princeton University Press, Princeton, NJ, 1960.

[33]

S. Niculescu, Delay Effects on Stability, Lecture Notes in Control and Information Sciences, 269, Springer-Verlag, Berlin, Heidelberg, 2001.

[34]

S. Novo and R. Obaya, Non-autonomous Functional Differential Equations and Applications. in Stability and Bifurcation for Non-Autonomous Differential Equations, Lecture Notes in Math., 2065, Springer-Verlag, Berlin, Heidelberg, 2013, 185-264. doi: 10.1007/978-3-642-32906-7_4.

[35]

S. Novo, R. Obaya and A. M. Sanz, Stability and extensibility results for abstract skew-product semiflows, J. Differential Equations, 235 (2007), 623-646. doi: 10.1016/j.jde.2006.12.009.

[36]

S. Novo, R. Obaya and A. M. Sanz, Topological dynamics for monotone skew-product semiflows with applications, J. Dyn. Diff. Equat., 25 (2013), 1201-1231. doi: 10.1007/s10884-013-9337-y.

[37]

S. Novo, R. Obaya and A. M. Sanz, Uniform persistence and upper Lyapunov exponents for monotone skew-product semiflows, Nonlinearity, 26 (2013), 2409-2440. doi: 10.1088/0951-7715/26/9/2409.

[38]

S. Novo, C. Nuñez, R. Obaya and A. M. Sanz, Skew-product semiflows for non-autonomous partial functional differential equations with delay, Discret. Contin. Dyn. S., 34 (2014), 4291-4321. doi: 10.3934/dcds.2014.34.4291.

[39]

C. Núñez, R. Obaya and A. M. Sanz, Minimal sets in monotone and concave skew-product semiflows I: A general theory, J. Differential Equations, 252 (2012), 5492-5517. doi: 10.1016/j.jde.2012.02.008.

[40]

C. Núñez, R. Obaya and A. M. Sanz, Minimal sets in monotone and concave skew-product semiflows II: Two-dimensional systems of differential equations, J. Differential Equations, 252 (2012), 3575-3607. doi: 10.1016/j.jde.2011.11.016.

[41]

P. Poláčik and I. Tereščák, Exponential separation and invariant bundles for maps in ordered Banach spaces with applications to parabolic equations, J. Dynamics Differential Equations, 5 (1993), 279-303. doi: 10.1007/BF01053163.

[42]

R. J. Sacker and G. R. Sell, Lifting properties in skew-products flows with applications to differential equations, Mem. Amer. Math. Soc., 11 (1977), iv+67 pp.

[43]

R. J. Sacker and G. R. Sell, A spectral theory for linear differential systems, J. Differential Equations, 27 (1978), 320-358. doi: 10.1016/0022-0396(78)90057-8.

[44]

R. J. Sacker and G. R. Sell, Dichotomies for linear evolutionary equations in Banach spaces, J. Differential Equations, 113 (1994), 17-67. doi: 10.1006/jdeq.1994.1113.

[45]

L. F. Shampine and M. W. Reichelt, The Matlab ODE suite, SIAM J. Sci. Comput., 18 (1997), 1-22. doi: 10.1137/S1064827594276424.

[46]

L. F. Shampine, I. Gladwell and S. Thompson, Solving ODEs with Matlab, Cambridge University Press, Cambridge, New York, Port Melbourne, Madrid, Cape Town, 2003. doi: 10.1017/CBO9780511615542.

[47]

L. F. Shampine, Solving odes and ddes with residual control, Appl. Numer. Math., 52 (2005), 113-127. doi: 10.1016/j.apnum.2004.07.003.

[48]

W. Shen and Y. Yi, Almost Automorphic and Almost Periodic Dynamics in Skew-Product Semiflows, Mem. Amer. Math. Soc., 136, Amer. Math. Soc., Providence, 1998. doi: 10.1090/memo/0647.

[49]

D. E. Sigeti, Exponentia decay of power spectra at high frequency and positive Lyapunov exponents, Phys. D, 82 (1995), 136-153. doi: 10.1016/0167-2789(94)00225-F.

[50]

C. Skokos, The Lyapunov Characteristic Exponents and their computation, Lecture Notes in Phys., 790 (2010), 63-135. doi: 10.1007/978-3-642-04458-8_2.

[51]

H. L. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems, Amer. Math. Soc., Providence, 1995.

[52]

H. L. Smith and P. Waltman, The Theory of the Chemostat. Dynamics of Microbial Competition, Cambridge Studies in Mathematical Biology, 13, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511530043.

[53]

H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, Graduate Studies in Mathematics, 118, American Mathematical Society, Providence, 2011.

[54]

P. van den Driessche and X. Zou, Global attractivity in delayed Hopfield neural networks models, SIAM J. Appl. Math., 58 (1998), 1878-1890. doi: 10.1137/S0036139997321219.

[55]

P. van den Driessche, J. Wu and X. Zou, Stabilization role of inhibitory self-connections in a delayed neural network, Physica D, 150 (2001), 84-90. doi: 10.1016/S0167-2789(00)00216-5.

[56]

J. Wu, Introduction to Neural Dynamics and Signal Transmission Delay, Nonlinear Analysis and Aplications, 6, Walter de Gruyter, Berlin, New York, 2001. doi: 10.1515/9783110879971.

show all references

References:
[1]

H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976), 620-709. doi: 10.1137/1018114.

[2]

O. Arratia, R. Obaya and M. E. Sansaturio, Applications of the exponential ordering in the study of almost-periodic delayed Hopfield neural networks, Physica D, 241 (2012), 1551-1566. doi: 10.1016/j.physd.2012.06.007.

[3]

G. Benettin, L. Galgani, A. Giorgilli and J. M. Strelcyn, Lyapunov exponents for smooth dynamical systems and for hamiltonian systems; a method for computing all of them. Part 1: Theory, Meccanica, 15 (1980), 9-20. doi: 10.1007/BF02128236.

[4]

G. Benettin, L. Galgani, A. Giorgilli and J. M. Strelcyn, Lyapunov exponents for smooth dynamical systems and for hamiltonian systems; a method for computing all of them. Part 1: numerical applications, Meccanica, 15 (1980), 21-30.

[5]

D. Breda, S. Maset and R. Vermiglio, Approximation of eigenvalues of evolution operators for lineal retarded functional differential equations, SIAM J. Numer. Anal., 50 (2012), 1456-1483. doi: 10.1137/100815505.

[6]

D. Breda and E. Van Vleck, Approximating Lyapunov exponents and Sacker-Sell spectrum for retarded functional differential equations, Numer. Math, 126 (2014), 225-257. doi: 10.1007/s00211-013-0565-1.

[7]

D. Breda, S. Maset and R. Vermiglio, Pseudospectral approximation of eigenvalues of derivative operators with non-local boundary conditions, Appl. Numer. Math., 56 (2006), 318-331. doi: 10.1016/j.apnum.2005.04.011.

[8]

L.O. Chua and L. Yang, Cellular neural network: Theory, IEEE Trans. Circuits. Syst., 35 (1988), 1257-1272. doi: 10.1109/31.7600.

[9]

L. O. Chua and L. Yang, Cellular neural network: Applications, IEEE Trans. Circuits. Syst., 35 (1988), 1273-1290. doi: 10.1109/31.7601.

[10]

I. Chueshov, Monotone Random Systems: Theory and Applications, Springer-Verlag, Berlin, 2002. doi: 10.1007/b83277.

[11]

L. Dieci, R. D. Russell and E. S. Van Vleck, On the computation of Lyapunov exponents for continuous dynamical systems, SIAM J. Numer. Anal. 34 (1997), 402-423. doi: 10.1137/S0036142993247311.

[12]

L. Dieci and E. S. Van Vleck, Perturbation theory for approximation of Lyapunov exponent by QR methods, J. Dyn. Diff. Equat., 18 (2006), 815-840. doi: 10.1007/s10884-006-9024-3.

[13]

C. Elia and R. Fabbri, Rotation number and exponential dichotomy for linear hamiltonian systems: From theoretical to numerical results, J. Dynam. Differential Equations, 25 (2013), 95-120. doi: 10.1007/s10884-013-9290-9.

[14]

R. Ellis, Lectures on Topological Dynamics, Benjamin, New York, 1969.

[15]

K. Engelborghs and D. Roose, On stability of LMS methods and characteristic roots of delay differential equations, SIAM J. Numer. Anal., 40 (2002), 629-650. doi: 10.1137/S003614290037472X.

[16]

W. H. Enright, A new error-control for initial value solvers, Appl. Math. Comput., 31 (1989), 288-301. doi: 10.1016/0096-3003(89)90123-9.

[17]

J. D. Farmer, Chaotic attractors of an infinite-dimensional system, Physica D, 4 (1981/82), 366-393. doi: 10.1016/0167-2789(82)90042-2.

[18]

M. Fiedler, Special Matrices and Their Applications in Numerical Mathematics, Dover Publications, Inc., Mineola, New York, 2008.

[19]

K. Geist, U. Parlitz and W. Lauterborn, Comparison of different methods for computing Lyapunov exponents, Prog. Theor. Phys., 83 (1990), 875-893. doi: 10.1143/PTP.83.875.

[20]

K. Gopalsamy and X. Z. He, Stability in asymmetric Hopfield nets with transmission delays, Physica D, 76 (1994), 344-358. doi: 10.1016/0167-2789(94)90043-4.

[21]

S. Grossberg, Nonlinear networks: Principles, mechanisms and architectures, Neural Netw., 1 (1988), 17-61. doi: 10.1016/0893-6080(88)90021-4.

[22]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, 99, Springer-Verlag, Berlin, Heidelberg, New York, 1993. doi: 10.1007/978-1-4612-4342-7.

[23]

J. J. Hopfield, Neural networks and physical systems with emergent collective computational abilities, Proc. Nat. Acad. Sci. USA, 79 (1982), 2554-2558. doi: 10.1073/pnas.79.8.2554.

[24]

J. J. Hopfield, Neurons with graded response have collective computational properties like those of two-state neurons, Proc. Nat. Acad. Sci. USA, 81 (1984), 2554-2558. doi: 10.1073/pnas.79.8.2554.

[25]

T. Insperger and G. Stépán, Semi-Discretization for Time-Delay Systems. Stability and Engineering Applications, Applied Mathematical Sciences, 178, Springer, New York, 2011. doi: 10.1007/978-1-4614-0335-7.

[26]

R. Johnson and F. Mantellini, Non-autonomous differential equations, in Dynamical Systems, Lecture Notes in Math., 1822, Springer, Berlin, 2003, 173-229. doi: 10.1007/978-3-540-45204-1_3.

[27]

R. Johnson, K. J. Palmer and G. R. Sell, Ergodic properties of linear dynamical systems, SIAM J. Math. Anal., 18 (1987), 1-33. doi: 10.1137/0518001.

[28]

W. Liu and E. S. Van Vleck, Exponential dichotomy for asymptotically hyperbolic two-dimensional linear systems, J. Dyn. Diff. Equat., 22 (2010), 697-722. doi: 10.1007/s10884-010-9170-5.

[29]

T. Luzyanina, K. Engelborghs and D. Roose, Computing stability of differential equations with bounded distributed delays, Numer. Algorithms, 34 (2003), 41-66. doi: 10.1023/A:1026194503720.

[30]

C. M. Marcus and R. M. Westervelt, Stability of analog neural networks with delay, Phys. Rev. A, 39 (2009), 347-359. doi: 10.1103/PhysRevA.39.347.

[31]

J. Mierczyński and W. Shen, Lyapunov exponents and asymptotic dynamics in random Kolmogorov models, J. Evol. Equ., 4 (2004), 371-390. doi: 10.1007/s00028-004-0160-0.

[32]

V. Nemytskii and V. Stepanoff, Qualitative Theory of Differential Equations, Princeton University Press, Princeton, NJ, 1960.

[33]

S. Niculescu, Delay Effects on Stability, Lecture Notes in Control and Information Sciences, 269, Springer-Verlag, Berlin, Heidelberg, 2001.

[34]

S. Novo and R. Obaya, Non-autonomous Functional Differential Equations and Applications. in Stability and Bifurcation for Non-Autonomous Differential Equations, Lecture Notes in Math., 2065, Springer-Verlag, Berlin, Heidelberg, 2013, 185-264. doi: 10.1007/978-3-642-32906-7_4.

[35]

S. Novo, R. Obaya and A. M. Sanz, Stability and extensibility results for abstract skew-product semiflows, J. Differential Equations, 235 (2007), 623-646. doi: 10.1016/j.jde.2006.12.009.

[36]

S. Novo, R. Obaya and A. M. Sanz, Topological dynamics for monotone skew-product semiflows with applications, J. Dyn. Diff. Equat., 25 (2013), 1201-1231. doi: 10.1007/s10884-013-9337-y.

[37]

S. Novo, R. Obaya and A. M. Sanz, Uniform persistence and upper Lyapunov exponents for monotone skew-product semiflows, Nonlinearity, 26 (2013), 2409-2440. doi: 10.1088/0951-7715/26/9/2409.

[38]

S. Novo, C. Nuñez, R. Obaya and A. M. Sanz, Skew-product semiflows for non-autonomous partial functional differential equations with delay, Discret. Contin. Dyn. S., 34 (2014), 4291-4321. doi: 10.3934/dcds.2014.34.4291.

[39]

C. Núñez, R. Obaya and A. M. Sanz, Minimal sets in monotone and concave skew-product semiflows I: A general theory, J. Differential Equations, 252 (2012), 5492-5517. doi: 10.1016/j.jde.2012.02.008.

[40]

C. Núñez, R. Obaya and A. M. Sanz, Minimal sets in monotone and concave skew-product semiflows II: Two-dimensional systems of differential equations, J. Differential Equations, 252 (2012), 3575-3607. doi: 10.1016/j.jde.2011.11.016.

[41]

P. Poláčik and I. Tereščák, Exponential separation and invariant bundles for maps in ordered Banach spaces with applications to parabolic equations, J. Dynamics Differential Equations, 5 (1993), 279-303. doi: 10.1007/BF01053163.

[42]

R. J. Sacker and G. R. Sell, Lifting properties in skew-products flows with applications to differential equations, Mem. Amer. Math. Soc., 11 (1977), iv+67 pp.

[43]

R. J. Sacker and G. R. Sell, A spectral theory for linear differential systems, J. Differential Equations, 27 (1978), 320-358. doi: 10.1016/0022-0396(78)90057-8.

[44]

R. J. Sacker and G. R. Sell, Dichotomies for linear evolutionary equations in Banach spaces, J. Differential Equations, 113 (1994), 17-67. doi: 10.1006/jdeq.1994.1113.

[45]

L. F. Shampine and M. W. Reichelt, The Matlab ODE suite, SIAM J. Sci. Comput., 18 (1997), 1-22. doi: 10.1137/S1064827594276424.

[46]

L. F. Shampine, I. Gladwell and S. Thompson, Solving ODEs with Matlab, Cambridge University Press, Cambridge, New York, Port Melbourne, Madrid, Cape Town, 2003. doi: 10.1017/CBO9780511615542.

[47]

L. F. Shampine, Solving odes and ddes with residual control, Appl. Numer. Math., 52 (2005), 113-127. doi: 10.1016/j.apnum.2004.07.003.

[48]

W. Shen and Y. Yi, Almost Automorphic and Almost Periodic Dynamics in Skew-Product Semiflows, Mem. Amer. Math. Soc., 136, Amer. Math. Soc., Providence, 1998. doi: 10.1090/memo/0647.

[49]

D. E. Sigeti, Exponentia decay of power spectra at high frequency and positive Lyapunov exponents, Phys. D, 82 (1995), 136-153. doi: 10.1016/0167-2789(94)00225-F.

[50]

C. Skokos, The Lyapunov Characteristic Exponents and their computation, Lecture Notes in Phys., 790 (2010), 63-135. doi: 10.1007/978-3-642-04458-8_2.

[51]

H. L. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems, Amer. Math. Soc., Providence, 1995.

[52]

H. L. Smith and P. Waltman, The Theory of the Chemostat. Dynamics of Microbial Competition, Cambridge Studies in Mathematical Biology, 13, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511530043.

[53]

H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, Graduate Studies in Mathematics, 118, American Mathematical Society, Providence, 2011.

[54]

P. van den Driessche and X. Zou, Global attractivity in delayed Hopfield neural networks models, SIAM J. Appl. Math., 58 (1998), 1878-1890. doi: 10.1137/S0036139997321219.

[55]

P. van den Driessche, J. Wu and X. Zou, Stabilization role of inhibitory self-connections in a delayed neural network, Physica D, 150 (2001), 84-90. doi: 10.1016/S0167-2789(00)00216-5.

[56]

J. Wu, Introduction to Neural Dynamics and Signal Transmission Delay, Nonlinear Analysis and Aplications, 6, Walter de Gruyter, Berlin, New York, 2001. doi: 10.1515/9783110879971.

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