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 & 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.  doi: 10.1137/1018114.  Google Scholar

[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.  doi: 10.1016/j.physd.2012.06.007.  Google Scholar

[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.  doi: 10.1007/BF02128236.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1137/100815505.  Google Scholar

[6]

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

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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.  doi: 10.1016/j.apnum.2005.04.011.  Google Scholar

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

[9]

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

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[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), 34 (1997), 402.  doi: 10.1137/S0036142993247311.  Google Scholar

[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.  doi: 10.1007/s10884-006-9024-3.  Google Scholar

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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.  doi: 10.1137/S003614290037472X.  Google Scholar

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[23]

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

[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.  doi: 10.1073/pnas.79.8.2554.  Google Scholar

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[26]

R. Johnson and F. Mantellini, Non-autonomous differential equations,, in Dynamical Systems, (1822), 173.  doi: 10.1007/978-3-540-45204-1_3.  Google Scholar

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

[32]

V. Nemytskii and V. Stepanoff, Qualitative Theory of Differential Equations,, Princeton University Press, (1960).   Google Scholar

[33]

S. Niculescu, Delay Effects on Stability,, Lecture Notes in Control and Information Sciences, (2001).   Google Scholar

[34]

S. Novo and R. Obaya, Non-autonomous Functional Differential Equations and Applications., in Stability and Bifurcation for Non-Autonomous Differential Equations, (2065), 185.  doi: 10.1007/978-3-642-32906-7_4.  Google Scholar

[35]

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

[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.  doi: 10.1007/s10884-013-9337-y.  Google Scholar

[37]

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

[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.  doi: 10.3934/dcds.2014.34.4291.  Google Scholar

[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.  doi: 10.1016/j.jde.2012.02.008.  Google Scholar

[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.  doi: 10.1016/j.jde.2011.11.016.  Google Scholar

[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.  doi: 10.1007/BF01053163.  Google Scholar

[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).   Google Scholar

[43]

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

[44]

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

[45]

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

[46]

L. F. Shampine, I. Gladwell and S. Thompson, Solving ODEs with Matlab,, Cambridge University Press, (2003).  doi: 10.1017/CBO9780511615542.  Google Scholar

[47]

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

[48]

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

[49]

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

[50]

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

[51]

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

[52]

H. L. Smith and P. Waltman, The Theory of the Chemostat. Dynamics of Microbial Competition,, Cambridge Studies in Mathematical Biology, (1995).  doi: 10.1017/CBO9780511530043.  Google Scholar

[53]

H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence,, Graduate Studies in Mathematics, (2011).   Google Scholar

[54]

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

[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.  doi: 10.1016/S0167-2789(00)00216-5.  Google Scholar

[56]

J. Wu, Introduction to Neural Dynamics and Signal Transmission Delay,, Nonlinear Analysis and Aplications, (2001).  doi: 10.1515/9783110879971.  Google Scholar

show all references

References:
[1]

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

[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.  doi: 10.1016/j.physd.2012.06.007.  Google Scholar

[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.  doi: 10.1007/BF02128236.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1137/100815505.  Google Scholar

[6]

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

[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.  doi: 10.1016/j.apnum.2005.04.011.  Google Scholar

[8]

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

[9]

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

[10]

I. Chueshov, Monotone Random Systems: Theory and Applications,, Springer-Verlag, (2002).  doi: 10.1007/b83277.  Google Scholar

[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), 34 (1997), 402.  doi: 10.1137/S0036142993247311.  Google Scholar

[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.  doi: 10.1007/s10884-006-9024-3.  Google Scholar

[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.  doi: 10.1007/s10884-013-9290-9.  Google Scholar

[14]

R. Ellis, Lectures on Topological Dynamics,, Benjamin, (1969).   Google Scholar

[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.  doi: 10.1137/S003614290037472X.  Google Scholar

[16]

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

[17]

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

[18]

M. Fiedler, Special Matrices and Their Applications in Numerical Mathematics,, Dover Publications, (2008).   Google Scholar

[19]

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

[20]

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

[21]

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

[22]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations,, Applied Mathematical Sciences, 99 (1993).  doi: 10.1007/978-1-4612-4342-7.  Google Scholar

[23]

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

[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.  doi: 10.1073/pnas.79.8.2554.  Google Scholar

[25]

T. Insperger and G. Stépán, Semi-Discretization for Time-Delay Systems. Stability and Engineering Applications,, Applied Mathematical Sciences, (2011).  doi: 10.1007/978-1-4614-0335-7.  Google Scholar

[26]

R. Johnson and F. Mantellini, Non-autonomous differential equations,, in Dynamical Systems, (1822), 173.  doi: 10.1007/978-3-540-45204-1_3.  Google Scholar

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

[32]

V. Nemytskii and V. Stepanoff, Qualitative Theory of Differential Equations,, Princeton University Press, (1960).   Google Scholar

[33]

S. Niculescu, Delay Effects on Stability,, Lecture Notes in Control and Information Sciences, (2001).   Google Scholar

[34]

S. Novo and R. Obaya, Non-autonomous Functional Differential Equations and Applications., in Stability and Bifurcation for Non-Autonomous Differential Equations, (2065), 185.  doi: 10.1007/978-3-642-32906-7_4.  Google Scholar

[35]

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

[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.  doi: 10.1007/s10884-013-9337-y.  Google Scholar

[37]

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

[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.  doi: 10.3934/dcds.2014.34.4291.  Google Scholar

[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.  doi: 10.1016/j.jde.2012.02.008.  Google Scholar

[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.  doi: 10.1016/j.jde.2011.11.016.  Google Scholar

[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.  doi: 10.1007/BF01053163.  Google Scholar

[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).   Google Scholar

[43]

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

[44]

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

[45]

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

[46]

L. F. Shampine, I. Gladwell and S. Thompson, Solving ODEs with Matlab,, Cambridge University Press, (2003).  doi: 10.1017/CBO9780511615542.  Google Scholar

[47]

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

[48]

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

[49]

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

[50]

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

[51]

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

[52]

H. L. Smith and P. Waltman, The Theory of the Chemostat. Dynamics of Microbial Competition,, Cambridge Studies in Mathematical Biology, (1995).  doi: 10.1017/CBO9780511530043.  Google Scholar

[53]

H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence,, Graduate Studies in Mathematics, (2011).   Google Scholar

[54]

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

[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.  doi: 10.1016/S0167-2789(00)00216-5.  Google Scholar

[56]

J. Wu, Introduction to Neural Dynamics and Signal Transmission Delay,, Nonlinear Analysis and Aplications, (2001).  doi: 10.1515/9783110879971.  Google Scholar

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