doi: 10.3934/dcdss.2021165
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Stochastic energy balance climate models with Legendre weighted diffusion and an additive cylindrical Wiener process forcing

1. 

Depto. Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain

2. 

Instituto Matemático Interdisplinar

*Corresponding author: Jesús Ildefonso Díaz

Dedicated to Georg Hetzer on occasion of his 75th birthday

Received  October 2021 Revised  November 2021 Early access December 2021

Fund Project: Partially supported the UCM Research Group MOMAT (ref. 910480) and the projects MTM2017-85449-P and PID2020-112517GB-I00 of the DGISPI, Spain

We consider a class of one-dimensional nonlinear stochastic parabolic problems associated to Sellers and Budyko diffusive energy balance climate models with a Legendre weighted diffusion and an additive cylindrical Wiener processes forcing. Our results use in an important way that, under suitable assumptions on the Wiener processes, a suitable change of variables leads the problem to a pathwise random PDE, hence an essentially "deterministic" formulation depending on a random parameter. Two applications are also given: the stability of solutions when the Wiener process converges to zero and the asymptotic behaviour of solutions for large time.

Citation: Gregorio Díaz, Jesús Ildefonso Díaz. Stochastic energy balance climate models with Legendre weighted diffusion and an additive cylindrical Wiener process forcing. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021165
References:
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L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar

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V. Barbu, Nonlinear Differential Equations of MonotoneType in Banach Spaces, SpringerMonographs in Mathematics. Springer, 2010. doi: 10.1007/978-1-4419-5542-5.  Google Scholar

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V. Barbu and M. Röckner, An operational approach to stochastic differential equations driven by linear multiplicative noise, J. Eur. Math. Soc., 17 (2015), 1789-1815.  doi: 10.4171/JEMS/545.  Google Scholar

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P. Bénilan, M. G. Crandall and A. Pazy, Nonlinear Evolution Equations Governed by Accretive Operators, Manuscript of Book in Preparation. Google Scholar

[6]

S. Bensid and J. I. Díaz, On the exact number of monotone solutions of a simplified Budyko climate model and their different stability, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 1033-1047.  doi: 10.3934/dcdsb.2019005.  Google Scholar

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A. Bensoussan and R. Temam, ćquations aux derivées partielles stochastiques non lineaires, Israel J. Math., 11 (1972), 95-129.  doi: 10.1007/BF02761449.  Google Scholar

[8]

W. J. BeynB. GessP. Lescot and M. Röckner, The global random attractor for a class of stochastic porous media equations, Comm. Partial Differential Equations, 36 (2011), 446-469.  doi: 10.1080/03605302.2010.523919.  Google Scholar

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H. Brezis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions Dans Les Espaces de Hilbert, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973.  Google Scholar

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Z. Brzezniak and J. van Neerven, Stochastic convolution is separable Bancah spaces and the stochastic lineal Cauchy problem, Studia Math., 143 (2000), 43-74.  doi: 10.4064/sm-143-1-43-74.  Google Scholar

[11]

R. Buckdahn and É. Pardoux, Monotonicity methods for white noise driven quasilinear SPDEs, Diffusion Processes and Related Problems in Analysis, I, M. Pinsky, ed., Birkhäuser Boston, MA, 22 (1990), 219–233.  Google Scholar

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M. I. Budyko, The effect of solar radiation variations on the climate of the Earth, Tellus, 21 (1969), 611-619.   Google Scholar

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T. CaraballoJ. A. Langa and J. Valero, Global attractors for multivalued random dynamical systems, Nonlinear Anal., 48 (2002), 805-829.  doi: 10.1016/S0362-546X(00)00216-9.  Google Scholar

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T. CaraballoJ. A. Langa and J. Valero, On the relationship between solutions of stochastic and random differential inclusions, Stoch. Anal. Appl., 21 (2003), 545-557.  doi: 10.1081/SAP-120020425.  Google Scholar

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G. Da Prato and H. Frankowska, A stochastic Filippov theorem, Stochastic Anal. Appl., 12 (1994), 409-426.  doi: 10.1080/07362999408809361.  Google Scholar

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G. Díaz and J. I. Díaz, On a stochastic parabolic PDE arising in Climatology, RACSAM. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat., 96 (2002), 123-128.   Google Scholar

[23]

J. I. Díaz, Mathematical analysis of some diffusive energy balance climate models, In Mathematics, Climate and Environment (J.I. Díaz and J.-L. Lions, eds.), Masson, Paris, 28–56, 1993.  Google Scholar

[24]

J. I. DíazJ. Hernández and L. Tello, On the multiplicity of equilibrium solutions to a nonlinear diffusion equation on a manifold arising in climatology, J. Math. Anal. Appl., 216 (1997), 593-613.  doi: 10.1006/jmaa.1997.5691.  Google Scholar

[25]

J. I. Díaz and G. Hetzer, A functional quasilinear reaction-diffusion equation arising in climatology, In ćquations Aux Dérivées Partielles et Applications, Articles dédiés à J.-L. Lions, Gauthier-Villars, Elsevier, Paris (1998), 461–480.  Google Scholar

[26]

J. I. DíazG. Hetzer and L. Tello, An energy balance climate model with hysteresis, Nonlinear Anal., 64 (2006), 2053-2074.  doi: 10.1016/j.na.2005.07.038.  Google Scholar

[27]

J. I. DíazJ. A. Langa and J. Valero, On the asymptotic behaviour of solutions of a stochastic energy balance climate model, Phys. D, 238 (2009), 880-887.  doi: 10.1016/j.physd.2009.02.010.  Google Scholar

[28]

J. I. Díaz and L. Tello, On a nonlinear parabolic problem on a Riemannian manifold without boundary arising in climatology, Collect. Math., 50 (1999), 19-51.   Google Scholar

[29]

J. I. Díaz and I. I. Vrabie, Existence for reaction-diffusion systems. A compactness method approach, J. Math. Anal. Appl., 188 (1994), 521-540.  doi: 10.1006/jmaa.1994.1443.  Google Scholar

[30]

H. Doss, Liens entre équations différentielles stochastiques et ordinaires, Ann. Inst. H. Poincaré Sect. B (N. S.), 13 (1977), 99-125.   Google Scholar

[31]

I. Gyöngy and E. Pardoux, On the regularization effecto to space-time white noise on quasi-linear stochastic partial differential equations, Probab. Theory Relat. Fields, 97 (1993), 211-229.  doi: 10.1007/BF01199321.  Google Scholar

[32]

X. Han and P. E. Kloeden, Stochastic Ordinary Differential Equations and Their Numerical Solutions, Springer Singapore, 2017. doi: 10.1007/978-981-10-6265-0.  Google Scholar

[33]

X. Han and P. E. Kloeden, Sigmoidal approximations of Heaviside functions in neural lattice models, J. Differential Equations, 268 (2020), 5283-5300.  doi: 10.1016/j.jde.2019.11.010.  Google Scholar

[34]

X. Han and P. E. Kloeden, Corrigendum to "Sigmoidal approximations of Heaviside functions in neural lattice models", J. Differential Equations, 274 (2020), 1214-1220.  doi: 10.1016/j.jde.2020.11.017.  Google Scholar

[35]

G. Hetzer, The structure of the principal component for semilinear diffusion equations from energy balance climate models, Houston J. Math., 16 (1990), 203-216.   Google Scholar

[36]

G. Hetzer, S-shapedness for energy balance climate models of Sellers-type, In The Mathematics of Models for Climatology and Environment (J. I. Díaz, ed.), Springer, Berlin, (1997), 253–287.  Google Scholar

[37]

G. Hetzer, The number of stationary solntions for one-dimensional Budyko-type climate models, Nonlinear Anal. Real World Appl., 2 (2001), 259-272.  doi: 10.1016/S0362-546X(00)00103-6.  Google Scholar

[38]

G. Hetzer and P. Schmidt, Analysis of energy balance models, World Congress of Nonlinear Analysts '92, (1996), 1609–1618.  Google Scholar

[39]

P. Imkeller, Energy balance models-viewed from stochastic dynamics, In Stochastic Climate Models (P. Imkeller and J.-S. von Storch, eds.), Birkhäuser, Basel, (2001), 213–240.  Google Scholar

[40]

H. G. Kaper and H. Engler, Mathematics and Climate, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pennsylvania, 2013. doi: 10.1137/1.9781611972610.  Google Scholar

[41]

A. V. Kapustyan, A random attractor of a stochastically perturbed evolution inclusion, Differ. Equ., 40 (2004), 1383-1388.  doi: 10.1007/s10625-005-0060-2.  Google Scholar

[42]

K. Liu, Stochastic Stability of Differential Equations in Abstract Spaces, Cambridge University Pres, 2019. doi: 10.1017/9781108653039.  Google Scholar

[43]

W. Liu and M. Röckner, Stochastic Partial Differential Equations: An Introduction, Universitext. Springer, Cham, 2015. doi: 10.1007/978-3-319-22354-4.  Google Scholar

[44]

V. Lucarini, L. Serdukova and G. Margazoglou, Lévy-noise versus Gaussian-noise-induced Transitions in the Ghil-Sellers Energy Balance Model, Nonlinear Processes in Geophysics, 2021. doi: 10.5194/npg-2021-34.  Google Scholar

[45]

A. F. Nikiforov, S. K. Suslov and V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable, Springer-Verlag, New York, 1991 doi: 10.1007/978-3-642-74748-9.  Google Scholar

[46]

G. R. North and R. F. Cahalan, Predictability in a solvable stochastic climate model, J. Atmospheric Science, 38 (1981), 504-513.  doi: 10.1175/1520-0469(1981)038<0504:PIASSC>2.0.CO;2.  Google Scholar

[47]

G. R. North and K. Y. Kim, Energy Balance Climate Models, Wiley-VCH, Weinheim, Germany, 2017. Google Scholar

[48]

B. Schmidt, Bifurcation of Stationary Solutions for Legendre-Type Boundary Value Problems Arising From Climate Modeling, Thesis (Ph.D.)–Auburn University, 1994.  Google Scholar

[49]

W. S. Sellers, A global climatic model based on the energy balance of the earth-atmosphere system, J. Appl. Meteorol, 8 (1969), 392-400.  doi: 10.1175/1520-0450(1969)008<0392:AGCMBO>2.0.CO;2.  Google Scholar

[50]

H. J. Sussmann, On the gap between deterministic and stochastic differential equations, Ann. Probability, 6 (1978), 19-41.   Google Scholar

[51]

J. van Neerven and M. Veraar, Maximal inequalities for stochastic convolutions and pathwise uniform convergence of time discretisation schemes, Stoch PDE: Anal Comp, 2021. doi: 10.1007/s40072-021-00204-y.  Google Scholar

[52]

I. I. Vrabie, Compactness Methods for Nonlinear Evolutions, 2$^{nd}$ edition, Pitman Monographs and Surveys in Pure and Applied Mathematics, New York, 1995.  Google Scholar

[53]

S. Yotsutani, Evolution equations associated with the subdifferentials, J. Math. Soc. Japan, 31 (1979), 623-646.  doi: 10.2969/jmsj/03140623.  Google Scholar

show all references

References:
[1]

D. ArcoyaJ. I. Díaz and L. Tello, S-Shaped bifurcation branch in a quasilinear multivalued model arising in climatology, J. Differential Equations, 150 (1998), 215-225.  doi: 10.1006/jdeq.1998.3502.  Google Scholar

[2]

L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar

[3]

V. Barbu, Nonlinear Differential Equations of MonotoneType in Banach Spaces, SpringerMonographs in Mathematics. Springer, 2010. doi: 10.1007/978-1-4419-5542-5.  Google Scholar

[4]

V. Barbu and M. Röckner, An operational approach to stochastic differential equations driven by linear multiplicative noise, J. Eur. Math. Soc., 17 (2015), 1789-1815.  doi: 10.4171/JEMS/545.  Google Scholar

[5]

P. Bénilan, M. G. Crandall and A. Pazy, Nonlinear Evolution Equations Governed by Accretive Operators, Manuscript of Book in Preparation. Google Scholar

[6]

S. Bensid and J. I. Díaz, On the exact number of monotone solutions of a simplified Budyko climate model and their different stability, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 1033-1047.  doi: 10.3934/dcdsb.2019005.  Google Scholar

[7]

A. Bensoussan and R. Temam, ćquations aux derivées partielles stochastiques non lineaires, Israel J. Math., 11 (1972), 95-129.  doi: 10.1007/BF02761449.  Google Scholar

[8]

W. J. BeynB. GessP. Lescot and M. Röckner, The global random attractor for a class of stochastic porous media equations, Comm. Partial Differential Equations, 36 (2011), 446-469.  doi: 10.1080/03605302.2010.523919.  Google Scholar

[9]

H. Brezis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions Dans Les Espaces de Hilbert, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973.  Google Scholar

[10]

Z. Brzezniak and J. van Neerven, Stochastic convolution is separable Bancah spaces and the stochastic lineal Cauchy problem, Studia Math., 143 (2000), 43-74.  doi: 10.4064/sm-143-1-43-74.  Google Scholar

[11]

R. Buckdahn and É. Pardoux, Monotonicity methods for white noise driven quasilinear SPDEs, Diffusion Processes and Related Problems in Analysis, I, M. Pinsky, ed., Birkhäuser Boston, MA, 22 (1990), 219–233.  Google Scholar

[12]

M. I. Budyko, The effect of solar radiation variations on the climate of the Earth, Tellus, 21 (1969), 611-619.   Google Scholar

[13]

T. CaraballoJ. A. Langa and J. Valero, Global attractors for multivalued random dynamical systems, Nonlinear Anal., 48 (2002), 805-829.  doi: 10.1016/S0362-546X(00)00216-9.  Google Scholar

[14]

T. CaraballoJ. A. Langa and J. Valero, On the relationship between solutions of stochastic and random differential inclusions, Stoch. Anal. Appl., 21 (2003), 545-557.  doi: 10.1081/SAP-120020425.  Google Scholar

[15]

A. N. Carvalho, J. A. Langa and J. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4.  Google Scholar

[16] P.-L. Chow, Stochastic Partial Differential Equation, CRC Press, Boca Raton, 2015.   Google Scholar
[17]

H. Crauel and F. Flandolfi, Attractors for random dynamical systems, Prob. Theory Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.  Google Scholar

[18]

G. Da Prato, Kolmogorov Equations for Stochastic PDEs, Advanced Courses in Mathematics. CRM Barcelona. Birkhäuser Verlag, Basel, 2004. doi: 10.1007/978-3-0348-7909-5.  Google Scholar

[19]

G. Da Prato and H. Frankowska, A stochastic Filippov theorem, Stochastic Anal. Appl., 12 (1994), 409-426.  doi: 10.1080/07362999408809361.  Google Scholar

[20]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Presss, 1992. doi: 10.1017/CBO9780511666223.  Google Scholar

[21]

G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge University Presss, 1996. doi: 10.1017/CBO9780511662829.  Google Scholar

[22]

G. Díaz and J. I. Díaz, On a stochastic parabolic PDE arising in Climatology, RACSAM. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat., 96 (2002), 123-128.   Google Scholar

[23]

J. I. Díaz, Mathematical analysis of some diffusive energy balance climate models, In Mathematics, Climate and Environment (J.I. Díaz and J.-L. Lions, eds.), Masson, Paris, 28–56, 1993.  Google Scholar

[24]

J. I. DíazJ. Hernández and L. Tello, On the multiplicity of equilibrium solutions to a nonlinear diffusion equation on a manifold arising in climatology, J. Math. Anal. Appl., 216 (1997), 593-613.  doi: 10.1006/jmaa.1997.5691.  Google Scholar

[25]

J. I. Díaz and G. Hetzer, A functional quasilinear reaction-diffusion equation arising in climatology, In ćquations Aux Dérivées Partielles et Applications, Articles dédiés à J.-L. Lions, Gauthier-Villars, Elsevier, Paris (1998), 461–480.  Google Scholar

[26]

J. I. DíazG. Hetzer and L. Tello, An energy balance climate model with hysteresis, Nonlinear Anal., 64 (2006), 2053-2074.  doi: 10.1016/j.na.2005.07.038.  Google Scholar

[27]

J. I. DíazJ. A. Langa and J. Valero, On the asymptotic behaviour of solutions of a stochastic energy balance climate model, Phys. D, 238 (2009), 880-887.  doi: 10.1016/j.physd.2009.02.010.  Google Scholar

[28]

J. I. Díaz and L. Tello, On a nonlinear parabolic problem on a Riemannian manifold without boundary arising in climatology, Collect. Math., 50 (1999), 19-51.   Google Scholar

[29]

J. I. Díaz and I. I. Vrabie, Existence for reaction-diffusion systems. A compactness method approach, J. Math. Anal. Appl., 188 (1994), 521-540.  doi: 10.1006/jmaa.1994.1443.  Google Scholar

[30]

H. Doss, Liens entre équations différentielles stochastiques et ordinaires, Ann. Inst. H. Poincaré Sect. B (N. S.), 13 (1977), 99-125.   Google Scholar

[31]

I. Gyöngy and E. Pardoux, On the regularization effecto to space-time white noise on quasi-linear stochastic partial differential equations, Probab. Theory Relat. Fields, 97 (1993), 211-229.  doi: 10.1007/BF01199321.  Google Scholar

[32]

X. Han and P. E. Kloeden, Stochastic Ordinary Differential Equations and Their Numerical Solutions, Springer Singapore, 2017. doi: 10.1007/978-981-10-6265-0.  Google Scholar

[33]

X. Han and P. E. Kloeden, Sigmoidal approximations of Heaviside functions in neural lattice models, J. Differential Equations, 268 (2020), 5283-5300.  doi: 10.1016/j.jde.2019.11.010.  Google Scholar

[34]

X. Han and P. E. Kloeden, Corrigendum to "Sigmoidal approximations of Heaviside functions in neural lattice models", J. Differential Equations, 274 (2020), 1214-1220.  doi: 10.1016/j.jde.2020.11.017.  Google Scholar

[35]

G. Hetzer, The structure of the principal component for semilinear diffusion equations from energy balance climate models, Houston J. Math., 16 (1990), 203-216.   Google Scholar

[36]

G. Hetzer, S-shapedness for energy balance climate models of Sellers-type, In The Mathematics of Models for Climatology and Environment (J. I. Díaz, ed.), Springer, Berlin, (1997), 253–287.  Google Scholar

[37]

G. Hetzer, The number of stationary solntions for one-dimensional Budyko-type climate models, Nonlinear Anal. Real World Appl., 2 (2001), 259-272.  doi: 10.1016/S0362-546X(00)00103-6.  Google Scholar

[38]

G. Hetzer and P. Schmidt, Analysis of energy balance models, World Congress of Nonlinear Analysts '92, (1996), 1609–1618.  Google Scholar

[39]

P. Imkeller, Energy balance models-viewed from stochastic dynamics, In Stochastic Climate Models (P. Imkeller and J.-S. von Storch, eds.), Birkhäuser, Basel, (2001), 213–240.  Google Scholar

[40]

H. G. Kaper and H. Engler, Mathematics and Climate, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pennsylvania, 2013. doi: 10.1137/1.9781611972610.  Google Scholar

[41]

A. V. Kapustyan, A random attractor of a stochastically perturbed evolution inclusion, Differ. Equ., 40 (2004), 1383-1388.  doi: 10.1007/s10625-005-0060-2.  Google Scholar

[42]

K. Liu, Stochastic Stability of Differential Equations in Abstract Spaces, Cambridge University Pres, 2019. doi: 10.1017/9781108653039.  Google Scholar

[43]

W. Liu and M. Röckner, Stochastic Partial Differential Equations: An Introduction, Universitext. Springer, Cham, 2015. doi: 10.1007/978-3-319-22354-4.  Google Scholar

[44]

V. Lucarini, L. Serdukova and G. Margazoglou, Lévy-noise versus Gaussian-noise-induced Transitions in the Ghil-Sellers Energy Balance Model, Nonlinear Processes in Geophysics, 2021. doi: 10.5194/npg-2021-34.  Google Scholar

[45]

A. F. Nikiforov, S. K. Suslov and V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable, Springer-Verlag, New York, 1991 doi: 10.1007/978-3-642-74748-9.  Google Scholar

[46]

G. R. North and R. F. Cahalan, Predictability in a solvable stochastic climate model, J. Atmospheric Science, 38 (1981), 504-513.  doi: 10.1175/1520-0469(1981)038<0504:PIASSC>2.0.CO;2.  Google Scholar

[47]

G. R. North and K. Y. Kim, Energy Balance Climate Models, Wiley-VCH, Weinheim, Germany, 2017. Google Scholar

[48]

B. Schmidt, Bifurcation of Stationary Solutions for Legendre-Type Boundary Value Problems Arising From Climate Modeling, Thesis (Ph.D.)–Auburn University, 1994.  Google Scholar

[49]

W. S. Sellers, A global climatic model based on the energy balance of the earth-atmosphere system, J. Appl. Meteorol, 8 (1969), 392-400.  doi: 10.1175/1520-0450(1969)008<0392:AGCMBO>2.0.CO;2.  Google Scholar

[50]

H. J. Sussmann, On the gap between deterministic and stochastic differential equations, Ann. Probability, 6 (1978), 19-41.   Google Scholar

[51]

J. van Neerven and M. Veraar, Maximal inequalities for stochastic convolutions and pathwise uniform convergence of time discretisation schemes, Stoch PDE: Anal Comp, 2021. doi: 10.1007/s40072-021-00204-y.  Google Scholar

[52]

I. I. Vrabie, Compactness Methods for Nonlinear Evolutions, 2$^{nd}$ edition, Pitman Monographs and Surveys in Pure and Applied Mathematics, New York, 1995.  Google Scholar

[53]

S. Yotsutani, Evolution equations associated with the subdifferentials, J. Math. Soc. Japan, 31 (1979), 623-646.  doi: 10.2969/jmsj/03140623.  Google Scholar

Figure 1.  Values of the parameter $ {{\rm{Q}}} $ with different multiplicity
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