June  2022, 12(2): 343-370. doi: 10.3934/mcrf.2021025

A nonzero-sum risk-sensitive stochastic differential game in the orthant

1. 

Department of Mathematics, Indian Institute of Science, Bangalore 560012, India

2. 

Department of Mathematics, Indian Institute of Science Education and Research, Pune, Maharashtra 411008, India

* Corresponding author: Somnath Pradhan

Received  July 2020 Revised  January 2021 Published  June 2022 Early access  April 2021

Fund Project: This work is partially supported by UGC Center for Advanced Study

We study a nonzero-sum risk-sensitive stochastic differential game for controlled reflecting diffusion processes in the nonnegative orthant. We treat two cost evaluation criteria, namely, discounted cost and ergodic cost. Under certain assumptions, we establish the existence of Nash equilibria. Also, we completely characterize a Nash equilibrium for the ergodic cost criterion in the space of stationary Markov strategies.

Citation: Mrinal K. Ghosh, Somnath Pradhan. A nonzero-sum risk-sensitive stochastic differential game in the orthant. Mathematical Control and Related Fields, 2022, 12 (2) : 343-370. doi: 10.3934/mcrf.2021025
References:
[1]

R. A. Adams, Sobolev Spaces, Academic Press, New York 1975.

[2]

A. Arapostathis and A. Biswas, Infinite horizon risk-sensitive control of diffusions without any blanket stability assumptions, Stochastic Process. Appl., 128 (2018), 1485-1524.  doi: 10.1016/j.spa.2017.08.001.

[3]

A. ArapostathisA. BiswasV. S. Borkar and K. S. Kumar, A variational characterization of the risk-sensitive average reward for controlled diffusions on $\mathbb{R}^{d}$, SIAM J. Control Optim., 58 (2020), 3785-3813.  doi: 10.1137/20M1329202.

[4]

A. Arapostathis, A. Biswas and S. Saha, Strict monotonicity of principal eigenvalues of elliptic operators in $\mathbb{R}^{d}$ and risk-sensitive control, J. Math. Pures Appl., 124 (2019), 169–219. doi: 10.1016/j.matpur.2018.05.008.

[5]

A. Arapostathis, V. S. Borkar and M. K. Ghosh, Ergodic Control of Diffusion Processes, Cambridge University Press, Cambridge, U.K. 2012.

[6]

A. ArapostathisV. S. Borkar and K. S. Kumar, Risk-sensitive control and an abstract Collatz-Wielandt formula, J. Theoret. Probab., 29 (2016), 1458-1484.  doi: 10.1007/s10959-015-0616-x.

[7]

A. Bagchi and K. Suresh Kumar, Dynamic asset management with risk-sensitive criterion and non-negative factor constraints: A differential game approach, Stochastics, 81 (2009), 503-530.  doi: 10.1080/17442500902774917.

[8]

A. Basu and M. K. Ghosh, Zero-sum risk-sensitive stochastic differential games, Math. Oper. Res., 37 (2012), 437-449.  doi: 10.1287/moor.1120.0542.

[9]

V. E. Bene$\breve{s}$, Existence of optimal strategies based on specified information of a class of stochastic decision problems, SIAM J. Control, 8 (1970), 179-188.  doi: 10.1137/0308012.

[10]

A. Biswas, An eigenvalue approach to the risk-sensitive control problem in near monotone case, Systems Control Lett., 60 (2011), 181-184.  doi: 10.1016/j.sysconle.2010.12.002.

[11]

A. Biswas, V. S. Borkar and K. Suresh Kumar, Risk-sensitive control with near monotone cost, Appl. Math. Optim., 62 (2009), 145–163. Errata corriege Ⅰ. ibid 62 (2010), 165–167. Errata corriege Ⅱ. ibid 62 (2010), 435–438. doi: 10.1007/s00245-010-9119-4.

[12]

A. Biswas and S. Saha, Zero-sum stochastic differential games with risk-sensitive cost, App. Math. Optim., 81 (2020), 113-140.  doi: 10.1007/s00245-018-9479-8.

[13]

V. Borkar and A. Budhiraja, Ergodic control for constained diffusions: Characterization using HJB equation, SIAM J. Control Optim., 43 (2005), 1467-1492.  doi: 10.1137/S0363012902417619.

[14]

V. S. Borkar and M. K. Ghosh, Stochastic differential games: occupation measure based approach, J. Optim. Theory Appl., 73 (1992), 359-385. Errata corriege. ibid 88 (1996), 251–252. doi: 10.1007/BF02192034.

[15]

A. Budhiraja, An ergodic control problem for constrained diffusion processes: Existence of optimal Markov control, SIAM J. Control Optim., 42 (2003), 532-558.  doi: 10.1137/S0363012901379073.

[16]

K. L. Chung and Z. X. Zhao, From Brownian Motion to Schrödinger's Equation, Grundlehren der Mathematischen Wissenschaften, 312, Springer, Berlin, 1995. doi: 10.1007/978-3-642-57856-4.

[17]

K. Fan, Fixed-point and minimax theorems in locally convex topological linear spaces, Proc. Nat. Acad. Sci. U.S.A., 38 (1952), 121-126.  doi: 10.1073/pnas.38.2.121.

[18]

J. Filar and K. Vrieze, Competitive Markov Decision Processes, Springer-Verlag, New York, 1997.

[19]

S. K. GauttamK. S. Kumar and C. Pal, Risk-sensitive control of reflected diffusion process on orthrant, Pure Appl. Funct. Anal., 2 (2017), 477-510. 

[20]

M. K. Ghosh and A. Bagchi, Stochastic games with average payoff criterion, Appl. Math. Optim., 38 (1998), 283-301.  doi: 10.1007/s002459900092.

[21]

M. K. Ghosh and S. Pradhan, Risk-sensitive stochastic differential game with reflecting diffusions, Stoch. Anal. Appl., 36 (2018), 1-27.  doi: 10.1080/07362994.2017.1356732.

[22]

M. K. Ghosh and S. Pradhan, Zero-sum risk-sensitive stochastic differential games with reflecting diffusions in the orthant, ESAIM Control Optim. Calc. Var., 26 (2020), Paper No. 114, 33 pp. doi: 10.1051/cocv/2020029.

[23]

M. K. Ghosh and K. Suresh Kumar, A stochastic differential game in the orthant, J. Math. Anal. Appl., 265 (2002), 12-37.  doi: 10.1006/jmaa.2001.7679.

[24]

M. K. Ghosh and K. Suresh Kumar, A nonzero-sum stochastic differential game in the orthant, J. Math. Anal. Appl., 305 (2005), 158-174.  doi: 10.1016/j.jmaa.2004.11.002.

[25]

M. K. Ghosh and K. Suresh Kumar, Nonzero-sum risk-sensitive stochastic differential games with reflecting diffusions, Comput. Appl. Math., 18 (1999), 355-368. 

[26]

M. K. Ghosh, K. Suresh Kumar and C. Pal, Nonzero-sum risk-sensitive stochastic differential games, arXiv: 1604.01142

[27]

M. K. Ghosh, K. Suresh Kumar, C. Pal and S. Pradhan, Nonzero-sum risk-sensitive stochastic differential games with discounted costs, Stochastic Analysis and Applications, 39 (2021), 306-326. doi: 10.1080/07362994.2020.1796707.

[28]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 224, 2nd ed., Springer, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.

[29]

C. J. HimmelbergT. ParthasarathyT. E. S. Raghavan and F. S. Van Fleck, Existence of $p$-equilibrium and optimal stationary strategies in stochastic games, Proc. Amer. Math. Soc., 60 (1976), 245-251.  doi: 10.2307/2041151.

[30]

D. L. Iglehart and W. Whitt, Multiple channel queues in heavy traffic, Adv. in Appl. Prob., 2 (1970), 150-177.  doi: 10.2307/3518347.

[31]

H. J. Kushner and L. F. Martins, Limit theorems for pathwise average cost per unit time problems for controlled queues in heavy traffic, Stochastics Stochastics Rep., 42 (1993), 25-51.  doi: 10.1080/17442509308833808.

[32]

H. J. Kushner and K. M. Ramachandran, Optimal and approximately optimal control policies for queues in heavy traffic, SIAM J. Control Optim., 27 (1989), 1293–1318. doi: 10.1137/0327066.

[33]

A. J. Lemoine, Network of queues - A survey of weak convergence results, Management Science, 24 (1978), 1175-1193.  doi: 10.1287/mnsc.24.11.1175.

[34]

G. Leoni, A First Course in Sobolev Spaces, Graduate Studies in Mathematics, 105, AMS, Rhode Island USA, 2009. doi: 10.1090/gsm/105.

[35]

G. M. Lieberman, Oblique Derivative Problems for Elliptic Equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013. doi: 10.1142/8679.

[36]

G. M. Lieberman, Local estimates for subsolutions and supersolutions of oblique derivative problems for general second order elliptic equations, Trans. Amer. Math. Soc., 304 (1987), 343-353.  doi: 10.1090/S0002-9947-1987-0906819-0.

[37]

P.-L. Lions and A.-S. Sznitman, Stochastic differential equations with reflecting boundary conditions, Comm. Pure Appl. Math., 37 (1984), 511-537.  doi: 10.1002/cpa.3160370408.

[38]

J.-L. Menaldi and M. Robin, Remarks on risk-sensitive control problems, Appl. Math. Optim., 52 (2005), 297-310.  doi: 10.1007/s00245-005-0829-y.

[39]

A. S. Nowak, Notes on risk-sensitive Nash equilibria. Advances in dynamic games, Ann. International. Soc. Dynam. Games, 7 (2005), 95-109.  doi: 10.1007/0-8176-4429-6_5.

[40]

S. Pradhan, Risk-sensitive ergodic control of reflected diffusion processes in orthant, Appl. Math. Optim., (2019), to appear. doi: 10.1007/s00245-019-09606-w.

[41]

M. I. Reiman, Open queueing networks in heavy traffic, Math. Oper. Res., 9 (1984), 441-458.  doi: 10.1287/moor.9.3.441.

[42]

A. Yu. Veretennikov, On strong and weak solutions of one-dimensional stochastic equations with boundary conditions, Theory Probab. Appl., 26 (1981), 670-686. 

[43]

J. Warga, Functions of relaxed controls, SIAM J. Control, 5 (1967), 628-641.  doi: 10.1137/0305042.

[44]

Z. Wu, J. Yin and C. Wang, Elliptic and Parabolic Equations, World Scientific, Singapore, 2006. doi: 10.1142/6238.

show all references

References:
[1]

R. A. Adams, Sobolev Spaces, Academic Press, New York 1975.

[2]

A. Arapostathis and A. Biswas, Infinite horizon risk-sensitive control of diffusions without any blanket stability assumptions, Stochastic Process. Appl., 128 (2018), 1485-1524.  doi: 10.1016/j.spa.2017.08.001.

[3]

A. ArapostathisA. BiswasV. S. Borkar and K. S. Kumar, A variational characterization of the risk-sensitive average reward for controlled diffusions on $\mathbb{R}^{d}$, SIAM J. Control Optim., 58 (2020), 3785-3813.  doi: 10.1137/20M1329202.

[4]

A. Arapostathis, A. Biswas and S. Saha, Strict monotonicity of principal eigenvalues of elliptic operators in $\mathbb{R}^{d}$ and risk-sensitive control, J. Math. Pures Appl., 124 (2019), 169–219. doi: 10.1016/j.matpur.2018.05.008.

[5]

A. Arapostathis, V. S. Borkar and M. K. Ghosh, Ergodic Control of Diffusion Processes, Cambridge University Press, Cambridge, U.K. 2012.

[6]

A. ArapostathisV. S. Borkar and K. S. Kumar, Risk-sensitive control and an abstract Collatz-Wielandt formula, J. Theoret. Probab., 29 (2016), 1458-1484.  doi: 10.1007/s10959-015-0616-x.

[7]

A. Bagchi and K. Suresh Kumar, Dynamic asset management with risk-sensitive criterion and non-negative factor constraints: A differential game approach, Stochastics, 81 (2009), 503-530.  doi: 10.1080/17442500902774917.

[8]

A. Basu and M. K. Ghosh, Zero-sum risk-sensitive stochastic differential games, Math. Oper. Res., 37 (2012), 437-449.  doi: 10.1287/moor.1120.0542.

[9]

V. E. Bene$\breve{s}$, Existence of optimal strategies based on specified information of a class of stochastic decision problems, SIAM J. Control, 8 (1970), 179-188.  doi: 10.1137/0308012.

[10]

A. Biswas, An eigenvalue approach to the risk-sensitive control problem in near monotone case, Systems Control Lett., 60 (2011), 181-184.  doi: 10.1016/j.sysconle.2010.12.002.

[11]

A. Biswas, V. S. Borkar and K. Suresh Kumar, Risk-sensitive control with near monotone cost, Appl. Math. Optim., 62 (2009), 145–163. Errata corriege Ⅰ. ibid 62 (2010), 165–167. Errata corriege Ⅱ. ibid 62 (2010), 435–438. doi: 10.1007/s00245-010-9119-4.

[12]

A. Biswas and S. Saha, Zero-sum stochastic differential games with risk-sensitive cost, App. Math. Optim., 81 (2020), 113-140.  doi: 10.1007/s00245-018-9479-8.

[13]

V. Borkar and A. Budhiraja, Ergodic control for constained diffusions: Characterization using HJB equation, SIAM J. Control Optim., 43 (2005), 1467-1492.  doi: 10.1137/S0363012902417619.

[14]

V. S. Borkar and M. K. Ghosh, Stochastic differential games: occupation measure based approach, J. Optim. Theory Appl., 73 (1992), 359-385. Errata corriege. ibid 88 (1996), 251–252. doi: 10.1007/BF02192034.

[15]

A. Budhiraja, An ergodic control problem for constrained diffusion processes: Existence of optimal Markov control, SIAM J. Control Optim., 42 (2003), 532-558.  doi: 10.1137/S0363012901379073.

[16]

K. L. Chung and Z. X. Zhao, From Brownian Motion to Schrödinger's Equation, Grundlehren der Mathematischen Wissenschaften, 312, Springer, Berlin, 1995. doi: 10.1007/978-3-642-57856-4.

[17]

K. Fan, Fixed-point and minimax theorems in locally convex topological linear spaces, Proc. Nat. Acad. Sci. U.S.A., 38 (1952), 121-126.  doi: 10.1073/pnas.38.2.121.

[18]

J. Filar and K. Vrieze, Competitive Markov Decision Processes, Springer-Verlag, New York, 1997.

[19]

S. K. GauttamK. S. Kumar and C. Pal, Risk-sensitive control of reflected diffusion process on orthrant, Pure Appl. Funct. Anal., 2 (2017), 477-510. 

[20]

M. K. Ghosh and A. Bagchi, Stochastic games with average payoff criterion, Appl. Math. Optim., 38 (1998), 283-301.  doi: 10.1007/s002459900092.

[21]

M. K. Ghosh and S. Pradhan, Risk-sensitive stochastic differential game with reflecting diffusions, Stoch. Anal. Appl., 36 (2018), 1-27.  doi: 10.1080/07362994.2017.1356732.

[22]

M. K. Ghosh and S. Pradhan, Zero-sum risk-sensitive stochastic differential games with reflecting diffusions in the orthant, ESAIM Control Optim. Calc. Var., 26 (2020), Paper No. 114, 33 pp. doi: 10.1051/cocv/2020029.

[23]

M. K. Ghosh and K. Suresh Kumar, A stochastic differential game in the orthant, J. Math. Anal. Appl., 265 (2002), 12-37.  doi: 10.1006/jmaa.2001.7679.

[24]

M. K. Ghosh and K. Suresh Kumar, A nonzero-sum stochastic differential game in the orthant, J. Math. Anal. Appl., 305 (2005), 158-174.  doi: 10.1016/j.jmaa.2004.11.002.

[25]

M. K. Ghosh and K. Suresh Kumar, Nonzero-sum risk-sensitive stochastic differential games with reflecting diffusions, Comput. Appl. Math., 18 (1999), 355-368. 

[26]

M. K. Ghosh, K. Suresh Kumar and C. Pal, Nonzero-sum risk-sensitive stochastic differential games, arXiv: 1604.01142

[27]

M. K. Ghosh, K. Suresh Kumar, C. Pal and S. Pradhan, Nonzero-sum risk-sensitive stochastic differential games with discounted costs, Stochastic Analysis and Applications, 39 (2021), 306-326. doi: 10.1080/07362994.2020.1796707.

[28]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 224, 2nd ed., Springer, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.

[29]

C. J. HimmelbergT. ParthasarathyT. E. S. Raghavan and F. S. Van Fleck, Existence of $p$-equilibrium and optimal stationary strategies in stochastic games, Proc. Amer. Math. Soc., 60 (1976), 245-251.  doi: 10.2307/2041151.

[30]

D. L. Iglehart and W. Whitt, Multiple channel queues in heavy traffic, Adv. in Appl. Prob., 2 (1970), 150-177.  doi: 10.2307/3518347.

[31]

H. J. Kushner and L. F. Martins, Limit theorems for pathwise average cost per unit time problems for controlled queues in heavy traffic, Stochastics Stochastics Rep., 42 (1993), 25-51.  doi: 10.1080/17442509308833808.

[32]

H. J. Kushner and K. M. Ramachandran, Optimal and approximately optimal control policies for queues in heavy traffic, SIAM J. Control Optim., 27 (1989), 1293–1318. doi: 10.1137/0327066.

[33]

A. J. Lemoine, Network of queues - A survey of weak convergence results, Management Science, 24 (1978), 1175-1193.  doi: 10.1287/mnsc.24.11.1175.

[34]

G. Leoni, A First Course in Sobolev Spaces, Graduate Studies in Mathematics, 105, AMS, Rhode Island USA, 2009. doi: 10.1090/gsm/105.

[35]

G. M. Lieberman, Oblique Derivative Problems for Elliptic Equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013. doi: 10.1142/8679.

[36]

G. M. Lieberman, Local estimates for subsolutions and supersolutions of oblique derivative problems for general second order elliptic equations, Trans. Amer. Math. Soc., 304 (1987), 343-353.  doi: 10.1090/S0002-9947-1987-0906819-0.

[37]

P.-L. Lions and A.-S. Sznitman, Stochastic differential equations with reflecting boundary conditions, Comm. Pure Appl. Math., 37 (1984), 511-537.  doi: 10.1002/cpa.3160370408.

[38]

J.-L. Menaldi and M. Robin, Remarks on risk-sensitive control problems, Appl. Math. Optim., 52 (2005), 297-310.  doi: 10.1007/s00245-005-0829-y.

[39]

A. S. Nowak, Notes on risk-sensitive Nash equilibria. Advances in dynamic games, Ann. International. Soc. Dynam. Games, 7 (2005), 95-109.  doi: 10.1007/0-8176-4429-6_5.

[40]

S. Pradhan, Risk-sensitive ergodic control of reflected diffusion processes in orthant, Appl. Math. Optim., (2019), to appear. doi: 10.1007/s00245-019-09606-w.

[41]

M. I. Reiman, Open queueing networks in heavy traffic, Math. Oper. Res., 9 (1984), 441-458.  doi: 10.1287/moor.9.3.441.

[42]

A. Yu. Veretennikov, On strong and weak solutions of one-dimensional stochastic equations with boundary conditions, Theory Probab. Appl., 26 (1981), 670-686. 

[43]

J. Warga, Functions of relaxed controls, SIAM J. Control, 5 (1967), 628-641.  doi: 10.1137/0305042.

[44]

Z. Wu, J. Yin and C. Wang, Elliptic and Parabolic Equations, World Scientific, Singapore, 2006. doi: 10.1142/6238.

[1]

Chandan Pal, Somnath Pradhan. Zero-sum games for pure jump processes with risk-sensitive discounted cost criteria. Journal of Dynamics and Games, 2022, 9 (1) : 13-25. doi: 10.3934/jdg.2021020

[2]

Beatris A. Escobedo-Trujillo. Discount-sensitive equilibria in zero-sum stochastic differential games. Journal of Dynamics and Games, 2016, 3 (1) : 25-50. doi: 10.3934/jdg.2016002

[3]

Alejandra Fonseca-Morales, Onésimo Hernández-Lerma. A note on differential games with Pareto-optimal NASH equilibria: Deterministic and stochastic models. Journal of Dynamics and Games, 2017, 4 (3) : 195-203. doi: 10.3934/jdg.2017012

[4]

Adel Chala, Dahbia Hafayed. On stochastic maximum principle for risk-sensitive of fully coupled forward-backward stochastic control of mean-field type with application. Evolution Equations and Control Theory, 2020, 9 (3) : 817-843. doi: 10.3934/eect.2020035

[5]

Getachew K. Befekadu, Panos J. Antsaklis. On noncooperative $n$-player principal eigenvalue games. Journal of Dynamics and Games, 2015, 2 (1) : 51-63. doi: 10.3934/jdg.2015.2.51

[6]

Jasmina Djordjević, Svetlana Janković. Reflected backward stochastic differential equations with perturbations. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 1833-1848. doi: 10.3934/dcds.2018075

[7]

Rui Mu, Zhen Wu. Nash equilibrium points of recursive nonzero-sum stochastic differential games with unbounded coefficients and related multiple\\ dimensional BSDEs. Mathematical Control and Related Fields, 2017, 7 (2) : 289-304. doi: 10.3934/mcrf.2017010

[8]

Athanasios Kehagias. A note on the Nash equilibria of some multi-player reachability/safety games. Journal of Dynamics and Games, 2022, 9 (1) : 117-122. doi: 10.3934/jdg.2021028

[9]

Yannick Viossat. Game dynamics and Nash equilibria. Journal of Dynamics and Games, 2014, 1 (3) : 537-553. doi: 10.3934/jdg.2014.1.537

[10]

Ying Hu, Shanjian Tang. Switching game of backward stochastic differential equations and associated system of obliquely reflected backward stochastic differential equations. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5447-5465. doi: 10.3934/dcds.2015.35.5447

[11]

Jun Moon. Linear-quadratic mean-field type stackelberg differential games for stochastic jump-diffusion systems. Mathematical Control and Related Fields, 2022, 12 (2) : 371-404. doi: 10.3934/mcrf.2021026

[12]

Juan Li, Wenqiang Li. Controlled reflected mean-field backward stochastic differential equations coupled with value function and related PDEs. Mathematical Control and Related Fields, 2015, 5 (3) : 501-516. doi: 10.3934/mcrf.2015.5.501

[13]

Bin Zhou, Hailin Sun. Two-stage stochastic variational inequalities for Cournot-Nash equilibrium with risk-averse players under uncertainty. Numerical Algebra, Control and Optimization, 2020, 10 (4) : 521-535. doi: 10.3934/naco.2020049

[14]

Jingzhen Liu, Ka-Fai Cedric Yiu. Optimal stochastic differential games with VaR constraints. Discrete and Continuous Dynamical Systems - B, 2013, 18 (7) : 1889-1907. doi: 10.3934/dcdsb.2013.18.1889

[15]

Alain Bensoussan, Jens Frehse, Christine Grün. Stochastic differential games with a varying number of players. Communications on Pure and Applied Analysis, 2014, 13 (5) : 1719-1736. doi: 10.3934/cpaa.2014.13.1719

[16]

Jiequn Han, Ruimeng Hu, Jihao Long. Convergence of deep fictitious play for stochastic differential games. Frontiers of Mathematical Finance, , () : -. doi: 10.3934/fmf.2021011

[17]

Carlos Hervés-Beloso, Emma Moreno-García. Market games and walrasian equilibria. Journal of Dynamics and Games, 2020, 7 (1) : 65-77. doi: 10.3934/jdg.2020004

[18]

Yahia Zare Mehrjerdi. A novel methodology for portfolio selection in fuzzy multi criteria environment using risk-benefit analysis and fractional stochastic. Numerical Algebra, Control and Optimization, 2021  doi: 10.3934/naco.2021019

[19]

J. R. L. Webb. Uniqueness of the principal eigenvalue in nonlocal boundary value problems. Discrete and Continuous Dynamical Systems - S, 2008, 1 (1) : 177-186. doi: 10.3934/dcdss.2008.1.177

[20]

Fei-Ying Yang, Wan-Tong Li, Jian-Wen Sun. Principal eigenvalues for some nonlocal eigenvalue problems and applications. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 4027-4049. doi: 10.3934/dcds.2016.36.4027

2020 Impact Factor: 1.284

Metrics

  • PDF downloads (334)
  • HTML views (433)
  • Cited by (0)

Other articles
by authors

[Back to Top]