doi: 10.3934/dcdsb.2021065

Single-target networks

1. 

Department of Mathematics and Department of Biomolecular Chemistry, University of Wisconsin-Madison, 53706

2. 

Department of Mathematics, University of Wisconsin-Madison, 53706

* Corresponding author: Jiaxin Jin

Received  June 2020 Revised  December 2020 Published  March 2021

Fund Project: The authors are supported by NSF grant DMS-1816238

Reaction networks can be regarded as finite oriented graphs embedded in Euclidean space. Single-target networks are reaction networks with an arbitrarily set of source vertices, but only one sink vertex. We completely characterize the dynamics of all mass-action systems generated by single-target networks, as follows: either (i) the system is globally stable for all choice of rate constants (in fact, is dynamically equivalent to a detailed-balanced system with a single linkage class), or (ii) the system has no positive steady states for any choice of rate constants and all trajectories must converge to the boundary of the positive orthant or to infinity. Moreover, we show that global stability occurs if and only if the target vertex of the network is in the relative interior of the convex hull of the source vertices.

Citation: Gheorghe Craciun, Jiaxin Jin, Polly Y. Yu. Single-target networks. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021065
References:
[1]

D. F. Anderson, A proof of the global attractor conjecture in the single linkage class case, SIAM Journal on Applied Mathematics, 71 (2011), 1487-1508.  doi: 10.1137/11082631X.  Google Scholar

[2]

D. F. Anderson, J. D. Brunner, G. Craciun and M. D. Johnston, On classes of reaction networks and their associated polynomial dynamical systems, (2020). Google Scholar

[3]

D. Angeli, A tutorial on chemical reaction network dynamics, European Journal of Control, 15 (2009), 398-406.  doi: 10.3166/ejc.15.398-406.  Google Scholar

[4]

M. W. Birch, Maximum likelihood in three-way contingency tables, Journal of the Royal Statistical Society. Series B (Methodological), 25 (1963), 220-233.  doi: 10.1111/j.2517-6161.1963.tb00504.x.  Google Scholar

[5]

B. Boros, Existence of positive steady states for weakly reversible mass-action systems, SIAM Journal on Mathematical Analysis, 51 (2019), 435-449.  doi: 10.1137/17M115534X.  Google Scholar

[6]

B. Boros and J. Hofbauer, Permanence of weakly reversible mass-action systems with a single linkage class, SIAM Journal on Applied Dynamical Systems, 19 (2020), 352-365.  doi: 10.1137/19M1248431.  Google Scholar

[7]

M. L. Brustenga, G. Craciun and M-S Sorea, Disguised toric dynamical systems, (2020). Google Scholar

[8]

G. Craciun, Toric differential inclusions and a proof of the global attractor conjecture, (2015), arXiv: 1501.02860 [math.DS]. Google Scholar

[9]

G. Craciun, Polynomial dynamical systems, reaction networks, and toric differential inclusions, SIAM Journal on Applied Algebra and Geometry, 3 (2019), 87-106.  doi: 10.1137/17M1129076.  Google Scholar

[10]

G. CraciunA. DickensteinB. Sturmfels and A. Shiu, Toric dynamical systems, Journal of Symbolic Computation, 44 (2009), 1551-1565.  doi: 10.1016/j.jsc.2008.08.006.  Google Scholar

[11]

G. CraciunJ. Jin and P. Y. Yu, An efficient characterization of complex-balanced, detailed-balanced, and weakly reversible systems, SIAM Journal on Applied Mathematics, 80 (2020), 183-205.  doi: 10.1137/19M1244494.  Google Scholar

[12]

G. Craciun, J. Jin and P. Y. Yu, Dynamical equivalence to complex balancing as an open condition in parameter space, in Preparation. Google Scholar

[13]

G. CraciunF. Nazarov and C. Pantea, Persistence and permanence of mass-action and power-law dynamical systems, SIAM Journal on Applied Mathematics, 73 (2013), 305-329.  doi: 10.1137/100812355.  Google Scholar

[14]

G. Craciun and C. Pantea, Identifiability of chemical reaction networks, Journal of Mathematical Chemistry, 44 (2008), 244-259.  doi: 10.1007/s10910-007-9307-x.  Google Scholar

[15]

A. Dickenstein and M. Pérez Millán, How far is complex balancing from detailed balancing?, Bulletin of Mathematical Biology, 73 (2011), 811-828.  doi: 10.1007/s11538-010-9611-7.  Google Scholar

[16]

M. Feinberg, Complex balancing in general kinetic systems, Archive for Rational Mechanics and Analysis, 49 (1972), 187-194.  doi: 10.1007/BF00255665.  Google Scholar

[17]

M. Feinberg, Chemical reaction network structure and the stability of complex isothermal reactors - I. The Deficiency Zero and the Deficiency One Theorems, Chemical Engineering Science, 42 (1987), 2229-2268.   Google Scholar

[18]

M. Feinberg, Necessary and sufficient conditions for detailed balancing in mass action systems of arbitrary complexity, Chemical Engineering Science, 44 (1989), 1819-1827.   Google Scholar

[19]

M. Feinberg, Foundations of Chemical Reaction Network Theory, Applied Mathematical Sciences, Springer International Publishing, 2019.  Google Scholar

[20]

M. GopalkrishnanE. Miller and A. Shiu, A geometric approach to the global attractor conjecture, SIAM Journal on Applied Dynamical Systems, 13 (2014), 758-797.  doi: 10.1137/130928170.  Google Scholar

[21]

J. Gunawardena, Chemical reaction network theory for in-silico biologists, (2003), http://vcp.med.harvard.edu/papers/crnt.pdf, Google Scholar

[22]

F. Horn, Necessary and sufficient conditions for complex balancing in chemical kinetics, Archive for Rational Mechanics and Analysis, 49 (1972), 172-186.  doi: 10.1007/BF00255664.  Google Scholar

[23]

F. Horn and R. Jackson, General mass action kinetics, Archive for Rational Mechanics and Analysis, 47 (1972), 81-116.  doi: 10.1007/BF00251225.  Google Scholar

[24]

M. D. Johnston, Translated chemical reaction networks, Bulletin of Mathematical Biology, 76 (2014), 1081-1116.  doi: 10.1007/s11538-014-9947-5.  Google Scholar

[25]

M. D. Johnston and E. Burton, Computing weakly reversible deficiency zero network translations using elementary flux modes, Bulletin of Mathematical Biology, 81 (2019), 1613-1644.  doi: 10.1007/s11538-019-00579-z.  Google Scholar

[26]

M. D. JohnstonD. Siegel and G. Szederkényi, Computing weakly reversible linearly conjugate chemical reaction networks with minimal deficiencys, Mathematical Biosciences, 241 (2013), 88-98.  doi: 10.1016/j.mbs.2012.09.008.  Google Scholar

[27]

G. LiptákG. Szederkényi and K. M. Hangos, Computing zero deficiency realizations of kinetic systems, Systems & Control Letters, 81 (2015), 24-30.  doi: 10.1016/j.sysconle.2015.05.001.  Google Scholar

[28]

S. Müller and G. Regensburger, Generalized mass action systems: Complex balancing equilibria and sign vectors of the stoichiometric and kinetic-order subspaces, SIAM Journal on Applied Mathematics, 72 (2012), 1926-1947.  doi: 10.1137/110847056.  Google Scholar

[29]

L. Onsager, Reciprocal relations in irreversible processes I., Physical Review, 37 (1931), 405-426.   Google Scholar

[30] L. Pachter and B. Sturmfels, Algebraic Statistics for Computational Biology, Cambridge University Press, 2005.  doi: 10.1017/CBO9780511610684.007.  Google Scholar
[31]

C. Pantea, On the persistence and global stability of mass-action systems, SIAM Journal on Mathematical Analysis, 44 (2012), 1636-1673.  doi: 10.1137/110840509.  Google Scholar

[32]

J. RudanG. SzederkényiK. M. Hangos and T. Péni, Polynomial time algorithms to determine weakly reversible realizations of chemical reaction networks, Journal of Mathematical Chemistry, 52 (2014), 1386-1404.  doi: 10.1007/s10910-014-0318-0.  Google Scholar

[33]

S. Schuster and R. Schuster, A generalization of Wegscheider's condition. Implications for properties of steady states and for quasi-steady-state approximation, Journal of Mathematical Chemistry, 3 (1989), 25-42.  doi: 10.1007/BF01171883.  Google Scholar

[34]

G. Szederkényi, Comment on "Identifiability of chemical reaction networks" by G. Craciun and C. Pantea, Journal of Mathematical Chemistry, 45 (2009), 1172-1174.  doi: 10.1007/s10910-008-9499-8.  Google Scholar

[35]

G. SzederkényiJ. R. Banga and A. A. Alonso, CRNreals: A toolbox for distinguishability and identifiability analysis of biochemical reaction networks, Bioinformatics, 28 (2012), 1549-1550.   Google Scholar

[36]

G. Szederkényi and K. M. Hangos, Finding complex balanced and detailed balanced realizations of chemical reaction networks, Journal of Mathematical Chemistry, 49 (2011), 1163-1179.  doi: 10.1007/s10910-011-9804-9.  Google Scholar

[37]

A. I. Vol'pert, Differential equations on graphs, Math. USSR-Sb, 88 (1972), 578-588.   Google Scholar

[38]

R. Wegscheider, Über simultane gleichgewichte und die beziehungen zwischen thermodynamik und reactionskinetik homogener systeme, Monatshefte für Chemie und verwandte Teile anderer Wissenschaften, 22 (1901), 849-906.   Google Scholar

[39]

P. Y. Yu and G. Craciun, Mathematical analysis of chemical reaction systems, Israel Journal of Chemistry, 58 (2018), 733-741.   Google Scholar

show all references

References:
[1]

D. F. Anderson, A proof of the global attractor conjecture in the single linkage class case, SIAM Journal on Applied Mathematics, 71 (2011), 1487-1508.  doi: 10.1137/11082631X.  Google Scholar

[2]

D. F. Anderson, J. D. Brunner, G. Craciun and M. D. Johnston, On classes of reaction networks and their associated polynomial dynamical systems, (2020). Google Scholar

[3]

D. Angeli, A tutorial on chemical reaction network dynamics, European Journal of Control, 15 (2009), 398-406.  doi: 10.3166/ejc.15.398-406.  Google Scholar

[4]

M. W. Birch, Maximum likelihood in three-way contingency tables, Journal of the Royal Statistical Society. Series B (Methodological), 25 (1963), 220-233.  doi: 10.1111/j.2517-6161.1963.tb00504.x.  Google Scholar

[5]

B. Boros, Existence of positive steady states for weakly reversible mass-action systems, SIAM Journal on Mathematical Analysis, 51 (2019), 435-449.  doi: 10.1137/17M115534X.  Google Scholar

[6]

B. Boros and J. Hofbauer, Permanence of weakly reversible mass-action systems with a single linkage class, SIAM Journal on Applied Dynamical Systems, 19 (2020), 352-365.  doi: 10.1137/19M1248431.  Google Scholar

[7]

M. L. Brustenga, G. Craciun and M-S Sorea, Disguised toric dynamical systems, (2020). Google Scholar

[8]

G. Craciun, Toric differential inclusions and a proof of the global attractor conjecture, (2015), arXiv: 1501.02860 [math.DS]. Google Scholar

[9]

G. Craciun, Polynomial dynamical systems, reaction networks, and toric differential inclusions, SIAM Journal on Applied Algebra and Geometry, 3 (2019), 87-106.  doi: 10.1137/17M1129076.  Google Scholar

[10]

G. CraciunA. DickensteinB. Sturmfels and A. Shiu, Toric dynamical systems, Journal of Symbolic Computation, 44 (2009), 1551-1565.  doi: 10.1016/j.jsc.2008.08.006.  Google Scholar

[11]

G. CraciunJ. Jin and P. Y. Yu, An efficient characterization of complex-balanced, detailed-balanced, and weakly reversible systems, SIAM Journal on Applied Mathematics, 80 (2020), 183-205.  doi: 10.1137/19M1244494.  Google Scholar

[12]

G. Craciun, J. Jin and P. Y. Yu, Dynamical equivalence to complex balancing as an open condition in parameter space, in Preparation. Google Scholar

[13]

G. CraciunF. Nazarov and C. Pantea, Persistence and permanence of mass-action and power-law dynamical systems, SIAM Journal on Applied Mathematics, 73 (2013), 305-329.  doi: 10.1137/100812355.  Google Scholar

[14]

G. Craciun and C. Pantea, Identifiability of chemical reaction networks, Journal of Mathematical Chemistry, 44 (2008), 244-259.  doi: 10.1007/s10910-007-9307-x.  Google Scholar

[15]

A. Dickenstein and M. Pérez Millán, How far is complex balancing from detailed balancing?, Bulletin of Mathematical Biology, 73 (2011), 811-828.  doi: 10.1007/s11538-010-9611-7.  Google Scholar

[16]

M. Feinberg, Complex balancing in general kinetic systems, Archive for Rational Mechanics and Analysis, 49 (1972), 187-194.  doi: 10.1007/BF00255665.  Google Scholar

[17]

M. Feinberg, Chemical reaction network structure and the stability of complex isothermal reactors - I. The Deficiency Zero and the Deficiency One Theorems, Chemical Engineering Science, 42 (1987), 2229-2268.   Google Scholar

[18]

M. Feinberg, Necessary and sufficient conditions for detailed balancing in mass action systems of arbitrary complexity, Chemical Engineering Science, 44 (1989), 1819-1827.   Google Scholar

[19]

M. Feinberg, Foundations of Chemical Reaction Network Theory, Applied Mathematical Sciences, Springer International Publishing, 2019.  Google Scholar

[20]

M. GopalkrishnanE. Miller and A. Shiu, A geometric approach to the global attractor conjecture, SIAM Journal on Applied Dynamical Systems, 13 (2014), 758-797.  doi: 10.1137/130928170.  Google Scholar

[21]

J. Gunawardena, Chemical reaction network theory for in-silico biologists, (2003), http://vcp.med.harvard.edu/papers/crnt.pdf, Google Scholar

[22]

F. Horn, Necessary and sufficient conditions for complex balancing in chemical kinetics, Archive for Rational Mechanics and Analysis, 49 (1972), 172-186.  doi: 10.1007/BF00255664.  Google Scholar

[23]

F. Horn and R. Jackson, General mass action kinetics, Archive for Rational Mechanics and Analysis, 47 (1972), 81-116.  doi: 10.1007/BF00251225.  Google Scholar

[24]

M. D. Johnston, Translated chemical reaction networks, Bulletin of Mathematical Biology, 76 (2014), 1081-1116.  doi: 10.1007/s11538-014-9947-5.  Google Scholar

[25]

M. D. Johnston and E. Burton, Computing weakly reversible deficiency zero network translations using elementary flux modes, Bulletin of Mathematical Biology, 81 (2019), 1613-1644.  doi: 10.1007/s11538-019-00579-z.  Google Scholar

[26]

M. D. JohnstonD. Siegel and G. Szederkényi, Computing weakly reversible linearly conjugate chemical reaction networks with minimal deficiencys, Mathematical Biosciences, 241 (2013), 88-98.  doi: 10.1016/j.mbs.2012.09.008.  Google Scholar

[27]

G. LiptákG. Szederkényi and K. M. Hangos, Computing zero deficiency realizations of kinetic systems, Systems & Control Letters, 81 (2015), 24-30.  doi: 10.1016/j.sysconle.2015.05.001.  Google Scholar

[28]

S. Müller and G. Regensburger, Generalized mass action systems: Complex balancing equilibria and sign vectors of the stoichiometric and kinetic-order subspaces, SIAM Journal on Applied Mathematics, 72 (2012), 1926-1947.  doi: 10.1137/110847056.  Google Scholar

[29]

L. Onsager, Reciprocal relations in irreversible processes I., Physical Review, 37 (1931), 405-426.   Google Scholar

[30] L. Pachter and B. Sturmfels, Algebraic Statistics for Computational Biology, Cambridge University Press, 2005.  doi: 10.1017/CBO9780511610684.007.  Google Scholar
[31]

C. Pantea, On the persistence and global stability of mass-action systems, SIAM Journal on Mathematical Analysis, 44 (2012), 1636-1673.  doi: 10.1137/110840509.  Google Scholar

[32]

J. RudanG. SzederkényiK. M. Hangos and T. Péni, Polynomial time algorithms to determine weakly reversible realizations of chemical reaction networks, Journal of Mathematical Chemistry, 52 (2014), 1386-1404.  doi: 10.1007/s10910-014-0318-0.  Google Scholar

[33]

S. Schuster and R. Schuster, A generalization of Wegscheider's condition. Implications for properties of steady states and for quasi-steady-state approximation, Journal of Mathematical Chemistry, 3 (1989), 25-42.  doi: 10.1007/BF01171883.  Google Scholar

[34]

G. Szederkényi, Comment on "Identifiability of chemical reaction networks" by G. Craciun and C. Pantea, Journal of Mathematical Chemistry, 45 (2009), 1172-1174.  doi: 10.1007/s10910-008-9499-8.  Google Scholar

[35]

G. SzederkényiJ. R. Banga and A. A. Alonso, CRNreals: A toolbox for distinguishability and identifiability analysis of biochemical reaction networks, Bioinformatics, 28 (2012), 1549-1550.   Google Scholar

[36]

G. Szederkényi and K. M. Hangos, Finding complex balanced and detailed balanced realizations of chemical reaction networks, Journal of Mathematical Chemistry, 49 (2011), 1163-1179.  doi: 10.1007/s10910-011-9804-9.  Google Scholar

[37]

A. I. Vol'pert, Differential equations on graphs, Math. USSR-Sb, 88 (1972), 578-588.   Google Scholar

[38]

R. Wegscheider, Über simultane gleichgewichte und die beziehungen zwischen thermodynamik und reactionskinetik homogener systeme, Monatshefte für Chemie und verwandte Teile anderer Wissenschaften, 22 (1901), 849-906.   Google Scholar

[39]

P. Y. Yu and G. Craciun, Mathematical analysis of chemical reaction systems, Israel Journal of Chemistry, 58 (2018), 733-741.   Google Scholar

Figure 1.  (a) A single-target network that is globally stable under mass-action kinetics. (b)–(c) Single-target networks with no positive steady states. (d) Not a single-target network
Figure 2.  Consider subnetworks of (a) under mass-action kinetics, whose associated dynamics is given by (9). If the coefficient of $ {\boldsymbol{x}}^{ {\boldsymbol{y}}_1} $ in $ \dot{x} $ is positive and the coefficient of $ {\boldsymbol{x}}^{ {\boldsymbol{y}}_3} $ in $ \dot{x} $ is negative, then the system (9) can be realized by a single-target network, determined by the sign of $ {\boldsymbol{x}}^{ {\boldsymbol{y}}_2} $ in $ \dot{x} $. If the net directions are as shown in (b), then (9) can be realized by the single-target network in (c). Similarly, if the net directions appear as in (d), then (9) can be realized by the network in (e)
Figure 3.  Reversible systems in (a) Example 3.12 and (b) Example 3.13 that are dynamically equivalent to detailed-balanced systems. Each undirected edge represents a pair of reversible edges
Figure 4.  Geometric argument for dynamically equivalence to single-target network in Example 3.12. Shown are the edges with $ \mathrm{{X}}_1 + \mathrm{{X}}_2 $ as their source. The centre $ \left(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2} \right)^\top $ of the tetrahedron is marked in blue. With rate constants given in the example, the weighted sum of reaction vectors points from the source to the centre
Figure 5.  (a) The mass-action system with two target vertices from Example 4.2, which is dynamically equivalent to a complex-balanced system using a subnetwork of (b) if and only if $ \frac{1}{25} \leq \frac{ \kappa_1 \kappa_3}{ \kappa_2 \kappa_4} \leq 25 $
Figure 5(a) is dynamically equivalent to a complex-balanced system if and only if $ \frac{1}{25} \leq \frac{ \kappa_1 \kappa_3}{ \kappa_2 \kappa_4} \leq 25 $. The system is equivalent to (a) when $ \kappa_2 \kappa_4 = 25 \kappa_1 \kappa_3 $ and (b) when $ 25 \kappa_2 \kappa_4 = \kappa_1 \kappa_3 $. For $ \frac{1}{25} \leq \frac{ \kappa_1 \kappa_3}{ \kappa_2 \kappa_4} \leq 25 $, the dynamically equivalent system is an appropriate convex combination of (a) and (b)">Figure 6.  The system in Figure 5(a) is dynamically equivalent to a complex-balanced system if and only if $ \frac{1}{25} \leq \frac{ \kappa_1 \kappa_3}{ \kappa_2 \kappa_4} \leq 25 $. The system is equivalent to (a) when $ \kappa_2 \kappa_4 = 25 \kappa_1 \kappa_3 $ and (b) when $ 25 \kappa_2 \kappa_4 = \kappa_1 \kappa_3 $. For $ \frac{1}{25} \leq \frac{ \kappa_1 \kappa_3}{ \kappa_2 \kappa_4} \leq 25 $, the dynamically equivalent system is an appropriate convex combination of (a) and (b)
[1]

Mauricio Achigar. Extensions of expansive dynamical systems. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3093-3108. doi: 10.3934/dcds.2020399

[2]

Monica Conti, Lorenzo Liverani, Vittorino Pata. A note on the energy transfer in coupled differential systems. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021042

[3]

Khosro Sayevand, Valeyollah Moradi. A robust computational framework for analyzing fractional dynamical systems. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021022

[4]

Xinyuan Liao, Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1087-1111. doi: 10.3934/cpaa.2007.6.1087

[5]

Alessandro Fonda, Rodica Toader. A dynamical approach to lower and upper solutions for planar systems "To the memory of Massimo Tarallo". Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3683-3708. doi: 10.3934/dcds.2021012

[6]

Tianhu Yu, Jinde Cao, Chuangxia Huang. Finite-time cluster synchronization of coupled dynamical systems with impulsive effects. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3595-3620. doi: 10.3934/dcdsb.2020248

[7]

Emma D'Aniello, Saber Elaydi. The structure of $ \omega $-limit sets of asymptotically non-autonomous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 903-915. doi: 10.3934/dcdsb.2019195

[8]

Yila Bai, Haiqing Zhao, Xu Zhang, Enmin Feng, Zhijun Li. The model of heat transfer of the arctic snow-ice layer in summer and numerical simulation. Journal of Industrial & Management Optimization, 2005, 1 (3) : 405-414. doi: 10.3934/jimo.2005.1.405

[9]

Mirela Kohr, Sergey E. Mikhailov, Wolfgang L. Wendland. Dirichlet and transmission problems for anisotropic stokes and Navier-Stokes systems with L tensor coefficient under relaxed ellipticity condition. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021042

[10]

Samir Adly, Oanh Chau, Mohamed Rochdi. Solvability of a class of thermal dynamical contact problems with subdifferential conditions. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 91-104. doi: 10.3934/naco.2012.2.91

[11]

Wenmin Gong, Guangcun Lu. On coupled Dirac systems. Discrete & Continuous Dynamical Systems, 2017, 37 (8) : 4329-4346. doi: 10.3934/dcds.2017185

[12]

Haiyan Wang. Existence and nonexistence of positive radial solutions for quasilinear systems. Conference Publications, 2009, 2009 (Special) : 810-817. doi: 10.3934/proc.2009.2009.810

[13]

Tuvi Etzion, Alexander Vardy. On $q$-analogs of Steiner systems and covering designs. Advances in Mathematics of Communications, 2011, 5 (2) : 161-176. doi: 10.3934/amc.2011.5.161

[14]

Alberto Bressan, Ke Han, Franco Rampazzo. On the control of non holonomic systems by active constraints. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3329-3353. doi: 10.3934/dcds.2013.33.3329

[15]

Lekbir Afraites, Abdelghafour Atlas, Fahd Karami, Driss Meskine. Some class of parabolic systems applied to image processing. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1671-1687. doi: 10.3934/dcdsb.2016017

[16]

Graziano Crasta, Philippe G. LeFloch. Existence result for a class of nonconservative and nonstrictly hyperbolic systems. Communications on Pure & Applied Analysis, 2002, 1 (4) : 513-530. doi: 10.3934/cpaa.2002.1.513

[17]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1717-1746. doi: 10.3934/dcdss.2020451

[18]

Yu-Hsien Liao. Solutions and characterizations under multicriteria management systems. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021041

[19]

Zhang Chen, Xiliang Li, Bixiang Wang. Invariant measures of stochastic delay lattice systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3235-3269. doi: 10.3934/dcdsb.2020226

[20]

Jicheng Liu, Meiling Zhao. Normal deviation of synchronization of stochastic coupled systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021079

2019 Impact Factor: 1.27

Article outline

Figures and Tables

[Back to Top]