# American Institute of Mathematical Sciences

October  2015, 20(8): 2497-2526. doi: 10.3934/dcdsb.2015.20.2497

## Separable Lyapunov functions for monotone systems: Constructions and limitations

 1 Institut für Mathematik, Universität Würzburg, Campus Hubland Nord, Emil-Fischer-Str. 40, 97074 Würzburg, Germany 2 Department of Systems Design and Informatics, Kyushu Institute of Technology, 680-4 Kawazu, Iizuka 820-8502, Japan 3 Automatic Control LTH, Lund University, Box 118, SE-221 00 Lund, Sweden 4 School of Mathematical & Physical Sciences, Faculty of Science & IT, The University of Newcastle (UON), University Drive, Callaghan NSW 2308, Australia

Received  July 2014 Revised  January 2015 Published  August 2015

For monotone systems evolving on the positive orthant of $\mathbb{R}^n_+$ two types of Lyapunov functions are considered: Sum- and max-separable Lyapunov functions. One can be written as a sum, the other as a maximum of functions of scalar arguments. Several constructive existence results for both types are given. Notably, one construction provides a max-separable Lyapunov function that is defined at least on an arbitrarily large compact set, based on little more than the knowledge about one trajectory. Another construction for a class of planar systems yields a global sum-separable Lyapunov function, provided the right hand side satisfies a small-gain type condition. A number of examples demonstrate these methods and shed light on the relation between the shape of sublevel sets and the right hand side of the system equation. Negative examples show that there are indeed globally asymptotically stable systems that do not admit either type of Lyapunov function.
Citation: Gunther Dirr, Hiroshi Ito, Anders Rantzer, Björn S. Rüffer. Separable Lyapunov functions for monotone systems: Constructions and limitations. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2497-2526. doi: 10.3934/dcdsb.2015.20.2497
##### References:
 [1] D. Angeli and A. Astolfi, A tight small-gain theorem for not necessarily ISS systems,, Systems Control Lett., 56 (2007), 87. doi: 10.1016/j.sysconle.2006.08.003. Google Scholar [2] D. Angeli, E. D. Sontag and Y. Wang, A characterization of integral input-to-state stability,, IEEE Trans. Autom. Control, 45 (2000), 1082. doi: 10.1109/9.863594. Google Scholar [3] A. Bacciotti, F. Ceragioli and L. Mazzi, Differential inclusions and monotonicity conditions for nonsmooth Lyapunov functions,, Set-Valued Analysis, 8 (2000), 299. doi: 10.1023/A:1008763931789. Google Scholar [4] A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences,, Academic Press, (1979). Google Scholar [5] N. P. Bhatia and G. P. Szegö, Stability Theory of Dynamical Systems,, Die Grundlehren der mathematischen Wissenschaften, (1970). Google Scholar [6] F. H. Clarke, Y. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory,, Springer, (1998). Google Scholar [7] G. Como, E. Lovisari and K. Savla, Throughput optimality and overload behavior of dynamical flow networks under monotone distributed routing,, IEEE Trans. Contr. Network Systems, 2 (2014), 57. doi: 10.1109/TCNS.2014.2367361. Google Scholar [8] S. N. Dashkovskiy, B. S. Rüffer and F. R. Wirth, Small gain theorems for large scale systems and construction of ISS Lyapunov functions,, SIAM J. Control Optim., 48 (2010), 4089. doi: 10.1137/090746483. Google Scholar [9] P. De Leenheer, D. Angeli and E. D. Sontag, Monotone chemical reaction networks,, Journal of Mathematical Chemistry, 41 (2007), 295. doi: 10.1007/s10910-006-9075-z. Google Scholar [10] D. Hinrichsen and A. J. Pritchard, Mathematical Systems Theory I - Modelling, State Space Analysis, Stability and Robustness,, Springer, (2005). doi: 10.1007/b137541. Google Scholar [11] M. W. Hirsch and H. Smith, Monotone dynamical systems,, in Handbook of Differential Equations: Ordinary Differential Equations, (2005), 239. Google Scholar [12] M. W. Hirsch, Systems of differential equations which are competitive or cooperative. I. Limit sets,, SIAM J. Math. Anal., 13 (1982), 167. doi: 10.1137/0513013. Google Scholar [13] M. W. Hirsch, Systems of differential equations that are competitive or cooperative. II. Convergence almost everywhere,, SIAM J. Math. Anal., 16 (1985), 423. doi: 10.1137/0516030. Google Scholar [14] M. W. Hirsch, Systems of differential equations which are competitive or cooperative. III. Competing species,, Nonlinearity, 1 (1988), 51. doi: 10.1088/0951-7715/1/1/003. Google Scholar [15] M. W. Hirsch, Systems of differential equations that are competitive or cooperative. V. Convergence in $3$-dimensional systems,, J. Diff. Eqns., 80 (1989), 94. doi: 10.1016/0022-0396(89)90097-1. Google Scholar [16] M. W. Hirsch, Systems of differential equations that are competitive or cooperative. IV. Structural stability in three-dimensional systems,, SIAM J. Math. Anal., 21 (1990), 1225. doi: 10.1137/0521067. Google Scholar [17] M. W. Hirsch, Systems of differential equations that are competitive or cooperative. VI. A local $C^r$ closing lemma for 3-dimensional systems,, Ergodic Theory Dynam. Systems, 11 (1991), 443. doi: 10.1017/S014338570000626X. Google Scholar [18] M. W. Hirsch and H. L. Smith, Competitive and cooperative systems: Mini-review,, in Positive systems (Rome, (2003), 183. doi: 10.1007/978-3-540-44928-7_25. Google Scholar [19] H. Ito, A Lyapunov approach to cascade interconnection of integral input-to-state stable systems,, IEEE Trans. Autom. Control, 55 (2010), 702. doi: 10.1109/TAC.2009.2037457. Google Scholar [20] H. Ito and Z.-P. Jiang, Necessary and sufficient small gain conditions for integral input-to-state stable systems: A Lyapunov perspective,, IEEE Trans. Autom. Control, 54 (2009), 2389. doi: 10.1109/TAC.2009.2028980. Google Scholar [21] H. Ito, State-dependent scaling problems and stability of interconnected iISS and ISS systems,, IEEE Trans. Autom. Control, 51 (2006), 1626. doi: 10.1109/TAC.2006.882930. Google Scholar [22] H. Ito, S. Dashkovskiy and F. Wirth, Capability and limitation of max- and sum-type construction of Lyapunov functions for networks of iISS systems,, Automatica J. IFAC, 48 (2012), 1197. doi: 10.1016/j.automatica.2012.03.018. Google Scholar [23] H. Ito, Z.-P. Jiang, S. Dashkovskiy and B. S. Rüffer, Robust stability of networks of iISS systems: Construction of sum-type Lyapunov functions,, IEEE Trans. Autom. Control, 58 (2013), 1192. doi: 10.1109/TAC.2012.2231552. Google Scholar [24] H. Ito, B. S. Rüffer and A. Rantzer, Max- and sum-separable Lyapunov functions for monotone systems and their level sets,, in Proc. 53rd IEEE Conf. Decis. Control, (2014), 2371. doi: 10.1109/CDC.2014.7039750. Google Scholar [25] Z.-P. Jiang, I. M. Y. Mareels and Y. Wang, A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems,, Automatica J. IFAC, 32 (1996), 1211. doi: 10.1016/0005-1098(96)00051-9. Google Scholar [26] H. K. Khalil, Nonlinear systems,, 3rd edition, (2002). Google Scholar [27] V. Lakshmikantham and S. Leela, Differential and Integral Inequalities: Theory and Applications, Vol. I: Ordinary Differential Equations,, Academic Press, (1969). Google Scholar [28] M. Margaliot and T. Tuller, Stability analysis of the ribosome flow model,, IEEE/ACM Trans. Comp. Biology & Bioinformatics, 9 (2012), 1545. doi: 10.1109/TCBB.2012.88. Google Scholar [29] A. Rantzer, Distributed control of positive systems,, in Proc. 50th IEEE Conf. Decis. Control and Europ. Contr. Conf., (2011), 6608. doi: 10.1109/CDC.2011.6161293. Google Scholar [30] A. Rantzer, Optimizing positively dominated systems,, in Proc. 51st IEEE Conf. Decis. Control, (2012), 272. doi: 10.1109/CDC.2012.6426312. Google Scholar [31] A. Rantzer, B. S. Rüffer and G. Dirr, Separable Lyapunov functions for monotone systems,, in Proc. 52nd IEEE Conf. Decis. Control, (2013), 4590. doi: 10.1109/CDC.2013.6760604. Google Scholar [32] B. S. Rüffer, Monotone inequalities, dynamical systems, and paths in the positive orthant of Euclidean $n$-space,, Positivity, 14 (2010), 257. doi: 10.1007/s11117-009-0016-5. Google Scholar [33] B. S. Rüffer, Small-gain conditions and the comparison principle,, IEEE Trans. Autom. Control, 55 (2010), 1732. doi: 10.1109/TAC.2010.2048053. Google Scholar [34] B. S. Rüffer, P. M. Dower and H. Ito, Computational comparison principles for large-scale system stability analysis,, in Proc. of the 10th SICE Annual Conference on Control Systems, (2010). Google Scholar [35] B. S. Rüffer, H. Ito and P. M. Dower, Computing asymptotic gains of large-scale interconnections,, in Proc. 49th IEEE Conf. Decis. Control, (2010), 7413. Google Scholar [36] B. S. Rüffer, C. M. Kellett and S. R. Weller, Connection between cooperative positive systems and integral input-to-state stability of large-scale systems,, Automatica J. IFAC, 46 (2010), 1019. doi: 10.1016/j.automatica.2010.03.012. Google Scholar [37] H. L. Smith, Monotone Dynamical Systems,, Mathematical Surveys and Monographs, (1995). Google Scholar [38] E. D. Sontag, Input to state stability,, in The Control Systems Handbook: Control System Advanced Methods (ed. W. S. Levine), (2010), 1. Google Scholar [39] E. D. Sontag, Smooth stabilization implies coprime factorization,, IEEE Trans. Autom. Control, 34 (1989), 435. doi: 10.1109/9.28018. Google Scholar [40] E. D. Sontag, Comments on integral variants of ISS,, Systems Control Lett., 34 (1998), 93. doi: 10.1016/S0167-6911(98)00003-6. Google Scholar [41] E. D. Sontag, Monotone and near-monotone biochemical networks,, Systems and Synthetic Biology, 1 (2007), 59. doi: 10.1007/s11693-007-9005-9. Google Scholar [42] E. D. Sontag and Y. Wang, On characterizations of the input-to-state stability property,, Systems Control Lett., 24 (1995), 351. doi: 10.1016/0167-6911(94)00050-6. Google Scholar [43] J. A. Yorke, Extending Liapunov's second method to non-Lipschitz Liapunov functions,, Bull. Amer. Math. Soc., 74 (1968), 322. doi: 10.1090/S0002-9904-1968-11940-8. Google Scholar [44] T. Yoshizawa, Stability Theory by Liapunov's Second Method,, Publications of the Mathematical Society of Japan, (1966). Google Scholar

show all references

##### References:
 [1] D. Angeli and A. Astolfi, A tight small-gain theorem for not necessarily ISS systems,, Systems Control Lett., 56 (2007), 87. doi: 10.1016/j.sysconle.2006.08.003. Google Scholar [2] D. Angeli, E. D. Sontag and Y. Wang, A characterization of integral input-to-state stability,, IEEE Trans. Autom. Control, 45 (2000), 1082. doi: 10.1109/9.863594. Google Scholar [3] A. Bacciotti, F. Ceragioli and L. Mazzi, Differential inclusions and monotonicity conditions for nonsmooth Lyapunov functions,, Set-Valued Analysis, 8 (2000), 299. doi: 10.1023/A:1008763931789. Google Scholar [4] A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences,, Academic Press, (1979). Google Scholar [5] N. P. Bhatia and G. P. Szegö, Stability Theory of Dynamical Systems,, Die Grundlehren der mathematischen Wissenschaften, (1970). Google Scholar [6] F. H. Clarke, Y. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory,, Springer, (1998). Google Scholar [7] G. Como, E. Lovisari and K. Savla, Throughput optimality and overload behavior of dynamical flow networks under monotone distributed routing,, IEEE Trans. Contr. Network Systems, 2 (2014), 57. doi: 10.1109/TCNS.2014.2367361. Google Scholar [8] S. N. Dashkovskiy, B. S. Rüffer and F. R. Wirth, Small gain theorems for large scale systems and construction of ISS Lyapunov functions,, SIAM J. Control Optim., 48 (2010), 4089. doi: 10.1137/090746483. Google Scholar [9] P. De Leenheer, D. Angeli and E. D. Sontag, Monotone chemical reaction networks,, Journal of Mathematical Chemistry, 41 (2007), 295. doi: 10.1007/s10910-006-9075-z. Google Scholar [10] D. Hinrichsen and A. J. Pritchard, Mathematical Systems Theory I - Modelling, State Space Analysis, Stability and Robustness,, Springer, (2005). doi: 10.1007/b137541. Google Scholar [11] M. W. Hirsch and H. Smith, Monotone dynamical systems,, in Handbook of Differential Equations: Ordinary Differential Equations, (2005), 239. Google Scholar [12] M. W. Hirsch, Systems of differential equations which are competitive or cooperative. I. Limit sets,, SIAM J. Math. Anal., 13 (1982), 167. doi: 10.1137/0513013. Google Scholar [13] M. W. Hirsch, Systems of differential equations that are competitive or cooperative. II. Convergence almost everywhere,, SIAM J. Math. Anal., 16 (1985), 423. doi: 10.1137/0516030. Google Scholar [14] M. W. Hirsch, Systems of differential equations which are competitive or cooperative. III. Competing species,, Nonlinearity, 1 (1988), 51. doi: 10.1088/0951-7715/1/1/003. Google Scholar [15] M. W. Hirsch, Systems of differential equations that are competitive or cooperative. V. Convergence in $3$-dimensional systems,, J. Diff. Eqns., 80 (1989), 94. doi: 10.1016/0022-0396(89)90097-1. Google Scholar [16] M. W. Hirsch, Systems of differential equations that are competitive or cooperative. IV. Structural stability in three-dimensional systems,, SIAM J. Math. Anal., 21 (1990), 1225. doi: 10.1137/0521067. Google Scholar [17] M. W. Hirsch, Systems of differential equations that are competitive or cooperative. VI. A local $C^r$ closing lemma for 3-dimensional systems,, Ergodic Theory Dynam. Systems, 11 (1991), 443. doi: 10.1017/S014338570000626X. Google Scholar [18] M. W. Hirsch and H. L. Smith, Competitive and cooperative systems: Mini-review,, in Positive systems (Rome, (2003), 183. doi: 10.1007/978-3-540-44928-7_25. Google Scholar [19] H. Ito, A Lyapunov approach to cascade interconnection of integral input-to-state stable systems,, IEEE Trans. Autom. Control, 55 (2010), 702. doi: 10.1109/TAC.2009.2037457. Google Scholar [20] H. Ito and Z.-P. Jiang, Necessary and sufficient small gain conditions for integral input-to-state stable systems: A Lyapunov perspective,, IEEE Trans. Autom. Control, 54 (2009), 2389. doi: 10.1109/TAC.2009.2028980. Google Scholar [21] H. Ito, State-dependent scaling problems and stability of interconnected iISS and ISS systems,, IEEE Trans. Autom. Control, 51 (2006), 1626. doi: 10.1109/TAC.2006.882930. Google Scholar [22] H. Ito, S. Dashkovskiy and F. Wirth, Capability and limitation of max- and sum-type construction of Lyapunov functions for networks of iISS systems,, Automatica J. IFAC, 48 (2012), 1197. doi: 10.1016/j.automatica.2012.03.018. Google Scholar [23] H. Ito, Z.-P. Jiang, S. Dashkovskiy and B. S. Rüffer, Robust stability of networks of iISS systems: Construction of sum-type Lyapunov functions,, IEEE Trans. Autom. Control, 58 (2013), 1192. doi: 10.1109/TAC.2012.2231552. Google Scholar [24] H. Ito, B. S. Rüffer and A. Rantzer, Max- and sum-separable Lyapunov functions for monotone systems and their level sets,, in Proc. 53rd IEEE Conf. Decis. Control, (2014), 2371. doi: 10.1109/CDC.2014.7039750. Google Scholar [25] Z.-P. Jiang, I. M. Y. Mareels and Y. Wang, A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems,, Automatica J. IFAC, 32 (1996), 1211. doi: 10.1016/0005-1098(96)00051-9. Google Scholar [26] H. K. Khalil, Nonlinear systems,, 3rd edition, (2002). Google Scholar [27] V. Lakshmikantham and S. Leela, Differential and Integral Inequalities: Theory and Applications, Vol. I: Ordinary Differential Equations,, Academic Press, (1969). Google Scholar [28] M. Margaliot and T. Tuller, Stability analysis of the ribosome flow model,, IEEE/ACM Trans. Comp. Biology & Bioinformatics, 9 (2012), 1545. doi: 10.1109/TCBB.2012.88. Google Scholar [29] A. Rantzer, Distributed control of positive systems,, in Proc. 50th IEEE Conf. Decis. Control and Europ. Contr. Conf., (2011), 6608. doi: 10.1109/CDC.2011.6161293. Google Scholar [30] A. Rantzer, Optimizing positively dominated systems,, in Proc. 51st IEEE Conf. Decis. Control, (2012), 272. doi: 10.1109/CDC.2012.6426312. Google Scholar [31] A. Rantzer, B. S. Rüffer and G. Dirr, Separable Lyapunov functions for monotone systems,, in Proc. 52nd IEEE Conf. Decis. Control, (2013), 4590. doi: 10.1109/CDC.2013.6760604. Google Scholar [32] B. S. Rüffer, Monotone inequalities, dynamical systems, and paths in the positive orthant of Euclidean $n$-space,, Positivity, 14 (2010), 257. doi: 10.1007/s11117-009-0016-5. Google Scholar [33] B. S. Rüffer, Small-gain conditions and the comparison principle,, IEEE Trans. Autom. Control, 55 (2010), 1732. doi: 10.1109/TAC.2010.2048053. Google Scholar [34] B. S. Rüffer, P. M. Dower and H. Ito, Computational comparison principles for large-scale system stability analysis,, in Proc. of the 10th SICE Annual Conference on Control Systems, (2010). Google Scholar [35] B. S. Rüffer, H. Ito and P. M. Dower, Computing asymptotic gains of large-scale interconnections,, in Proc. 49th IEEE Conf. Decis. Control, (2010), 7413. Google Scholar [36] B. S. Rüffer, C. M. Kellett and S. R. Weller, Connection between cooperative positive systems and integral input-to-state stability of large-scale systems,, Automatica J. IFAC, 46 (2010), 1019. doi: 10.1016/j.automatica.2010.03.012. Google Scholar [37] H. L. Smith, Monotone Dynamical Systems,, Mathematical Surveys and Monographs, (1995). Google Scholar [38] E. D. Sontag, Input to state stability,, in The Control Systems Handbook: Control System Advanced Methods (ed. W. S. Levine), (2010), 1. Google Scholar [39] E. D. Sontag, Smooth stabilization implies coprime factorization,, IEEE Trans. Autom. Control, 34 (1989), 435. doi: 10.1109/9.28018. Google Scholar [40] E. D. Sontag, Comments on integral variants of ISS,, Systems Control Lett., 34 (1998), 93. doi: 10.1016/S0167-6911(98)00003-6. Google Scholar [41] E. D. Sontag, Monotone and near-monotone biochemical networks,, Systems and Synthetic Biology, 1 (2007), 59. doi: 10.1007/s11693-007-9005-9. Google Scholar [42] E. D. Sontag and Y. Wang, On characterizations of the input-to-state stability property,, Systems Control Lett., 24 (1995), 351. doi: 10.1016/0167-6911(94)00050-6. Google Scholar [43] J. A. Yorke, Extending Liapunov's second method to non-Lipschitz Liapunov functions,, Bull. Amer. Math. Soc., 74 (1968), 322. doi: 10.1090/S0002-9904-1968-11940-8. Google Scholar [44] T. Yoshizawa, Stability Theory by Liapunov's Second Method,, Publications of the Mathematical Society of Japan, (1966). Google Scholar
 [1] Christopher M. Kellett. Classical converse theorems in Lyapunov's second method. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2333-2360. doi: 10.3934/dcdsb.2015.20.2333 [2] Frédéric Mazenc, Christophe Prieur. Strict Lyapunov functions for semilinear parabolic partial differential equations. Mathematical Control & Related Fields, 2011, 1 (2) : 231-250. doi: 10.3934/mcrf.2011.1.231 [3] Luis Barreira, Claudia Valls. Stability of nonautonomous equations and Lyapunov functions. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2631-2650. doi: 10.3934/dcds.2013.33.2631 [4] Volodymyr Pichkur. On practical stability of differential inclusions using Lyapunov functions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1977-1986. doi: 10.3934/dcdsb.2017116 [5] Jóhann Björnsson, Peter Giesl, Sigurdur F. Hafstein, Christopher M. Kellett. Computation of Lyapunov functions for systems with multiple local attractors. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4019-4039. doi: 10.3934/dcds.2015.35.4019 [6] Sigurdur Freyr Hafstein. A constructive converse Lyapunov theorem on exponential stability. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 657-678. doi: 10.3934/dcds.2004.10.657 [7] Lars Grüne, Peter E. Kloeden, Stefan Siegmund, Fabian R. Wirth. Lyapunov's second method for nonautonomous differential equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 375-403. doi: 10.3934/dcds.2007.18.375 [8] Peter Giesl, Sigurdur Hafstein. Computational methods for Lyapunov functions. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : i-ii. doi: 10.3934/dcdsb.2015.20.8i [9] Janusz Mierczyński, Wenxian Shen. Formulas for generalized principal Lyapunov exponent for parabolic PDEs. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1189-1199. doi: 10.3934/dcdss.2016048 [10] Michael Schönlein. Asymptotic stability and smooth Lyapunov functions for a class of abstract dynamical systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 4053-4069. doi: 10.3934/dcds.2017172 [11] Sigurdur F. Hafstein, Christopher M. Kellett, Huijuan Li. Computing continuous and piecewise affine lyapunov functions for nonlinear systems. Journal of Computational Dynamics, 2015, 2 (2) : 227-246. doi: 10.3934/jcd.2015004 [12] Antonio Siconolfi, Gabriele Terrone. A metric proof of the converse Lyapunov theorem for semicontinuous multivalued dynamics. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4409-4427. doi: 10.3934/dcds.2012.32.4409 [13] Jean Mawhin, James R. Ward Jr. Guiding-like functions for periodic or bounded solutions of ordinary differential equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 39-54. doi: 10.3934/dcds.2002.8.39 [14] Peter Giesl, Sigurdur Hafstein. Review on computational methods for Lyapunov functions. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2291-2331. doi: 10.3934/dcdsb.2015.20.2291 [15] Sergey Zelik. On the Lyapunov dimension of cascade systems. Communications on Pure & Applied Analysis, 2008, 7 (4) : 971-985. doi: 10.3934/cpaa.2008.7.971 [16] Antonio Cañada, Salvador Villegas. Lyapunov inequalities for partial differential equations at radial higher eigenvalues. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 111-122. doi: 10.3934/dcds.2013.33.111 [17] Xiaofei He, X. H. Tang. Lyapunov-type inequalities for even order differential equations. Communications on Pure & Applied Analysis, 2012, 11 (2) : 465-473. doi: 10.3934/cpaa.2012.11.465 [18] Dimitri Breda, Sara Della Schiava. Pseudospectral reduction to compute Lyapunov exponents of delay differential equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2727-2741. doi: 10.3934/dcdsb.2018092 [19] Marat Akhmet, Duygu Aruğaslan. Lyapunov-Razumikhin method for differential equations with piecewise constant argument. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 457-466. doi: 10.3934/dcds.2009.25.457 [20] Bin Wang, Arieh Iserles. Dirichlet series for dynamical systems of first-order ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 281-298. doi: 10.3934/dcdsb.2014.19.281

2018 Impact Factor: 1.008

## Metrics

• HTML views (0)
• Cited by (6)

• on AIMS