January  2021, 26(1): 299-336. doi: 10.3934/dcdsb.2020331

Computing complete Lyapunov functions for discrete-time dynamical systems

1. 

Department of Mathematics, University of Sussex, Falmer BN1 9QH, United Kingdom

2. 

Faculty of Physical Sciences, University of Iceland, 107 Reykjavik, Iceland

Received  May 2020 Revised  October 2020 Published  November 2020

Fund Project: The research in this paper was supported by the Icelandic Research Fund (Rannís) grant number 163074-052, Complete Lyapunov functions: Efficient numerical computation

A complete Lyapunov function characterizes the behaviour of a general discrete-time dynamical system. In particular, it divides the state space into the chain-recurrent set where the complete Lyapunov function is constant along trajectories and the part where the flow is gradient-like and the complete Lyapunov function is strictly decreasing along solutions. Moreover, the level sets of a complete Lyapunov function provide information about attractors, repellers, and basins of attraction.

We propose two novel classes of methods to compute complete Lyapunov functions for a general discrete-time dynamical system given by an iteration. The first class of methods computes a complete Lyapunov function by approximating the solution of an ill-posed equation for its discrete orbital derivative using meshfree collocation. The second class of methods computes a complete Lyapunov function as solution of a minimization problem in a reproducing kernel Hilbert space. We apply both classes of methods to several examples.

Citation: Peter Giesl, Zachary Langhorne, Carlos Argáez, Sigurdur Hafstein. Computing complete Lyapunov functions for discrete-time dynamical systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 299-336. doi: 10.3934/dcdsb.2020331
References:
[1]

E. Akin, The General Topology of Dynamical Systems, American Mathematical Society, 1993. doi: 10.1090/gsm/001.  Google Scholar

[2]

C. Argáez, P. Giesl and S. Hafstein, Analysing dynamical systems towards computing complete Lyapunov functions, In Proceedings of the 7th International Conference on Simulation and Modeling Methodologies, Technologies and Applications, Madrid, Spain, pages 323–330, 2017. Google Scholar

[3]

C. Argáez, P. Giesl and S. Hafstein, Computation of complete Lyapunov functions for three-dimensional systems, In Proceedings of the 57rd IEEE Conference on Decision and Control (CDC), pages 4059–4064, 2018. Google Scholar

[4]

C. Argáez, P. Giesl and S. Hafstein, Dynamical systems in theoretical perspective, Chapter Computational Approach for Complete Lyapunov Functions, 248 (2018), 1–11. Springer, Springer Proceedings in Mathematics and Statistics. doi: 10.1007/978-3-319-96598-7_1.  Google Scholar

[5]

C. Argáez, P. Giesl and S. Hafstein, Iterative construction of complete Lyapunov functions, In Proceedings of the 8th International Conference on Simulation and Modeling Methodologies, Technologies and Applications, (2018), 211–222. Google Scholar

[6]

N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc., 68 (1950), 337-404.  doi: 10.1090/S0002-9947-1950-0051437-7.  Google Scholar

[7]

J. Auslander, Generalized recurrence in dynamical systems, Contributions to Differential Equations, 3 (1964), 65-74.   Google Scholar

[8]

H. Ban and W. D. Kalies, A computational approach to Conley's decomposition theorem, J. Comput. Nonlinear Dynam, 1 (2006), 312-319.  doi: 10.1115/1.2338651.  Google Scholar

[9]

P. Bernard and S. Suhr, Lyapunov functions of closed cone fields: From Conley theory to time functions, Commun. Math. Phys., 359 (2018), 467-498.  doi: 10.1007/s00220-018-3127-7.  Google Scholar

[10]

J. BjörnssonP. GieslS. F. Hafstein and C. M. Kellett, Computation of Lyapunov functions for systems with multiple attractors, Discrete Contin. Dyn. Syst., 35 (2015), 4019-4039.  doi: 10.3934/dcds.2015.35.4019.  Google Scholar

[11]

S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics. Springer New York, 2008. doi: 10.1007/978-0-387-75934-0.  Google Scholar

[12]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Spinger, 2011.  Google Scholar

[13]

C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series in Mathematics, 38. American Mathematical Society, 1978.  Google Scholar

[14]

M. Dellnitz, G. Froyland and O. Junge, The algorithms behind GAIO – set oriented numerical methods for dynamical systems, Ergodic theory, Analysis, and Efficient Simulation of Dynamical Systems, (2001), 145–174,805–807.  Google Scholar

[15]

A. Fathi and P. Pageault, Smoothing Lyapunov functions, Trans. Amer. Math. Soc., 371 (2019), 1677-1700.  doi: 10.1090/tran/7329.  Google Scholar

[16]

J. Franks, Generalizations of the Poincaré-Birkhoff theorem, Ann. of Math., 128 (1998), 139–151. Erratum: arXiv: math/0410316. doi: 10.2307/1971464.  Google Scholar

[17]

P. Giesl, Construction of Global Lyapunov Functions Using Radial Basis Functions, Lecture Notes in Math. 1904, Springer, 2007.  Google Scholar

[18]

P. GieslC. ArgáezS. Hafstein and H. Wendland, Construction of a complete Lyapunov function using quadratic programming, Proceedings of the 15th International Conference on Informatics in Control, Automation and Robotics, 1 (2018), 560-568.   Google Scholar

[19]

P. Giesl, C. Argáez, S. Hafstein and H. Wendland, Convergence of discretized minimization problems with applications to complete Lyapunov functions, submitted, 2020. Google Scholar

[20]

P. Giesl and H. Wendland, Meshless collocation: Error estimates with application to dynamical systems, SIAM J. Numer. Anal., 45 (2007), 1723-1741.  doi: 10.1137/060658813.  Google Scholar

[21]

S. V. GonchenkoI. I. OvsyannikovC. Simó and D. Turaev, Three-dimensional Hénon-like maps and wild Lorenz-like attractors, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 3493-3508.  doi: 10.1142/S0218127405014180.  Google Scholar

[22]

A. GoulletS. HarkerK. MischaikowW. D. Kalies and D. Kasti, Efficient computation of Lyapunov functions for Morse decompositions, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2418-2451.  doi: 10.3934/dcdsb.2015.20.2419.  Google Scholar

[23]

M. Hurley, Chain recurrence and attraction in non-compact spaces, Ergodic Theory Dynam. Systems, 11 (1991), 709-729.  doi: 10.1017/S014338570000643X.  Google Scholar

[24]

M. Hurley, Noncompact chain recurrence and attraction, Proc. Amer. Math. Soc., 115 (1992), 1139-1148.  doi: 10.1090/S0002-9939-1992-1098401-X.  Google Scholar

[25]

M. Hurley, Chain recurrence, semiflows, and gradients, J. Dynam. Differential Equations, 7 (1995), 437-456.  doi: 10.1007/BF02219371.  Google Scholar

[26]

M. Hurley, Lyapunov functions and attractors in arbitrary metric spaces, Proc. Amer. Math. Soc., 126 (1998), 245-256.  doi: 10.1090/S0002-9939-98-04500-6.  Google Scholar

[27]

A. Iske, Perfect centre placement for radial basis function methods, Technical Report TUM-M9809, TU Munich, Germany, 1998. Google Scholar

[28]

W. D. KaliesK. Mischaikow and R. C. A. M. VanderVorst, An algorithmic approach to chain recurrence, Found. Comput. Math., 5 (2005), 409-449.  doi: 10.1007/s10208-004-0163-9.  Google Scholar

[29]

D. E. Norton, The fundamental theorem of dynamical systems, Comment. Math. Univ. Carolin., 36 (1995), 585-597.   Google Scholar

[30]

M. Patrão, Existence of complete Lyapunov functions for semiflows on separable metric spaces, Far East J. Dyn. Syst., 17 (2011), 49-54.   Google Scholar

[31] C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, Studies in Advanced Mathematics. CRC Press, 2. edition, 1999.   Google Scholar
[32]

S. Suhr and S. Hafstein, Smooth complete Lyapunov functions for ODEs, submitted, 2020. Google Scholar

[33]

H. Wendland, Error estimates for interpolation by compactly supported radial basis functions of minimal degree, J. Approx. Theory, 93 (1998), 258-272.  doi: 10.1006/jath.1997.3137.  Google Scholar

[34] H. Wendland, Scattered Data Approximation, volume 17 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2005.   Google Scholar

show all references

References:
[1]

E. Akin, The General Topology of Dynamical Systems, American Mathematical Society, 1993. doi: 10.1090/gsm/001.  Google Scholar

[2]

C. Argáez, P. Giesl and S. Hafstein, Analysing dynamical systems towards computing complete Lyapunov functions, In Proceedings of the 7th International Conference on Simulation and Modeling Methodologies, Technologies and Applications, Madrid, Spain, pages 323–330, 2017. Google Scholar

[3]

C. Argáez, P. Giesl and S. Hafstein, Computation of complete Lyapunov functions for three-dimensional systems, In Proceedings of the 57rd IEEE Conference on Decision and Control (CDC), pages 4059–4064, 2018. Google Scholar

[4]

C. Argáez, P. Giesl and S. Hafstein, Dynamical systems in theoretical perspective, Chapter Computational Approach for Complete Lyapunov Functions, 248 (2018), 1–11. Springer, Springer Proceedings in Mathematics and Statistics. doi: 10.1007/978-3-319-96598-7_1.  Google Scholar

[5]

C. Argáez, P. Giesl and S. Hafstein, Iterative construction of complete Lyapunov functions, In Proceedings of the 8th International Conference on Simulation and Modeling Methodologies, Technologies and Applications, (2018), 211–222. Google Scholar

[6]

N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc., 68 (1950), 337-404.  doi: 10.1090/S0002-9947-1950-0051437-7.  Google Scholar

[7]

J. Auslander, Generalized recurrence in dynamical systems, Contributions to Differential Equations, 3 (1964), 65-74.   Google Scholar

[8]

H. Ban and W. D. Kalies, A computational approach to Conley's decomposition theorem, J. Comput. Nonlinear Dynam, 1 (2006), 312-319.  doi: 10.1115/1.2338651.  Google Scholar

[9]

P. Bernard and S. Suhr, Lyapunov functions of closed cone fields: From Conley theory to time functions, Commun. Math. Phys., 359 (2018), 467-498.  doi: 10.1007/s00220-018-3127-7.  Google Scholar

[10]

J. BjörnssonP. GieslS. F. Hafstein and C. M. Kellett, Computation of Lyapunov functions for systems with multiple attractors, Discrete Contin. Dyn. Syst., 35 (2015), 4019-4039.  doi: 10.3934/dcds.2015.35.4019.  Google Scholar

[11]

S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics. Springer New York, 2008. doi: 10.1007/978-0-387-75934-0.  Google Scholar

[12]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Spinger, 2011.  Google Scholar

[13]

C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series in Mathematics, 38. American Mathematical Society, 1978.  Google Scholar

[14]

M. Dellnitz, G. Froyland and O. Junge, The algorithms behind GAIO – set oriented numerical methods for dynamical systems, Ergodic theory, Analysis, and Efficient Simulation of Dynamical Systems, (2001), 145–174,805–807.  Google Scholar

[15]

A. Fathi and P. Pageault, Smoothing Lyapunov functions, Trans. Amer. Math. Soc., 371 (2019), 1677-1700.  doi: 10.1090/tran/7329.  Google Scholar

[16]

J. Franks, Generalizations of the Poincaré-Birkhoff theorem, Ann. of Math., 128 (1998), 139–151. Erratum: arXiv: math/0410316. doi: 10.2307/1971464.  Google Scholar

[17]

P. Giesl, Construction of Global Lyapunov Functions Using Radial Basis Functions, Lecture Notes in Math. 1904, Springer, 2007.  Google Scholar

[18]

P. GieslC. ArgáezS. Hafstein and H. Wendland, Construction of a complete Lyapunov function using quadratic programming, Proceedings of the 15th International Conference on Informatics in Control, Automation and Robotics, 1 (2018), 560-568.   Google Scholar

[19]

P. Giesl, C. Argáez, S. Hafstein and H. Wendland, Convergence of discretized minimization problems with applications to complete Lyapunov functions, submitted, 2020. Google Scholar

[20]

P. Giesl and H. Wendland, Meshless collocation: Error estimates with application to dynamical systems, SIAM J. Numer. Anal., 45 (2007), 1723-1741.  doi: 10.1137/060658813.  Google Scholar

[21]

S. V. GonchenkoI. I. OvsyannikovC. Simó and D. Turaev, Three-dimensional Hénon-like maps and wild Lorenz-like attractors, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 3493-3508.  doi: 10.1142/S0218127405014180.  Google Scholar

[22]

A. GoulletS. HarkerK. MischaikowW. D. Kalies and D. Kasti, Efficient computation of Lyapunov functions for Morse decompositions, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2418-2451.  doi: 10.3934/dcdsb.2015.20.2419.  Google Scholar

[23]

M. Hurley, Chain recurrence and attraction in non-compact spaces, Ergodic Theory Dynam. Systems, 11 (1991), 709-729.  doi: 10.1017/S014338570000643X.  Google Scholar

[24]

M. Hurley, Noncompact chain recurrence and attraction, Proc. Amer. Math. Soc., 115 (1992), 1139-1148.  doi: 10.1090/S0002-9939-1992-1098401-X.  Google Scholar

[25]

M. Hurley, Chain recurrence, semiflows, and gradients, J. Dynam. Differential Equations, 7 (1995), 437-456.  doi: 10.1007/BF02219371.  Google Scholar

[26]

M. Hurley, Lyapunov functions and attractors in arbitrary metric spaces, Proc. Amer. Math. Soc., 126 (1998), 245-256.  doi: 10.1090/S0002-9939-98-04500-6.  Google Scholar

[27]

A. Iske, Perfect centre placement for radial basis function methods, Technical Report TUM-M9809, TU Munich, Germany, 1998. Google Scholar

[28]

W. D. KaliesK. Mischaikow and R. C. A. M. VanderVorst, An algorithmic approach to chain recurrence, Found. Comput. Math., 5 (2005), 409-449.  doi: 10.1007/s10208-004-0163-9.  Google Scholar

[29]

D. E. Norton, The fundamental theorem of dynamical systems, Comment. Math. Univ. Carolin., 36 (1995), 585-597.   Google Scholar

[30]

M. Patrão, Existence of complete Lyapunov functions for semiflows on separable metric spaces, Far East J. Dyn. Syst., 17 (2011), 49-54.   Google Scholar

[31] C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, Studies in Advanced Mathematics. CRC Press, 2. edition, 1999.   Google Scholar
[32]

S. Suhr and S. Hafstein, Smooth complete Lyapunov functions for ODEs, submitted, 2020. Google Scholar

[33]

H. Wendland, Error estimates for interpolation by compactly supported radial basis functions of minimal degree, J. Approx. Theory, 93 (1998), 258-272.  doi: 10.1006/jath.1997.3137.  Google Scholar

[34] H. Wendland, Scattered Data Approximation, volume 17 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2005.   Google Scholar
Figure 1.  Example (24) with solving $ \Delta v(x,y) = -1 $. Chain-recurrent set (top) approximated by the set $ \{(x,y)\, \mid\, \Delta v(x,y)\ge \gamma\} $, see Table 2, and the orbital derivative (middle) $ \Delta v(x,y) $ of the constructed complete Lyapunov function $ v $. $ \Delta v $ is approximately zero on the chain-recurrent set (origin) and negative everywhere else. Bottom: Constructed complete Lyapunov function $ v(x,y) $, which has a minimum at the origin
Figure 2.  Example (24) with equality and inequality constraints. Chain-recurrent set (top) approximated by the set $ \{(x,y)\,\mid\,\Delta v(x,y)\ge \gamma\} $, see Table 2, and the orbital derivative $ \Delta v(x,y) $ of the constructed complete Lyapunov function $ v $ (middle). Again, the orbital derivative $ \Delta v $ is correctly approximated being zero on the chain-recurrent set (origin) and negative everywhere else. Bottom: Constructed complete Lyapunov function $ v(x,y) $, which has a minimum at the origin. The point with equality constraint was $ (0.5,0) $
Figure 3.  Example (24) with inequality constraints. Chain recurrent set (top) approximated by the set $ \{(x,y)\,\mid\,\Delta v(x,y)\ge \gamma\} $, see Table 2, and the orbital derivative $ \Delta v(x,y) $ of the constructed complete Lyapunov function $ v $ (middle). $ \Delta v $ is approximately zero on the chain-recurrent set (origin) and negative everywhere else. Bottom: The constructed complete Lyapunov function $ v(x,y) $, which has a minimum at the origin
Figure 4.  Example (25) with solving $ \Delta v(x,y) = -1 $. Chain-recurrent set (top) approximated by the set $ \{(x,y)\,\mid\,\Delta v(x,y)\ge \gamma\} $, see Table 2, and the orbital derivative (second) $ \Delta v(x,y) $ of the constructed complete Lyapunov function $ v $ over the chain-recurrent set. The third figure shows the orbital derivative in a larger set. The approximated chain-recurrent set includes the equilibria at the origin and $ (\pm 1,0) $, but is much larger, in particular around the origin. Bottom: Constructed complete Lyapunov function $ v(x,y) $, which has a saddle point at the origin and a local maximum at the unstable equilibria $ (\pm 1,0) $
Figure 5.  Example (25) with equality and inequality constraints. Chain-recurrent set (top) approximated by the set $ \{(x,y)\,\mid\,\Delta v(x,y)\ge \gamma\} $, see Table 2, and the orbital derivative (second) $ \Delta v(x,y) $ of the constructed complete Lyapunov function $ v $ over the chain-recurrent set. The third figure shows the orbital derivative in a larger set. $ \Delta v $ is approximately zero on the chain-recurrent set, consisting of three equilibria at the origin and $ (\pm 1,0) $, and negative everywhere else. The approximation of the chain-recurrent set (the equilibria) is much better than when solving the equation $ \Delta v(x,y) = -1 $. Bottom: Constructed complete Lyapunov function $ v(x,y) $, which has a saddle point at the origin and local maxima at the unstable equilibria $ (\pm 1,0) $. Note that they have different levels, which is due to the extra point with equality constraint at $ (0.5,0) $, resulting in an unsymmetric approximation
Figure 6.  Example (25) with inequality constraints. Chain-recurrent set (top) approximated by the set $ \{(x,y)\,\mid\,\Delta v(x,y)\ge \gamma\} $, see Table 2, and the orbital derivative (second) $ \Delta v(x,y) $ of the constructed complete Lyapunov function $ v $ over the chain-recurrent set. The third figure shows the orbital derivative in a larger set. $ \Delta v $ is approximately zero on the chain-recurrent set, consisting of three equilibria at the origin and $ (\pm 1,0) $, and negative everywhere else. The approximation of the chain-recurrent set (the equilibria) is much better than when solving the equation $ \Delta v(x,y) = -1 $. Bottom: Constructed complete Lyapunov function $ v(x,y) $, which has a saddle point at the origin and local maxima at the unstable equilibria $ (\pm 1,0) $
Figure 7.  Example (26) with solving $ \Delta v(x,y) = -1 $. Chain-recurrent set (top) approximated by the set $ \{(x,y)\,\mid\,\Delta v(x,y)\ge \gamma\} $, see Table 2, and the orbital derivative (second) $ \Delta v(x,y) $ of the constructed complete Lyapunov function $ v $ over the chain-recurrent set. The third figure shows the orbital derivative in a larger set. Bottom: Constructed complete Lyapunov function $ v(x,y) $. The approximated chain-recurrent set does not resemble the Hénon attractor very well, neither using the orbital derivative nor as the local minimum of the constructed function
Figure 8.  Example (26) with equality and inequality constraints. Chain-recurrent set (top) approximated by the set $ \{(x,y)\,\mid\,\Delta v(x,y)\ge \gamma\} $, see Table 2, and the orbital derivative (second) $ \Delta v(x,y) $ of the constructed complete Lyapunov function $ v $ over the chain-recurrent set. The third figure shows the orbital derivative in a larger set. The characteristic shape of the Hénon attractor is clearly visible. Bottom: Constructed complete Lyapunov function $ v(x,y) $ with a local minimum at the Hénon attractor. The point with equality constraint was $ (0.5,0) $
Figure 9.  Example (26) with inequality constraints. Chain-recurrent set (top) approximated by the set $ \{(x,y)\,\mid\,\Delta v(x,y)\ge \gamma\} $, see Table 2, and the orbital derivative (second) $ \Delta v(x,y) $ of the constructed complete Lyapunov function $ v $ over the chain-recurrent set. The third figure shows the orbital derivative in a larger set. The characteristic shape of the Hénon attractor is clearly visible. Bottom: Constructed complete Lyapunov function $ v(x,y) $ with a local minimum at the Hénon attractor
Figure 10.  Example (27) with solving $ \Delta v(x,y) = -1 $. Chain-recurrent set (top) approximated by the set $ \{(x,y)\,\mid\,\Delta v(x,y)\ge \gamma\} $, see Table 2, and the orbital derivative (second) $ \Delta v(x,y) $ of the constructed complete Lyapunov function $ v $ over the chain-recurrent set. The third figure shows the orbital derivative in a larger set. Bottom: Constructed complete Lyapunov function $ v(x,y) $. The approximated chain-recurrent set shows the Hénon repeller better than the Hénon attractor in the previous example, but still not very clearly. It is not clearly visible as local maximum of the constructed function either
Figure 11.  Example (27) with equality and inequality constraints. Chain-recurrent set (top) approximated by the set $ \{(x,y)\,\mid\,\Delta v(x,y)\ge \gamma\} $, see Table 2, and the orbital derivative (second) $ \Delta v(x,y) $ of the constructed complete Lyapunov function $ v $ over the chain-recurrent set. The third figure shows the orbital derivative in a larger set. Bottom: Constructed complete Lyapunov function $ v(x,y) $, showing the Hénon repeller as a local maximum. The repeller is clearly visible in all figures. The point with equality constraint was $ (0.5,0) $
Figure 12.  Example (27) with inequality constraints. Chain-recurrent set (top) approximated by the set $ \{(x,y)\,\mid\,\Delta v(x,y)\ge \gamma\} $, see Table 2, and the orbital derivative (second) $ \Delta v(x,y) $ of the constructed complete Lyapunov function $ v $ over the chain-recurrent set. The third figure shows the orbital derivative in a larger set. Bottom: Constructed complete Lyapunov function $ v(x,y) $, showing the Hénon repeller as a local maximum. The repeller is clearly visible in all figures
Figure 13.  Example (28) with solving $ \Delta v(x,y) = -1 $. Chain-recurrent set (top) approximated by the set $ \{(x,y)\,\mid\,\Delta v(x,y)\ge \gamma\} $, see Table 2, and the orbital derivative (second) $ \Delta v(x,y) $ of the constructed complete Lyapunov function $ v $ over the chain-recurrent set. The third figure shows the orbital derivative in a larger set. Bottom: Constructed complete Lyapunov function $ v(x,y) $. The approximated chain-recurrent set shows the attractor relatively well in the orbital derivative, but not very clearly as local minimum of the constructed function
Figure 14.  Example (28) with equality and inequality constraints. Chain-recurrent set (top) approximated by the set $ \{(x,y)\,\mid\,\Delta v(x,y)\ge \gamma\} $, see Table 2, and the orbital derivative (second) $ \Delta v(x,y) $ of the constructed complete Lyapunov function $ v $ over the chain-recurrent set. The third figure shows the orbital derivative in a larger set. Bottom: Constructed complete Lyapunov function $ v(x,y) $, showing the attractor as a local minimum. The attractor is clearer than in the previous method, both using the orbital derivative and as local minimum of the constructed function. The point with equality constraint was $ (0.5,0) $, where the orbital derivative is fixed at $ -1 $
Figure 15.  Example (28) with inequality constraints. Chain-recurrent set (top) approximated by the set $ \{(x,y)\,\mid\,\Delta v(x,y)\ge \gamma\} $, see Table 2, and the orbital derivative (second) $ \Delta v(x,y) $ of the constructed complete Lyapunov function $ v $ over the chain-recurrent set. The third figure shows the orbital derivative in a larger set. Bottom: Constructed complete Lyapunov function $ v(x,y) $, showing the attractor as a local minimum. The attractor is clearer than in the first method, both using the orbital derivative and as local minimum of the constructed function
Figure 16.  Example (29) with solving $ \Delta v(x,y,z) = -1 $. Top: Chain-recurrent set approximated by the set $ \{(x,y,z)\,\mid\,\Delta v(x,y,z)\ge \gamma\} $, see Table 2. The other figures show projections of this set: projections to the $ xy- $ (second), $ yz- $ (third) and $ xz- $plane (bottom)
Figure 17.  Example (29) with equality-inequality constrains. Top: Chain-recurrent set approximated by the set $ \{(x,y,z)\,\mid\,\Delta v(x,y,z)\ge \gamma\} $, see Table 2. The other figures show projections of this set: projections to the $ xy- $ (second), $ yz- $ (third) and $ xz- $plane (bottom). The figures are not as good as with the previous method. The point with equality constraint is $ (0.4,0.4,0) $
Figure 18.  Example (29) with inequality constrains. Top: Chain-recurrent set approximated by the set $ \{(x,y,z)\,\mid\,\Delta v(x,y,z)\ge \gamma\} $, see Table 2. The other figures show projections of this set: projections to the $ xy- $ (second), $ yz- $ (third) and $ xz- $plane (bottom)
Table 1.  Collocation points $ X $ for the examples. We have used $ N $ collocation points in a hexagonal grid with parameter $ \alpha_{\text{Hexa-basis}} $ within a rectangle $ (x,y)\in [x_{\rm min},x_{\rm max}]\times [y_{\rm min},y_{\rm max}] $ or $ (x,y,z)\in [x_{\rm min},x_{\rm max}]\times [y_{\rm min},y_{\rm max}]\times [z_{\rm min},z_{\rm max}] $ for the three-dimensional example (29). The number of evaluation points is also displayed
Example $ N $ $ \alpha_{\text{Hexa-basis}} $ #-evaluation $ x_{\rm min} $ $ x_{\rm max} $ $ y_{\rm min} $ $ y_{\rm max} $ $ z_{\rm min} $ $ z_{\rm max} $
(24) $ 3,584 $ $ 0.072 $ 1,779,556 $ -2 $ $ 2 $ $ -2 $ $ 2 $
(25) $ 10,108 $ $ 0.03 $ 2,003,001 $ -2 $ $ 2 $ $ -1 $ $ 1 $
(26) $ 1,440 $ $ 0.05 $ 1,334,000 $ -1.5 $ $ 1.5 $ $ -0.5 $ $ 0.5 $
(27) $ 5,520 $ $ 0.025 $ 1,334,000 $ -1.5 $ $ 1.5 $ $ -0.5 $ $ 0.5 $
(28) $ 2,900 $ $ 0.08 $ 1,779,556 $ -2 $ $ 2 $ $ -2 $ $ 2 $
(29) $ 5,301 $ $ 0.07 $ 1,030,301 $ -0.2 $ $ 0.9 $ $ -0.2 $ $ 0.9 $ $ -0.2 $ $ 0.9 $
Example $ N $ $ \alpha_{\text{Hexa-basis}} $ #-evaluation $ x_{\rm min} $ $ x_{\rm max} $ $ y_{\rm min} $ $ y_{\rm max} $ $ z_{\rm min} $ $ z_{\rm max} $
(24) $ 3,584 $ $ 0.072 $ 1,779,556 $ -2 $ $ 2 $ $ -2 $ $ 2 $
(25) $ 10,108 $ $ 0.03 $ 2,003,001 $ -2 $ $ 2 $ $ -1 $ $ 1 $
(26) $ 1,440 $ $ 0.05 $ 1,334,000 $ -1.5 $ $ 1.5 $ $ -0.5 $ $ 0.5 $
(27) $ 5,520 $ $ 0.025 $ 1,334,000 $ -1.5 $ $ 1.5 $ $ -0.5 $ $ 0.5 $
(28) $ 2,900 $ $ 0.08 $ 1,779,556 $ -2 $ $ 2 $ $ -2 $ $ 2 $
(29) $ 5,301 $ $ 0.07 $ 1,030,301 $ -0.2 $ $ 0.9 $ $ -0.2 $ $ 0.9 $ $ -0.2 $ $ 0.9 $
Table 2.  The value of the parameter $ \gamma\le 0 $, close to $ 0 $, for all examples. The chain-recurrent set is approximated by the set $ \left(\Delta v \right)^{-1}([\gamma,\infty)) $
$ \gamma $ for each method
System $ \Delta v(x)=-1 $ equality-inequality inequality
(24) $ -0.1 $ $ -10^{-5} $ $ 0 $
(25) $ -0.2 $ $ -10^{-5} $ $ -10^{-4} $
(26) $ -0.1 $ $ 0 $ $ -10^{-2} $
(27) $ -0.1 $ $ 0 $ $ 0 $
(28) $ -0.1 $ $ 0 $ $ 0 $
(29) $ -0.1 $ $ 0 $ $ 0 $
$ \gamma $ for each method
System $ \Delta v(x)=-1 $ equality-inequality inequality
(24) $ -0.1 $ $ -10^{-5} $ $ 0 $
(25) $ -0.2 $ $ -10^{-5} $ $ -10^{-4} $
(26) $ -0.1 $ $ 0 $ $ -10^{-2} $
(27) $ -0.1 $ $ 0 $ $ 0 $
(28) $ -0.1 $ $ 0 $ $ 0 $
(29) $ -0.1 $ $ 0 $ $ 0 $
[1]

Angelica Pachon, Federico Polito, Costantino Ricciuti. On discrete-time semi-Markov processes. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1499-1529. doi: 10.3934/dcdsb.2020170

[2]

Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020339

[3]

Veena Goswami, Gopinath Panda. Optimal customer behavior in observable and unobservable discrete-time queues. Journal of Industrial & Management Optimization, 2021, 17 (1) : 299-316. doi: 10.3934/jimo.2019112

[4]

Ming Chen, Hao Wang. Dynamics of a discrete-time stoichiometric optimal foraging model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 107-120. doi: 10.3934/dcdsb.2020264

[5]

Stefan Siegmund, Petr Stehlík. Time scale-induced asynchronous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1011-1029. doi: 10.3934/dcdsb.2020151

[6]

Haixiang Yao, Ping Chen, Miao Zhang, Xun Li. Dynamic discrete-time portfolio selection for defined contribution pension funds with inflation risk. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020166

[7]

Guangjun Shen, Xueying Wu, Xiuwei Yin. Stabilization of stochastic differential equations driven by G-Lévy process with discrete-time feedback control. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 755-774. doi: 10.3934/dcdsb.2020133

[8]

Ting Liu, Guo-Bao Zhang. Global stability of traveling waves for a spatially discrete diffusion system with time delay. Electronic Research Archive, , () : -. doi: 10.3934/era.2021003

[9]

Peter Giesl, Sigurdur Hafstein. System specific triangulations for the construction of CPA Lyapunov functions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020378

[10]

Xu Zhang, Chuang Zheng, Enrique Zuazua. Time discrete wave equations: Boundary observability and control. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 571-604. doi: 10.3934/dcds.2009.23.571

[11]

Madhurima Mukhopadhyay, Palash Sarkar, Shashank Singh, Emmanuel Thomé. New discrete logarithm computation for the medium prime case using the function field sieve. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020119

[12]

Fanni M. Sélley. A self-consistent dynamical system with multiple absolutely continuous invariant measures. Journal of Computational Dynamics, 2021, 8 (1) : 9-32. doi: 10.3934/jcd.2021002

[13]

Mikhail I. Belishev, Sergey A. Simonov. A canonical model of the one-dimensional dynamical Dirac system with boundary control. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021003

[14]

Mohammed Abdulrazaq Kahya, Suhaib Abduljabbar Altamir, Zakariya Yahya Algamal. Improving whale optimization algorithm for feature selection with a time-varying transfer function. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 87-98. doi: 10.3934/naco.2020017

[15]

Lars Grüne, Roberto Guglielmi. On the relation between turnpike properties and dissipativity for continuous time linear quadratic optimal control problems. Mathematical Control & Related Fields, 2021, 11 (1) : 169-188. doi: 10.3934/mcrf.2020032

[16]

Klemens Fellner, Jeff Morgan, Bao Quoc Tang. Uniform-in-time bounds for quadratic reaction-diffusion systems with mass dissipation in higher dimensions. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 635-651. doi: 10.3934/dcdss.2020334

[17]

Juntao Sun, Tsung-fang Wu. The number of nodal solutions for the Schrödinger–Poisson system under the effect of the weight function. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021011

[18]

Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046

[19]

Xiaoping Zhai, Yongsheng Li. Global large solutions and optimal time-decay estimates to the Korteweg system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1387-1413. doi: 10.3934/dcds.2020322

[20]

Maoli Chen, Xiao Wang, Yicheng Liu. Collision-free flocking for a time-delay system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1223-1241. doi: 10.3934/dcdsb.2020251

2019 Impact Factor: 1.27

Article outline

Figures and Tables

[Back to Top]