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  January 2021 Early access  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 $
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