# American Institute of Mathematical Sciences

January  2021, 8(1): 59-97. doi: 10.3934/jcd.2021004

## A general framework for validated continuation of periodic orbits in systems of polynomial ODEs

 1 VU Amsterdam, Department of Mathematics, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands 2 Rutgers University, Department of Mathematics, 110 Frelinghusen Rd, Piscataway, NJ 08854-8019, USA

Received  February 2019 Revised  November 2019 Published  October 2020

Fund Project: The authors are partially supported by NWO-VICI grant 639033109

In this paper a parametrized Newton-Kantorovich approach is applied to continuation of periodic orbits in arbitrary polynomial vector fields. This allows us to rigorously validate numerically computed branches of periodic solutions. We derive the estimates in full generality and present sample continuation proofs obtained using an implementation in Matlab. The presented approach is applicable to any polynomial vector field of any order and dimension. A variety of examples is presented to illustrate the efficacy of the method.

Citation: Jan Bouwe van den Berg, Elena Queirolo. A general framework for validated continuation of periodic orbits in systems of polynomial ODEs. Journal of Computational Dynamics, 2021, 8 (1) : 59-97. doi: 10.3934/jcd.2021004
##### References:
 [1] G. Arioli and H. Koch, Computer-assisted methods for the study of stationary solutions in dissipative systems, applied to the Kuramoto-Sivashinski equation, Arch. Ration. Mech. Anal., 197 (2010), 1033-1051.  doi: 10.1007/s00205-010-0309-7. [2] J. B. van den Berg, M. Breden, J.-P. Lessard and M. Murray, Continuation of homoclinic orbits in the suspension bridge equation: A computer-assisted proof, J. Differential Equations, 264 (2018), 3086-3130.  doi: 10.1016/j.jde.2017.11.011. [3] J. B. van den Berg, C. Groothedde and J.-P. Lessard, A general method for computer-assisted proofs of periodic solutions in delay differential problems, 2018, Preprint. [4] J. B. van den Berg and J.-P. Lessard, Chaotic braided solutions via rigorous numerics: chaos in the Swift-Hohenberg equation, SIAM J. Appl. Dyn. Syst., 7 (2008), 988-1031.  doi: 10.1137/070709128. [5] J. B. van den Berg, J.-P. Lessard and K. Mischaikow, Global smooth solution curves using rigorous branch following, Math. Comp., 79 (2010), 1565-1584.  doi: 10.1090/S0025-5718-10-02325-2. [6] J. B. van den Berg, and J.-P. Lessard and E. Queirolo, Rigorous verification of Hopf bifurcations in ODEs, 2020, Preprint [7] J. B. van den Berg and E. Queirolo, MATLABcode for "A general framework for validated continuation of periodic orbits in systems of polynomial ODEs", 2018, https://www.math.vu.nl/ janbouwe/code/continuation/. [8] J. B. van den Berg and R. Sheombarsing, Validated computations for connecting orbits in polynomial vector fields, Indag. Math. (N.S.), 31 (2020), 310-373. doi: 10.1016/j.indag.2020.01.007. [9] J. B. van den Berg and J. F. Williams, Validation of the bifurcation diagram in the 2D Ohta-Kawasaki problem, Nonlinearity, 30 (2017), 1584-1638.  doi: 10.1088/1361-6544/aa60e8. [10] M. Breden and R. Castelli, Existence and instability of steady states for a triangular cross-diffusion system: A computer-assisted proof, Journal of Differential Equations, 264 (2018), 6418-6458.  doi: 10.1016/j.jde.2018.01.033. [11] M. Breden, J.-P. Lessard and M. Vanicat, Global bifurcation diagrams of steady states of systems of pdes via rigorous numerics: A 3-component reaction-diffusion system, Acta applicandae mathematicae, 128 (2013), 113-152.  doi: 10.1007/s10440-013-9823-6. [12] R. Castelli, Rigorous computation of non-uniform patterns for the 2-dimensional gray-scott reaction-diffusion equation, Acta Applicandae Mathematicae, 151 (2017), 27-52.  doi: 10.1007/s10440-017-0101-x. [13] A. R. Champneys and B. Sandstede, Numerical computation of coherent structures, In Numerical continuation methods for dynamical systems, Underst. Complex Syst., Springer, Dordrecht, 2007, pages 331-358. doi: 10.1007/978-1-4020-6356-5_11. [14] R. H. Clewley, W. E. Sherwood, M. D. LaMar and J. M. Guckenheimer, PyDSTool, a software environment for dynamical systems modeling, 2007, https://sourceforge.net/projects/pydstool/. [15] A. Correc and J.-P. Lessard, Coexistence of nontrivial solutions of the one-dimensional Ginzburg-Landau equation: A computer-assisted proof, European J. Appl. Math., 26 (2015), 33-60.  doi: 10.1017/S0956792514000308. [16] H. Dankowicz and F. Schilder, Recipes for continuation, volume 11 of Computational Science & Engineering, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2013. https://sourceforge.net/projects/cocotools/. doi: 10.1137/1.9781611972573. [17] S. Day, J.-P. Lessard and K. Mischaikow, Validated continuation for equilibria of PDEs, SIAM J. Numer. Anal., 45 (2007), 1398-1424.  doi: 10.1137/050645968. [18] A. Dhooge, W. Govaerts, Yu. A. Kuznetsov, H. G. E. Meijer and B. Sautois, New features of the software MatCont for bifurcation analysis of dynamical systems, Math. Comput. Model. Dyn. Syst., 14 (2008), 147-175. https://sourceforge.net/projects/matcont/. doi: 10.1080/13873950701742754. [19] E. J. Doedel, B. E. Oldeman, A. R. Champneys, F. Dercole, T. F. Fairgrieve, Y. Kuznetsov, R. C. Paffenroth, B. Sandstede, X. J. Wang and C. H. Zhang, AUTO-07p : Continuation and bifurcation software for ordinary differential equations, 2012, http://sourceforge.net/projects/auto-07p/. [20] B. Ermentrout, Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students, volume 14 of Software, Environments, and Tools, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. http://www.math.pitt.edu/ bard/xpp/xpp.html. doi: 10.1137/1.9780898718195. [21] J.-Ll. Figueras A. Haro and A. Luque, Rigorous computer-assisted application of KAM theory: A modern approach, Found. Comput. Math., 17 (2017), 1123-1193.  doi: 10.1007/s10208-016-9339-3. [22] M. Gameiro, J.-P. Lessard and A. Pugliese, Computation of smooth manifolds via rigorous multi-parameter continuation in infinite dimensions, Found. Comput. Math., 16 (2016), 531-575.  doi: 10.1007/s10208-015-9259-7. [23] A. Gasull, H. Giacomini and M. Grau, Effective construction of Poincaré-Bendixson regions, J. Appl. Anal. Comput., 7 (2017), 1549-1569.  doi: 10.11948/2017094. [24] A. Hungria, J.-P. Lessard and J. D Mireles James, Rigorous numerics for analytic solutions of differential equations: The radii polynomial approach, Mathematics of Computation, 85 (2016), 1427-1459.  doi: 10.1090/mcom/3046. [25] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, volume 112 of Applied Mathematical Sciences, Springer-Verlag, New York, third edition, 2004. doi: 10.1007/978-1-4757-3978-7. [26] J.-P. Lessard, Recent advances about the uniqueness of the slowly oscillating periodic solutions of Wright's equation, J. Differential Equations, 248 (2010), 992-1016.  doi: 10.1016/j.jde.2009.11.008. [27] J.-P. Lessard and J. D. Mireles James, Computer assisted Fourier analysis in sequence spaces of varying regularity, SIAM J. Math. Anal., 49 (2017), 530-561.  doi: 10.1137/16M1056006. [28] J.-P. Lessard, J. D. Mireles James and J. Ransford, Automatic differentiation for Fourier series and the radii polynomial approach, Phys. D, 334 (2016), 174-186.  doi: 10.1016/j.physd.2016.02.007. [29] K. Makino and M. Berz, Rigorous integration of flows and odes using taylor models, In Proceedings of the 2009 conference on Symbolic Numeric Computation, pages 79-84, 2009. doi: 10.1145/1577190.1577206. [30] S. M Rump, Intlab—interval laboratory, In Developments in reliable computing, pages 77-104. Springer, 1999. doi: 10.1007/978-94-017-1247-7_7. [31] G. S. Rychkov, The maximum number of limit cycles of polynomial liénard systems of degree five is equal to two, Differential Equations, 11 (1975), 301-302. [32] T. Wanner, Computer-assisted bifurcation diagram validation and applications in materials science, In Rigorous numerics in dynamics, volume 74 of Proc. Sympos. Appl. Math., pages 123-174. Amer. Math. Soc., Providence, RI, 2018. [33] P. Zgliczyński, Steady state bifurcations for the Kuramoto-Sivashinsky equation: a computer assisted proof, J. Comput. Dyn., 2 (2015), 95-142.  doi: 10.3934/jcd.2015.2.95. [34] CAPD: Computer assisted proofs in dynamics, a package for rigorous numerics, http://capd.ii.uj.edu.pl/.

show all references

##### References:
 [1] G. Arioli and H. Koch, Computer-assisted methods for the study of stationary solutions in dissipative systems, applied to the Kuramoto-Sivashinski equation, Arch. Ration. Mech. Anal., 197 (2010), 1033-1051.  doi: 10.1007/s00205-010-0309-7. [2] J. B. van den Berg, M. Breden, J.-P. Lessard and M. Murray, Continuation of homoclinic orbits in the suspension bridge equation: A computer-assisted proof, J. Differential Equations, 264 (2018), 3086-3130.  doi: 10.1016/j.jde.2017.11.011. [3] J. B. van den Berg, C. Groothedde and J.-P. Lessard, A general method for computer-assisted proofs of periodic solutions in delay differential problems, 2018, Preprint. [4] J. B. van den Berg and J.-P. Lessard, Chaotic braided solutions via rigorous numerics: chaos in the Swift-Hohenberg equation, SIAM J. Appl. Dyn. Syst., 7 (2008), 988-1031.  doi: 10.1137/070709128. [5] J. B. van den Berg, J.-P. Lessard and K. Mischaikow, Global smooth solution curves using rigorous branch following, Math. Comp., 79 (2010), 1565-1584.  doi: 10.1090/S0025-5718-10-02325-2. [6] J. B. van den Berg, and J.-P. Lessard and E. Queirolo, Rigorous verification of Hopf bifurcations in ODEs, 2020, Preprint [7] J. B. van den Berg and E. Queirolo, MATLABcode for "A general framework for validated continuation of periodic orbits in systems of polynomial ODEs", 2018, https://www.math.vu.nl/ janbouwe/code/continuation/. [8] J. B. van den Berg and R. Sheombarsing, Validated computations for connecting orbits in polynomial vector fields, Indag. Math. (N.S.), 31 (2020), 310-373. doi: 10.1016/j.indag.2020.01.007. [9] J. B. van den Berg and J. F. Williams, Validation of the bifurcation diagram in the 2D Ohta-Kawasaki problem, Nonlinearity, 30 (2017), 1584-1638.  doi: 10.1088/1361-6544/aa60e8. [10] M. Breden and R. Castelli, Existence and instability of steady states for a triangular cross-diffusion system: A computer-assisted proof, Journal of Differential Equations, 264 (2018), 6418-6458.  doi: 10.1016/j.jde.2018.01.033. [11] M. Breden, J.-P. Lessard and M. Vanicat, Global bifurcation diagrams of steady states of systems of pdes via rigorous numerics: A 3-component reaction-diffusion system, Acta applicandae mathematicae, 128 (2013), 113-152.  doi: 10.1007/s10440-013-9823-6. [12] R. Castelli, Rigorous computation of non-uniform patterns for the 2-dimensional gray-scott reaction-diffusion equation, Acta Applicandae Mathematicae, 151 (2017), 27-52.  doi: 10.1007/s10440-017-0101-x. [13] A. R. Champneys and B. Sandstede, Numerical computation of coherent structures, In Numerical continuation methods for dynamical systems, Underst. Complex Syst., Springer, Dordrecht, 2007, pages 331-358. doi: 10.1007/978-1-4020-6356-5_11. [14] R. H. Clewley, W. E. Sherwood, M. D. LaMar and J. M. Guckenheimer, PyDSTool, a software environment for dynamical systems modeling, 2007, https://sourceforge.net/projects/pydstool/. [15] A. Correc and J.-P. Lessard, Coexistence of nontrivial solutions of the one-dimensional Ginzburg-Landau equation: A computer-assisted proof, European J. Appl. Math., 26 (2015), 33-60.  doi: 10.1017/S0956792514000308. [16] H. Dankowicz and F. Schilder, Recipes for continuation, volume 11 of Computational Science & Engineering, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2013. https://sourceforge.net/projects/cocotools/. doi: 10.1137/1.9781611972573. [17] S. Day, J.-P. Lessard and K. Mischaikow, Validated continuation for equilibria of PDEs, SIAM J. Numer. Anal., 45 (2007), 1398-1424.  doi: 10.1137/050645968. [18] A. Dhooge, W. Govaerts, Yu. A. Kuznetsov, H. G. E. Meijer and B. Sautois, New features of the software MatCont for bifurcation analysis of dynamical systems, Math. Comput. Model. Dyn. Syst., 14 (2008), 147-175. https://sourceforge.net/projects/matcont/. doi: 10.1080/13873950701742754. [19] E. J. Doedel, B. E. Oldeman, A. R. Champneys, F. Dercole, T. F. Fairgrieve, Y. Kuznetsov, R. C. Paffenroth, B. Sandstede, X. J. Wang and C. H. Zhang, AUTO-07p : Continuation and bifurcation software for ordinary differential equations, 2012, http://sourceforge.net/projects/auto-07p/. [20] B. Ermentrout, Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students, volume 14 of Software, Environments, and Tools, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. http://www.math.pitt.edu/ bard/xpp/xpp.html. doi: 10.1137/1.9780898718195. [21] J.-Ll. Figueras A. Haro and A. Luque, Rigorous computer-assisted application of KAM theory: A modern approach, Found. Comput. Math., 17 (2017), 1123-1193.  doi: 10.1007/s10208-016-9339-3. [22] M. Gameiro, J.-P. Lessard and A. Pugliese, Computation of smooth manifolds via rigorous multi-parameter continuation in infinite dimensions, Found. Comput. Math., 16 (2016), 531-575.  doi: 10.1007/s10208-015-9259-7. [23] A. Gasull, H. Giacomini and M. Grau, Effective construction of Poincaré-Bendixson regions, J. Appl. Anal. Comput., 7 (2017), 1549-1569.  doi: 10.11948/2017094. [24] A. Hungria, J.-P. Lessard and J. D Mireles James, Rigorous numerics for analytic solutions of differential equations: The radii polynomial approach, Mathematics of Computation, 85 (2016), 1427-1459.  doi: 10.1090/mcom/3046. [25] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, volume 112 of Applied Mathematical Sciences, Springer-Verlag, New York, third edition, 2004. doi: 10.1007/978-1-4757-3978-7. [26] J.-P. Lessard, Recent advances about the uniqueness of the slowly oscillating periodic solutions of Wright's equation, J. Differential Equations, 248 (2010), 992-1016.  doi: 10.1016/j.jde.2009.11.008. [27] J.-P. Lessard and J. D. Mireles James, Computer assisted Fourier analysis in sequence spaces of varying regularity, SIAM J. Math. Anal., 49 (2017), 530-561.  doi: 10.1137/16M1056006. [28] J.-P. Lessard, J. D. Mireles James and J. Ransford, Automatic differentiation for Fourier series and the radii polynomial approach, Phys. D, 334 (2016), 174-186.  doi: 10.1016/j.physd.2016.02.007. [29] K. Makino and M. Berz, Rigorous integration of flows and odes using taylor models, In Proceedings of the 2009 conference on Symbolic Numeric Computation, pages 79-84, 2009. doi: 10.1145/1577190.1577206. [30] S. M Rump, Intlab—interval laboratory, In Developments in reliable computing, pages 77-104. Springer, 1999. doi: 10.1007/978-94-017-1247-7_7. [31] G. S. Rychkov, The maximum number of limit cycles of polynomial liénard systems of degree five is equal to two, Differential Equations, 11 (1975), 301-302. [32] T. Wanner, Computer-assisted bifurcation diagram validation and applications in materials science, In Rigorous numerics in dynamics, volume 74 of Proc. Sympos. Appl. Math., pages 123-174. Amer. Math. Soc., Providence, RI, 2018. [33] P. Zgliczyński, Steady state bifurcations for the Kuramoto-Sivashinsky equation: a computer assisted proof, J. Comput. Dyn., 2 (2015), 95-142.  doi: 10.3934/jcd.2015.2.95. [34] CAPD: Computer assisted proofs in dynamics, a package for rigorous numerics, http://capd.ii.uj.edu.pl/.
Representation of the matrix introduced in Equation (25)
The shape of the operator $Q$, as defined in Lemma 5.1. The vertical columns represent the vectors $y^{[k]}$, where $k$ denotes the column index
The boundaries of the validation interval $(r_{\min},r_{\max})$ versus the number of modes $K$ for various values of $\nu$
On the right: the minimal number of modes $K$ necessary for the validation of a periodic orbit of (52) depending on $\mu$. The stepsize in $\mu$ is not chosen uniformly, because the problem is more sensitive to changes in $\mu$ for $\mu$ bigger, therefore requesting a steeper change in the number of modes. One the left: several of the validated periodic orbits
The computation time versus the number of modes $K$. The reference line has a slope corresponding to $K^3$
The computation time versus the dimension $N$ of the system (53). The reference lines have slopes corresponding to quadratic and cubic dependence on $N$
The computation time versus the order $D$ of the system (54). The reference lines have slopes corresponding to quadratic and cubic dependence on $D$
Validation of a periodic solution for the Lorenz system (55) with the classical parameter values $\sigma = 10$, $\beta = 8/3$ and $\rho = 28$. The solution has a period close to 25. This solution was validated with $\nu = 1+10^{-6}$ and $K = 800$
Step size versus $\mu$. Some oscillation is noticeable in the step size. This is due to the fact that the stepsize is decreased when the number of modes is changed, see Section 9. During this validation, the number of modes has been increased automatically from 4 to about 400
Validated continuation for the Lorenz system (55) from $\rho = 28$ to $\rho = 14.8$. In the left graph, the lower orbit still has two distinct swirls in the left lobe, but they are too close to each other to be seen in this graph
Validated continuation for $10$ coupled Lorenz systems (77) from $\epsilon = 0.14205$ to $\epsilon = 0.14252$. In the left graph we have depicted in blue the orbit of the first three component, in green the orbit of the last three component, with the biggest validated $\epsilon$, in the right graph we depicted the values of $\epsilon$, where we can notice the adaptation of the stepsize
Continuation through the fold of the Rychkov system (78). The validation started at the top, where a small initial stepsize was chosen. One can see how the stepsize was automatically adjusted along the validated curve. As can be seen from the values along the horizontal axis, we only depict a validated continuation for parameter values very close to saddle-node
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