American Institute of Mathematical Sciences

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December  2020, 40(12): 6681-6707. doi: 10.3934/dcds.2020162

Computer assisted proofs of two-dimensional attracting invariant tori for ODEs

Maciej J. Capiński 1,, , Emmanuel Fleurantin 2, and  J. D. Mireles James 2,

1. 

AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Kraków, Poland
2. 

Florida Atlantic University, 777 Glades Rd., Boca Raton, FL 33431, USA

* Corresponding author: maciej.capinski@agh.edu.pl

Received  May 2019 Revised  December 2019 Published  December 2020 Early access  March 2020

Fund Project: The first author is supported by by the NCN grant 2018/29/B/ST1/00109. The work has been conducted during his visit to FAU sponsored by the Fulbright Foundation. The third author was partially supported by National Science Foundation grant DMS 1813501

This work studies existence and regularity questions for attracting invariant tori in three dimensional dissipative systems of ordinary differential equations. Our main result is a constructive method of computer assisted proof which applies to explicit problems in non-perturbative regimes. We obtain verifiable lower bounds on the regularity of the attractor in terms of the ratio of the expansion rate on the torus with the contraction rate near the torus. We consider separately two important cases of rotational and resonant tori. In the rotational case we obtain $ C^k $ lower bounds on the regularity of the embedding. In the resonant case we verify the existence of tori which are only $ C^0 $ and neither star-shaped nor Lipschitz.

Keywords: Attracting invariant tori, computer assisted proof, dynamics of ordinary differential equations.
Mathematics Subject Classification: 34C45, 70K43, 37G35, 65P20, 65G20.
Citation: Maciej J. Capiński, Emmanuel Fleurantin, J. D. Mireles James. Computer assisted proofs of two-dimensional attracting invariant tori for ODEs. Discrete and Continuous Dynamical Systems, 2020, 40 (12) : 6681-6707. doi: 10.3934/dcds.2020162
References:
[1]

G. Alefeld, Inclusion methods for systems of nonlinear equations–the interval Newton method and modifications, in Topics in Validated Computations, Stud. Comput. Math., 5, North-Holland, Amsterdam, 1994, 7–26.

[2]

G. Arioli and H. Koch, Existence and stability of traveling pulse solutions of the FitzHugh-Nagumo equation, Nonlinear Anal., 113 (2015), 51-70.  doi: 10.1016/j.na.2014.09.023.

[3]

C. Baesens, J. Guckenheimer, S. Kim and R. S. MacKay, Three coupled oscillators: Mode-locking, global bifurcations and toroidal chaos, Phys. D, 49 (1991), 387-475.  doi: 10.1016/0167-2789(91)90155-3.

[4]

T. Bakri, Y. A. Kuznetsov and F. Verhulst, Torus bifurcations in a mechanical system, J. Dynam. Differential Equations, 27 (2015), 371-403.  doi: 10.1007/s10884-013-9339-9.

[5]

T. Bakri and F. Verhulst, Bifurcations of quasi-periodic dynamics: Torus breakdown, Z. Angew. Math. Phys., 65 (2014), 1053-1076.  doi: 10.1007/s00033-013-0363-8.

[6]

J. B. van den Berg and J.-P. Lessard, Rigorous numerics in dynamics, Notices Amer. Math. Soc., 62 (2015), 1057-1061.  doi: 10.1090/noti1276.

[7]

H. W. Broer, H. M. Osinga and G. Vegter, On the computation of normally hyperbolic invariant manifolds, in Nonlinear Dynamical Systems and Chaos, Progr. Nonlinear Differential Equations Appl., 19, Birkhäuser, Basel, 1996,423–447. doi: 10.1007/978-3-0348-7518-9_20.

[8]

H. W. Broer, H. M. Osinga and G. Vegter, Algorithms for computing normally hyperbolic invariant manifolds, Z. Angew. Math. Phys., 48 (1997), 480-524.  doi: 10.1007/s000330050044.

[9]

M. Canadell and À. Haro, Computation of quasi-periodic normally hyperbolic invariant tori: Algorithms, numerical explorations and mechanisms of breakdown, J. Nonlinear Sci., 27 (2017), 1829-1868.  doi: 10.1007/s00332-017-9388-z.

[10]

M. Canadell and À. Haro, Computation of quasiperiodic normally hyperbolic invariant tori: Rigorous results, J. Nonlinear Sci., 27 (2017), 1869-1904.  doi: 10.1007/s00332-017-9389-y.

[11]

M. Canadell and À. Haro, Parameterization method for computing quasi-periodic reducible normally hyperbolic invariant tori, in Advances in Differential Equations and Applications, SEMA SIMAI Springer Ser., 4, Springer, Cham, 2014, 85–94. doi: 10.1007/978-3-319-06953-1_9.

[12]

M. J. Capiński, Computer assisted existence proofs of Lyapunov orbits at $L_2$ and transversal intersections of invariant manifolds in the Jupiter-Sun PCR3BP, SIAM J. Appl. Dyn. Syst., 11 (2012), 1723-1753.  doi: 10.1137/110847366.

[13]

M. J. Capiński and H. Kubica, Persistence of normally hyperbolic invariant manifolds in the absence of rate conditions, preprint, arXiv: 1804.05580.

[14]

M. J. Capiński and P. Zgliczyński, Cone conditions and covering relations for topologically normally hyperbolic invariant manifolds, Discrete Contin. Dyn. Syst., 30 (2011), 641-670.  doi: 10.3934/dcds.2011.30.641.

[15]

M. J. Capiński and P. Zgliczyński, Geometric proof for normally hyperbolic invariant manifolds, J. Differential Equations, 259 (2015), 6215-6286.  doi: 10.1016/j.jde.2015.07.020.

[16]

A. Celletti and L. Chierchia, Rigorous estimates for a computer-assisted KAM theory, J. Math. Phys., 28 (1987), 2078-2086.  doi: 10.1063/1.527418.

[17]

A. Celletti and L. Chierchia, A computer-assisted approach to small-divisors problems arising in Hamiltonian mechanics, in Computer Aided Proofs in Analysis, IMA Vol. Math. Appl., 28, Springer, New York, 1991, 43–51. doi: 10.1007/978-1-4613-9092-3_6.

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M. Cercek, T. Gyergyek and M. Stanojevic, On the nonlinear dynamics of an instability in front of a positively biased electrode in a magnetized plasma, Nuclear Energy in Central Europe, Portoroz, Slovenia, 1996,531–538.

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L. Dieci, J. Lorenz and R. D. Russell, Numerical calculation of invariant tori, SIAM J. Sci. Statist. Comput., 12 (1991), 607-647.  doi: 10.1137/0912033.

[20]

J.-P. Eckmann and P. Wittwer, A complete proof of the Feigenbaum conjectures, J. Statist. Phys., 46 (1987), 455-475.  doi: 10.1007/BF01013368.

[21]

N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J., 21 (1971/72), 193-226.  doi: 10.1512/iumj.1972.21.21017.

[22]

J.-Ll. Figueras, À. 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.

[23]

J. E. Flaherty and F. C. Hoppensteadt, Frequency entrainment of a forced van der Pol oscillator, Studies in Appl. Math., 58 (1978), 5–15. doi: 10.21236/ADA039211.

[24]

E. Fleurantin and J. D. Mireles-James, Resonant tori, transport barriers, and chaos in a vector field with a Neimark-Sacker bifurcation, Commun. Nonlinear Sci. Numer. Simul., 85 (2020). doi: 10.1016/j.cnsns.2020.105226.

[25]

J. Gómez-Serrano, Computer-assisted proofs in PDE: A survey, SeMA J., 76 (2019), 459-484.  doi: 10.1007/s40324-019-00186-x.

[26]

J. Guckenheimer, K. Hoffman and W. Weckesser, The forced van der Pol equation. I. The slow flow and its bifurcations, SIAM J. Appl. Dyn. Syst., 2 (2003), 1-35.  doi: 10.1137/S1111111102404738.

[27]

À. Haro and R. de la Llave, A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Rigorous results, J. Differential Equations, 228 (2006), 530-579.  doi: 10.1016/j.jde.2005.10.005.

[28]

À. Haro and R. de la Llave, A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Numerical algorithms, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1261-1300.  doi: 10.3934/dcdsb.2006.6.1261.

[29]

À. Haro and R. de la Llave, A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: explorations and mechanisms for the breakdown of hyperbolicity, SIAM J. Appl. Dyn. Syst., 6 (2007), 142-207.  doi: 10.1137/050637327.

[30]

À. Haro and A. Luque, A-posteriori KAM theory with optimal estimates for partially integrable systems, J. Differential Equations, 266 (2019), 1605-1674.  doi: 10.1016/j.jde.2018.08.003.

[31]

M. W. Hirsch and C. C. Pugh, Stable manifolds and hyperbolic sets, in Global Analysis (Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 1970,133–163.

[32]

M. W. Hirsch, C. C. Pugh and M. Shub, Invariant manifolds, Bull. Amer. Math. Soc., 76 (1970), 1015-1019.  doi: 10.1090/S0002-9904-1970-12537-X.

[33]

M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, 583, Springer-Verlag, Berlin-New York, 1977. doi: 10.1007/BFb0092042.

[34]

K. Kaneko, Transition from torus to chaos accompanied by frequency lockings with symmetry breaking. In connection with the coupled-logistic map, Progr. Theoret. Phys., 69 (1983), 1427-1442.  doi: 10.1143/PTP.69.1427.

[35]

S.-H. Kim, R. S. MacKay and J. Guckenheimer, Resonance regions for families of torus maps, Nonlinearity, 2 (1989), 391-404.  doi: 10.1088/0951-7715/2/3/001.

[36]

B. Krauskopf, H. M. Osinga, E. J. Doedel, M. E. Henderson, J. Guckenheimer, A. Vladimirsky, M. Dellnitz and O. Junge, A survey of methods for computing (un)stable manifolds of vector fields, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 763-791.  doi: 10.1142/S0218127405012533.

[37]

O. E. Lanford, A computer-assisted proof of the Feigenbaum conjectures, Bull. Amer. Math. Soc. (N.S.), 6 (1982), 427-434.  doi: 10.1090/S0273-0979-1982-15008-X.

[38]

W. F. Langford, Numerical studies of torus bifurcations, in Numerical Methods for Bifurcation Problems, Internat. Schriftenreihe Numer. Math., 70, Birkhäuser, Basel, 1984,285–295. doi: 10.1007/978-3-0348-6256-1_19.

[39]

J.-P. Lessard, J. D. Mireles James and C. Reinhardt, Computer assisted proof of transverse saddle-to-saddle connecting orbits for first order vector fields, J. Dynam. Differential Equations, 26 (2014), 267-313.  doi: 10.1007/s10884-014-9367-0.

[40]

R. de la Llave, Invariant manifolds associated to nonresonant spectral subspaces, J. Statist. Phys., 87 (1997), 211-249.  doi: 10.1007/BF02181486.

[41]

R. de la Llave and D. Rana, Accurate strategies for small divisor problems, Bull. Amer. Math. Soc. (N.S.), 22 (1990), 85-90.  doi: 10.1090/S0273-0979-1990-15848-3.

[42]

R. de la Llave and D. Rana, Accurate strategies for K.A.M. bounds and their implementation, in Computer Aided Proofs in Analysis, IMA Vol. Math. Appl, 28, Springer, New York, 1991,127–146. doi: 10.1007/978-1-4613-9092-3_12.

[43]

J. Llibre, R. Martínez and C. Simó, Tranversality of the invariant manifolds associated to the Lyapunov family of periodic orbits near $L_2$ in the restricted three-body problem, J. Differential Equations, 58 (1985), 104-156.  doi: 10.1016/0022-0396(85)90024-5.

[44]

T. Matsumoto, L. O. Chua and R. Tokunaga, Chaos via torus breakdown, IEEE Trans. Circuits and Systems, 34 (1987), 240-253.  doi: 10.1109/TCS.1987.1086135.

[45]

J. D. Mireles James and K. Mischaikow, Rigorous a posteriori computation of (un)stable manifolds and connecting orbits for analytic maps, SIAM J. Appl. Dyn. Syst., 12 (2013), 957-1006.  doi: 10.1137/12088224X.

[46]

J. I. Neĭmark, Some cases of the dependence of periodic motions on parameters, Dokl. Akad. Nauk SSSR, 129 (1959), 736-739. 

[47]

H. M. Osinga, Computing global invariant manifolds: Techniques and applications, Proceedings of the International Congress of Mathematicians, 4, Kyung Moon Sa, Seoul, 2014, 1101–1123.

[48]

B. van der Pol, A theory of the amplitude of free and forced triode vibrations, Radio Review, 1 (1920), 701–710,754–762.

[49]

B. van der Pol, Frequency demultiplication, Nature, 120 (1927), 363-364.  doi: 10.1038/120363a0.

[50]

R. J. Sacker, On Invariant Surfaces and Bifurcation of Periodic Solutions of Ordinary Differential Equations, Ph.D. thesis, New York University, 1964.

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C. Simó, Connection of invariant manifolds in the $n$-body problem, $n>3$, In Proceedings of the Sixth Conference of Portuguese and Spanish Mathematicians, Rev. Univ. Santander, 1979, 1257–1261.

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O. Sosnovtseva and E. Mosekilde, Torus destruction and chaos-chaos intermittency in a commodity distribution chain, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 7 (1997), 1225-1242.  doi: 10.1142/S0218127497000996.

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D. Wilczak and P. Zgliczyński, $C^n$ Lohner algorithm, Scheade Informaticae, 20 (2011), 9-46. 

[56]

D. Wilczak, S. Serrano and R. Barrio, Coexistence and dynamical connections between hyperchaos and chaos in the 4D Rössler system: A computer-assisted proof, SIAM J. Appl. Dyn. Syst., 15 (2016), 356-390.  doi: 10.1137/15M1039201.

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P. Zgliczyński, Covering relations, cone conditions and the stable manifold theorem, J. Differential Equations, 246 (2009), 1774-1819.  doi: 10.1016/j.jde.2008.12.019.

show all references

References:
[1]

G. Alefeld, Inclusion methods for systems of nonlinear equations–the interval Newton method and modifications, in Topics in Validated Computations, Stud. Comput. Math., 5, North-Holland, Amsterdam, 1994, 7–26.

[2]

G. Arioli and H. Koch, Existence and stability of traveling pulse solutions of the FitzHugh-Nagumo equation, Nonlinear Anal., 113 (2015), 51-70.  doi: 10.1016/j.na.2014.09.023.

[3]

C. Baesens, J. Guckenheimer, S. Kim and R. S. MacKay, Three coupled oscillators: Mode-locking, global bifurcations and toroidal chaos, Phys. D, 49 (1991), 387-475.  doi: 10.1016/0167-2789(91)90155-3.

[4]

T. Bakri, Y. A. Kuznetsov and F. Verhulst, Torus bifurcations in a mechanical system, J. Dynam. Differential Equations, 27 (2015), 371-403.  doi: 10.1007/s10884-013-9339-9.

[5]

T. Bakri and F. Verhulst, Bifurcations of quasi-periodic dynamics: Torus breakdown, Z. Angew. Math. Phys., 65 (2014), 1053-1076.  doi: 10.1007/s00033-013-0363-8.

[6]

J. B. van den Berg and J.-P. Lessard, Rigorous numerics in dynamics, Notices Amer. Math. Soc., 62 (2015), 1057-1061.  doi: 10.1090/noti1276.

[7]

H. W. Broer, H. M. Osinga and G. Vegter, On the computation of normally hyperbolic invariant manifolds, in Nonlinear Dynamical Systems and Chaos, Progr. Nonlinear Differential Equations Appl., 19, Birkhäuser, Basel, 1996,423–447. doi: 10.1007/978-3-0348-7518-9_20.

[8]

H. W. Broer, H. M. Osinga and G. Vegter, Algorithms for computing normally hyperbolic invariant manifolds, Z. Angew. Math. Phys., 48 (1997), 480-524.  doi: 10.1007/s000330050044.

[9]

M. Canadell and À. Haro, Computation of quasi-periodic normally hyperbolic invariant tori: Algorithms, numerical explorations and mechanisms of breakdown, J. Nonlinear Sci., 27 (2017), 1829-1868.  doi: 10.1007/s00332-017-9388-z.

[10]

M. Canadell and À. Haro, Computation of quasiperiodic normally hyperbolic invariant tori: Rigorous results, J. Nonlinear Sci., 27 (2017), 1869-1904.  doi: 10.1007/s00332-017-9389-y.

[11]

M. Canadell and À. Haro, Parameterization method for computing quasi-periodic reducible normally hyperbolic invariant tori, in Advances in Differential Equations and Applications, SEMA SIMAI Springer Ser., 4, Springer, Cham, 2014, 85–94. doi: 10.1007/978-3-319-06953-1_9.

[12]

M. J. Capiński, Computer assisted existence proofs of Lyapunov orbits at $L_2$ and transversal intersections of invariant manifolds in the Jupiter-Sun PCR3BP, SIAM J. Appl. Dyn. Syst., 11 (2012), 1723-1753.  doi: 10.1137/110847366.

[13]

M. J. Capiński and H. Kubica, Persistence of normally hyperbolic invariant manifolds in the absence of rate conditions, preprint, arXiv: 1804.05580.

[14]

M. J. Capiński and P. Zgliczyński, Cone conditions and covering relations for topologically normally hyperbolic invariant manifolds, Discrete Contin. Dyn. Syst., 30 (2011), 641-670.  doi: 10.3934/dcds.2011.30.641.

[15]

M. J. Capiński and P. Zgliczyński, Geometric proof for normally hyperbolic invariant manifolds, J. Differential Equations, 259 (2015), 6215-6286.  doi: 10.1016/j.jde.2015.07.020.

[16]

A. Celletti and L. Chierchia, Rigorous estimates for a computer-assisted KAM theory, J. Math. Phys., 28 (1987), 2078-2086.  doi: 10.1063/1.527418.

[17]

A. Celletti and L. Chierchia, A computer-assisted approach to small-divisors problems arising in Hamiltonian mechanics, in Computer Aided Proofs in Analysis, IMA Vol. Math. Appl., 28, Springer, New York, 1991, 43–51. doi: 10.1007/978-1-4613-9092-3_6.

[18]

M. Cercek, T. Gyergyek and M. Stanojevic, On the nonlinear dynamics of an instability in front of a positively biased electrode in a magnetized plasma, Nuclear Energy in Central Europe, Portoroz, Slovenia, 1996,531–538.

[19]

L. Dieci, J. Lorenz and R. D. Russell, Numerical calculation of invariant tori, SIAM J. Sci. Statist. Comput., 12 (1991), 607-647.  doi: 10.1137/0912033.

[20]

J.-P. Eckmann and P. Wittwer, A complete proof of the Feigenbaum conjectures, J. Statist. Phys., 46 (1987), 455-475.  doi: 10.1007/BF01013368.

[21]

N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J., 21 (1971/72), 193-226.  doi: 10.1512/iumj.1972.21.21017.

[22]

J.-Ll. Figueras, À. 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.

[23]

J. E. Flaherty and F. C. Hoppensteadt, Frequency entrainment of a forced van der Pol oscillator, Studies in Appl. Math., 58 (1978), 5–15. doi: 10.21236/ADA039211.

[24]

E. Fleurantin and J. D. Mireles-James, Resonant tori, transport barriers, and chaos in a vector field with a Neimark-Sacker bifurcation, Commun. Nonlinear Sci. Numer. Simul., 85 (2020). doi: 10.1016/j.cnsns.2020.105226.

[25]

J. Gómez-Serrano, Computer-assisted proofs in PDE: A survey, SeMA J., 76 (2019), 459-484.  doi: 10.1007/s40324-019-00186-x.

[26]

J. Guckenheimer, K. Hoffman and W. Weckesser, The forced van der Pol equation. I. The slow flow and its bifurcations, SIAM J. Appl. Dyn. Syst., 2 (2003), 1-35.  doi: 10.1137/S1111111102404738.

[27]

À. Haro and R. de la Llave, A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Rigorous results, J. Differential Equations, 228 (2006), 530-579.  doi: 10.1016/j.jde.2005.10.005.

[28]

À. Haro and R. de la Llave, A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Numerical algorithms, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1261-1300.  doi: 10.3934/dcdsb.2006.6.1261.

[29]

À. Haro and R. de la Llave, A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: explorations and mechanisms for the breakdown of hyperbolicity, SIAM J. Appl. Dyn. Syst., 6 (2007), 142-207.  doi: 10.1137/050637327.

[30]

À. Haro and A. Luque, A-posteriori KAM theory with optimal estimates for partially integrable systems, J. Differential Equations, 266 (2019), 1605-1674.  doi: 10.1016/j.jde.2018.08.003.

[31]

M. W. Hirsch and C. C. Pugh, Stable manifolds and hyperbolic sets, in Global Analysis (Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 1970,133–163.

[32]

M. W. Hirsch, C. C. Pugh and M. Shub, Invariant manifolds, Bull. Amer. Math. Soc., 76 (1970), 1015-1019.  doi: 10.1090/S0002-9904-1970-12537-X.

[33]

M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, 583, Springer-Verlag, Berlin-New York, 1977. doi: 10.1007/BFb0092042.

[34]

K. Kaneko, Transition from torus to chaos accompanied by frequency lockings with symmetry breaking. In connection with the coupled-logistic map, Progr. Theoret. Phys., 69 (1983), 1427-1442.  doi: 10.1143/PTP.69.1427.

[35]

S.-H. Kim, R. S. MacKay and J. Guckenheimer, Resonance regions for families of torus maps, Nonlinearity, 2 (1989), 391-404.  doi: 10.1088/0951-7715/2/3/001.

[36]

B. Krauskopf, H. M. Osinga, E. J. Doedel, M. E. Henderson, J. Guckenheimer, A. Vladimirsky, M. Dellnitz and O. Junge, A survey of methods for computing (un)stable manifolds of vector fields, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 763-791.  doi: 10.1142/S0218127405012533.

[37]

O. E. Lanford, A computer-assisted proof of the Feigenbaum conjectures, Bull. Amer. Math. Soc. (N.S.), 6 (1982), 427-434.  doi: 10.1090/S0273-0979-1982-15008-X.

[38]

W. F. Langford, Numerical studies of torus bifurcations, in Numerical Methods for Bifurcation Problems, Internat. Schriftenreihe Numer. Math., 70, Birkhäuser, Basel, 1984,285–295. doi: 10.1007/978-3-0348-6256-1_19.

[39]

J.-P. Lessard, J. D. Mireles James and C. Reinhardt, Computer assisted proof of transverse saddle-to-saddle connecting orbits for first order vector fields, J. Dynam. Differential Equations, 26 (2014), 267-313.  doi: 10.1007/s10884-014-9367-0.

[40]

R. de la Llave, Invariant manifolds associated to nonresonant spectral subspaces, J. Statist. Phys., 87 (1997), 211-249.  doi: 10.1007/BF02181486.

[41]

R. de la Llave and D. Rana, Accurate strategies for small divisor problems, Bull. Amer. Math. Soc. (N.S.), 22 (1990), 85-90.  doi: 10.1090/S0273-0979-1990-15848-3.

[42]

R. de la Llave and D. Rana, Accurate strategies for K.A.M. bounds and their implementation, in Computer Aided Proofs in Analysis, IMA Vol. Math. Appl, 28, Springer, New York, 1991,127–146. doi: 10.1007/978-1-4613-9092-3_12.

[43]

J. Llibre, R. Martínez and C. Simó, Tranversality of the invariant manifolds associated to the Lyapunov family of periodic orbits near $L_2$ in the restricted three-body problem, J. Differential Equations, 58 (1985), 104-156.  doi: 10.1016/0022-0396(85)90024-5.

[44]

T. Matsumoto, L. O. Chua and R. Tokunaga, Chaos via torus breakdown, IEEE Trans. Circuits and Systems, 34 (1987), 240-253.  doi: 10.1109/TCS.1987.1086135.

[45]

J. D. Mireles James and K. Mischaikow, Rigorous a posteriori computation of (un)stable manifolds and connecting orbits for analytic maps, SIAM J. Appl. Dyn. Syst., 12 (2013), 957-1006.  doi: 10.1137/12088224X.

[46]

J. I. Neĭmark, Some cases of the dependence of periodic motions on parameters, Dokl. Akad. Nauk SSSR, 129 (1959), 736-739. 

[47]

H. M. Osinga, Computing global invariant manifolds: Techniques and applications, Proceedings of the International Congress of Mathematicians, 4, Kyung Moon Sa, Seoul, 2014, 1101–1123.

[48]

B. van der Pol, A theory of the amplitude of free and forced triode vibrations, Radio Review, 1 (1920), 701–710,754–762.

[49]

B. van der Pol, Frequency demultiplication, Nature, 120 (1927), 363-364.  doi: 10.1038/120363a0.

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R. J. Sacker, On Invariant Surfaces and Bifurcation of Periodic Solutions of Ordinary Differential Equations, Ph.D. thesis, New York University, 1964.

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Figure 1.  On the left we have a cone attached at the point $ \gamma _U^{-1}(q) $ in the case when $ k = 2 $ and $ n = 3 $. Note that the cone is not the blue (cone shaped) set. The cone $ \mathbf{Q}(\gamma_U^{-1}(q)) $ is the complement of the blue set in $ \mathbb{R}^3 $; i.e. the white region outside of the blue set. On the right we have an example of a Lipschitz manifold
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Figure 2.  The set $ U $ is a collection of boxes, and we prove the existence of a star-shaped invariant closed curve around $ q^{\ast} $ which satisfies the cone conditions
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Figure 3.  A well aligned cone
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Figure 4.  Construction of $ \mathcal{G}(h) $
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Figure 5.  Since $ \mathcal{G}^{n}(h) $ and $ \mathcal{G}^{n-1}(h) $ satisfy cone conditions, we can find an angle $ \alpha $, such that the isosceles triangle, with base joining $ q_{n} $ and $ q_{n-1} $, as in above plot, will fit between $ \mathcal{G}^{n}(h) $ and $ \mathcal{G}^{n-1}(h) $. By compactness of $ U $ and the fact that we have a finite number of $ C^{1} $ local maps $ \gamma_{i} $, the $ \alpha $ can be chosen independently of $ n, \theta, q_{n} $ and of $ q_{n-1} $. This means that the area between $ \mathcal{G}^{n}(h) $ and $ \mathcal{G}^{n-1}(h) $ is bounded from below by $ C\Vert q_{n}-q_{n-1}\Vert^{2} $, where $ C>0 $ is some constant independent from $ n $ and $ \theta $
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Figure 6.  Generalization to a vector bundle setting. The vector bundle $ E $ is in grey, its base is the curve $ p^* $, which is in black, with the fibers $ E_{\theta} $ represented as the grey lines. The set $ U $ consists of the union of the small rectangles. Note that in this picture $ p^* $ is not the invariant curve, rather it is the base of the vector bundle
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Figure 7.  Intersections of the invariant Lipschitz tori for the Van der Pol system with the $ t = 0 $ section for each parameter from (22). The smaller the $ \mu $ the more circular/smooth the curve
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Figure 8.  At $ \alpha = 0.75 $ we have an attracting limit cycle of $ P^2 $ on $ \Sigma $ (figure on the left) which is the intersection of the two dimensional $ C^k $ torus of the ODE with $ \Sigma \cap \{y>0\} $. On the right we plot half of the torus. In black we have both components of the torus intersection with $ \Sigma $; one for $ y<0 $ and the other for $ y>0 $
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Figure 10.  Colors have the same meaning as in Figure 9. At $ \alpha = 0.8225 $ we have transverse intersections of $ W^{u}(c_{h}) $ and $ W^{s}(c_{h}) $ which leads to chaotic dynamics
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Figure 9.  The plot of the periodic orbits $ c_{h} $ and $ c_{s} $ on $ \Sigma \cap \{y>0\} $. (The orbits are of period $ 6 $, but we plot only half of the points with $ y>0 $.) The hyperbolic orbit $ c_{u} $ is in blue, and the attracting orbit $ c_{s} $ is in green. The manifold $ W^{s}(c_{h}) $ is in red and $ W^{u}(c_{h}) $ is blue. (Left) at $ \alpha = 0.815 $, we see that a branch of $ W^{u}(c_{h}) $ goes inside and wraps around the attracting invariant circle. (Right) at $ \alpha = 0.835 $, we observe that a branch of $ W^{u}(c_{h}) $ goes to the other side of $ W^{s}(c_{h}) $ and gets caught in the basin of attraction of $ c_{s} $
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Figure 11.  A resonance torus for $ \alpha = 0.85 $. In red we plot the periodic orbit from which the tori have initially originated through the Hopf type bifurcation
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