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July  2021, 8(3): 309-323. doi: 10.3934/jcd.2021013

On polynomial forms of nonlinear functional differential equations

Olivier Hénot ,

McGill University, Department of Mathematics and Statistics, 805 Sherbrooke Street West, Montréal, Québec, H3A 0B9, Canada

* Corresponding author: Olivier Hénot

Received  April 2020 Revised  April 2021 Published  July 2021 Early access  June 2021

In this paper we study nonlinear autonomous retarded functional differential equations; that is, functional equations where the time derivative may depend on the past values of the variables. When the nonlinearities in such equations are comprised of elementary functions, we give a constructive proof of the existence of an embedding of the original coordinates yielding a polynomial differential equation. This embedding is a topological conjugacy between the semi-flow of the original differential equation and the semi-flow of the auxiliary polynomial differential equation. Further dynamical features are investigated; notably, for an equilibrium or a periodic orbit and its embedded counterpart, the stable and unstable eigenvalues have the same algebraic and geometric multiplicity.

Keywords: Functional differential equations, automatic differentiation, equilibria, periodic orbits, invariant manifolds.
Mathematics Subject Classification: Primary: 34Kxx, 34K17, 34K12, 34K20, 37C25; Secondary: 37C86.
Citation: Olivier Hénot. On polynomial forms of nonlinear functional differential equations. Journal of Computational Dynamics, 2021, 8 (3) : 309-323. doi: 10.3934/jcd.2021013
References:
[1]

X. Cabre, E. Fontich and R. de la Llave, The parameterization method for invariant manifolds Ⅰ: Manifolds associated to non-resonant subspaces, Indiana Univ. Math. J., 52 (2003), 283-328.  doi: 10.1512/iumj.2003.52.2245.

[2]

X. Cabre, E. Fontich and R. de la Llave, The parameterization method for invariant manifolds Ⅱ: Regularity with respect to parameters, Indiana Univ. Math. J., 52 (2003), 329-360.  doi: 10.1512/iumj.2003.52.2407.

[3]

X. Cabre, E. Fontich and R. de la Llave, The parameterization method for invariant manifolds Ⅲ: Overview and applications, Journal of Differential Equations, 218 (2005), 444-515.  doi: 10.1016/j.jde.2004.12.003.

[4]

A. Deprit and J. F. Price, The computation of characteristic exponents in the planar restricted problem of three bodies, Astronomical Journal, 70 (1965), 836-846.  doi: 10.21236/AD0622985.

[5]

E. Fehlberg, Zur numerischen Integration von Differentialgleichungen durch Potenzreihen-Ansätze, dargestellt an Hand physikalischer Beispiele, ZAMM, 44 (1964), 83-88.  doi: 10.1002/zamm.19640440303.

[6]

J. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977. doi: 10.1007/978-1-4612-9892-2.

[7]

À. Haro, M. Canadell, J.-L. Figueras, A. Luque and J.-M. Mondelo, The Parameterization Method for Invariant Manifolds, Springer International Publishing, Switzerland, 2016. doi: 10.1007/978-3-319-29662-3.

[8]

S. Kepley and J. D. Mireles, Chaotic motions in the restricted four body problem via Devaney's saddle-focus homoclinic tangle theorem, Journal of Differential Equations, 266 (2019), 1709-1755.  doi: 10.1016/j.jde.2018.08.007.

[9]

J.-P. Lessard, J. D. Mireles and J. Ransford, Automatic differentiation for Fourier series and the radii polynomial approach, Physica D Nonlinear Phenomena, 334 (2016), 174-186.  doi: 10.1016/j.physd.2016.02.007.

[10]

E. Rabe, Determination and survey of periodic trojan orbits in the restricted problem of three bodies, Astronomical Journal, 66 (1961), 500-513.  doi: 10.1086/108451.

[11]

E. Rabe, Additional periodic trojan orbits and further studies of their stability features, Astronomical Journal, 67 (1962), 382-390.  doi: 10.1086/108744.

[12]

E. Rabe and A. F. Schanzle, Periodic librations about the triangular solutions of the restricted earth-moon problem and their orbital stabilities, Astronomical Journal, 67 (1962), 732-739.  doi: 10.1086/108802.

[13]

L. M. Rauch and W. C. Riddell, The iterative solutions of the analytical $N$-body problem, Journal of the Society for Industrial and Applied Mathematics, 8 (1960), 568-581.  doi: 10.1137/0108043.

[14]

J. F. Steffensen, On the restricted problem of three bodies, Mat. Fys. Medd. Kgl. Danske Videnskab. Selskab., 30 (1956).

[15]

J. B. van den Berg, C. Groothedde and J.-P. Lessard, A general method for computer-assisted proofs of periodic solutions in delay differential equations, Journal of Dynamics and Differential Equations, (2020), 1–44. doi: 10.1016/j.jde.2017.11.011.

show all references

References:
[1]

X. Cabre, E. Fontich and R. de la Llave, The parameterization method for invariant manifolds Ⅰ: Manifolds associated to non-resonant subspaces, Indiana Univ. Math. J., 52 (2003), 283-328.  doi: 10.1512/iumj.2003.52.2245.

[2]

X. Cabre, E. Fontich and R. de la Llave, The parameterization method for invariant manifolds Ⅱ: Regularity with respect to parameters, Indiana Univ. Math. J., 52 (2003), 329-360.  doi: 10.1512/iumj.2003.52.2407.

[3]

X. Cabre, E. Fontich and R. de la Llave, The parameterization method for invariant manifolds Ⅲ: Overview and applications, Journal of Differential Equations, 218 (2005), 444-515.  doi: 10.1016/j.jde.2004.12.003.

[4]

A. Deprit and J. F. Price, The computation of characteristic exponents in the planar restricted problem of three bodies, Astronomical Journal, 70 (1965), 836-846.  doi: 10.21236/AD0622985.

[5]

E. Fehlberg, Zur numerischen Integration von Differentialgleichungen durch Potenzreihen-Ansätze, dargestellt an Hand physikalischer Beispiele, ZAMM, 44 (1964), 83-88.  doi: 10.1002/zamm.19640440303.

[6]

J. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977. doi: 10.1007/978-1-4612-9892-2.

[7]

À. Haro, M. Canadell, J.-L. Figueras, A. Luque and J.-M. Mondelo, The Parameterization Method for Invariant Manifolds, Springer International Publishing, Switzerland, 2016. doi: 10.1007/978-3-319-29662-3.

[8]

S. Kepley and J. D. Mireles, Chaotic motions in the restricted four body problem via Devaney's saddle-focus homoclinic tangle theorem, Journal of Differential Equations, 266 (2019), 1709-1755.  doi: 10.1016/j.jde.2018.08.007.

[9]

J.-P. Lessard, J. D. Mireles and J. Ransford, Automatic differentiation for Fourier series and the radii polynomial approach, Physica D Nonlinear Phenomena, 334 (2016), 174-186.  doi: 10.1016/j.physd.2016.02.007.

[10]

E. Rabe, Determination and survey of periodic trojan orbits in the restricted problem of three bodies, Astronomical Journal, 66 (1961), 500-513.  doi: 10.1086/108451.

[11]

E. Rabe, Additional periodic trojan orbits and further studies of their stability features, Astronomical Journal, 67 (1962), 382-390.  doi: 10.1086/108744.

[12]

E. Rabe and A. F. Schanzle, Periodic librations about the triangular solutions of the restricted earth-moon problem and their orbital stabilities, Astronomical Journal, 67 (1962), 732-739.  doi: 10.1086/108802.

[13]

L. M. Rauch and W. C. Riddell, The iterative solutions of the analytical $N$-body problem, Journal of the Society for Industrial and Applied Mathematics, 8 (1960), 568-581.  doi: 10.1137/0108043.

[14]

J. F. Steffensen, On the restricted problem of three bodies, Mat. Fys. Medd. Kgl. Danske Videnskab. Selskab., 30 (1956).

[15]

J. B. van den Berg, C. Groothedde and J.-P. Lessard, A general method for computer-assisted proofs of periodic solutions in delay differential equations, Journal of Dynamics and Differential Equations, (2020), 1–44. doi: 10.1016/j.jde.2017.11.011.

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