Renato C. Calleja
Dynamical Systems • Mathematical Physics • Computation

Renato C. Calleja

I am a mathematician at the Department of Mathematics and Mechanics, IIMAS, UNAM. My research focuses on dynamical systems and mathematical physics, especially invariant objects, a-posteriori KAM theory, conformally symplectic systems, celestial mechanics, and constructive computational methods.

a-posteriori KAM theory Conformally symplectic dynamics Invariant tori Celestial mechanics Parameterization method Computer-assisted methods
Curriculum Vitae Google Scholar Citation Summary (PDF) ORCID Current Teaching Animations

Research

My work lies in dynamical systems and mathematical physics, with an emphasis on the existence, persistence, computation, and validation of invariant structures and on their role in applications.

Conformally symplectic dynamics

Quasi-periodic invariant tori, dissipative perturbations of Hamiltonian systems, whiskered tori, domains of analyticity, breakdown phenomena, and efficient validated algorithms.

a-posteriori KAM and parameterization methods

Constructive KAM theorems, Newton-type schemes, numerical KAM algorithms, and geometric formulations for flows and maps with quasi-periodic dependence.

Celestial mechanics and choreographies

Periodic and quasi-periodic solutions, symmetry, bifurcation, choreographies in the n-body problem, and constructive approaches connected to Marchal’s conjecture.

Applied and computational dynamics

Symplectic maps, transport problems, delay differential equations, nonlinear lattice equations, and computational techniques for invariant objects in models from mathematical physics.

Selected publications

A short list of representative recent work appears below. A complete publication list is included further down.

From the Lagrange Triangle to the Figure Eight Choreography: Proof of Marchal’s Conjecture

With C. García-Azpeitia, O. Hénot, J.-P. Lessard, and J. Mireles-James. Accepted for publication in Transactions of the AMS (2026). ArXiv

KAM algorithms for Lagrangian tori in time-quasi-periodic Hamiltonian systems

With A. Haro and P. Porras. Communications in Nonlinear Science and Numerical Simulation 153 (2026), 109445. DOI

Constructive Approaches to QP-Time-Dependent KAM Theory for Lagrangian Tori in Hamiltonian Systems

With A. Haro and P. Porras. Journal of Differential Equations 449 (2025), 113681. ArXiv

Accurate computations up to break-down of quasi-periodic attractors in the dissipative spin-orbit problem

With A. Celletti, J. Gimeno, and R. de la Llave. Journal of Nonlinear Science 34 (2024), Paper No. 12. DOI

Non-twist invariant circles in conformally symplectic systems

With M. Canadell and A. Haro. Communications in Nonlinear Science and Numerical Simulation 96 (2021), 105695. ArXiv

Local Behavior Near Quasi-Periodic Solutions of Conformally Symplectic Systems

With A. Celletti and R. de la Llave. Journal of Dynamical and Differential Equations 25 (2013), no. 3, 821–841. Preprint

Complete publication list
  1. Calleja, R., García-Azpeitia, C., Hénot, O., Lessard, J.-P., Mireles-James, J., From the Lagrange Triangle to the Figure Eight Choreography: Proof of Marchal’s Conjecture, accepted for publication in Transactions of the AMS (2026), preprint: ArXiv_2406.17564
  2. Calleja, R., Haro, A., Porras, P., KAM algorithms for Lagrangian tori in time-quasi-periodic Hamiltonian systems, Commun. Nonlinear Sci. Numer. Simul. 153 (2026), 109445, pdf, preprint: ArXiv_2504.00335
  3. Calleja, R., Haro, A., Porras, P., Constructive Approaches to QP-Time-Dependent KAM Theory for Lagrangian Tori in Hamiltonian Systems, J. Differential Equations 449 (2025), 113681, pdf, preprint: ArXiv_2503.09740
  4. R. Calleja, A. Celletti, J. Gimeno, R. de la Llave, Accurate computations up to break-down of quasi-periodic attractors in the dissipative spin-orbit problem, J. Nonlinear Sci. 34 (2024), no. 1, Paper No. 12, pdf, preprint: ArXiv_2210.05796
  5. R. Calleja, A. Celletti, R. de la Llave, KAM quasi-periodic solutions for the dissipative standard map, Commun. Nonlinear Sci. Numer. Simul. 106 (2022), 106111, pdf, preprint: ArXiv_2002.10647
  6. R. Calleja, A. Celletti, J. Gimeno, R. de la Llave, KAM quasi-periodic tori for the dissipative spin-orbit problem, Commun. Nonlinear Sci. Numer. Simul. 106 (2022), 106099, pdf, preprint: ArXiv_2107.02853
  7. R. Calleja, A. Celletti, J. Gimeno, R. de la Llave, Efficient and accurate KAM tori construction for the dissipative spin-orbit problem using a map reduction, Journal of Nonlinear Science 32 (2022), Article No. 4, pdf, preprint: ArXiv_2106.09175
  8. Calleja, R., Celletti, A., de la Llave, R., KAM Theory for Some Dissipative Systems, in: New Frontiers of Celestial Mechanics: Theory and Applications, Springer Proceedings in Mathematics & Statistics, vol. 399 (2022), pdf, preprint: ArXiv_2007.08394
  9. R. Calleja, D. del-Castillo-Negrete, D. Martínez-del-Río, A. Olvera, A new method to compute periodic orbits in general symplectic maps, Commun. Nonlinear Sci. Numer. Simul. 99 (2021), 105838, pdf, preprint: ArXiv_2003.02788
  10. R. Calleja, C. García-Azpeitia, J.-P. Lessard, J. D. Mireles-James, From the Lagrange polygon to the figure eight I: Numerical evidence extending a conjecture of Marchal, Celestial Mechanics and Dynamical Astronomy 133 (2021), Article No. 10, pdf, preprint: ArXiv_2012.10759
  11. R. Calleja, M. Canadell, A. Haro, Non-twist invariant circles in conformally symplectic systems, Commun. Nonlinear Sci. Numer. Simul. 96 (2021), 105695, pdf, preprint: ArXiv_2005.09754
  12. A. P. Bustamante, R. Calleja, Corrigendum and Addendum to “Computation of domains of analyticity for the dissipative standard map in the limit of small dissipation”, Physica D 417 (2021), 132837, pdf, preprint: ArXiv_2010.07500
  13. R. Calleja, C. García-Azpeitia, J.-P. Lessard, J. D. Mireles-James, Torus knot choreographies in the n-body problem, Nonlinearity 34 (2021), 313, pdf, preprint: FAU
  14. Calleja, R., Celletti, A., de la Llave, R., Existence of whiskered KAM tori of conformally symplectic systems, Nonlinearity 33 (2020), no. 1, 538–597, pdf, preprint: ArXiv_1901.07483
  15. Calleja, R., Celletti, A., de la Llave, R., Whiskered KAM Tori of Conformally Symplectic Systems, Mathematics Research Reports 1 (2020), 15–29, pdf, preprint: ArXiv_1901.06059
  16. A. P. Bustamante, R. Calleja, Computation of domains of analyticity for the dissipative standard map in the limit of small dissipation, Physica D 395 (2019), 15–23, pdf, preprint: ArXiv_1712.05476
  17. R. Calleja, E. Doedel, C. García-Azpeitia, Choreographies in the n-vortex problem, Regular and Chaotic Dynamics 23 (2018), no. 5, 595–612, pdf, view only, preprint: ArXiv_1807.08212
  18. R. Calleja, E. Doedel, C. García-Azpeitia, Carlos L. Pando, Choreographies in the Discrete Nonlinear Schrödinger Equations, Eur. Phys. J. Special Topics 227 (2018), 615–624, pdf, preprint: ArXiv_1801.03187
  19. R. Calleja, E. Doedel, C. García-Azpeitia, Symmetries and choreographies in families that bifurcate from the polygonal relative equilibrium of the n-body problem, Celestial Mech. Dynam. Astronom. 130 (2018), no. 7, 48, pdf, preprint: ArXiv_1702.03990, animations
  20. R. Calleja, A. Celletti, R. de la Llave, Domains of analyticity of Lindstedt expansions of KAM tori in dissipative perturbations of Hamiltonian systems, Nonlinearity 30 (2017), 3151–3202, pdf, preprint: mp_arc 15-44
  21. R. C. Calleja, A. R. Humphries, B. Krauskopf, Resonance phenomena in a scalar delay differential equation with two state-dependent delays, SIAM J. Appl. Dyn. Syst. 16 (2017), no. 3, 1474–1513, pdf, preprint: ArXiv_1607.02683
  22. R. Calleja, D. del-Castillo-Negrete, D. Martínez-del-Río, A. Olvera, Global transport in a nonautonomous standard map, Commun. Nonlinear Sci. Numer. Simul. 51 (2017), 198–215, pdf, preprint: ArXiv_1601.04824
  23. R. Calleja, A. Celletti, L. Corsi, R. de la Llave, Response solutions for quasi-periodic forced, dissipative wave equations, SIAM J. Math. Anal. 49 (2017), no. 4, 3161–3207, pdf, preprint: ArXiv_1501.05979
  24. Calleja, R., Doedel, E., García-Azpeitia, C., Symmetry-breaking for a restricted n-body problem in the Maxwell-ring configuration, Eur. Phys. J. Special Topics 225 (2016), 2741–2750, pdf, preprint: ArXiv_1601.06160
  25. A. R. Humphries, D. A. Bernucci, R. Calleja, N. Homayounfar, M. Snarski, Periodic Solutions of a Singularly Perturbed Delay Differential Equation With Two State-Dependent Delays, J. Dynam. Differential Equations 28 (2016), no. 3, 1215–1263, pdf, preprint: ArXiv_1411.6060
  26. D. Martínez, D. del-Castillo-Negrete, A. Olvera, R. Calleja, Self-consistent chaotic transport in a high-dimensional mean-field Hamiltonian map model, Qual. Theory Dyn. Sys. 14 (2015), no. 2, 313–335, pdf, preprint: ArXiv_1601.00942
  27. Calleja, R., Celletti, A., Falcolini, C., de la Llave, R., An Extension of Greene's Criterion for Conformally Symplectic Systems and a Partial Justification, SIAM J. Math. Anal. 46 (2014), no. 4, 2350–2384, pdf, preprint: mp_arc 13-60
  28. Calleja, R., Celletti, A., de la Llave, R., Local Behavior Near Quasi-Periodic Solutions of Conformally Symplectic Systems, J. Dynam. Differential Equations 25 (2013), no. 3, 821–841, pdf, preprint: mp_arc 12-16
  29. Calleja, R., Celletti, A., de la Llave, R., KAM theory for conformally symplectic systems: Efficient algorithms and their validation, J. Differential Equations 255 (2013), no. 5, 978–1049, pdf, preprint: mp_arc 11-188
  30. Calleja, R., Celletti, A., de la Llave, R., Construction of response functions in forced strongly dissipative systems, DCDS-A 33 (2013), no. 10, 4411–4433, pdf, preprint: mp_arc 12-79
  31. Calleja, R., Doedel, E. J., Humphries, A. R., Lemus-Rodriguez, A., Oldeman, B. E., Boundary-value problem formulations for computing invariant manifolds and connecting orbits in the circular restricted three body problem, Celestial Mechanics and Dynamical Astronomy 114 (2012), no. 1–2, 77–106, pdf, preprint: ArXiv 1111.0032
  32. Calleja, R., Figueras, J.-Ll., Collision of invariant bundles of quasi-periodic attractors in the dissipative standard map, Chaos 22 (2012), 033114, pdf, preprint: mp_arc 11-182
  33. Calleja, R., de la Llave, R., Computation of the breakdown of analyticity in statistical mechanics models: numerical results and a renormalization group explanation, Journal of Statistical Physics 141 (2010), 940–951, pdf, preprint: mp_arc 09-56
  34. Calleja, R., de la Llave, R., A numerically accessible criterion for the breakdown of quasi-periodic solutions and its rigorous justification, Nonlinearity 23 (2010), 2029–2058, pdf, preprint: mp_arc 09-150
  35. Calleja, R., Celletti, A., Breakdown of invariant attractors for the dissipative standard map, Chaos 20 (2010), 013121, pdf, preprint: mp_arc 09-124
  36. Calleja, R., Sire, Y., Travelling waves in discrete nonlinear systems with non-nearest neighbour interactions, Nonlinearity 22 (2009), 2583–2605, pdf, preprint: mp_arc 08-188
  37. Calleja, R., de la Llave, R., Fast numerical computation of quasi-periodic equilibrium states in 1-D statistical mechanics, including twist maps, Nonlinearity 22 (2009), 1311–1336, pdf, preprint: mp_arc 08-189
  38. Lomelí, H. E., Calleja, R., Heteroclinic bifurcations and chaotic transport in the two-harmonic Standard-Map, Chaos 16 (2006), 023117, pdf

Teaching

Recent course pages are listed first, followed by archived material.

Current and recent courses

  • Ecuaciones Diferenciales Ordinarias 2026-2 — EDO
  • Introducción a la Mecánica Analítica 2026-1 — IMA
  • Sistemas Dinámicos Hamiltonianos 2025-2 — SDH
  • Introducción a la Mecánica Analítica 2025-1 — IMA
  • Ecuaciones Diferenciales Ordinarias 2024-2 — EDO
  • Introducción a la Mecánica Analítica 2024-1 — IMA

Archive

  • Métodos Numéricos en Teoría KAM, Mexican HAT (con Pedro Porras Flores) — NumKAM
  • Ecuaciones Diferenciales Ordinarias 2023-2 — EDO
  • Sistemas Dinámicos Hamiltonianos 2021-2 — SDH
  • Introducción a la Mecánica Analítica 2021-1 — IMA
  • Sistemas Dinámicos No Lineales 2020-2 — 6011
  • Computer Assisted Proofs in Nonlinear Dynamics Course — CAPNLD

Preprints

  1. Calleja, R., Padilla, P., Rodriguez, E., Biological network dynamics: Lindstedt series and the effect of delays, preprint: ArXiv_2504.01322

Theses

Ph.D. Dissertation

Existence and persistence of invariant objects in dynamical systems and mathematical physics
The University of Texas at Austin, Ph.D. Dissertation (May 2009)
Supervisor: Rafael de la Llave
Frank Gerth III Dissertation Award
FGDA · mp_arc 09-77

B.Sc. Thesis

Dinámica simpléctica y approximaciones numéricas de separatrices y transporte caótico
Instituto Tecnológico Autónomo de México, B.Sc. Thesis (June 2004)
Supervisor: Hector Lomelí

Animations and visualizations

n-body problem animations

Visual material related to choreographies and celestial mechanics: Animations for the n-body problem