{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T13:32:01Z","timestamp":1787232721337,"version":"build-2736575974"},"reference-count":49,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Appl. Dyn. Syst."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>Let $f:\\mathbb{R}^3\\rightarrow\\mathbb{R}^3$ be a diffeomorphism with $p_0,p_1\\in\\mathbb{R}^3$ distinct hyperbolic fixed points. Assume that $W^u(p_0)$ and $W^s(p_1)$ are two-dimensional manifolds which intersect transversally at a point q. Then the intersection is locally a one-dimensional smooth arc $\\tilde{\\gamma}$ through q, and points on $\\tilde{\\gamma}$ are orbits heteroclinic from $p_0$ to $p_1$. We describe and implement a numerical scheme for computing the jets of $\\tilde{\\gamma}$ to arbitrary order. We begin by computing high order polynomial approximations of some functions $P_u,P_s:\\mathbb{R}^2\\rightarrow\\mathbb{R}^3$, and domain disks $D_u,D_s\\subset\\mathbb{R}^2$, such that $W_{loc}^u(p_0)=P_u(D_u)$ and $W_{loc}^s(p_1)=P_s(D_s)$ with $W_{loc}^u(p_0)\\cap W_{loc}^s(p_1)\\neq\\emptyset$. Then the intersection arc $\\tilde{\\gamma}$ solves a functional equation involving $P_s$ and $P_u$. We develop an iterative numerical scheme for solving the functional equation, resulting in a high order Taylor expansion of the arc $\\tilde{\\gamma}$. We present numerical example computations for the volume preserving H\u00e9non family and compute some global invariant branched manifolds.<\/jats:p>","DOI":"10.1137\/090776329","type":"journal-article","created":{"date-parts":[[2010,8,5]],"date-time":"2010-08-05T18:13:42Z","timestamp":1281032022000},"page":"919-953","source":"Crossref","is-referenced-by-count":29,"title":["Computation of Heteroclinic Arcs with Application to the Volume Preserving H\u00e9non Family"],"prefix":"10.1137","volume":"9","author":[{"given":"J. D. Mireles","family":"James","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hector","family":"Lomel\u00ed","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,8,5]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.96.244103"},{"key":"R2","first-page":"1","volume":"65","author":"Alligood K. T.","year":"2007","journal-title":"Rend. Semin. Mat. Univ. Politec. Torino"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1007\/s00332-005-0736-z"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/15\/5\/313"},{"key":"R5","unstructured":"M. Berz and K. Makino,\n                      Cosy Infinity\n                      , http:\/\/www.cosyinfinity.org."},{"key":"R6","unstructured":"M. Berz and K. Makino,\n                      Suppression of the Wrapping Effect by Taylor Model-Based Validated Integrators\n                      , MSU HEP Report 40910, Michigan State University, East Lansing, MI, 2003; available online from http:\/\/bt.pa.msu.edu\/pub\/papers\/VIRC03\/VIRC03.pdf."},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1023\/A:1009958918582"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142995281693"},{"key":"R9","doi-asserted-by":"crossref","unstructured":"H. Broer,\n                      Formal normal form theorems for vector fields and some consequences for bifurcations in the volume preserving case\n                      , in Dynamical Systems and Turbulence, Warwick 1980 (Coventry, 1979\/1980), Lecture Notes in Math. 898, Springer, Berlin, 1981, pp. 54\u201374.","DOI":"10.1007\/BFb0091907"},{"key":"R10","doi-asserted-by":"crossref","unstructured":"H. M. B\u00fccker and G. F. Corliss,\n                      A bibliography of automatic differentiation\n                      , in Automatic Differentiation: Applications, Theory, and Implementations, Lect. Notes Comput. Sci. Eng. 50, Springer, Berlin, 2006, pp. 321\u2013322.","DOI":"10.1007\/3-540-28438-9_28"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1512\/iumj.2003.52.2245"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2004.12.003"},{"key":"R13","doi-asserted-by":"crossref","unstructured":"W. Cheney,\n                      Analysis for Applied Mathematics\n                      , Grad. Texts in Math. 208, Springer, Berlin, 2001.","DOI":"10.1007\/978-1-4757-3559-8"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1007\/s10884-005-3146-x"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/5\/6\/006"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127408021439"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1016\/0377-0427(89)90153-2"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1137\/080728160"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1137\/05062408X"},{"key":"R20","unstructured":"V. Guillemin and A. Pollack,\n                      Differential Topology\n                      , Prentice-Hall, Englewood Cliffs, NJ, 1974."},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1063\/1.525415"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1137\/040614670"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/17\/2\/019"},{"key":"R24","unstructured":"D. E. Knuth,\n                      The Art of Computer Programming. Vol.\n                      2.\n                      Seminumerical Algorithms\n                      , 3rd ed., Addison-Wesley Longman, Boston, 1997."},{"key":"R25","doi-asserted-by":"crossref","unstructured":"S. G. Krantz,\n                      Implicit Function Theorem: History, Theory, and Applications\n                      , Birkh\u00e4user, Basel, 2002.","DOI":"10.1007\/978-1-4612-0059-8"},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127498000310"},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127405012533"},{"key":"R28","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/21\/8\/001"},{"key":"R29","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/11\/3\/009"},{"key":"R30","doi-asserted-by":"publisher","DOI":"10.1016\/S0375-9601(00)00264-4"},{"key":"R31","doi-asserted-by":"publisher","DOI":"10.1063\/1.166480"},{"key":"R32","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/22\/8\/001"},{"key":"R33","first-page":"239","volume":"6","author":"Makino K.","year":"2003","journal-title":"Int. J. Pure Appl. Math."},{"key":"R34","first-page":"3","volume":"12","author":"Mel$'$nikov V. K.","year":"1963","journal-title":"Tr. Mosk. Mat. Ob\u0161.","ISSN":"https:\/\/id.crossref.org\/issn\/0134-8663","issn-type":"print"},{"key":"R35","unstructured":"J. Milnor,\n                      Morse Theory\n                      , Ann. of Math. Stud. 51, Princeton University Press, Princeton, NJ, 1963."},{"key":"R36","unstructured":"J. D. Mireles James,\n                      VPHLab: MATLAB Implementation of Newton Scheme for Taylor Expansion of Heteroclinic Arcs for the Volume Preserving H\u00e9non Map\n                      , http:\/\/www.math.utexas.edu\/users\/jjames\/VPHLab http:\/\/www.math.utexas.edu\/users\/jjames\/VPHLab."},{"key":"R37","doi-asserted-by":"crossref","unstructured":"S. Newhouse, M. Berz, J. Grote, and K. Makino,\n                      On the estimation of topological entropy on surfaces\n                      , in Geometric and Probabilistic Structures in Dynamics, Contemp. Math. 469, AMS, Providence, RI, 2008, pp. 243\u2013270.","DOI":"10.1090\/conm\/469\/09170"},{"key":"R38","doi-asserted-by":"crossref","unstructured":"J. M. Ortega and W. C. Rheinboldt,\n                      Iterative Solution of Nonlinear Equations in Several Variables\n                      , Classics Appl. Math. 30, SIAM, Philadelphia, 2000.","DOI":"10.1137\/1.9780898719468"},{"key":"R39","doi-asserted-by":"crossref","unstructured":"J. Palis, Jr., and W. de Melo,\n                      Geometric Theory of Dynamical Systems. An Introduction\n                      , translated from the Portuguese by A. K. Manning, Springer-Verlag, New York, 1982.","DOI":"10.1007\/978-1-4612-5703-5"},{"key":"R40","unstructured":"H. Poincar\u00e9,\n                      Les m\u00e9thodes nouvelles de la m\u00e9canique c\u00e9leste. Tome\n                      I\u2013III, Les Grands Classiques Gauthier-Villars, Librairie Scientifique et Technique Albert Blanchard, Paris, 1987."},{"key":"R41","doi-asserted-by":"publisher","DOI":"10.1007\/s10884-006-9011-8"},{"key":"R42","doi-asserted-by":"publisher","DOI":"10.1016\/S0898-1221(96)00204-0"},{"key":"R43","unstructured":"W. C. Rheinboldt,\n                      Numerical Analysis of Parametrized Nonlinear Equations\n                      , University of Arkansas Lecture Notes in the Mathematical Sciences 7, John Wiley & Sons, New York, 1986."},{"key":"R44","doi-asserted-by":"publisher","DOI":"10.1007\/BF01395883"},{"key":"R45","doi-asserted-by":"publisher","DOI":"10.1112\/blms\/22.2.113"},{"key":"R46","doi-asserted-by":"crossref","unstructured":"M. Schwarz,\n                      Morse Homology\n                      , Progr. Math. 111, Birkh\u00e4user, Basel, 1993.","DOI":"10.1007\/978-3-0348-8577-5"},{"key":"R47","unstructured":"G. Strang,\n                      Linear Algebra and Its Applications\n                      , 3rd ed., Thomson (Brooks\/Cole), New York, 1988."},{"key":"R48","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/18\/6\/010"},{"key":"R49","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/17\/6\/014"}],"container-title":["SIAM Journal on Applied Dynamical Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/090776329","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T13:20:17Z","timestamp":1787232017000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/090776329"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,1]]},"references-count":49,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2010,1]]}},"alternative-id":["10.1137\/090776329"],"URL":"https:\/\/doi.org\/10.1137\/090776329","relation":{},"ISSN":["1536-0040"],"issn-type":[{"value":"1536-0040","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,1]]}}}