### Abstract

A rigorous numerical method for establishing the existence of a transversal connecting orbit from one hyperbolic periodic orbit to another of a differential equation in ℝ is presented. As the first component of this method, a general shadowing theorem that guarantees the existence of such a connecting orbit near a suitable pseudo connection orbit given the invertibility of a certain operator is proved. The second component consists of a refinement procedure for numerically computing a pseudo connecting orbit between two pseudo periodic orbits with sufficiently small local errors so as to satisfy the hypothesis of the theorem. The third component consists of a numerical procedure to verify the invertibility of the operator and obtain a rigorous upper bound for the norm of its inverse. Using this method, existence of chaos is demonstrated on examples with transversal homoclinic orbits, and with cycles of transversal heteroclinic orbits.

Original language | English (US) |
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Pages (from-to) | 427-469 |

Number of pages | 43 |

Journal | Numerische Mathematik |

Volume | 106 |

Issue number | 3 |

DOIs | |

State | Published - May 2007 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics
- Computational Mathematics

### Cite this

*Numerische Mathematik*,

*106*(3), 427-469. https://doi.org/10.1007/s00211-007-0065-2

**Transversal connecting orbits from shadowing.** / Coomes, Brian A; Kocak, Huseyin; Palmer, Kenneth J.

Research output: Contribution to journal › Article

*Numerische Mathematik*, vol. 106, no. 3, pp. 427-469. https://doi.org/10.1007/s00211-007-0065-2

}

TY - JOUR

T1 - Transversal connecting orbits from shadowing

AU - Coomes, Brian A

AU - Kocak, Huseyin

AU - Palmer, Kenneth J.

PY - 2007/5

Y1 - 2007/5

N2 - A rigorous numerical method for establishing the existence of a transversal connecting orbit from one hyperbolic periodic orbit to another of a differential equation in ℝ is presented. As the first component of this method, a general shadowing theorem that guarantees the existence of such a connecting orbit near a suitable pseudo connection orbit given the invertibility of a certain operator is proved. The second component consists of a refinement procedure for numerically computing a pseudo connecting orbit between two pseudo periodic orbits with sufficiently small local errors so as to satisfy the hypothesis of the theorem. The third component consists of a numerical procedure to verify the invertibility of the operator and obtain a rigorous upper bound for the norm of its inverse. Using this method, existence of chaos is demonstrated on examples with transversal homoclinic orbits, and with cycles of transversal heteroclinic orbits.

AB - A rigorous numerical method for establishing the existence of a transversal connecting orbit from one hyperbolic periodic orbit to another of a differential equation in ℝ is presented. As the first component of this method, a general shadowing theorem that guarantees the existence of such a connecting orbit near a suitable pseudo connection orbit given the invertibility of a certain operator is proved. The second component consists of a refinement procedure for numerically computing a pseudo connecting orbit between two pseudo periodic orbits with sufficiently small local errors so as to satisfy the hypothesis of the theorem. The third component consists of a numerical procedure to verify the invertibility of the operator and obtain a rigorous upper bound for the norm of its inverse. Using this method, existence of chaos is demonstrated on examples with transversal homoclinic orbits, and with cycles of transversal heteroclinic orbits.

UR - http://www.scopus.com/inward/record.url?scp=34248210709&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34248210709&partnerID=8YFLogxK

U2 - 10.1007/s00211-007-0065-2

DO - 10.1007/s00211-007-0065-2

M3 - Article

AN - SCOPUS:34248210709

VL - 106

SP - 427

EP - 469

JO - Numerische Mathematik

JF - Numerische Mathematik

SN - 0029-599X

IS - 3

ER -