東華大學圖書館 |

語系: 繁體中文

說明(常見問題)

回圖書館首頁

手機版館藏查詢

登入

回首頁

切換: 標籤 | MARC模式 | ISBD

Advances in Computer-Assisted Proofs...

Henot, Olivier,

FindBook

Google Book

Amazon

博客來

Advances in Computer-Assisted Proofs: Invariant Manifolds and Connecting Orbits /

紀錄類型:	書目-電子資源 : Monograph/item
正題名/作者:	Advances in Computer-Assisted Proofs: Invariant Manifolds and Connecting Orbits // Olivier Henot.
作者:	Henot, Olivier,
面頁冊數:	1 electronic resource (196 pages)
附註:	Source: Dissertations Abstracts International, Volume: 85-11, Section: B.
Contained By:	Dissertations Abstracts International85-11B.
標題:	Physiology. -
電子資源:	https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=31078795
ISBN:	9798382614779

Advances in Computer-Assisted Proofs: Invariant Manifolds and Connecting Orbits /
Henot, Olivier,

Advances in Computer-Assisted Proofs: Invariant Manifolds and Connecting Orbits /Olivier Henot. - 1 electronic resource (196 pages)

Source: Dissertations Abstracts International, Volume: 85-11, Section: B.

The thesis compiles four articles on computer-assisted proofs in dynamical systems. Individually, each one answers their own set of questions within different topics. Together, they pave the way for computer-assisted proofs on the existence of transverse homoclinic orbits for periodic solutions of delay differential equations. These works further illustrate how rigorous numerics constitute a powerful, flexible and accessible tool for a wide range of problems.First, we briefly introduce the open-source software Radii Polynomial: an innovative library to conduct rigorous computations.Practical considerations regarding the nonlinearities of differential equations lead to the formulation of a general framework to best exploit the Banach algebra structure induced by spectral methods. In the first article, we show how to construct an auxiliary system of polynomial differential equations from an original system of differential equations comprised of elementary functions (i.e. exponential, logarithmic, algebraic functions and compositions thereof).For delay differential equations, the unstable manifold of an equilibrium or periodic orbit is finite dimensional. In the second article, we show how to rigorously compute a parameterization of the local unstable manifold for equilibria of delay differential equations.On the other hand, the stable manifold is infinite dimensional and requires a different strategy to rigorously enclose it, such as a Lyapunov-Perron method. In the third article, we give a general method to prove the existence of localized radial solutions of semilinear elliptic systems on Rd. The particularity of the approach is the incorporation of the Lyapunov-Perron method to control the local graph of a centre-stable manifold.In the fourth and last article, we present a numerical method to compute transverse homoclinic orbits for periodic solutions of delay differential equations. Notably, we compute numerically such a connection for the Mackey-Glass equation.

English

ISBN: 9798382614779Subjects--Topical Terms:

518431
Physiology.

Advances in Computer-Assisted Proofs: Invariant Manifolds and Connecting Orbits /
LDR:05539nmm a22003853i 4500 001 2434069
005 20250905091337.5
006 m o d
007 cr|nu||||||||
008 261011s2023 miu||||||m |||||||eng d
020 $a 9798382614779
035 $a (MiAaPQD)AAI31078795
035 $a (MiAaPQD)McGill_bg257m072
035 $a AAI31078795
040 $a MiAaPQD $b eng $c MiAaPQD $e rda
100 1 $a Henot, Olivier, $e author. $3 3823926
245 1 0 $a Advances in Computer-Assisted Proofs: Invariant Manifolds and Connecting Orbits / $c Olivier Henot.
264 1 $a Ann Arbor : $b ProQuest Dissertations & Theses, $c 2023
300 $a 1 electronic resource (196 pages)
336 $a text $b txt $2 rdacontent
337 $a computer $b c $2 rdamedia
338 $a online resource $b cr $2 rdacarrier
500 $a Source: Dissertations Abstracts International, Volume: 85-11, Section: B.
500 $a Advisors: Lessard, Jean-Philippe.
502 $b Ph.D. $c McGill University (Canada) $d 2023.
520 $a The thesis compiles four articles on computer-assisted proofs in dynamical systems. Individually, each one answers their own set of questions within different topics. Together, they pave the way for computer-assisted proofs on the existence of transverse homoclinic orbits for periodic solutions of delay differential equations. These works further illustrate how rigorous numerics constitute a powerful, flexible and accessible tool for a wide range of problems.First, we briefly introduce the open-source software Radii Polynomial: an innovative library to conduct rigorous computations.Practical considerations regarding the nonlinearities of differential equations lead to the formulation of a general framework to best exploit the Banach algebra structure induced by spectral methods. In the first article, we show how to construct an auxiliary system of polynomial differential equations from an original system of differential equations comprised of elementary functions (i.e. exponential, logarithmic, algebraic functions and compositions thereof).For delay differential equations, the unstable manifold of an equilibrium or periodic orbit is finite dimensional. In the second article, we show how to rigorously compute a parameterization of the local unstable manifold for equilibria of delay differential equations.On the other hand, the stable manifold is infinite dimensional and requires a different strategy to rigorously enclose it, such as a Lyapunov-Perron method. In the third article, we give a general method to prove the existence of localized radial solutions of semilinear elliptic systems on Rd. The particularity of the approach is the incorporation of the Lyapunov-Perron method to control the local graph of a centre-stable manifold.In the fourth and last article, we present a numerical method to compute transverse homoclinic orbits for periodic solutions of delay differential equations. Notably, we compute numerically such a connection for the Mackey-Glass equation.
520 $a La these comprend quatre articles sur les preuves assistees par ordinateur dans les systemes dynamiques. Individuellement, chacun repond a une problematique propre dans differents sujets. Ensemble, ils frayent la voie vers des preuves assistees par ordinateur sur l'existence d'orbites homoclines transverses pour les solutions periodiques des equations differentielles retardees. En outre, ces travaux illustrent comment les calculs numeriques rigoureux constituent un outil puissant, flexible et accessible pour un bon nombre de problemes.Premierement, nous introduisons brievement le logicielopen-sourceRadiiPolynomial: une bibliotheque innovante pour realiser des calculs rigoureux.Des considerations pratiques sur les nonlinearites des equations differentielles amenent a la formulation d'un cadre general afin d'exploiter au mieux l'algebre de Banach induit par des methodes spectrales. Dans le premier article, nous montrons comment construire un systeme auxiliaire d'equations differentielles polynomiales a partir d'un systeme d'equations differentielles constituees de fonctions elementaires (i.e. fonctions exponentielles, logarithmiques, algebriques et compositions de celles-ci).Pour les equations differentielles retardees, la variete instable d'un point d'equilibre ou d'une solution periodique est de dimension finie. Dans le second article, nous montrons comment calculer rigoureusement une parametrisation de la variete instable locale pour les points d'equilibres des equations differentielles retardees.D'autre part, la variete stable est de dimension infinie et necessite une strategie differente pour l'envelopper rigoureusement, telle que la methode de Lyapunov-Perron.Dans le troisieme article, nous donnons une methode generale pour prouver l'existence de solutions radiales locales de systemes elliptiques semilineaires sur Rd. La particularite de l'approche est l'incorporation de la methode de Lyapunov-Perron pour controler le graphe local d'une variete centre-stable.Dans le quatrieme et dernier article, nous presentons une methode numerique pour calculer les orbites homoclines transverses pour les solutions periodiques des equations differentielles retardees. Notamment, nous calculons une telle connexion pour l'equation de Mackey-Glass.
546 $a English
590 $a School code: 0781
650 4 $a Physiology. $3 518431
650 4 $a Eigenvalues. $3 631789
650 4 $a Partial differential equations. $3 2180177
650 4 $a Banach spaces. $3 579190
650 4 $a 19th century. $3 1972810
650 4 $a Dynamical systems. $3 3219613
650 4 $a Orbits. $3 534192
650 4 $a Mathematics. $3 515831
690 $a 0719
690 $a 0405
710 2 $a McGill University (Canada). $e degree granting institution. $3 3759190
720 1 $a Lessard, Jean-Philippe $e degree supervisor.
773 0 $t Dissertations Abstracts International $g 85-11B.
790 $a 0781
791 $a Ph.D.
792 $a 2023
856 4 0 $u https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=31078795