A Shadowing Theorem for ordinary differential equations

Coomes, Brian A.; Koçak, Hüseyin; Palmer, Kenneth J.

doi:10.1007/bf00952258

Cited by 21 publications

(16 citation statements)

References 11 publications

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“…Furthermore, the hypotheses of the theorem are presented with an eye toward computer verification. The theorem is similar to the finite time shadowing theorems of Coomes, Koçak, and Palmer [7], [10], but takes into account information provided by g and allows one to say more about the shadowing orbit, namely that it lies in Λ.…”

Section: Dg(y)f (Y) =mentioning

confidence: 97%

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Shadowing orbits of ordinary differential equations on invariant submanifolds

Coomes

1997

Trans. Amer. Math. Soc.

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Abstract. A finite time shadowing theorem for autonomous ordinary differential equations is presented. Under consideration is the case were there exists a twice continuously differentiable function g mapping phase space into R m with the property that for a particular regular value c of g the submanifold g −1 (c) is invariant under the flow. The main theorem gives a condition which implies that an approximate solution lying close to g −1 (c) is uniformly close to a true solution lying in g −1 (c). Applications of this theorem to computer generated approximate orbits are discussed.

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Section: Dg(y)f (Y) =mentioning

confidence: 97%

“…. , N − 1, x − y k ≤ ε 0 and 0 ≤ t ≤ h k + ε 0 it follows from (3.9), (3.10), (3.11), Gronwall's Lemma, and the variation of constants formula (see [7] for details) that…”

Section: Lemma 53 the Local Boundmentioning

confidence: 99%

Shadowing orbits of ordinary differential equations on invariant submanifolds

Coomes

1997

Trans. Amer. Math. Soc.

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“…In particular, at each step we could use ϕ h i as defined in equation (1.4) with h = h i being the length of the ODE integration timestep taken at step i. The resulting method for shadowing numerical ODE integrations has been dubbed the Map Method by Coomes, Koçak, and Palmer (1994b, 1995a, 1995b. As described in section 2.2.5, however, ODE integrations suffer from errors in time.…”

Section: Rescaling Time 361 Informal Descriptionmentioning

confidence: 99%

“…For large |t − t 0 | this can be a significant difference, so a shadowing method which does not take the rescaling of time into account is likely to grossly underestimate the length of the shadow. Coomes, Koçak, and Palmer (1994b, 1995a, 1995b dramatically demonstrate this when they show that a rescaling of time allows the Lorenz equations to be shadowed for almost 10 5 time units, while the map method, which does not rescale time, finds shadows lasting only 10 time units-an astounding increase in shadow length of a factor of 10 4 ! Finally, note that the non-shadowable example given in the tutorial (y = 0, page 11) is shadowable if time is rescaled.…”

Section: Introductionmentioning

confidence: 98%

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Rigorous Shadowing of Numerical Solutions of Ordinary Differential Equations by Containment

Hayes¹,

Jackson²

2003

SIAM J. Numer. Anal.

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Rigorous Shadowing of Numerical Solutions of Ordinary Differential Equations by Containment Wayne Brian Hayes Doctor of Philosophy Graduate Department of Computer Science University of Toronto 2001An exact trajectory of a dynamical system lying close to a numerical trajectory is called ashadow. We present a general-purpose method for proving the existence of finite-time shadows of numerical ODE integrations of arbitrary dimension in which some measure of hyperbolicity is present and there is either 0 or 1 expanding modes, or 0 or 1 contracting modes. Much of the rigor is provided automatically by interval arithmetic and validated ODE integration software that is freely available. The method is a generalization of a previously published containment process that was applicable only to two-dimensional maps. We extend it to handle maps of arbitrary dimension with the above restrictions, and finally to ODEs. The method involves building n-cubes around each point of the discrete numerical trajectory through which the shadow is guaranteed to pass at appropriate times. The proof consists of two steps: first, the rigorous computational verification of an inductive containment property; and second, a simple geometric argument showing that this property implies the existence of a shadow. The computational step is almost entirely automated and easily adaptable to any ODE problem.The method allows for the rescaling of time, which is a necessary ingredient for successfully shadowing ODEs. Finally, the method is local, in the sense that it builds the shadow inductively, requiring information only from the most recent integration step, rather than more global information typical of several other methods. The method produces shadows of comparable length and distance to all currently published results.iii AcknowledgementsOur strength as a species comes from our ability to communicate with each other. Very few feats, scholarly or otherwise, can be accomplished in a vacuum. Without the ideas, help, and challenges from professional collegues, and without the warmth, care, and compassion of friends and family, our work would be non-existent or meaningless.On the professional side, I thank my committee. Their doors were always open, both for professional consultation, and for occasional personal discussion. The guidance of my supervisor, Ken Jackson, both around obstacles and out of dead ends, is much appreciated. On many occasions Wayne Enright provided much-needed guidance when my understanding of accuracy and stability issues went awry, and his keen eye for precise use of terms tightened my presentation in several places in the thesis. Tom Fairgrieve's experience with nonlinear chaotic systems helped me to view the larger picture, and his probing questions always made me think carefully about what was important. Ted Shepherd's practical experience in numerical techniques for physics problems and deep understanding of classical mechanics provided me with an even wider view than I would have ever thought possible. I cannot t...

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Transversal connecting orbits from shadowing

2007

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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 n 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.

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A Shadowing Theorem for ordinary differential equations

Cited by 21 publications

References 11 publications

Shadowing orbits of ordinary differential equations on invariant submanifolds

Shadowing orbits of ordinary differential equations on invariant submanifolds

Rigorous Shadowing of Numerical Solutions of Ordinary Differential Equations by Containment

Transversal connecting orbits from shadowing

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