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Tuesday, July 21, 2020 | History

4 edition of computer-assisted proof of universality for area-preserving maps found in the catalog.

# computer-assisted proof of universality for area-preserving maps

## by Jean Pierre Eckmann

Written in English

Subjects:
• Hamiltonian systems -- Data processing.,
• Mappings (Mathematics) -- Data processing.,
• Error analysis (Mathematics)

• Edition Notes

Classifications The Physical Object Statement J.-P. Eckmann, H. Koch, and P. Wittwer. Series Memoirs of the American Mathematical Society,, no. 289 (Jan. 1984), Memoirs of the American Mathematical Society ;, no. 289. Contributions Koch, H., Wittwer, P. LC Classifications QA3 .A57 no. 289, QA614.83 .A57 no. 289 Pagination vi, 121 p. ; Number of Pages 121 Open Library OL3179850M ISBN 10 0821822896 LC Control Number 83022456

The description of the spectral properties of the linearised maps at the fixed points is justified by ref. [18] in the dissipative case and [10] for the area-preserving fixed point - in ref. [10] the spectrum of DT is studied on the space of (infinitesimal) reversible area-preserving maps, which has codimension one in the space of constant. A 26 ) and Eckmann et al ( Mem. Am. Math. Soc. 47 ) in a computer-assisted proof of existence of a 'universal' area-preserving map F-*-a map with orbits of all binary periods 2(k), k is an element of N. In this paper, we consider maps in some neighbourhood of F-* and study their dynamics.

In the late 's, P. Coullet and C. Tresser and M. Feigenbaum. independently found striking, unexpected features of the transition from simple to chaotic dynamics in one-dimensional dynamical systems (cf. also Routes to chaos).By the example of the family of quadratic mappings $f _ \mu (x)= 1- \mu x ^ {2}$ acting (for $0 \leq \mu \leq 2$) on the interval $x \in [- 1, 1]$, the period. It is known that the famous Feigenbaum-Coullet-Tresser period doubling universality has a counterpart for area-preserving maps of R2. A renormalization approach has been used in (Eckmann et al ) and (Eckmann et al ) in a computer-assisted proof of existence of a “universal ” area-preserving map F ∗ — a map with orbits of all.

The other extreme case is when the maps preserve area, for example, when the average Jacobian is one. Indeed, the main result presented here is that the period-doubling Cantor sets of area-preserving maps in the universality class of the Eckmann–Koch–Wittwer renormalization fixed . Dynamics of the Universal Area-Preserving Map Associated with Period Doubling: Stable Sets It is known that the famous Feigenbaum-Coullet-Tresser period doubling universality has a counterpart for area-preserving maps of ${\fR}^2$. A renormalization approach has been used in \cite{EKW1} and \cite{EKW2} in a computer-assisted proof of.

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### Computer-assisted proof of universality for area-preserving maps by Jean Pierre Eckmann Download PDF EPUB FB2

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Genre/Form: Electronic books: Additional Physical Format: Print version: Eckmann, Jean Pierre. Computer-assisted proof of universality for area-preserving maps /. A Computer-Assisted Proof of Universality for Area-Preserving Maps Share this page Jean-Pierre Eckmann; Hans Koch; Peter Wittwer.

Table of Contents. Search. Go > Advanced search. Table of Contents A Computer-Assisted Proof of Universality for Area-Preserving Maps Base Product Code Keyword List: memo; MEMO; memo/47 Book Series Name: Memoirs. Eckmann, J.P., Koch, H., Wittwer, P.: () A computer-assisted proof of universality for area preserving maps Memoirs AMS 47 () Google Scholar Eckmann, J.P., Wittwer, P.: Computer Methods and Bored Summability Applied to Feigenbaum's Equation (Springer, Berlin Heidelberg) Springer Lecture Notes in Physics Vol () Google ScholarCited by: 2.

Abstract It has been observed that the famous Feigenbaum-Coullet-Tresser period doubling universality has a counterpart for area-preserving maps of ${\fR}^2$. A renormalization approach. Area-preserving maps have been observed to undergo a universal period-doubling cascade, analogous to the famous Feigenbaum-Coullet-Tresser period doubling cascade in one-dimensional dynamics.

A renormalization approach has been used by Eckmann, Koch and Wittwer in a computer-assisted proof of existence of a conservative renormalization fixed point. It has been observed that the famous Feigenbaum-Coullet-Tresser period doubling universality has a counterpart for area-preserving maps of ${\fR}^2$.

A renormalization approach has been used in a "hard" computer-assisted proof of existence of an area-preserving map with orbits of all binary periods in Eckmann et al (). A renormalization approach has been used in a computer-assisted proof of existence of an area-preserving map with orbits of all binary periods in [EKW1] and [EKW2].

As it. The universal area-preserving map is a map with hyperbolic periodic points of all binary periods 2n. Its existence, as many other universality phenomena, is best described in the in a computer-assisted.

No elliptic islands for the universal area-preserving map proof of existence of a reversible renormalization ﬁxed point F.

doubling universality has a counterpart for area-preserving maps of R2. A renormalization approach has been used in (Eckmann et al ) and (Eckmann et al ) in a computer-assisted proof of existence of a \universal" area-preserving map F | a map with.

A renormalization approach has been used in a computer-assisted proof of existence of an area-preserving map with orbits of all binary periods by J.-P. Eckmann, H. Koch and P. Wittwer ( and. We gain tight rigorous bounds on the renormalisation fixed point for period doubling in families of unimodal maps with degree 4 critical point.

We prove that the fixed point is hyperbolic and use a contraction mapping argument to bound essential eigenfunctions and eigenvalues for the linearisation and for the scaling of additive noise. We find analytic extensions of the fixed point function to.

J.-P. Eckmann, H. Koch, and P. Wittwer, A computer-assisted proof of universality for area-preserving maps, Mem. AMS (). It has been observed that the famous Feigenbaum-Coullet-Tresser period doubling universality has a counterpart for area-preserving maps of R 2.

A renormalization approach has been used in a computer-assisted proof of existence of an area-preserving map with orbits of all binary periods in [EKW1] and [EKW2]. A renormalization approach has been used in a computer-assisted proof of existence of an area-preserving map with orbits of all binary periods by J.-P.

Eckmann, H. Koch and P. Wittwer. ISBNISBNprijs: EURO (incl. BTW & verzendkosten), A Computer-Assisted Proof of Universality for Area-Preserving Maps van Jean-Pierre Eckmann. A renormalization approach has been used in (Eckmann et al ) and (Eckmann et al ) in a computer-assisted proof of existence of a “universal ” area-preserving map F ∗ — a map with orbits of all binary periods 2k, k ∈ N.

In this paper, we consider maps in. J.-P. Eckmann, H. Koch and P. Wittwer, A computer-assisted proof of universality for area-preserving maps, Mem. Amer. Math. Soc., 47 (), 1. doi: /memo/ Phys. Rev. A 26(1) (), –; A computer-assisted proof of universality for area-preserving maps.

Mem. Amer. Math. Soc. 47 (), 1–]. As is the case with all non-trivial universality problems in non-dissipative systems in dimensions more than one, no analytic proof of this period-doubling universality exists to date.

A computer-assisted proof of universality for area-preserving maps, Mem. Amer. Math. Soc. 47(), 1– Galias Z., Computer assisted proof of chaos in the Muthuswamy-Chua memristor circuit, Nonlinear Theory Appl.

IEICE 5(), –. It is known that the famous Feigenbaum–Coullet–Tresser period doubling universality has a counterpart for area-preserving maps of.

A renormalization approach has been used in.doubling universality has a counterpart for area-preserving maps of R2. A renormalization approach has been used in (Eckmann et al ) and (Eckmann et al ) in a computer-assisted proof of existence of a “universal” area-preserving map F∗ — a map with orbits of all binary periods 2k,k ∈ N.

In. More recently Lanford [11] has proved the Feigenbaum conjectures, while universality of the doubling equation has been extended to area-preserving maps of the plane [8]. However, for dimension greater than one the situation appears far more complicated: for example, some Hon maps can bifurcate from stability directly to a homoclinic structure.