scholarly journals Regularity Properties of Critical Invariant Circles of Twist Maps, and Their Universality

2008 ◽  
Vol 7 (3) ◽  
pp. 962-987 ◽  
Author(s):  
Arturo Olvera ◽  
Nikola P. Petrov
Keyword(s):  
Regularity Properties ◽  
Invariant Circles ◽  
Twist Maps

A method for proving that monotone twist maps have no invariant circles

1991 ◽  
Vol 11 (1) ◽  
pp. 79-84 ◽  
Author(s):  
Irwin Jungreis
Keyword(s):  
Existence Theorem ◽  
Computer Assisted ◽  
Twist Map ◽  
Standard Map ◽  
Rotation Numbers ◽  
Invariant Circles ◽  
Twist Maps ◽  
Parameter Values

AbstractWe present an existence theorem for certain kinds of orbits of a monotone twist map and use it to obtain a criterion for proving that there are no invariant circles with a certain range of rotation numbers. We have used this criterion to prove (computer assisted) that the standard map has no invariant circles for several parameter values includingk= 0.9718.

Analytic destruction of invariant circles

1994 ◽  
Vol 14 (2) ◽  
pp. 267-298 ◽  
Author(s):  
Giovanni Forni
Keyword(s):  
Small Perturbations ◽  
Invariant Circles ◽  
Twist Maps ◽  
Area Preserving

AbstractWe give destruction results under analytic small perturbations for invariant circles of exact area-preserving monotone twist maps, applying methods developed by M. Herman and J. Mather.

scholarly journals Invariant circles and the order structure of periodic orbits in monotone twist maps

Topology ◽  
1987 ◽  
Vol 26 (1) ◽  
pp. 21-35 ◽  
Author(s):  
Philip L. Boyland ◽  
Glen Richard Hall
Keyword(s):  
Periodic Orbits ◽  
Order Structure ◽  
Invariant Circles ◽  
Twist Maps

scholarly journals About periodic and quasi-periodic orbits of a new type for twist maps of the torus

2002 ◽  
Vol 74 (1) ◽  
pp. 25-31
Author(s):  
SALVADOR ADDAS-ZANATA
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Classical Result ◽  
Important Class ◽  
Twist Map ◽  
Positive Topological Entropy ◽  
Invariant Circles ◽  
Twist Maps ◽  
New Type ◽  
Rotational Invariant

We prove that for a large and important class of C¹ twist maps of the torus periodic and quasi-periodic orbits of a new type exist, provided that there are no rotational invariant circles (R.I.C's). These orbits have a non-zero "vertical rotation number'' (V.R.N.), in contrast to what happens to Birkhoff periodic orbits and Aubry-Mather sets. The V.R.N. is rational for a periodic orbit and irrational for a quasi-periodic. We also prove that the existence of an orbit with a V.R.N = a > 0, implies the existence of orbits with V.R.N = b, for all 0 < b < a. And as a consequence of the previous results we get that a twist map of the torus with no R.I.C's has positive topological entropy, which is a very classical result. In the end of the paper we present some applications and examples, like the Standard map, such that our results apply.

scholarly journals Quantitative destruction of invariant circles

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Lin Wang
Keyword(s):  
Trigonometric Polynomial ◽  
Integrable System ◽  
Arithmetic Property ◽  
Invariant Circles ◽  
Twist Maps ◽  
Area Preserving

<p style='text-indent:20px;'>For area-preserving twist maps on the annulus, we consider the problem on quantitative destruction of invariant circles with a given frequency <inline-formula><tex-math id="M1">\begin{document}$ \omega $\end{document}</tex-math></inline-formula> of an integrable system by a trigonometric polynomial of degree <inline-formula><tex-math id="M2">\begin{document}$ N $\end{document}</tex-math></inline-formula> perturbation <inline-formula><tex-math id="M3">\begin{document}$ R_N $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M4">\begin{document}$ \|R_N\|_{C^r}&lt;\epsilon $\end{document}</tex-math></inline-formula>. We obtain a relation among <inline-formula><tex-math id="M5">\begin{document}$ N $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ r $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> and the arithmetic property of <inline-formula><tex-math id="M8">\begin{document}$ \omega $\end{document}</tex-math></inline-formula>, for which the area-preserving map admit no invariant circles with <inline-formula><tex-math id="M9">\begin{document}$ \omega $\end{document}</tex-math></inline-formula>.</p>

Degenerate Diffusion Operators Arising in Population Biology (AM-185)

2017 ◽  
Author(s):  
Charles L. Epstein ◽  
Rafe Mazzeo
Keyword(s):  
Adjoint Operator ◽  
Integral Kernel ◽  
Mathematical Finance ◽  
Elliptic Operators ◽  
Complete Analysis ◽  
Time Behavior ◽  
Degenerate Elliptic Operators ◽  
Regularity Properties ◽  
Degenerate Elliptic ◽  
Mathematical Foundations

This book provides the mathematical foundations for the analysis of a class of degenerate elliptic operators defined on manifolds with corners, which arise in a variety of applications such as population genetics, mathematical finance, and economics. The results discussed in this book prove the uniqueness of the solution to the martingale problem and therefore the existence of the associated Markov process. The book uses an “integral kernel method” to develop mathematical foundations for the study of such degenerate elliptic operators and the stochastic processes they define. The precise nature of the degeneracies of the principal symbol for these operators leads to solutions of the parabolic and elliptic problems that display novel regularity properties. Dually, the adjoint operator allows for rather dramatic singularities, such as measures supported on high codimensional strata of the boundary. The book establishes the uniqueness, existence, and sharp regularity properties for solutions to the homogeneous and inhomogeneous heat equations, as well as a complete analysis of the resolvent operator acting on Hölder spaces. It shows that the semigroups defined by these operators have holomorphic extensions to the right half plane. The book also demonstrates precise asymptotic results for the long-time behavior of solutions to both the forward and backward Kolmogorov equations.

Wave propagation dynamics in a fractional Zener model with stochastic excitation

2020 ◽  
Vol 23 (6) ◽  
pp. 1570-1604
Author(s):  
Teodor Atanacković ◽  
Stevan Pilipović ◽  
Dora Seleši
Keyword(s):  
Wave Propagation ◽  
Constitutive Equation ◽  
Equations Of Motion ◽  
Weak Form ◽  
Stochastic Excitation ◽  
Expansion Theorem ◽  
Zener Model ◽  
Dissipation Inequality ◽  
Regularity Properties ◽  
Fractional Zener Model

Abstract Equations of motion for a Zener model describing a viscoelastic rod are investigated and conditions ensuring the existence, uniqueness and regularity properties of solutions are obtained. Restrictions on the coefficients in the constitutive equation are determined by a weak form of the dissipation inequality. Various stochastic processes related to the Karhunen-Loéve expansion theorem are presented as a model for random perturbances. Results show that displacement disturbances propagate with an infinite speed. Some corrections of already published results for a non-stochastic model are also provided.

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