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Evelyn Sander

Other affiliations: University of Minnesota
Bio: Evelyn Sander is an academic researcher from George Mason University. The author has contributed to research in topics: Quasiperiodic function & Dynamical systems theory. The author has an hindex of 16, co-authored 49 publications receiving 637 citations. Previous affiliations of Evelyn Sander include University of Minnesota.


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TL;DR: In this paper, a general theory of cascades for generic parametrized maps is presented, and it is shown that there is a close connection between the transition through innitely many cascades and the creation of a horseshoe.
Abstract: The appearance of numerous period-doubling cascades is among the most prominent features of parametrized maps, that is, smooth one-parameter families of maps F : R M ! M, where M is a smooth locally compact manifold without boundary, typically R N . Each cascade has innitely many period-doubling bifurcations, and it is typical to observe { such as in all the examples we investigate here { that whenever there are any cascades, there are innitely many cascades. We develop a general theory of cascades for generic F . We illustrate this theory with several examples. We show that there is a close connection between the transition through innitely many cascades and the creation of a horseshoe.

58 citations

Journal ArticleDOI
TL;DR: A mechanism which explains why most solutions for the Cahn--Hilliard equation starting near a homogeneous equilibrium within the spinodal interval exhibit phase separation with a characteristic wavelength when exiting a ball of radius R is proved.
Abstract: This paper gives theoretical results on spinodal decomposition for the Cahn--Hillard equation. We prove a mechanism which explains why most solutions for the Cahn--Hilliard equation starting near a homogeneous equilibrium within the spinodal interval exhibit phase separation with a characteristic wavelength when exiting a ball of radius R. Namely, most solutions are driven into a region of phase space in which linear behavior dominates for much longer than expected.The Cahn--Hilliard equation depends on a small parameter $\epsilon,$ modeling the (atomic scale) interaction length; we quantify the behavior of solutions as $\epsilon \rightarrow 0$. Specifically, we show that most solutions starting close to the homogeneous equilibrium remain close to the corresponding solution of the linearized equation with relative distance $O(\epsilon^{2-n/2})$ up to a ball of radius R in the $H^2(\Omega)$-norm, where R is proportional to $\epsilon^{-1+\rho+n/4}$ as $\epsilon \to 0$. Here, $n \in \{ 1,2,3 \}$ denotes the ...

46 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that often virtually all (i.e., all but finitely many) "regular" periodic orbits at μ2 are each connected to exactly one cascade by a path of regular periodic orbits; and virtually all cascades are either paired or solitary.
Abstract: The appearance of infinitely-many period-doubling cascades is one of the most prominent features observed in the study of maps depending on a parameter. They are associated with chaotic behavior, since bifurcation diagrams of a map with a parameter often reveal a complicated intermingling of period-doubling cascades and chaos. Period doubling can be studied at three levels of complexity. The first is an individual period-doubling bifurcation. The second is an infinite collection of period doublings that are connected together by periodic orbits in a pattern called a cascade. It was first described by Myrberg and later in more detail by Feigenbaum. The third involves infinitely many cascades and a parameter value μ2 of the map at which there is chaos. We show that often virtually all (i.e. all but finitely many) "regular" periodic orbits at μ2 are each connected to exactly one cascade by a path of regular periodic orbits; and virtually all cascades are either paired — connected to exactly one other cascade, or solitary — connected to exactly one regular periodic orbit at μ2. The solitary cascades are robust to large perturbations. Hence the investigation of infinitely many cascades is essentially reduced to studying the regular periodic orbits of F(μ2, ⋅). Examples discussed include the forced-damped pendulum and the double-well Duffing equation.

46 citations

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TL;DR: The investigation of infinitely many cascades is essentially reduced to studying the regular periodic orbits of F(μ2, ⋅), which shows that often virtually all "regular" periodic orbits at μ2 are each connected to exactly one cascade by a path ofregular periodic orbits.
Abstract: The appearance of infinitely-many period-doubling cascades is one of the most prominent features observed in the study of maps depending on a parameter. They are associated with chaotic behavior, since bifurcation diagrams of a map with a parameter often reveal a complicated intermingling of period-doubling cascades and chaos. Period doubling can be studied at three levels of complexity. The first is an individual period-doubling bifurcation. The second is an infinite collection of period doublings that are connected together by periodic orbits in a pattern called a cascade. It was first described by Myrberg and later in more detail by Feigenbaum. The third involves infinitely many cascades and a parameter value $\mu_2$ of the map at which there is chaos. We show that often virtually all (i.e., all but finitely many) ``regular'' periodic orbits at $\mu_2$ are each connected to exactly one cascade by a path of regular periodic orbits; and virtually all cascades are either paired -- connected to exactly one other cascade, or solitary -- connected to exactly one regular periodic orbit at $\mu_2$. The solitary cascades are robust to large perturbations. Hence the investigation of infinitely many cascades is essentially reduced to studying the regular periodic orbits of $F(\mu_2, \cdot)$. Examples discussed include the forced-damped pendulum and the double-well Duffing equation.

40 citations

Journal ArticleDOI
TL;DR: In this paper, the authors used Monte Carlo simulations to examine the dependence of R, the radius to which spinodal decomposition occurs, as a function of the parameter e of the governing equation and gave a description of the dominating regions on the surface of the ball by estimating certain densities of the distributions of the exit points.
Abstract: This paper addresses the phenomenon of spinodal decomposition for the Cahn–Hilliard equation Namely, we are interested in why most solutions to the Cahn–Hilliard equation which start near a homogeneous equilibrium u 0≡μ in the spinodal interval exhibit phase separation with a characteristic wavelength when exiting a ball of radius R in a Hilbert space centered at u 0 There are two mathematical explanations for spinodal decomposition, due to Grant and to Maier-Paape and Wanner In this paper, we numerically compare these two mathematical approaches In fact, we are able to synthesize the understanding we gain from our numerics with the approach of Maier-Paape and Wanner, leading to a better understanding of the underlying mechanism for this behavior With this new approach, we can explain spinodal decomposition for a longer time and larger radius than either of the previous two approaches A rigorous mathematical explanation is contained in a separate paper Our approach is to use Monte Carlo simulations to examine the dependence of R, the radius to which spinodal decomposition occurs, as a function of the parameter e of the governing equation We give a description of the dominating regions on the surface of the ball by estimating certain densities of the distributions of the exit points We observe, and can show rigorously, that the behavior of most solutions originating near the equilibrium is determined completely by the linearization for an unexpectedly long time We explain the mechanism for this unexpectedly linear behavior, and show that for some exceptional solutions this cannot be observed We also describe the dynamics of these exceptional solutions

38 citations


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TL;DR: The aim of this work is to provide the readers with the know how for the application of recurrence plot based methods in their own field of research, and detail the analysis of data and indicate possible difficulties and pitfalls.

2,993 citations

Journal ArticleDOI
TL;DR: 5. M. Green, J. Schwarz, and E. Witten, Superstring theory, and An interpretation of classical Yang-Mills theory, Cambridge Univ.
Abstract: 5. M. Green, J. Schwarz, and E. Witten, Superstring theory, Cambridge Univ. Press, 1987. 6. J. Isenberg, P. Yasskin, and P. Green, Non-self-dual gauge fields, Phys. Lett. 78B (1978), 462-464. 7. B. Kostant, Graded manifolds, graded Lie theory, and prequantization, Differential Geometric Methods in Mathematicas Physics, Lecture Notes in Math., vol. 570, SpringerVerlag, Berlin and New York, 1977. 8. C. LeBrun, Thickenings and gauge fields, Class. Quantum Grav. 3 (1986), 1039-1059. 9. , Thickenings and conformai gravity, preprint, 1989. 10. C. LeBrun and M. Rothstein, Moduli of super Riemann surfaces, Commun. Math. Phys. 117(1988), 159-176. 11. Y. Manin, Critical dimensions of string theories and the dualizing sheaf on the moduli space of (super) curves, Funct. Anal. Appl. 20 (1987), 244-245. 12. R. Penrose and W. Rindler, Spinors and space-time, V.2, spinor and twistor methods in space-time geometry, Cambridge Univ. Press, 1986. 13. R. Ward, On self-dual gauge fields, Phys. Lett. 61A (1977), 81-82. 14. E. Witten, An interpretation of classical Yang-Mills theory, Phys. Lett. 77NB (1978), 394-398. 15. , Twistor-like transform in ten dimensions, Nucl. Phys. B266 (1986), 245-264. 16. , Physics and geometry, Proc. Internat. Congr. Math., Berkeley, 1986, pp. 267302, Amer. Math. Soc, Providence, R.I., 1987.

1,252 citations

01 Jan 2016
TL;DR: The nonlinear functional analysis and its applications is universally compatible with any devices to read and is available in the book collection an online access to it is set as public so you can get it instantly.
Abstract: nonlinear functional analysis and its applications is available in our book collection an online access to it is set as public so you can get it instantly. Our books collection hosts in multiple countries, allowing you to get the most less latency time to download any of our books like this one. Merely said, the nonlinear functional analysis and its applications is universally compatible with any devices to read.

581 citations